[    {        "title": "2204.01459v1.Towards_high_all_optical_data_writing_rates_in_synthetic_ferrimagnets.pdf",        "content": "Towards high all-optical data writing rates in synthetic ferrimagnets\nYouri L.W. van Hees,a)Bert Koopmans, and Reinoud Lavrijsen\nDepartment of Applied Physics, Institute for Photonic Integration, Eindhoven University of Technology,\nP.O. Box 513 5600 MB Eindhoven, the Netherlands\n(Dated: 5 April 2022)\nAlthough all-optical magnetization switching with fs laser pulses has garnered much technological interest,\nthe ultimate data rates achievable have scarcely been investigated. Recently it has been shown that after a\nswitching event in a GdCo alloy, a second laser pulse arriving 7 ps later can consistently switch the magne-\ntization. However, it is as of yet unknown whether the same holds in layered ferrimagnetic systems, which\nhold much promise for applications. In this work we investigate the minimum time delay required between\ntwo subsequent switching events in synthetic ferrimagnetic Co/Gd bilayers using two fs laser pulses. We\nexperimentally demonstrate that the minimum time delay needed for consistent switching can be as low as 10\nps. Moreover, we demonstrate the importance of engineering heat di\u000busion away from the magnetic material,\nas well as control over the laser pulse power. This behavior is reproduced using modelling, where we \fnd\nthat the second switch can occur even when the magnetization is not fully recovered. We further con\frm\nthat heat di\u000busion is a critical factor in reducing the time delay for the second switch, while also con\frming\na critical dependence on laser power.\nAll-optical switching (AOS) of the magnetization of\nthin \flm ferrimagnets using single femtosecond laser\npulses has been demonstrated to be a robust, ultrafast,\nand energy e\u000ecient method to write data with promise\nfor future memory devices1. The mechanism was \frst\ndiscovered in ferrimagnetic GdFeCo alloys2{6, and was\nsoon followed by demonstrations in synthetic ferrimag-\nnets (Co/Gd and Co/Tb)7,8. The switching was shown\nto be symmetrical, with each subsequent laser pulse\ntoggling the magnetization between the 'up' and 'down'\nstates, a process which can be repeated over hundreds of\nmillions of cycles without failure9. The energy e\u000eciency\nof AOS is especially interesting for applications, with\ntypical energies of only tens of fJ needed to switch\nnanoscale bits7,10. Moreover, the magnetization only\ntakes a few picoseconds to cross zero11,12, potentially\nimplying near THz writing speeds. However, this is not\nthe most relevant timescale for determining the ultimate\ndata rates, as it is expected that the magnetization\nshould relax to the opposite state after each switching\nevent to facilitate the next switch. This remagnetization\nprocess can potentially take hundreds of ps9, and is\nexpected to be governed by heat di\u000busion away from the\nmagnetic layers13. Despite the large body of research\ninto AOS, understanding of the ultimate speed with\nwhich subsequent switches can actually take place\nremains scarce14. It has been shown experimentally\nthat a second fs laser pulse can consistently switch the\nmagnetization in a GdFeCo alloy again after 300 to 400\nps15. Here it was conjectured that this is likely not a\nfundamental limit of the switching process but rather\na limit imposed by heat di\u000busion. Very recently it was\ndemonstrated by Steinbach et al. that this is indeed\nthe case, with a minimum possible waiting time of 7\nps in a GdCo alloy when a substrate with proper heat\na)Electronic mail: y.l.w.v.hees@tue.nlconductivity is used16. The remagnetization time of the\nalloy was emphasized by the authors as a critical factor\nfor facilitating the second switch. In this light, one might\nexpect longer waiting times between switches in tech-\nnologically highly relevant synthetic ferrimagnets17{19,\nwhere the magnetic sublattices are less strongly coupled\nand therefore expected to remagnetize slower after AOS\nthan in alloys20.\nIn this work we experimentally demonstrate that\nby taking into account the heat di\u000busion of the sample,\nswitching events can take place with very short delays\neven in synthetic ferrimagnetic Co/Gd systems. The\nminimum time delay for the second pulse to consistently\nswitch the magnetization is found to be as low as 10\nps, yielding potential writing rates of up to 100 GHz.\nThe importance of the heat di\u000busion in this process is\nhighlighted by demonstrating a larger minimum time\nfor substrates with lower heat conductivity. Moreover,\nthe absence of rapid double switching when reversing\nthe order of the laser pulses or when slightly increasing\nthe power of one of the pulses highlights the need for\ncareful control of the irradiation conditions. Finally, we\npresent modelling results using the Microscopic Three\nTemperature Model (M3TM)21con\frming that heat\ndi\u000busion is the dominant factor in reducing the delay for\nthe second switch, and also illuminating the critical role\nof the laser pulse energy.\nThe experiments in this work are performed on\nTa(4)/Pt(4)/Co(1)/Gd(3)/Cap.(4) multilayer stacks\n(where numbers in parentheses indicate layer thickness\nin nm, and `Cap.' is Ta or TaN) known to exhibit\nAOS7, which are deposited on Si substrates using DC\nmagnetron sputter deposition. In order to achieve a high\nheat conductivity, a degenerately Boron doped substrate\nis used. As sketched in Fig. 1a, individual \u0018100 fs\nlaser pulses with a central wavelength of 700 nm arearXiv:2204.01459v1  [cond-mat.mes-hall]  4 Apr 20222\n250:50 BSSample\nPulse 1Pulse 2Δt(a)\n(b)\n1\n240 ps 50 ps 20 ps 10 ps\nFIG. 1. (a) Sketch of the experimental setup, colors are for\nillustrative purposes only. (b) Kerr microscopy images of the\nmagnetic state of a Si:B//Ta(4)/Pt(4)/Co(1)/Gd(3)/Ta(4)\nsample after exposure to two \u0018100 fs laser pulses with vary-\ning time delay. The green and blue lines enclose the areas\nthat would be switched by the two pulses individually. The\nscale bar represents 50 µm.\nsplit in two using a 50:50 beam splitter, after which one\npulse goes through a delay line so that the time delay\nbetween the two pulses can be adjusted. Both pulses are\nsubsequently focused onto the sample via the same ob-\njective. The magnetic state of a sample after exposure is\nimaged using an ex situ Kerr microscope. The magnetic\nstate of the sample after exposing di\u000berent regions to\nsets of two pulses with varying time delays is shown in\nFig. 1b. Here, the regions that would be switched by\nthe \frst and second pulse separately are indicated by\nthe solid green and dashed blue shapes, respectively.\nFor the longest time delay shown here (240 ps) a clear\nregion within the overlap of the two pulses is observed\nwhere the magnetization is switched twice, returning to\nthe initial state. This is comparable to previous work\non GdFeCo alloys, where the second switch was possible\nafter 300-400 ps15. Moreover, comparable to recent work\non GdCo16we \fnd that double switching also occurs\nwhen reducing the time delay to 50 ps (as indicated by\nthe red dotted circle), and even stays possible for time\ndelays as low as 10 ps. As will be discussed later, this\nresult is somewhat surprising, as magnetization recovery\nis expected to be faster in alloys than in synthetic\nferrimagnets, where the magnetic sublattices are only\ncoupled at the interface, and the Gd magnetization\nshows a stronger temperature dependence. Comparing\nthe images for 240 and 10 ps time delay, we note the\nshrinkage of the region where double switching occurs\nfor the shortest time delay, indicating a more critical\ndependence on the exact laser \ruence of the two pulses.\nSwitching in the region where the second pulse is\nbelow threshold, as was observed in similar experiments\non GdFeCo15, has not been observed in the present work.\nTo investigate the e\u000bect of heat di\u000busion on the\n150 ps 50 ps2\n1\n(a)\n– 50 ps\n10 ps(b)\n(c)\n20 ps 10 psFIG. 2. Kerr microscopy images of the magnetic state of a\nSi/SiO 2/Ta(4)/Pt(4)/Co(1)/Gd(3)/TaN(4) sample after ex-\nposure to two \u0018100 fs laser pulses with varying time delay.\nThe solid green and dashed blue lines enclose the areas that\nwould be switched by the two pulses individually. The scale\nbar represents 50 µm. In (a) and (c), pulse 1 arrives before\npulse 2, whereas in (b) pulse 2 arrives \frst. The intensity of\npulse 2 is 10% higher in (c) than in (a) and (b). The red\ndotted ellipse in (c) indicates the area where the e\u000bect of the\ncombined heating of both pulses was high enough to induce\nmagnetic and/or structural damage.\nminimum double switching time, we now turn\nto a substrate with lower heat conductivity. A\nTa(4)/Pt(4)/Co(1)/Gd(3)/TaN(4) stack is deposited on\na silicon substrate with a 100 nm coating of SiO 2. The\noxide layer is expected to be less e\u000ecient in conducting\nheat away from the metallic multilayer stack than the\nsemimetallic Si:B substrate. Remagnetization therefore\nis expected to be slower, and the minimum time needed\nbetween two pulses for consistent switching should be\nlarger. We perform the same experiment presented in\nFig. 1 on the multilayer stack grown on the Si/SiO 2\nsubstrate, the result of which is shown in Fig. 2a. The\nareas switched by the two pulses separately are again\nindicated with solid green and dashed blue lines in the\nKerr microscope image for 150 ps time delay. Note that\nthe intensity of the two laser pulses is identical, however\nthe power density in the \frst pulse is higher, leading to\na smaller switched area. Consistent double switching is\nagain observed for time delays down to 20 ps. However\nfor a time delay of 10 ps, where double switching was\npossible previously, di\u000berent behavior is observed. Only\na small part of the area where both pulses overlapped\nhas returned to the initial state, with this area being\nrather complexly bounded. This is an indication that\nthis area is actually not consistently switched twice,\nbut rather remagnetized in a random state after cooling\ndown. Such a process can occur when the temperature\nof the lattice exceeds the Curie temperature for a longer\ntime, leading to a complete loss of magnetization for\na short period of time6. This observation is consistent\nwith the expectation that heat remains in the system\nfor a longer time when heat transfer is impeded by the\noxide coating.3\nTo illustrate the sensitivity of the double switching\nprocess, Fig. 2b shows the result of exposure when the\norder of the two pulses is swapped. For a delay as large\nas 50 ps, a more random state is found in the overlapping\nregion, indicating a critical dependence on the power\ndistribution of the two laser pulses. More speci\fcally,\nwhen the pulse with higher intensity ('pulse 1') arrives\nlast, double switching is less consistent. This can again\nbe understood by realizing that by strongly heating\nthe sample with the second pulse before the heat from\nthe \frst pulse has signi\fcantly dissipated, the lattice\ntemperature can exceed the Curie temperature. As an\nadditional demonstration of the criticality of the laser\npower, we increase the intensity of pulse 1 by as little as\n10%, leading to the magnetic state shown in 2c. Here we\n\fnd a region (indicated by the red dashed ellipse) where\nthe magnetic properties of the sample have changed\ndue to laser irradiation. Both the small dark dot in the\ncenter of this region as well as the lighter area around it\nhave been annealed by the combined e\u000bect of both laser\npulses, and can not be switched again with either a laser\npulse or an external magnetic \feld. This indicates the\nhigh temperature of the sample, and further highlights\nthe importance of proper heat engineering.\nTo better understand double switching at these ul-\ntrashort timescales we turn to modelling, which has\nbeen successful in describing AOS in (synthetic)\nferrimagnets3,22{27. Here we use the simpli\fed Micro-\nscopic Three Temperature Model (M3TM) as introduced\nby Beens et al.21, which can describe AOS in layered\nferrimagnets. In this model, the system is split in four\ninteracting systems, namely separate spin systems for\nCo and Gd, mobile spinless electrons, and phonons. The\ntwo spin sublattices carry the magnetization, and are\ndescribed using a Weiss mean-\feld approach, whereas\nthe electrons and phonons are described in terms of tem-\nperatures. An incident laser pulse is initially absorbed\nby the electron system, raising the electron temperature\n(Te). Due to electron-phonon scattering, the electron\nand phonon temperature ( Tp) will equilibrate. Ultrafast\ndemagnetization occurs via electron-phonon scattering\nevents, which have a \fnite probability for an electron to\n\rip its spin. Exchange of angular momentum between\nthe magnetic sublattices is described using exchange\nscattering. The model parameters are taken from Beens\net al.21. Similar to previous work14, we include heat\ndi\u000busion to the substrate by adding a phenomenological\nterm to the phonon temperature, namely\ndTp\ndt/Tamb\u0000Tp\n\u001cd; (1)\nwhereTambis the ambient temperature (room temper-\nature), and \u001cda characteristic time constant for heat\ndi\u000busion.\nFigure 3a shows the simulated magnetization dy-\nnamics (top) as well as TeandTp(bottom) of a layered\n-1.0-0.50.00.51.0\nCo\nGdMag netization (norm.)Msat,GdPulse1 Pulse2\nP1=P2=50·108J/m3\nτd=10ps\n0 20 40 60 800500100015002000Te\nTpTemperatu re(K)\nTimedelay(ps)\n0 5 10 15 20 25 30 35 40020406080100\nP1,P2(108J/m3):\n40\n50Minimumseco ndpulsetime (ps)\nd(ps)(a)\n(b)FIG. 3. (a) M3TM simulation results of excitation of a Co/Gd\nbilayer (3 monolayers each) with two laser pulses (gray lines)\nseparated by 40 ps. The top graph shows the average mag-\nnetization response of both Co and Gd over time, with the\ndashed line indicating the Gd magnetization at saturation.\nThe bottom graph shows the electron and phonon tempera-\nture of the combined system, with the ambient temperature\nTamb(295 K) indicated by the dashed line. (b) Minimum time\ndelay needed between two pulses for consistent double switch-\ning as a function of the heat di\u000busion constant \u001cd, determined\nvia M3TM simulations.\nCo/Gd system with \u001cd= 10 ps after excitation with\ntwo laser pulses separated by 40 ps. After the \frst\npulse, the Co and Gd magnetization are switched,\nfollowed by remagnetization in the opposite direction.\nTeincreases rapidly due to heating by the laser pulse\nand equilibrates with Tpmore slowly, while heat is\nbeing removed from the system with the characteristic\ntimescale\u001cd. As the intrinsic remagnetization rate of Co\nis faster than \u001cd, the latter is dominant in determining\nthe remagnetization time. By the time the second pulse\narrives, the heat has nearly dissipated from the system,\nand the Co magnetization is very close to saturation.\nFor Gd however, the remagnetization rate is an order of4\nmagnitude slower28{30, such that Gd has not returned to\nsaturation (dashed line) when the second pulse arrives.\nThis is in contrast with the expected behavior in alloys,\nwhere the Gd has more transition metal neighbors with\nwhich it can exchange angular momentum and therefore\nremagnetize more rapidly. Nevertheless, this second\npulse also leads to the switching of the magnetization\nof both layers, after which remagnetization towards the\noriginal direction proceeds. Although it might seem\ncounter-intuitive that switching is still possible with a\nstrongly reduced Gd moment, this is in line with previ-\nous research on AOS in synthetic ferrimagnets where a\nchange in the relative magnetization of the sublattices\nwas found not to hinder switching21. Conversely, in\nferrimagnetic alloys it is known that AOS can only\noccur if the magnetizations of both sublattices (nearly)\ncancel each other31. As such, we conjecture that any\npossible detriment in remagnetization speed in synthetic\nferrimagnets could be compensated by the absence of a\nneed to wait for full magnetization recovery, leading to\nvery similar timescales for repeated AOS.\nFinally, we investigate the e\u000bect of heat di\u000busion\non double switching in the M3TM. For di\u000berent values\nof\u001cdwe model excitation with two laser pulses. By\nvarying the time delay between the pulses and evaluating\nthe \fnal magnetization state, we extract a value for\nthe minimum time delay needed for double switching\nas a function of \u001cd. Fig. 3b shows these values for two\ndi\u000berent values of the absorbed laser power density. For\n\u001cd>5 ps, the minimum waiting time between pulses\ndepends approximately linearly on \u001cd. In this regime\nheat di\u000busion seems to be the dominant factor, however\na non-trivial dependence on the power of both pulses\nis also found. Comparing the blue squares and green\ncircles, for 40 and 50 \u0002108J m\u00003respectively, it is\nfound that the second switch can happen faster if the\npower of both pulses is higher. Individual time traces\nfor di\u000berent pulse powers (not shown here) indicate\nthat the switch occurs faster for higher pulse powers,\ngiving more time for relaxation towards saturation.\nFor\u001cd<5 ps, the minimum time needed for the\nsecond pulse starts to increase, which is attributed to\na hindrance of the switching mechanism in general by\nthe unrealistically fast dissipation of heat. Here, the\nsystem already cools down enough for the sublattices to\nstart remagnetizing at the timescale at which the switch\nwould normally take place. From these results it is clear\nthat although e\u000ecient heat dissipation is essential for\nachieving high switching repetition rates, the power also\nneeds to be carefully controlled. Combining this with\nthe experimental observation that a slight increase in\nlaser power can already be detrimental, this highlights\nthe narrow range of laser powers for which consistent as\nwell as rapid double switching is possible. Finally, we\nnote that double switching within 10 ps, as observed in\n1b, is not reproduced in modelling even for very e\u000ecient\nheat dissipation. We believe this to be an intrinsic limitof the system modelled here, as this probably represents\na minimum time needed for su\u000ecient recovery of the\nGd magnetization. Past work has shown that including\nintermixing, which is undoubtedly present in the real\nsystem, leads to more e\u000ecient transfer of angular\nmomentum between the two sublattices20. Hence, we\nexpect that in an intermixed system both Co and Gd\ncould remagnetize more rapidly than in a bilayer with\nperfect interfaces, potentially reducing the waiting time\nneeded for the second switch. It should be noted that\nswitching back within a few picoseconds, as was reported\nusing a di\u000berent model14, does not seem to be pos-\nsible using the M3TM for any combination of parameters.\nIn conclusion, we have investigated the timescales\nfor repeated all-optical switching in synthetic ferri-\nmagnetic Co/Gd bilayers, and have demonstrated a\nminimum waiting time of 10 ps between two subsequent\nsuccessful switching events, implying writing speeds\nof up to 100 GHz. We have shown that the layered\nnature of these systems need not be a hindrance to\nachieve similar writing speeds as in alloys, explained\nby the notion that the slower remagnetization of Gd is\ncompensated by a less critical dependence of AOS on the\nGd moment. Furthermore, by changing the substrate\nwe have con\frmed the importance of engineering heat\ndi\u000busion away from the magnetic system. Finally, with\nmodelling e\u000borts using the M3TM we have resolved the\nrole of heat di\u000busion in ultrafast repeated switching, but\nwe also stress that controlling the laser power is critical\nto reliable integration in future optically written data\nstorage devices.\nACKNOWLEDGEMENTS\nWe gratefully acknowledge M. Beens for assistance on im-\nplementation of the M3TM and discussion of simulation\nresults. This work is part of the Gravitation programme\n`Research Centre for Integrated Nanophotonics', which\nis \fnanced by the Netherlands Organisation for Scien-\nti\fc Research (NWO).\nAUTHOR DECLARATIONS\nCon\rict of Interest\nThe authors have no con\ricts to disclose.\nData availability\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.5\n1A. V. Kimel and M. Li, \\Writing magnetic memory with ultra-\nshort light pulses,\" Nature Reviews Materials 4, 189{200 (2019).\n2C. D. Stanciu, F. Hansteen, A. V. 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Man-\ngin, \\Ultrafast magnetization manipulation using single fem-\ntosecond light and hot-electron pulses,\" Advanced Materials 29,\n1703474 (2017)."    },    {        "title": "2104.02198v3.Landau_Lifshitz_Bloch_equation_for_ferrimagnets_with_higher_order_interaction.pdf",        "content": "Landau-Lifshitz-Bloch equation for ferrimagnets with\nhigher-order interaction\nMarco Menarini\u0003and Vitaliy Lomakin\nDepartment of Electrical and Computer Engineering,\nCenter for Memory and Recording Research,\nUniversity of California, San Diego, La Jolla, California 92093\n(Dated: September 16, 2021)\nAbstract\nWe present a micromagnetic formulation for modeling the magnetization dynamics and ther-\nmal equilibrium in ferrimagnetic materials at low and elevated temperatures. The formulation\nis based on a mean \feld approximation (MFA). In this formulation, the ferrimagnet is described\nmicromagnetically by two coupled sublattices with corresponding interactions, including inter- and\nintra-sublattice micromagnetic exchange as well as four-spin interactions described as an inter-\nsublattice molecular \feld with a cubic dependence of the magnetization. The MFA is used to\nderive a Landau Lifshitz Bloch type equation for ferrimagnetic material, including cases with a\nferromagnetic - antiferromagnetic phase transitions. For validation, the results obtained via the\npresented model are compared with recent experimental data for phase transitions in FeRh.\nPACS numbers: 75.10.-b, 75.30.-m, 75.40.Gb, 75.78.Cd, 75.78.-n\n1arXiv:2104.02198v3  [cond-mat.mtrl-sci]  14 Sep 2021I. INTRODUCTION\nThere is an increased interest in using antiferromagnetic (AF) materials for creating re-\nliable and compact sources of coherent signals in the THz frequency. This is enabled due\nto the fact that the frequency of antiferromagnetic resonances !AFMR can reach the THz\nrange, signi\fcantly exceeding the frequency of ferromagnetic resonances[1, 2]. Several de-\nvices for spin torque oscillators have been proposed that leverage the strong inter-sublattice\nAF exchange as the source of the THz signal [3] and using the spin current to induce a cant-\ning angle between the two sublattices. Such devices have been proposed as possible THz\nfrequency comb-generators to be used as arti\fcial neurons for neuromorphic computing due\nto their fast response time and threshold behaviour [4].\nRecently, Medapalli et al. [5] showed that it is possible to optically generate a THz pulse\nin a FeRh/Pt bi-layer. In the experiment, an ultrafast laser pulse excites metamagnetic FeRh\ninjecting a spin-current into the non-magnetic Pt interface that is, then, converted into a\nspin-current via the inverse spin Hall e\u000bect [6, 7]. The spin current in the AF state can\noriginate from a precessional response of FeRh during a partial phase transition induced by\nthe laser [8]. Such transformation occurs on a sub picosend time scale, much faster than any\nlattice expansion [9]. The phase transition occurs due to the competition between bilinear\nand the Rh mediated biquadratic exchange interactions in an e\u000bective spin Hamiltonian\n[10]. Bilinear and biquadratic exchange energies strongly depend on the temperature. Using\natomistic simulations, it is possible to reproduce such phase transitions by including both\nthe bilinear and biquadratic exchanges [11].\nHowever, despite the computing power of modern computers, to model realistic structures,\na coarse-grained model for the dynamic of the magnetization is desirable. The Landau-\nLifshitz-Bloch (LLB) equation of motion for macroscopic magnetization vectors [12] has\nbeen used to accurately model the behaviour of complex magnetic structures at high tem-\nperatures. Its usability has been extended by Atxitia et al. [13] to ferrimagnets with two\nsublattices. However, this model cannot describe phase transition between ferromagnetic\nand antiferromagnetic states as observed in experiments [9, 14] and may miss additional\ne\u000bects related to the inter-sublattice micromagnetic exchange interactions.\nIn this paper, we present an LLB formulation for ferrimagnetic materials introducing\ne\u000bects of higher-order exchange and show that they are necessary to model a metamagnetic\n2AF/FM transitions driven by temperature. We derive a macroscopic equation for the mag-\nnetization dynamics of two-sublattice metamagnetic systems with higher order exchange\nvalid in the entire temperature range. As a concrete test case, we consider metamagnetic\nFeRh particles. FeRh is modelled as two sublattices, each with its length and direction,\ncoupled via an inter-sublattice exchange. We use the mean-\feld approximation (MFA) to\nderive a macroscopic equation for the magnetization of each sublattice. We study the mean\n\feld energy of the system to better understand the phase transition and validate the model\nagainst the experimental results.\nII. MEAN FIELD APPROXIMATION OF A TWO SUBLATTICE SYSTEM\nWITH HIGHER-ORDER INTERACTIONS\nWe start by consider an atomistic model for an FeRh ferrimagnet as used by Barker et\nal. [11]. The e\u000bective Hamiltonian Hcontains only the degrees of freedom of a simple cubic\n(sc) Fe lattice , with the e\u000bect of the induced Rh moment included into e\u000bective Fe-Rh-Fe\ninteractions. The Hamiltonian is augmented by the applied \feld Hand uniaxial anisotropy:\nH=\u0000X\ni\u0016iHSi+X\nijJij\u0011ij(Si;xSj;x+Si;ySj;y)\u0000X\nijJijSiSj\n+1\n3X\ni;j;k;lDijkl[(SiSj) (SkSl) + (SiSk) (SjSl) + (SiSl) (SkSj)]: (1)\nHere, Siis the normalized spin vector of the atoms iand\u0016iis its magnetic moment. Jij\nare the Heisenberg exchange interactions (bilinear), including the direct Fe-Fe and indirect\nFe-Rh-Fe contributions. Dijklare the four-spin exchange (biquadratic) coe\u000ecients, which\nonly have contributions from the Fe-Rh-Fe interactions. The parameter \u0011ij\u001c1 de\fnes\nthe strength of the anisotropy in the direction perpendicular to the easy axis [12]. For the\nHeisenberg exchange interactions, only the nearest neighbors and the second nearest neigh-\nbors inside the unit cell are considered (\fg. 1(a)). The cyclical four-spin interaction inside\neach unit cell is given by pairwise interactions between the 3 nearest neighbors converging\non one of the vertices of the sc lattice (\fg. 1(b)).\nThe free energy of the system described by Hin eq. (1) can be given as F=\u0000TlnZ,\nwhereZis the partition function and Tis the temperature. In the mean-\feld approximation\nwe consider each spin on a site ias an isolated spin subjected to the e\u000bective \feld due to\nthe mean values of the neighboring spins.\n3Figure 1: Simpli\fed model of the unit cell (a) with the nearest-neighbor exchange (red dashed\nline)Jh001iand the second nearest-neighbor exchange (blue dashed lines) Jh011i. In (b) eight 4-spin\ncyclical interactions inside the unit cell (thick dark lines) are shown.\nSince in the AF state the nearest neighbors tend to be antiparallel to each other and\nthe second nearest neighbors tend to be parallel and taking into account the symmetry of\nthe system, we can consider this mean \feld as the \feld produced by the two sublattices\nmA;i=hSA;iiandmB;i=hSB;ii. The mean-\feld Hamiltonian is then obtained from eq. (1)\nas:\nHMFA=H00\u0000X\niX\n\u0016=A;B\u0016\u0016HMFA\n\u0016;iS\u0016;i; (2)\nThe termH00is given by\nH00=Jh011i\n2X\nijX\n\u0016=A;B(m\u0016;im\u0016;j) +Jh011i\n2X\nijX\n\u0016=A;BX\nk=x;y\u0011\u0016(m\u0016;i\u0001^ ek) (m\u0016;j\u0001^ ek)\n+Jh001i\n2X\nij(mA;imB;j)\u000012DhQiX\niX\n\u0016=A;B\n\u00166=\u0017(m\u0016;im\u0016;i) (m\u0016;im\u0017;i);(3)\nwhereJh011iis the inter-sublattice exchange coe\u000ecient, Jh001iis the intra-sublattice exchange\ncoe\u000ecient, and ^ ekis the unit vector in the direction of k=x;y. The molecular \feld for the\n4two sublattices \u0016;\u0017=A;B is given by\n\u0016\u0016HMFA\n\u0016;i =\u0016\u0016H+Jh011iX\njm\u0016;j+Jh011iX\njX\nk=x;y\u0011\u0016(m\u0016;j\u0001^ ek)^ ek\n+Jh001i\n2X\nj(m\u0017;j)\u00008DhQi(m\u0016;im\u0017;i)m\u0016;i\u00004D0\u0000\nm2\n\u0016;i+m2\n\u0017;i\u0001\nm\u0017;i:(4)\nThe solution of the one-spin problem in eq. (2) leads to\nF=H00\u0000NTln(4\u0019)\u0000TX\niX\n\u0016\u0003 (\u0018\u0016;i); \u0003 (\u0018) = ln\u0012sinh (\u0018)\n\u0018\u0013\n; (5)\nwhereNis the total number of spins, \u0018\u0016;i=\f\f\u0018\u0016;i\f\fis the reduced \feld for the sublattice \u0016and\nspiniwith\u0018\u0016;i=\u0016\u0016\fHMFA\ni, and\f= 1=T, where the temperature Tis given in the units\nof energy. The MFA free energy in eq. (5) can be minimized with respect to the average\nmagnetization m\u0016;ito \fnd the equilibrium solution of the system.\nIf we consider the continuum limit we can go from the sums in eqs. (3) and (4) to volume\nintegrals. For small anisotropy and assuming small changes of the magnetization between\nspins in the same sublattices, we can rewrite the short-range interaction between the nearest\nneighbors and second nearest neighbors as:\nX\njJh001im\u0017;j\u0019J1m\u0017;i+Aex;\u0016\u0017\u0001m\u0017;i; (6)\nX\njJh011im\u0016;j\u0019J2m\u0016;i+Aex;\u0016\u0016\u0001m\u0016;i: (7)\nHere, \u0001 is the Laplace operator acting on the sublattice magnetization mA(r). In addition,\nJ1=zJh001iis the average of the exchange interactions for z= 6 nearest neighbors in the sc\nlattice and J2=qJh011iis the average over the second nearest neighbors with q= 12. For\nthe sc lattice, the exchange constants are given by Aex;\u0016\u0016 = 2J2a2\n0=qandAex;\u0016\u0017 =J1a2\n0=z,\nwherea0is the lattice spacing assumed to be the same in both directions.\nSubstituting eqs. (6) and (7) in eqs. (3) and (4) and taking the continuum limit in eq. (5),\none obtains:\nF\nJ2=1\nv0Z\ndrX\n\u0016=A;B\n\u00166=\u0017(\n1\u00006d(m\u0016m\u0017)\n2m2\n\u0016+jm\u0016m\u0017\n2+\u0000\nm\u0016;heff\n\u0016\u0000h\u0016\u0001\n2\u00001\n\fJ2\u0003(\u0018\u0016))\n\u0000NT\nJ2;(8)\n5wherev0is the unit-cell volume, j=J1=(2J2)<1 is the normalized inter-sublattice exchange\ncoe\u000ecient, and d= 4DhQi=J2<1=6 is the normalized four-spins coe\u000ecient. The reduced\n\feld and the normalized e\u000bective \felds for the sublattice \u0016are given by\n\u0018\u0016=\fJ2\u001a\n[1\u00002d(m\u0016m\u0017)]m\u0016+\u0014j\n2\u0000d\u0000\nm2\n\u0016+m2\n\u0017\u0001\u0015\nm\u0017+heff\n\u0016\u001b\n; (9)\nheff\n\u0016=h\u0016+Aex;\u0016\u0016\nJ2\u0001m\u0016+Aex;\u0016\u0017\nJ2\u0001m\u0017\u0000\u0011\u0016X\nk=x;y(m\u0016\u0001^ ek)^ ek; (10)\nwhere h\u0016=\u0016\u0016H=J2\u001c1 is the normalized applied \feld and heff\u0016;iacting on the sublattice\n\u0016.\nThe reduced \feld given in eq. (9) and the e\u000bective \feld given in eq. (10) can be used to\nformulate an LLB equation for ferrimagnets with higher order interaction, as we show in the\nnext section.\nIII. LLB EQUATION FOR HIGHER ORDER FERRIMAGNET\nTo derive a two component LLB model, we follow the procedure outlined by Atxitia et al.\n[13]. By substituting eqs. (6) and (7) into the eq. (4) we obtain the mean-\feld approximation\nof the molecular \feld:\nHMFA\n\u0016;i =Heff\n\u0016;i+Hk\nE\u0016;i+H?\nE\u0016;i; (11a)\n\u0016\u0016Heff\n\u0016;i=\u0016\u0016H+Aex;\u0016\u0016\u0001m\u0016;i+Aex;\u0016\u0016\u0001m\u0016;i\u0000\u0016\u0016HK;\u0016X\nk=x;y(m\u0016;i\u0001^ ek)^ ek; (11b)\nHk\nE\u0016;i=Jk\n\u0016;i\n\u0016\u0016m\u0016;i; (11c)\nH?\nE\u0016;i=\u0000J?\n\u0016;i\n\u0016\u0016m\u0017;i\u0002(m\u0017;i\u0002m\u0016;i)\nm2\n\u0017;i; (11d)\nJk\n\u0016;i=J2\u0014\n(1\u00002d(m\u0016;im\u0017;i)) +\u0012j\n2\u0000d\u0000\nm2\n\u0016;i+m2\n\u0017;i\u0001\u0013\n\u0002(m\u0017;i;m\u0016;i)\u0015\n; (11e)\nJ?\n\u0016;i=J2\u0014j\n2\u0000d\u0000\nm2\n\u0016;i+m2\n\u0017;i\u0001\u0015\n; (11f)\nwhered= 4DhQi=J2,j=J1=J2,HK;\u0016=J2\u0011\u0016=\u0016\u0016is the anisotropy \feld, and Hk\nE\u0016;iandH?\nE\u0016;i\nare the intra-sublattice parallel and perpendicular exchange, respectively. Given two vectors\nvAandvB, the function\n\u0002(vA;vB) =vA\u0001vB\nv2\nB; (12)\n6is the projection of the vector mAin the direction of the vector mB. We substitute the MFA\nfor the \feld in eq. (11) into the dynamic formulation of the mean magnetization obtained\nthrough the Fokker-Planck equation [12]. The corresponding set of coupled LLB equations\nfor each sublattice \u0016is given by\ndm\u0016\ndt=\r\u0016\u0002\nm\u0016\u0002HMFA\n\u0016\u0003\n\u0000\u0000\u0016;k\u0012\n1\u0000m\u0016m0;\u0016\nm2\n\u0016\u0013\nm\u0016\u0000\u0000\u0016;?[m\u0016\u0002(m\u0016\u0002m0;\u0016)]\nm2\n\u0016;(13)\nwhere\r\u0016is the gyromagnetic ratio, \u0000 \u0016;kand \u0000\u0016;?are the longitudinal and transverse relax-\nation rates, and the instantaneous equilibrium magnetization m0;\u0016is given by\nm0;\u0016=B(\u0018\u0016)\u0018\u0016\n\u0018\u0016;\u0018\u0016=\f\u0016\u0016HMFA\n\u0016: (14)\nHere,\u0018\u0016=\f\f\u0018\u0016\f\fis the reduced \feld and B(x) = coth(x)\u00001=xis the Langevin function.\nThe parallel and perpendicular relaxation rates are given by\n\u0000\u0016;k= \u0003\u0016B(\u0018\u0016)\n\u00180;\u0016B0(\u0018\u0016); (15)\n\u0000\u0016;?=\u0003\u0016\n2\u0014\u0018\u0016\nB(\u0018\u0016)\u00001\u0015\n; (16)\nwhere \u0003 \u0016= 2\r\u0016\u0015\u0016=\f\u0016\u0016is the characteristic di\u000busion relaxation rate given by the Neel\nattempt frequency with the atomistic damping constant \u0015.\nEquation eq. (13) with eq. (14) and eq. (11) can be directly used for numerical modeling.\nHowever, it is possible to rewrite it in a more compact form if the parallel intra-sublattice\nexchange is large in comparison with the other components of the MFA \feld (i.e.,\f\f\fHk\nE;\u0016\f\f\f\u001d\n\f\fHeff\n\u0016\f\fand max [j;4d\u0000j]\u001c2), which is valid in the entire range of temperatures for many\nferromagnetic and ferrimagnetic materials [12]. Using this approximation, we can expand\nthe Langevin equation to the \frst order of the Taylor series around Hk\nE;\u0016:\nm0;\u0016\u0019B(\u00180;\u0016)\nm\u0016m\u0016+\u00160;\u0016\fB0(\u00180;\u0016)\u0000\nm\u0016Heff\n\u0016\u0001\nm\u0016\nm2\n\u0016; (17)\nwhere\u00180;\u0016=\fJk\n\u0016m\u0016. Using eq. (17), we can write the LLB equation in the standard form:\ndm\u0016\ndt=\r\u0016h\nm\u0016\u0002\u0010\nHeff\n\u0016+H?\nE\u0016\u0011i\n\u0000\r\u0016\u000bk;\u0016 \n1\u0000B(\u00180;\u0016)=m\u0016\n\u00160;\u0016\fB0(\u00180;\u0016)\u0000m\u0016Heff\n\u0016\nm2\n\u0016!\nm\u0016\n\u0000\r\u0016\u000b?;\u0016m\u0016\u0002h\nm\u0016\u0002\u0010\nHeff\n\u0016+H?\nE\u0016\u0011i\nm2\n\u0016;(18)\n7where the parallel and perpendicular damping coe\u000ecients are functions of the temperature\nand the angle between the two sublattices:\n\u000bk;\u0016=2\u0015\u0016T\n\fJk; \u000b?;\u0016=\u0015\u0016\u0014\n1\u0000T\n\fJk\u0015\n: (19)\nSince the perpendicular intra-sublattice exchange in eq. (18) only appears in the precessional\nand the longitudinal damping terms, the contribution of m\u0017;iin the direction of m\u0016;iin the\ncross product m\u0016;i\u0002m\u0017;iis zero by geometrical reasoning, and equation eq. (11d) can be\nrewritten using the triple vector product identity and the function \u0002 de\fned in eq. (12):\nH?\nE\u0016;i=J?\n\u0016;i\n\u0016\u0016\u0002(m\u0016;i;m\u0017;i)m\u0017;i: (20)\nIV. RESULTS\nIn this section, we use the MFA of the energy and LLB formulations developed in sec-\ntions II and III to study the phase transition in an example material. We choose FeRh for\nthe readily available experimental literature [9, 14, 15] and atomistic simulations [11]. First,\nwe consider the equilibrium conditions by minimizing the free energy with respect to the\nmagnetization to obtain the critical point at which we have the transition between the AF\nand FM states. Then, we study the magnetization behaviour via the modi\fed LLB equation\nand compare it with experimentally results.\nA. Energy and thermal equilibrium analysis\nTo study the equilibrium conditions, we consider an isotropic case with heff\n\u0016= 0. This\ncase allows obtaining an analytical solution for the energy and demonstrating the model\nuse in a clear way, including the AF to FM transitions. An external \feld or anisotropy can\nalso be added. These additional \feld components only change the preferential direction of\nthe system and their e\u000bects can be studied numerically via the perturbation theory, e.g., as\ndone for the ferromagnetic case in Ref. [16].\nWe minimize the terms between the brackets in eq. (8) with respect to the magnetization\nvector. In the absence of an external \feld the system is symmetric with respect to the\npolar angle \u001e. The energy minimization can be accomplished by obtaining the values of\npcr=fmA;cr;mB;cr;\u0012A;cr;\u0012B;crgfor which@F=@ m\u0016jp=pcr=@F=@m\u0016^ r+ 1=m\u0001@F=@\u0012\u0016^\u0012= 0\n8Figure 2: Derivative of the free energy with respect to (a) the magnetization length and (b) the\nangle between the magnetization of the sublattices as a function of the magnetization length m\nand the angle \u0012.\nand@2F=@mi@mjjp=pcr>0. If we use one of the sublattices as the reference of our system,\nwe can set \u0012\u0017= 0 and obtain a solution with respect to the angle only for \u0012\u0016=\u0012, which\nallows reducing the system with 6 degrees of freedom to an equivalent system with 3 degrees\nof freedom for the vector p=fmA;mB;\u0012g. The \frst derivative of \u0003( x) is the Langevin\nfunctionB(x) = coth(x)\u00001=xand the reduced \feld is given by\n\u0018\u0016=\fJ2vuuut\u0002\nm\u0016(1\u00002dm\u0016m\u0017cos(\u0012\u0016\u0000\u0012\u0017)) cos(\u0012\u0016) +m\u0017\u0000j\n2\u00002d\u0000\nm2\n\u0016+m2\n\u0017\u0001\u0001\ncos(\u0012\u0017)\u00032+\n\u0002\nm\u0016(1\u00002dm\u0016m\u0017cos(\u0012\u0016\u0000\u0012\u0017)) sin(\u0012\u0016) +m\u0017\u0000j\n2\u00002d\u0000\nm2\n\u0016+m2\n\u0017\u0001\u0001\nsin(\u0012\u0017)\u00032:\n(21)\nDue to the symmetry of the system, at the equilibrium we expect to have mA=mB=me,\nand\u0018A=\u0018B=\u0018e. This is true when \u0012=n\u0019withn= 0;1;2;::: or whenj= 4dm2\ne.\nThe minimum condition of the energy eq. (5) for meand\u0018eleads to a modi\fed Curie-Weiss\nequation:\nme;\u0016=B(\u0018e(T;me;\u0012\u0016;\u0012\u0017))\u0018e(T;me;\u0012\u0016;\u0012\u0017)\n\u0018e(T;me;\u0012\u0016;\u0012\u0017): (22)\nWe de\fne the value of the critical equilibrium magnetization as mcr=p\nj=4d. When\nthe magnetization of the two sublattices is above the critical value m > mcrthe e\u000bective\n9Figure 3: Equilibrium magnetization meas a function of the angle \u0012for di\u000berent temperature.\nThe black dashed line is the critical equilibrium magnetization mcr=p\nj=4d\nexchange between the two sublattices is AF, and the equilibrium condition is reached for\n\u0012=\u0019. When the magnetization of the two sublattices is below the critical value m < mcr\nthe equilibrium is reached for \u0012= 0 and the material is in the FM state (\fg. 2).\nSince the equilibrium magnetization meand the the e\u000bective exchange are functions of\nthe angle between the two sublattices (\fg. 3), it is possible for the two sublattices to be\nin either the AF or FM con\fguration depending on the previous history of the system (i.e.\nhysteretic behaviour of the phase transition).\nB. LLB analysis\nTo validate the LLB model, we \frst study the phase transition observed in FeRh as a\nfunction of the temperature [14, 15] and, then, the timescale of phase transition as a function\nof\u0015. We conclude this section by presenting an application of our model, for a theoretical\nmaterial exhibiting a ferrimagnetic to ferromagnetic \frst order phase transition. A good\nexample of such material are the Heusler alloys, which show similar ferri- to ferromagnetic\ntransition close to room temperature [17]. Similarly, to FeRh, the phase transition in these\nalloy can be explained via the interaction between the bilinear and the biquadratic exchange\n10Value \u000fUnit\nJ22:44035\u000210\u0000200:7025 J\nj7:7743\u000210\u0000221:5202\nd3:2046\u000210\u0000221:7081\nmcr 0:7788\n\u0016Fe 3:15 \u0016b\nTable I: Corrected magnetic parameters and correction factor \u000f.\n[18, 19].\nWe de\fne the magnetization as the mean of the magnetization in the two sublattices\nM= (MA+MB)=2M0and the N\u0013 eel vector as MN= (MA\u0000MB)=2M0, whereM0=\n(MS;A+MS;B)=2 andMS;A;MS;Bare the saturation magnetization in the two sublattices\n[20]. For sublattices with the same magnetic moments, such as FeRh, the magnetization\nand N\u0013 eel vector are de\fned as:\nM=mA+mB\n2;MN=mA\u0000mB\n2; (23)\nwhere mA,mBare the magnetization vectors of the sublattice A and B, respectively, nor-\nmalized with respect to the saturation magnetization MS;A=MS;B=MS.\nSince by using the MFA, we neglected the higher order wave \ructuations, we update\nthe parameters obtained from the atomistic model for FeRh [11] by a correction factor \u000fto\nmatch the experimental results quantitatively. The correction factors are given in Tab. I. To\navoid using a correction factor, we can obtain J2,j, andddirectly from the experimental\ndata forTCand the phase transition temperatures.\nWe \frst consider an isotropic particle of 5nm \u00025nm\u00025nm initially in the AF state with\na critical atomistic constant \u0015= 1. The temperature is increased step-wise from 1K up to\n720K. At every thermal step, the system is let to relax for 40ps to reach the equilibrium.\nThe magnetization length and the antiferromagnetic N\u0013 eel vector length are obtained by\naveraging over a 20ps period after the system reaches the equilibrium.\nThe particle is let to evolve according to the dynamics described in eq. (18) augmented\nwith the uncorrelated thermal \feld acting on the longitudinal and perpendicular relaxation\nterms of each sublattices described in Ref. [21]. The system is integrated numerically using\na semi-implicit scheme [22] to accurately solve the stochastic di\u000berential equation in a way\n11Figure 4: Magnetization (solid line) and N\u0013 eel vector (dashed line) for an isotropic macrospin of\nFeRh as a function of the temperature.\nthat satis\fed the Stratonovich calculus [23].\nFigure 4 shows the equilibrium magnetization as a function of the temperature. Similar\nto what is done with FM materials, we can relate the Curie temperature of the material\nwith the e\u000bective exchange constant in each sublattice J2(1 +j) = 3kbTC[16]. As shown\nin section IV A, the material is susceptible to a phase transition when the magnetization in\nthe two sublattices is close to the critical value mcr=p\nj=4d. The magnetization of the\nmaterial in the region close to the transition temperature TM(i.e.,m\u0016;e\u0018mcr) is a function\nof the magnetization history of the material. This hysteretic behaviour, expected from the\nanalysis of the free energy and observed in the experiments [11, 14, 15] can be explained by\nlooking at the interaction between the reduced parameters jandd. Due to the presence of\nthe the four-spin exchange, the equilibrium magnetization in the two phases is a function of\nthe material state and it is given by\nme;AF=FM =B(\u0018AF=FM ); \u0018AM=FM =\fJ2\u0014\n1\u0007\u0012j\n2\u00003dm2\u0013\u0015\nm2: (24)\nAt lower temperature T\u001cTMthe contribution of the four-spin interactions is dominant (i.e.,\n12j\u001c6dm2) andme;AF> me;FM while at higher temperatures the cubic component of the\nfour-spin interactions drops faster than the linear component of the nearest neighbors (i.e.\nj\u001d6dm2), leading to me;AF< me;FM. Depending on the initial phase of the system, the\nmagnetization in the two sublattices reaches the critical point mcrat di\u000berent temperatures,\ndepending on the initial con\fguration of the system, hence the hysteresis loop. By controlling\nthe parameters jandd, it is possible to engineer the position and the width of the phase\ntransition.\nTo study the dynamical response of the macro-magnetic particle to a rapid change of\ntemperature, we consider the e\u000bect of a sub-picosecond laser pulse modelled as a Gaussian\nthermal pulse. In FeRh, the initial magnetization response due to an ultrafast thermal pulse\nis observed in the \frst 500fs, signi\fcantly faster than the lattice expansion time that is of the\norder of several ps [9, 24]. In the experiments, a bias \feld is applied in the direction of the\neasy axis for a particle displaying a weak uniaxial anisotropy and the change in longitudinal\nmagnetization Mzis measured through the transient magneto-optics Kerr e\u000bect (MOKE).\nTo simulate the response of such particle to an ultrafast thermal pulse we consider an\nanisotropy parameter \u0011= 0:0001 (equivalent to an HK\u00190:08T) and an applied \feld \feld\nofH= 0:1T, similar to what is used in Ref. [9]. We introduce the heating produced\nby the thermal pulse in our model via a two temperature model (2TM) [25], where the\nmagnetization of the particle is coupled via the e\u000bective electron temperature Te. The 2TM\nis de\fned as\nCe(T)dTe\ndt=\u0000Gel(Te\u0000Tl) +P(t); (25a)\nCldTl\ndt=Gel(Te\u0000Tl); (25b)\nwhereTlis the lattice temperature, Ce=\reTeis the electron speci\fc heat capacity and \re\nis the electron heat capacity constant, Clis the lattice speci\fc heat capacity, and Gelis the\nelectron-lattice exchange. The ultrafast laser pulse is introduced as a Gaussian pulse:\nP(t) =P0exp\u0014\n\u00002:77\u0012t\u00003\u001cpulse\n\u001cpulse\u0013\u0015\n; (26)\nwhere\u001cpulse is the duration of the laser pulse and P0is the nominal optical power. The\nparameters for the 2TM used in the simulations are given in Tab II. The power of the\npulse is chosen such that the electron temperature Terises above the Curie temperature\n13Value Unit\n\re3:5\u000210\u00003J mol\u00001K\u00002\nCl4:45\u0002101J mol\u00001K\u00001\nGel1:05\u00021012J mol\u00001K\u00001s\u00001\nP01:5\u00021016J mol\u00001s\u00001\n\u001cpulse 100 fs\nTable II: Two temperature model parameters for eq. (25).\n(TC= 715K) in the \frst 100fs when the pulse is applied, and Teequilibrates with Tlafter\n\u001ceq= 10ps, where Tl(\u001ceq) is below the phase transition temperature TM\u0019350K.\nThe results for di\u000berent values of the atomistic damping parameter \u0015= 0:01;0:05;0:1\nare shown in \fg. 5. The phase transition depends on the damping parameter. In the low-\ndamping regime ( \u0015= 0:01), the contribution of the transverse intra-sublattice exchange\nH?\nE\u0016to the perpendicular damping is not strong enough for the phase transition to occur in\nthe time scale of the temperature pulse, which is due to the low coupling with the magnetic\nsystem. Higher damping ( \u0015= 0:05) leads to a partial phase transition into the FM phase.\nThis FM phase transition lasts for approximately 20ps before decaying back to the AF\nphase. For still larger damping parameters ( \u0015= 0:10), the perpendicular \feld leads to a\ncomplete transition into the FM phase. The increased stability due to the larger equilibrium\nmagnetization eq. (14) after the cool down leads to the FM state to persists for hundreds\nof picoseconds. Increasing the damping further leads to a faster collapse into the AF phase\ndue to the increased magnitude of the force exercised by the perpendicular intra-sublattice\nexchange in the perpendicular relaxation. The results obtained are consistent with what has\nbeen observed in the experimental results [9] as well as the atomistic simulations [11].\nThe dynamics phase transition observed in the micromagnetic model shows a sharper\ntransition into the FM phase for \u0015= 0:05 than the one observed using the atomistic model.\nThese di\u000berences can be explained by the \fnite dimension e\u000bects in the computation of the\ne\u000bective damping for small particles shown both in theory [26, 27] and numerical simulations\n[28].\nThe presented framework is also applicable to materials with di\u000berent magnetic moments\nin the two sublattices, i.e., for ferrimagnetic materials. To demonstrate the model for such\n14Figure 5: Time dependence of the magnetization for an isotropic particle after laser heating with\na 100fs laser pulse for \u0015= 0:01 (red line), \u0015= 0:01 (green line), and \u0015= 0:1 (blue line). The\nred shaded area de\fnes the electron temperature pro\fle and the green shaded area de\fnes the\nsublattice temperature.\na case, we consider a ferrimagnetic material whith the magnetization moments in the two\nsublattices given by \u0016A= 3\u0016band\u0016B= 1:5\u0016b. We also assume for the two sublattices\ndi\u000berent Curie temperature TC;A= (J2;A+J1)=3kBandTC;B= (J2;B+J1)=3kB. The\nparameters used in the simulations are given in table III.\nFor the considered ferrimagnetic material, which has a magnetic moments \u0016A> \u0016B, if\nTC;A<TC;B, the magnetization in the the sublattice A decreases as a function of the tem-\nperature faster than the magnetization in the sublattice B, \fg. 6(a). The material, initially\nin the ferrimagnetic phase, transitions to the ferrimagnetic phase, \fg. 6(b), when the mag-\nnetization length is below the critical magnetization for each sublattice (i.e. me;\u0016<mcr;\u0016).\nThe Curie temperature of the material is given by the largest of the Curie temperatures of\nthe sublattices (i.e. TC\u0019TC;B).\nFor a material with J2;A6=J2;B, we cannot assume me;A=me;B=meas we did in\n15Value Unit\nTC;A 411 K\nTC;B 531 K\nJ2;A1:357\u000210\u000020J\nJ2;B1:764\u000210\u000020J\nJ13:751\u000210\u000022J\nDhQi0:808\u000210\u000022J\n\u0016A 3:0\u0016b\n\u0016B 1:5\u0016b\nTable III: material parameters for an hypothetical Heusler alloy showing a \frst order ferri- to\nferromagnetic transition near room temperature\nsection IV A, and we expect mcr;A6=mcr;B. However, the critical magnetization mcr=\np\nJ1=4DhQiobtained in the previous section can still be used to estimate the transition\ntemperature of the material, since it can be observed that the critical magnetization for the\nfaster decaying sublattice is mcr;A/mcr.\nThe N\u0013 eel vector decreases until it becomes zero at the compensation temperature TCP,\nwhereMA(TCP) =MB(TCP). It increases for T > TCPwhenMB(T)> MA(T) up to a\nmaximum before going back to zero at the Curie temperature.\nV. CONCLUSIONS\nWe presented a micromagnetic formulation for modeling ferrimagnetic materials at low\nand high temperatures, including cases with metamagnetic (AF to FM) phase transitions.\nThe model is based on a mean \feld approximation (MFA) of the system energy that is used\nto derive an LLB equation. The ferrimagnet is described micromagnetically by two coupled\nsublattices as in the previous work by Atxitia et al. [13]. However, our model includes one\ninter- and one intra-sublattice micromagnetic exchange. In addition, four-spin interactions\nare introduced via an inter-sublattice molecular \feld and a perpendicular molecular \feld\nwith a cubic dependence in the magnetization of the two sublattices. The LLB equation is\npresented in two forms: a general form and a form simpli\fed under the assumption of a strong\nhomogeneous exchange \feld, which is applicable to most ferromagnetic and ferrimagnetic\n16Figure 6: (a) Magnetization as a function of the temperature for the two sublattices of a ferrimag-\nnetic material described by the parameters in table III. (b) Normalized magnetization and N\u0013 eel\nvector for the ferrimagnetic material.\nmaterials.\nThe presented formulation was used for modeling the thermal equilibrium and metam-\nagnetic phase transitions in FeRh. The simulations show that the origin of such transitions\nis in the inter-sublattice molecular \feld obtained from the nearest-neighbors and second-\nnearest neighbors as well as the molecular \feld with cubic dependence in the magnetization\nobtained from the four-spin interactions [11, 29]. The formulation reproduces the hysteretic\nAF to FM transition behaviour and time scales observed in recent experiments [9, 14, 15]\n17and atomistic simulations [11].\nThe model we developed can be considered as an extension of previous micromagnetic\nmodels and it is able to simulate ferrimagnetic materials showing similar \frst-order phase\ntransitions, like Heusler alloys [17], and it can be used to model a wide range of ferrimagnetic\nmaterials and phenomena, including recently observed all-optical driven THz spintronic\ne\u000bects observed in FeRh [5, 8] as well as memory application that exploit phase transitions\n[30].\nVI. ACKNOWLEDGMENTS\nThis work was supported as part of the Quantum-Materials for Energy E\u000ecient\nNeuromorphic-Computing(Q-MEEN-C), an Energy Frontier Research Center funded by the\nU.S. Department of Energy, O\u000ece of Science, Basic Energy Sciences under Award No. DE-\nSC0019273. The authors thank Professor Prof Roy Chantrell and Dr. Mara Strungaru for\nthe help with the atomistic modelling as well as Dr. Joseph Barker for the helpful con-\nversation. For simulations, this work used the Extreme Science and Engineering Discovery\nEnvironment (XSEDE), which is supported by National Science Foundation grant number\nACI-1548562, speci\fcally, it used the Bridges and Comet systems supported by NSF Grant\n# ACI-1445506 at Pittsburgh and San Diego Supercomputer Centers.\n18\u0003Electronic address: menarini.marco@gmail.com\n1E. Gomonay and V. Loktev, Low Temperature Physics 40, 17 (2014).\n2V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Reviews of Modern\nPhysics 90, 015005 (2018).\n3R. Khymyn, I. Lisenkov, V. Tiberkevich, B. A. Ivanov, and A. Slavin, Scienti\fc reports 7, 1\n(2017).\n4R. Khymyn, I. Lisenkov, J. Voorheis, O. Sulymenko, O. Prokopenko, V. Tiberkevich, J. Aker-\nman, and A. Slavin, Scienti\fc reports 8, 1 (2018).\n5R. Medapalli, G. Li, S. K. Patel, R. Mikhaylovskiy, T. Rasing, A. Kimel, and E. Fullerton,\nApplied Physics Letters 117, 142406 (2020).\n6H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Takahashi, Y. Kajiwara, K.-i. Uchida, Y. Fu-\njikawa, and E. Saitoh, Physical Review B 85, 144408 (2012).\n7R. Cheng, D. Xiao, and A. Brataas, Physical review letters 116, 207603 (2016).\n8M. Menarini, R. Medapalli, E. E. Fullerton, and V. Lomakin, AIP Advances 9, 035040 (2019).\n9G. Ju, J. Hohlfeld, B. Bergman, R. J. M. van de Veerdonk, O. N. Mryasov, J.-Y. Kim, X. Wu,\nD. Weller, and B. Koopmans, Physical review letters 93, 197403 (2004).\n10O. N. Mryasov, Phase Transitions 78, 197 (2005).\n11J. Barker and R. W. Chantrell, Physical Review B 92, 094402 (2015).\n12D. A. Garanin, Physical Review B 55, 3050 (1997), URL https://link.aps.org/doi/10.\n1103/PhysRevB.55.3050 .\n13U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko, Physical Review B 86, 104414 (2012).\n14J.-U. Thiele, S. Maat, and E. E. Fullerton, Applied Physics Letters 82, 2859 (2003).\n15J. Kouvel and C. Hartelius, in Proceedings of the Seventh Conference on Magnetism and Mag-\nnetic Materials (Springer, 1962), pp. 1343{1344.\n16H. Kachkachi and D. Garanin, Physica A: Statistical Mechanics and its Applications 291, 485\n(2001).\n17M. Ovichi, M. Ghahremani, E. Della Torre, L. H. Bennett, F. Johnson, and V. Srivastava,\nJournal of Applied Physics 115, 17A906 (2014).\n18E. Simon, A. Donges, L. Szunyogh, and U. Nowak, Physical Review Materials 4, 084408 (2020).\n1919S. Bosu, Y. Sakuraba, K. Saito, H. Wang, S. Mitani, and K. Takanashi, IEEE Transactions on\nMagnetics 44, 2620 (2008).\n20C. C. Chiang, S. Y. Huang, D. Qu, P. H. Wu, and C. L. Chien, Physical review letters 123,\n227203 (2019).\n21M. Menarini and V. Lomakin, Physical Review B 102, 024428 (2020).\n22J. Mentink, M. Tretyakov, A. Fasolino, M. Katsnelson, and T. Rasing, Journal of Physics:\nCondensed Matter 22, 176001 (2010).\n23P. E. Kloeden and E. Platen, in Numerical Solution of Stochastic Di\u000berential Equations\n(Springer, 1992), pp. 103{160.\n24J.-U. Thiele, M. Buess, and C. H. Back, Applied Physics Letters 85, 2857 (2004).\n25J. Mendil, P. Nieves, O. Chubykalo-Fesenko, J. Walowski, T. Santos, S. Pisana, and M. M unzen-\nberg, Scienti\fc Reports 4, 3980 (2014), URL http://dx.doi.org/10.1038/srep03980 .\n26D. A. Garanin, Physica A: Statistical Mechanics and its Applications 172, 470 (1991).\n27D. A. Garanin and O. Chubykalo-Fesenko, Physical Review B 70, 212409 (2004).\n28M. Strungaru, S. Ruta, R. F. L. Evans, and R. W. Chantrell, Physical Review Applied 14,\n014077 (2020).\n29O. N. Mryasov, Phase Transitions 78, 197 (2005), https://doi.org/10.1080/01411590412331316591,\nURL https://doi.org/10.1080/01411590412331316591 .\n30I. Fina, N. Dix, E. Menendez, A. Crespi, M. Foerster, L. Aballe, F. Sanchez, and J. Fontcuberta,\nACS applied materials & interfaces 12, 15389 (2020).\n20"    },    {        "title": "1711.05808v1.Octahedral_tilt_independent_magnetism_in_confined_GdTiO__3__films.pdf",        "content": "arXiv:1711.05808v1  [cond-mat.mes-hall]  15 Nov 2017Octahedral tilt independent magnetism in confined GdTiO 3films\nR. F. Need,1B. J. Isaac,1B. J. Kirby,2J. A. Borchers,2S. Stemmer,1and S. D. Wilson1,∗\n1Materials Department, University of California, Santa Bar bara, California 93106, USA\n2NIST Center for Neutron Research, National Institute of Sta ndards and Technology, Gaithersburg, Maryland 20899 USA\nPolarized neutron reflectometry measurements are presente d exploring the evolution of ferrimag-\nnetism in GdTiO 3films as they are confined between SrTiO 3layers of variable thicknesses. As\nGdTiO 3films approach the thin layer limit and are confined within a su bstantially thicker SrTiO 3\nmatrix, the TiO 6octahedral tilts endemic to GdTiO 3coherently relax toward the undistorted, cubic\nphase of SrTiO 3. Our measurements reveal that the ferrimagnetic state with in the GdTiO 3layers\nsurvives as the TiO 6octahedral tilts in the GdTiO 3layers are suppressed. Furthermore, our data\nsuggest that a magnetic dead layer develops within the GdTiO 3layer at each GdTiO 3/ SrTiO 3\ninterface. The ferrimagnetic moment inherent to the core Gd TiO3layers is negligibly (in models\nwith dead layers) or only weakly (in models without dead laye rs) impacted as the octahedral tilt\nangles are suppressed by more than 50% and the t2gbandwidth is dramatically renormalized.\nComplex oxide thin films and interfaces continue\nto constitute an exciting frontier in condensed mat-\nter physics where layer thickness, interfacial strain, and\nchemistry can be used to tune competing interactions\nand generate emergent ground states1,2. This tunability,\nwhencombinedwith strongelectron-electroncorrelations\nin these systems, results in a rangeofelectronic andmag-\nnetic ground states unique from their bulk components,\nsuch as interfacial ferromagnetism3,4, metal-to-insulator\ntransitions5, and voltage-tunable superconductivity6,7.\nWithin the realm of engineered heterostructures,\nABO3perovskites have received considerable attention,\ndue in part to the wide range of possible chemistries and\nthe atomic precision with which multilayer films can be\nfabricated. For many bulk perovskites, the A-site cation\nis too small for the perovskite structure to retain cu-\nbic (Pm3m) symmetry. The consequence is a coopera-\ntive distortion (i.e. tilts and rotations) of the BO 6oc-\ntahedra network that may take one of multiple possible\npatterns8and is proportional in magnitude to the Gold-\nschmidt tolerance factor9. As the radius of the A-site\ncation decreases, the structural distortions increase lead-\ningtomovementoftheB-O-Bbondangleawayfrom180◦\nand a corresponding decrease in orbital overlap that can\neffect both electronic and magnetic properties10–12.\nThese cooperative distortions are altered from their\nbulk patterns near a heterointerface of two dissimi-\nlar perovskite films (i.e. ABO 3/A′B′O3)13. Which\noctahedral network distorts and the degree of distor-\ntion can be intentionally engineered by the choice layer\nthicknesses and interfacial strain to generate emergent\nproperties14,15. For example, interfacial octahedral engi-\nneering has been successfully employed to enhance ferro-\nelectric polarization in CaTiO 3/BiFeO 3superlattices16,\nmagnetism in LaMnO 3/SrTiO 3superlattices17, and to\nmanipulate quantum criticality in SmTiO 3/SrTiO 3and\nGdTiO 3/SrTiO 3quantum wells18,19.\nParticularly fascinating phenomena appear at engi-\n∗stephendwilson@engineering.ucsb.eduneered GdTiO 3/SrTiO 3interfaces. In the bulk, the Mott\ninsulator GdTiO 3(GTO) possesses GdFeO 3-type distor-\ntions in its TiO 6octahedra network while the band in-\nsulator SrTiO 3(STO) possesses the undistorted parent\ncubic structure at room temperature20,21. By interfacing\nthin epitaxial layers of GTO and STO, the octahedral\ntilts inherent to each layer can be coherently controlled\nwithdramaticeffectsonthefreecarriersgeneratedbythe\npolar discontinuity at the interface. For instance, trans-\nport measurements have shown that this system goes\nthrough a Mott-Hubbard-like metal-to-insulator transi-\ntion when carriers within STO quantum wells 2 uc (unit\ncells) thick or less are sandwiched between relatively\nthick GTO layers5. In samples with thin GTO lay-\ners, SQUID magnetometry has suggested a critical GTO\nthickness of 6 uc (2 nm), below which GTO transitions\nfromits bulk ferrimagneticstate22,23into aparamagnetic\nstate in conjunction with a 33% reduction in Ti-O-Ti\nbond angles in the center of the GTO layers24. This im-\nplies an ability to exert fine control over the magnetic\nstate of GTO through interfacial manipulation of its oc-\ntahedral tilts.\nIn this paper, we explore the coupling between octa-\nhedral tilting and magnetism in confined GTO films by\nusing polarized neutron reflectometry (PNR) to probe\ntheirinterplayin thin GTOlayers. Surprisingly, ourdata\nshow no evidence of a ferrimagnetic-paramagnetic tran-\nsition near the thin well limit, but rather that GTO re-\nmains ferrimagnetic down to layers as thin as 4 uc (1.6\nnm). The magnetization curves extracted from the PNR\ndata are analyzed using models both with and without\nmagnetic dead layers (MDLs) in the GTO. When exam-\nined using a model with no MDLs, the thinnest GTO\nlayers show ≤23% suppression in the apparent, satu-\nrated magnetization. Inclusion of MDLs into the PNR\nmodel results in better fits to the experimental data, and\na magnetization response that is independent of GTO\nlayer thickness. Our results indicate that the substantial\nrelaxation of TiO 6octahedral tilts in GTO/STO inter-\nfaces at the thin GTO layer limit has minimal impact on\nthe magnetically ordered state. More broadly, this im-\nplies that ferrimagnetism in GTO is largely independent2\n10-7 10-5 10-3 10-1 \nq [Å-1]10-7 10-5 10-3 10-1 \nReflectivity\n0.4 0.3 0.2 0.1 a \n c 4.4 nm GTO layers1.6 nm GTO layers\nFIG. 1. X-ray reflectometry data and calculated fits to two rep resentative GTO-STO superlattices measured in this study: (a)\n1.6 nm or 4 uc thick GdTiO 3layers, and (c) 4.4 nm or 11 uc thick GdTiO 3layers. The refined models from which the curve\nfits were calculated are shown in panels (b) and (d) for thin an d thick GTO samples respectively.\nof the interface-engineered t 2gbandwidth.\nSuperlattice samples ofalternatingGTO and STO lay-\ners were grown for this study using hybrid molecular\nbeam epitaxy as described elsewhere25–27. The degree\nof distortion/tilting within the GTO titania octahedra\nnetwork was controlled by varying the thickness of the\nGTOlayers. Previousscanningtransmissionelectronmi-\ncroscopy (STEM) measurements of Gd-O-Gd bond an-\ngles are used as a proxy for the relation between layer\nthickness and octahedra tilts24. The thin GTO superlat-\ntice contained 4 uc (1.6 nm) GTO layers, in which all of\natomic planes within the GTO were distorted from their\nbulk tilting pattern by 50% or more. The thick GTO su-\nperlattice had 11 uc (4.4 nm) GTO layers, in which the\nbulk GTO tilt structure was present throughout the en-\ntirety of the layers with the exception of the one unit cell\nat each interface where tilts are suppressed as the titania\nnetwork transitions into the neighboring STO. For this\nsample, thin STO spacers (0.6 nm) were used to reduce\ndistortions to the interfacial GTO tilts.\nPolarized neutron reflectometry measurements were\nperformed on the PBR reflectometer at the NIST Center\nfor Neutron Researchwith an incident wavelengthof4.75\n˚A. Samples were mounted in a cryostat with the film’s\nsurface normal to the scattering wavevector, q. PNR\nmeasurements were collected in a zero field cooled (ZFC)\nstate by cooling the sample from well above the Curie\ntemperature to 5 K under zero field, then polarizing the\nsample to µ0H = 3 T applied in the plane of the film and\ncollecting PNR scans as the field was stepped back to\nzero. The layer thicknesses and interface quality of these\nsamples were characterized using non-resonant, unpolar-\nized x-ray reflectometry (XRR) performed with a Cu K αlab diffractometer. XRR measurements were performed\nin air at room temperature. All reflectometry data sets\nwere refined to slab layer models using the Refl1D code\nthat implements an optical matrix formalism28,29.\nFigures1(a)and 1(c) containthe XRR dataand model\nfits for thin and thick GTO superlattices, respectively.\nThe refined structural models corresponding to these\nsamples are shown in the Figs. 1(b) and 1(d). During\nrefinement, all layers were allowed to have an indepen-\ndently refined thickness, but layer chemistry and inter-\nface roughness were confined to be uniform for all layers\nof a given type in order to reduce the number of free pa-\nrameters. The topmost layer in each sample was allowed\nto be unique in order to account for surface degradation\nknown to occur in rare earth transition metal oxides30.\nAverageGTOlayerthicknesseswererefined tobe 4.48(6)\nnm forthe thick GTOsampleand 1.57(6)nm forthe thin\nGTO sample and are in near perfect agreement with the\ndesigned structures. The apparent chemical roughnesses,\nwhich are effectively averaged over the entire x-ray beam\nspot (≈10 mm2), span a small range from 2.3 ˚A- 4.4˚A\nand attest to the excellent quality of these films. Pre-\nvious reflectometry and electron microscopy studies on\nthis system suggest local interfaces are in fact atomi-\ncally sharp4,21, and the apparent roughness values arise\nfrom steps on the substrate surface propagating upwards\nthrough the film, rather than chemical intermixing.\nWe begin by analyzing PNR data for the thin GTO\nsample with 4 uc GTO layers using a magnetization\nmodel without dead layers, similar to that previously\napplied to the GTO/STO system4. Figure 2(a) shows\nPNR data collected at 5 K after cooling under zero field\nthen applying µ0H = 3 T at base temperature. Data3\nFIG. 2. Polarized neutron reflectometry data and refined fits\nfor (a) thin GTO layer and (b) thick GTO layer superlattices\nmeasured in a ZFC state under a µ0H = 3 T applied field.\nrefinement shows that the thin GTO layers still exhibit\na net in-plane magnetization and reach 2.7(1) µB/fu in\nthe center of the GTO layers under the assumption of no\nmagnetic dead layers. While this is lower than the 3.5(1)\nµB/fu observed in the thick GTO superlattice (Fig. 2\n(b)), the 23% magnetization reduction observed here is\nsignificantly less than the 85% reduction observed using\na volume-averagedtechnique24. The survival of bulk-like\nmagnetism at this thin well limit where TiO 6octahedral\ntiltshavebeensuppressedbyover50%issurprising24and\ndeviates from the current picture of completely quenched\nmagnetism at this limit. This contradiction in the ap-\nparent suppression between depth-resolved and volume-\naveraged probes suggests the presence of magnetic dead\nlayers that create a finite thickness effect.\nTherefore, the data were reanalyzed incorporating\nMDLs into the layer model of the multilayer film. A\nnumber of different MDL models were compared with\nthe best models providing better visual and numerical\nfits to the PNR data than models without MDLs31. The\nmost descriptive model is shown in Fig. 3 where refined\nmagnetization profiles of the thin GTO superlattice with\nand without MDLs are overlaid on the chemical layers.\nThis model has matching MDLs on both sides of the\nGTO layer that begin at the chemical GTO/STO inter-\nfaceandextend2.5 ˚AintotheGTOlayer(i.e. noneofthe\nMDLiscontainedinthe STOlayers). Roughnessesofthe\nMDLs were constrained to be no smaller than the chem-\nical roughnesses of the interfaces where the MDLs were\nlocated. The justification for this roughness constraint\nstems from the interpretation of the local chemical in-\nterface roughnesses arising from the stepped substrate,FIG. 3. Refined GTO layer magnetization profiles for the\nthin GTO superlattice under the assumptions of no MDLs\nand 2.5˚AMDLs. These profiles are overlaid on a schematic\nrepresentation of the best MDL model, in which the MDLs\n(grey regions) begin at the chemical GTO/STO interface and\nextend 2.5 ˚Ainto the GTO layers.\nwhich implies these values represent lower limits below\nwhich roughness values lose physical meaning. Because\nof the small MDL thicknesses relative to the chemical\nroughness, roughnesses were propagated across multiple\ninterfaces when calculating reflectometry profiles.\nWithin this model, the moments in the center of the\nGTO layers increase to 3.9(1) µB/fu and 3.8(1) µB/fu\nfor samples with 1.6 nm and 4.4 nm GTO thicknesses,\nrespectively. The larger increase in moment seen in the\nthin GTO samplehighlights the proportionalrelationbe-\ntween the refined magnetization and the relative volume\nfraction of GTO layers lost due to the addition of MDLs.\nApplyingthe MDLmodeltothe entireZFCdatasetfor\nboth thick and thin GTO samples results in a field po-\nlarized magnetization that is independent of GTO layer\nthickness, as shown by the filled symbols in Fig. 4. The\nthin GTO superlattice refined to a model with no MDLs\nis also included as a reference. The magnetization data\non both films are characterized by little to no remnant\nmagnetization upon field removal and a slow onset of\nsaturation that agree well with previously reported mag-\nnetometry data from bulk GTO22. Single ion param-\nagnetism is ruled out as a possible explanation of this\ndata due to the well-defined order parameter measured\nin these films4,24and the temperature dependence of the\nmagnetization that disagreese with predictions from a\nBrillouin function31.\nThese combined results suggest that the apparent sup-\npression of magnetization, in this work and also the pre-4\nFIG. 4. GTO magnetic moment values determined via refine-\nment of PNR data measured in a ZFC state. Moments refined\nwith an MDL model are shown by filled symbols. Open sym-\nbols show the refinedmoments for the thin GTO sample when\nno MDL are included.\nvious SQUID magnetometry study24, is likely an effect\nof neglecting the magnetic dead layers at the GTO/STO\ninterface and instead averaging magnetization over the\nentire GTO layer. When these dead layers are incorpo-\nrated into a model of these systems, the two PNR data\nsets collapse onto one another, indicating that the mag-\nnetism in the center of the GTO layers is independent\nof the interface-induced octahedral tilting. From the re-\nportedbondanglesinGTO/STOheterostructures24, this\nis true up to at least a 50% change in distortion of the\noctahedral network from its preferred bulk pattern (Ti-\nO-Tiangle ≈144◦) towardsanundistortedstructure(Ti-\nO-Ti angle = 180◦).\nWe stress here that even absent the presence of mod-\neled MDLs, the observed ferrimagnetism in 4 uc thick\nGTO is only suppressed 23% relative to bulk-like, 11 uc\nthick GTO. This is a surprisingly weak perturbation to\nthe magnetism given the known alteration of the octa-\nhedral tilt structure in these thin GTO layers and an\nunambiguous demonstration that robust ferrimagnetism\npersists well below the previously reported bound of 6 uc\nthick GTO layers.\nAdditional support for the inclusion of MDLs into\nthe model of GTO/STO interfaces comes from the fre-\nquency with which heterointerfaces result in the for-\nmation of MDLs near the interface. MDLs are often\nobserved in both ferromagnetic metals32–34and oxides\nsuch as La 1−xSrxMnO3(LSMO) and La 1−xCaxMnO3\n(LCMO)35–39. The origins of these MDLs are typically\nunique to the interface in question. While structural dis-tortions are a common source of MDLs, that explanation\nis ruled out in the GTO/STO system because the in-\nterfacial, MDL-containing unit cell in thick ( ≥3.5 nm)\nGTO is distorted by approximately 50%, the same level\nof distortion that is present in the center of thin (1.6 nm)\nGTO layers that show unperturbed ferrimagnetism.\nAnother possible source of MDLs is orbital reconstruc-\ntion at the interface. This is particularly relevant for ox-\nide heterostructures where interfacial orbital reconstruc-\ntion is regularly observed40–43. In the case of thin LSMO\nlayers, x-ray measurements have shown that the 3z2-r2\norbital is preferentially occupied, leading to a weaken-\ning of the double exchange responsible for LSMO’s FM\nand resulting in its observed MDLs44,45. In bulk GTO,\nfirst principles calculations suggest both orbital ordering\nand FM are stabilized by a hybridization of the t 2g-eg\norbitals10. This hybridization is due to the GdFeO 3-type\noctahedraldistortionand, asthat distortionis decreased,\nFM exchange is weakened. Thus while evidence for or-\nbital reconstructionin GTO/STOhasyet to be reported,\nit is possible to speculate towards a case where, either\nvia compressive strain or symmetry breaking at the in-\nterface, anorbitalreconstructionoccurs. This mayresult\nin decreased t2g-egoverlapand hybridizationpushing the\nsystem towards a FM-AFM instability, but this is not\ndirectly reflected in the reported Gd-O-Gd bond angles\nthat have been used as a proxy for octahedral tilting and\nrotations in this study.\nIn summary, PNR was used to explore the relation-\nship between the cooperative structural distortion of the\nTiO6octahedra network and the ferrimagnetic state in\nGTO thin films. PNR measurements provide evidence\nthat ferrimagnetism in GTO layers survives as the sin-\ngle layer limit is approached. Specifically, the saturated\nmoment of the ferrimagnetic state in GTO layers as thin\nas 4 uc is reduced by only 23% relative to bulk-like lay-\ners in models neglecting the potential presence of MDLs\nand becomes identical to bulk-like layers once models in-\ncorporating MDLs are used. Incorporating thin MDLs\nat GTO/STO interfaces improves refined models of PNR\ndata; howeveranalysisofthe data within either approach\nrevealsthatthemagnetizationintheinteriorofGTOlay-\ners (excluding MDLs) is largely independent of changes\nin octahedral tilts and rotations as measured by Ti-O-Ti\nbond angles. Our data curiously point toward a picture\nofcorrelatedmagnetism in GTOwhich is decoupled from\nthe modified octahedral tilts thought to drive the metal-\ninsulator instability in this compound.\nACKNOWLEDGMENTS\nThe authors thank B.B. Maranville and A. Green for\ndevelopment of the Refl1D code with roughness propaga-\ntion. S.W., R. 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Lett. 105,\n131906 (2014)."    },    {        "title": "2005.03965v3.Sublattice_magnetizations_of_ultrathin_ferrimagnetic_lamellar_nanostructures_between_cobalt_leads.pdf",        "content": "arXiv:2005.03965v3  [cond-mat.mes-hall]  12 Apr 2023SPIN\nVol. 1, No. 1 (2022) 1–11\n©World Scientific Publishing Company\nSublattice magnetizations of ultrathin ferrimagnetic lam ellar\nnanostructures between cobalt leads\nVinod Ashokan*\nDepartment of Physics,\nDr. B. R. Ambedkar National Institute of Technology,\nJalandhar (Punjab) 144 027, India\nashokanv@nitj.ac.in\nA. Khater\nDepartment of Physics, Le Mans University, 72085 Le Mans, Fr ance;\nDepartment of Theoretical Physics, Jan Dlugosz University, Czestochowa, Poland\nM. Abou Ghantous\nScience Department, American University of Technology,\nFidar Campus, Halat, Lebanon\nIn this work we model the salient magnetic properties of the alloy lame llar ferrimagnetic nanos-\ntructures [ Co1−cGdc]ℓ′[Co]ℓ[Co1−cGdc]ℓ′between Cosemi-infinite leads. We have employed the\nIsing spin effective field theory (EFT) to compute the reliable magnet ic exchange constants for\nthe pure cobalt JCo−Coand gadolinium JGd−Gdmaterials, in complete agreement with their\nexperimental data. The sublattice magnetizations of the CoandGdsites on the individual hcp\natomic (0001) planes of the Co−Gdlayered nanostructures are computed for each plane and\ncorresponding sites, by using the combined EFT and mean field theor y (MFT) spin methods.\nThe sublattice magnetizations, effective site magnetic moments, an d ferrimagnetic compensa-\ntion characteristics for the individual hcp atomic planes of the embe dded nanostructures, are\ncomputed as a function of temperature, and for various stable eu tectic concentrations in the\nrangec≤0.5. The theoretical results for the sublattice magnetizations and the local magnetic\nvariablesof these ultrathin ferrimagneticlamellar nanostructured systems, between cobalt leads,\nare necessary for the study of their magnonic transport proper ties, and eventually their spin-\ntronic dynamic computations. The method developed in this work is ge neral and can be applied\nto comparable magnetic systems nanostructured with other mate rials.\nKeywords : effective field theory; mean field theory; sublattice magnetization ; exchange constant;\ncobalt-gadolinium alloy; ferrimagnetic nanojunction.\n1. Introduction\nThe multilayered lamellar nano-magnetic nanos-\ntructurehas madea tremendousprogress in prepar-\ning and analyzing the physical properties of nano-\nmagnetic layered nanostructures and magneticnano-junctions. These systems have a panoply of\nindustrial and technological applications in the ar-\neas of spin wave magnonics1;2;3;4, and spintron-\nics5;6;7. However, the study of nano-magnetic\nlamellar multilayered nanostructure and nanojunc-\n∗Corresponding author\n12Vinod Ashokan; A. Khater and M. Abou Ghantous\ntion composite of rare earth-transition metal alloy\nsystems are still in its infancy. A fundamental and\nintriguing interest associated with such rare earth-\ntransition metal systems is to understand the phe-\nnomenawhichmay ariseduetothedecreaseoftheir\nsize when surface and quantum mechanical effects\ncome into play. The atomic interfaces and sublat-\ntice magnetization in these nanostructures are be-\ncome critical components for the physics of embed-\nded nano-junctions.\nThe nano-magnetic properties of ferrimag-\nnetic alloy nanostructure of rare earth-transition\nmetal has been reported in the literature\n8;9;10;11;12;13;14. The nano fabrication tech-\nnique has made the experimental progress to real-\nize the thin multilayered nanostructures with novel\nphysical properties and promising applications in\nmagnonic devices. In this work we study in par-\nticular the Co-Gd rare earth-transition metal alloy\nsystem; the transition metal Coand rare earth Gd\nare ferromagnetic with their Curie temperatures of\n119.4 and 25.2 meV respectively. Nano-magnetic\nnanojunctions build from Co-Gd alloys in diverse\nmultilayer formats can present hence very useful\nproperties for technological applications at room\ntemperature. In this respect, the bulk Co 1−cGdcal-\nloy materials with different alloy concentrations c,\nhave been studied intensively in the past for diverse\napplications in sensors,magneto optical devices and\nmagnetic storage elements15;16.\nA greater understanding of lamellar Co/Gd\nmultilayers nanostructures has been achieved\n9;11;12. It is noted that when we form an amor-\nphous alloy Co1−cGdcin these systems there is a\nstrong asymmetric spontaneous diffusion of Cointo\ntheGdplane occurs, and interfaces for various con-\ncentration. The experimental techniques made pos-\nsible to control the interdiffusion for some stable\neutectic concentration c≤0.5 while preserving the\nferrimagnetic structure of the multilayer systems.\nThe properties of multilayers presenting alloy in-\nterfaces, with few lamellar atomic planes thick sig-\nnificantly depends on the degree of material inter-\ndiffusion9;11. Such interdiffusion may play crucial\nrole in determining the nano-magnetic properties\nof the multilayers systems8, because the individ-\nual planes are having different exchange couplings\nat their interfaces from the bulk. This has also been\nreported earlier by model calculations which show\nthat nano-atomic scale magnetic alloyed interfaces\ncan significantly modify the nano-magnetic proper-ties of multilayer systems17;18;19. The prepara-\ntion of alloy like and composition stable nanojunc-\ntions composed of CoandGdbetween cobalt leads\nis hence possible in principle experimentally due to\nthe method of controlled interdiffusion process.\nIt should be noted that there has been at-\ntempts in the past to model the magnetic proper-\nties of bulk and layered cobalt-gadolinium systems\n13;20;21usingthe MFT method. These model cal-\nculations have been performed by adjusting in gen-\neral the MFT results to fit the experimental data,\nusing the cobalt spin SCoas a fitting parameter. In\nsome of the calculations where SCois assigned its\nfundamental value, the overall fit function for the\nmagnetization with temperature does not give bet-\nter agreement with the experimental data. Further-\nmore, theasymmetricchoice of nearestneighbor ex-\nchange constants for cobalt JCo−Coand gadolinium\nJGd−Gdis made in these references to reduce the\nnumber of adjustable parameters, but without giv-\ning any fundamental justification. To add to this\ncomplex situation, there is a wide array of exper-\nimental values of exchange constant for the cobalt\nJCo−Coand gadolinium JGd−Gdare available in the\nliterature from different types of measurements22,\nwhich does not help to clarify the situation for ad-\nvanced modeling.\nIn our previous work23, we computed the\nballistic and scattering transport properties of\nspin waves (SW) incident from cobalt leads, on\nto the embedded ultrathin ferrimagnetic cobalt-\ngadolinium [ Co1−cGdc]ℓnanojunction systems be-\ntween the leads. The nanojunction [ Co1−cGdc]ℓis\nprinciple a randomly disordered alloy with varied\nhcp atomic palnes ℓbetween matching hcp planes\nof theColeads, at known stable concentrations\nc≤0.5 for this nano-alloy system. To be able to\ncarry out these computations it was necessary to\ncompute the sublattice magnetizations and mag-\nnetic exchange constants in this system24.\nIn the present work we have modeled the\nsublattice magnetizations and magnetic exchange\nconstants of the alloy layered nanostructures\n[Co1−cGdc]ℓ′[Co]ℓ[Co1−cGdc]ℓ′sandwiched between\nsemi-infinite cobalt leads, at concentrations c≤\n0.5. These triple-nanostructure systems are more\ncomplex than the previously studied single-\nnanostructure systems24. The present work is\nhence motivated by the objective to present a\nmore complex computational modeling of the\nsalient nano-magnetic properties of the triple-Sublattice magnetizations of nano-magnetic layered mater ials3\nnanostructures for fundamental interest, and the\nballistic transport and scattering of spin waves in-\ncident from cobalt leads on complex ferrimagnetic\ncobalt-gadolinium nanojunction systems25. Such\ncomplex systems have as it turns out a richer and\nwider range of spin wave filtering properties. The\ncomplex embedded triple-nanostructures are de-\nnoted symbolically henceforth by [ ℓ′ℓℓ′] for conve-\nnience, corresponding to the alternating alloy and\npure nanostructures. The basal hcp (0001) atomic\nplanesofthe ...Co][ℓ′ℓℓ′][Co...nanojunctionsystems\nare normal to the direction of the c-axis itself con-\nsidered to be along the direction of the leads.\nThe alloy layered [ ℓ′ℓℓ′] nanostructures under\nconsideration are ultrathin ∼1.5 nm composite\ncobalt-gadolinium alloy systems sandwiched be-\ntween Co leads, and are hence different from bulk\nalloy and multilayer systems8;21;13. Also, due to\nthe absence of first principle calculations for Co-\nCo and Co-Gd exchange in the alloy layered [ ℓ′ℓℓ′]\nnanostructures.There is hence effectively a need for\nreliable data for the exchange and sublattice mag-\nnetizations in these systems to be able to develop\nmodelingstudiesfor the ...Co][ℓ′ℓℓ′][Co...nanojunc-\ntions which are key elements for ballistic spin wave\ntransport in magnonic devices23;26;27;28. This\nneed has motivated our EFT calculations to deter-\nmine such data with no fitting parameters, using\nbasic values SCo= 1 and SGd= 7/2 as the spin\nreferences at absolute 0 K.\nThe structure of the paper is as follows. In sec-\ntion 2, the EFT Ising spin method with experimen-\ntal data is computed for the reliable JCo−Coand\nJGd−Gdexchange for the purecrystalline cobalt and\ngadolinium materials. These are then attributed to\nnearest neighbor Co−CoandGd−Gdinterac-\ntions in the ...Co][ℓ′ℓℓ′][Co...nanojunction systems\nfor eutectic stable concentrations c≤0.5. The com-\nbined EFT and MFT methods are presented in sec-\ntion 3, to compute sublattice magnetizations for the\ncobalt and gadolinium sites on the individual hcp\nbasal atomic planes of the alloy layered [ ℓ′ℓℓ′] lamel-\nlar nanostructures as a function of temperature,\nwith thicknesses [2′22′] and [3′33′], and for differ-\nent alloy concentrations c. The sublattice magne-\ntizations and corresponding ferrimagnetic compen-\nsation temperatures are shown in this section. The\noverall discussions and conclusions are presented in\nsection 4.2. EFT modeling of pure CoandGd\nsystems\nTheEFT modeling incorporates the contribution of\nthe single site spin correlations to the order param-\neter, and hence known to be superior to the MFT.\nWe use it in the present work to model and calcu-\nlate the exchange constant for CoandGdcrystals\nover their ordered nano-magnetic phase, by com-\nparing the EFT magnetization results and Curie\ntemperatures with the experimental data29. The\nexchange for CoandGdcrystals are then calcu-\nlated by using our EFT constitutive relations30\nkTc/zJS(S+ 1) = 0 .3127 and kTc/zJS(S+ 1) =\n0.3162, valid for cobalt and gadolinium, respec-\ntively. The EFT exchange JCo−CoandJGd−Gdare\ningoodagreementwiththemeanvaluesofexchange\nconstant obtainedfromextensiveexperimental data\n22forCoandGd. The calculated EFT magne-\ntization and exchange results for cobalt are then\nused for the calculations of sublattice magnetiza-\ntions of the alloy layered ...Co][ℓ′ℓℓ′][Co...nanos-\ntructure between Co leads, by using the MFT\nmethod20;21;13. The schematic representation\nfor the consecutive planes of cobalt-gadolinium\nVCA alloy and pure cobalt plane of nanojunction\n[Co1−cGdc]2[Co]2[Co1−cGdc]2between crystalline\ncobalt leads are shown in Fig.1.\nThe Ising spin Hamiltonian Hin the absence\nof local spin anisotropy and Zeeman effects may be\nexpressed as,\nH=−J/summationdisplay\n<i,j>Siz.Sjz, (1)\nwhere,/angb∇acketlefti,j/angb∇acket∇ightrepresents sum over nearest neighbors\nin the crystal, and Jis the nearest neighbors mag-\nnetic exchange constant that induces spin order\nalong a selected z-axis. The coordination number\nofCoandGdisz= 12, and present negligible\nanisotropy in the bulk material as compared to\ntheexchange. TheHamiltonian Eq.(1)forcomputa-\ntional purposes may be written in more useful form\nas,\nH=/summationdisplay\ni/summationdisplay\nj(−JSjz)Siz≡/summationdisplay\niHi(x)≡/summationdisplay\ni−x Siz,\n(2)\nThe thermodynamic canonical averages for any\ndesiredspin operator “Op” may be calculated by\nthe effective filed theory method using the Van der\nWearden’s (VdW) operator exp( JSz∇). TheMath-\nematica code formulation for any given characteris-4Vinod Ashokan; A. Khater and M. Abou Ghantous\nFig. 1. Schematic representation for the consecutive plane s of cobalt-gadolinium VCA alloy and pure cobalt plane of nan o-\njunction [ Co1−cGdc]2[Co]2[Co1−cGdc]2between crystalline cobalt leads. The hcp crystal c-axis is normal to the symmetry\n(0001) atomic planes.\ntic function fOp(x) of the spin system may be ex-\npressed as\nfOp(x) =Tr(Op.MatrixExp[ −Hi(x)/kT])\nTr(MatrixExp[ −Hi(x)/kT]),(3)\nwhere the Van der Wearden’s (VdW) operator for\ncobalt with S= 1 is given as31,\nexp(JSz∇) =S2\nzcosh(J∇)+Szsinh(J∇)+1−S2\nz.\n(4)\nThe differential operator ∇=∂/∂xoperate with\nproperty <exp(a∇)> fOp(x)|x→0=fOp(x+\na)|x→0=fop(a). The EFT method calcula-\ntions are discussed in detail in the earlier works\n31;32;33;34. With the help of Matrix quantum\nmechanics all required averages can be evaluated\nwith symbolic and numerical procedures.\nThe thermodynamic canonical averages are\nrepresented by <Op>=<(exp(JSz∇))z>\nfOp(x)|x→0. The canonical averages are confined to\nsingle-sitespinvariables σ=< Sz>andq=< S2\nz>\nwith reference to their basic spin values S= 1 for\nCoandS= 7/2 forGdatT= 0K. In the com-\nputational procedure desired decoupling approxi-\nmation is used, which is equivalent to neglecting\nthe site-site correlations < SizSjz>but preserv-\ning the single-site correlations < SizSiz>. Further-\nmore, this decoupling approximation makes EFT\napproachquiteeffectivetocomputethesalient mag-\nnetic properties of the systems.The comparison between the normalized mag-\nnetizations for the CoandGdcrystals calculated\nusing EFT method, and by using the EFT consti-\ntutive relations30,kTc/zJS(S+ 1) = 0 .3127 for\nCo(S= 1,z= 12), and kTc/zJS(S+1) = 0 .3162\nforGd(S= 7/2,z= 12), with their experimental\nmeasurements29;35;36;37, yield their respective\nCurie temperatures, that is 119.4 meV and 25.2\nmeV24. It is striking to note that for Cowith\nspinS= 1, the EFT calculated magnetization and\nCurie temperature agree with the experimentally\nobserved mean value of exchange constant JCo−Co\n= 15.9 meV22. Similarly, the hcp Gdwith the\nspinS= 7/2, the EFT calculated magnetization\nand Curie temperature agree with the correspond-\ning experimentally measured exchange JGd−Gd=\n0.42meV.29.TheEFTcalculated exchange JCo−Co\nand spin variable σCofor sites on the cobalt leads\nseed from the interfaces inwards, to calculate the\nsublattice magnetization using MFT of embedded\nalloy layered [ ℓ′ℓℓ′] nanostructure.\n3. Sublattice magnetizations of the\nalloy layered [ℓ′ℓℓ′]ferrimagnetic\nnanostructures between cobalt\nleads\nInthepresentcomputationalmodelingthealloyhcp\natomic planes of the layered [ ℓ′ℓℓ′] nanostructure\nsandwiched between cobalt leads, are modeled asSublattice magnetizations of nano-magnetic layered mater ials5\ncrystalline atomic planes. On their hcp lattice there\nis a random homogeneous distributions of Coand\nGdatoms. Any random site is considered to have\nthe usual six nearest neighbors in its hcp (0001)\nbasal plane, and another six neighbors on the two\nadjacent planes. The system is made of alternat-\ning hcp (0001) atomic planes, and the structural\nmorphology of the two interfaces between the leads\nand the layered nanostructure are abrupt and crys-\ntalline. The advent of advanced experimental tech-\nniques for Co/Gdmultilayer systems,12;11, per-\nmits minimizing the interface roughness, and the\ncontrol of the atomic interdiffusion towards stable\neutectic compositions c= 0.1 to 0.5.\nTo calculate the sublattice magnetization for\nthe individual basal atomic hcp planes of the al-\nloy layered [ ℓ′ℓℓ′] nanostructures by MFT, the Bril-\nlouin’s functions are used to calculate initially\nthe different spin variables σ(n′)\nαfor then′th lay-\nered atomic plane. To simplify the notation we\nsystematically call σ(n′)\nαas the thermodynamic\ncanonical spin variable so that σ(n′)\nα=Sα.BS≡\nBα(Sα,T,H(n′)\nα), where Sαrepresents the funda-\nmental atomic spin and αnamely for the Co and\nGd atoms, and BSis the Brillouin function. The\nthermodynamic canonical spin variable is given as,\nσ(n′)\nα=\n2Sα+1\n2Coth/parenleftigg\n2Sα+1\n2SαH(n′)\nα\nkT/parenrightigg\n−1\n2Coth/parenleftigg\n1\n2SαH(n′)\nα\nkT/parenrightigg\n(5)\nwhereH(n′)\nαrepresents the molecular field energy\nfor the element αin the atomic plane n′due to its\ninteraction with its z= 12 nearest neighbors. The\nkTrepresentsthermalenergyandtheeffective mag-\nnetic moment per site is ¯M(n′)in then′th plane, is\ngiven as in units of Bohr magnetons by\n¯M(n′)/µB= (1−c)g(n′)\nCoσ(n′)\nCo+cg(n′)\nGdσ(n′)\nGd.(6)\ng(n′)\nαare the g factors for the alloy element on the\nn′th plane. The magnetization for the n′th atomic\nplane is calculated by multiplying ¯M(n′)by the\nnumber of sites per unit volume for the atomic\nplane.3.1.Alloy layered\n[Co1−cGdc]2[Co]2[Co1−cGdc]2\nferrimagnetic nanostructure\nbetween cobalt leads\nConsider in this subsection the embedded layered\n[2′22′] nanostructure between cobalt leads. For the\nCoandGdatomintheferrimagneticalloyed atomic\nplanes are found with the respective probabilities\n(1−c) andc. The molecular field energy usingMFT\nfor aCoatom on the 1st hcp basal plane of the lay-\nered nanostructure at the interface with the cobalt\nlead, may hence be expressed as\nH(1)\nCo= (3σ(B)\nCoJcc)+6[(1−c)σ(1)\nCoJcc+cσ(1)\nGdJcg]\n+3[(1−c)σ(2)\nCoJcc+cσ(2)\nGdJcg]. (7)\nThe exchange interactions are denoted by the\nsimplified notation JCoCo≡Jcc,JGdGd≡Jgg, and\nJCoGd≡Jcg. Equally, the molecular field energy for\naGdatom on the 1st hcp basal plane of the al-\nloy layered nanostructure at the interface with the\ncobalt lead, is\n(a)\nCo\nGdc=0.1 c=0.5\nc=0.5 c=0.1\nstep 0.1\nmeV2|22|\n020406080100120-3-2-101\nkTσCo1σGd16Vinod Ashokan; A. Khater and M. Abou Ghantous\nCo\nGd(b)\nc=0.1\nc=0.5c=0.5\nc=0.1\nmeVstep 0.1\n2|22|\n020406080100120-3-2-101\nkTσCo2σGd2\nFig. 2. Calculated spin variables σCoandσGd, forCoand\nGdsites on the 1st (a) and 2nd (b) hcp basal (0001) planes,\nof the alloy layered [ Co1−cGdc]2[Co]2[Co1−cGdc]2ferrimag-\nnetic nanostructure between cobalt leads, for different all oy\nconcentrations c, as a function of kTin meV. The down (up)\narrows in each figure correspond to the trend of the σspin\nvariations for the Co(Gd) sites with cstep changes.\nH(1)\nGd= (3σ(B)\nCoJcg)+6[(1−c)σ(1)\nCoJcg+cσ(1)\nGdJgg]\n+3[(1−c)σ(2)\nCoJcg+cσ(2)\nGdJgg]. (8)\nIn the present formulation, the seeding spin\nvalue for the lead Coatom at the interface with the\nalloy layered nanostructure is represented by σ(B)\nCo,\nwhich is obtained singularly from the EFT calcula-\ntions described in detail in section 2.\nIn contrast, the molecular field for a Coatom\non the 2nd hcp basal plane of the layered nanos-\ntructure inwards from the 1st, is\nH(2)\nCo= 3[(1−c)σ(1)\nCoJcc+cσ(1)\nGdJcg]+6[(1−c)σ(2)\nCoJcc\n+cσ(2)\nGdJcg]+3σ(3)\nCoJcc (9)\nSimilarly, the corresponding molecular field for a\nGdatom on the 2nd hcp basal plane of the layered\nnanostructure, is\nH(2)\nGd= 3[(1−c)σ(1)\nCoJcg+cσ(1)\nGdJgg]+6[(1−c)σ(2)\nCoJcg\n+cσ(2)\nGdJgg]+3σ(3)\nCoJcg (10)M1/\u0001Bc=0.1\nc=0.5\nM2/\u0000B\n020406080100120-1.0-0.50.00.5\nkTM1,2(persite) / μB\nFig. 3. Calculated magnetic moments per site for sites on\nthe 1st (solid curves) and 2nd (dotted curves) hcp basal\n(0001) planes ofthe alloy layered ferrimagnetic nanostruc ture\n[Co1−cGdc]2[Co]2[Co1−cGdc]2between cobalt leads. They\npresent small differences only at the high temperature kT\nend of the ordered phase. The down arrows follow the varia-\ntion trend for the magnetic moments per site with the cstep\nchanges, on the 1st (solid arrow) and 2nd (dotted arrow) hcp\nbasal (0001) planes.\nThehcp basal planes of the pure[ Co]2layer be-\ntween the alloy layers [ Co1−cGdc]2, are designated\nrespectively as the 3rd and 4th atomic planes. Us-\ning the symmetry properties of the layered [2′22′]\nnanostructure, we note that σ(3)\nCo≡σ(4)\nCo. The corre-\nsponding molecular field for a Coatom on the 3rd\nhcp basal plane is hence\nH(3)\nCo= 3[(1−c)σ(2)\nCoJcc+cσ(2)\nGdJcg]+9σ(3)\nCoJcc(11)\nThe above equations can be put into matrix\nform\n\nH(1)\nCo\nH(1)\nGd\nH(2)\nCo\nH(2)\nGd\nH(3)\nCo\n=\nA1\nA2\n0\n0\n0\n+\nx1x2x3x4x5\ny1y2y3y4y5\nu1u2u3u4u5\nv1v2v3v4v5\nz1z2z3z4z5\n\nσ(1)\nCo\nσ(1)\nGd\nσ(2)\nCo\nσ(2)\nGd\nσ(3)\nCo\n\n(12)Sublattice magnetizations of nano-magnetic layered mater ials7\nand the coefficients matrix is identical to\nx1x2x3x4x5\ny1y2y3y4y5\nu1u2u3u4u5\nv1v2v3v4v5\nz1z2z3z4z5\n\n≡\n2βJcc6cJcgβJcc3cJcg0\n2βJcg6cJggβJcg3cJgg0\nβJcc3cJcg2βJcc6cJcg3Jcc\nβJcg3cJgg2βJcg6cJgg3Jcg\n0 0 βJcc3cJcg9Jcc\n(13)\nwhereA1= 3σ(B)\nCoJcc,A2= 3σ(B)\nCoJcgandβ=\n3(1−c). Using Eqs.(7) to (11), and the spin vari-\nablesσ(n′)\nαformat given by Eq.(5), it follows that\nEq.(12) represents a nonlinear equations and to be\nsolved for the spin variables\nM1/μB\nM2/μB\nM3/μB meVc=0.1\nc=0.5\n3|33|\n020406080100120-1.0-0.50.00.5\nkTM1,2,3(persite) / μB\nFig. 4. Calculated magnetic moments per site for the 1st,\n2nd, and 3rd hcp basal atomic planes of the alloy layered\n[Co1−cGdc]3[Co]3[Co1−cGdc]3ferrimagnetic nanostructure.\nThe magnetic moments per site on the 1st and 3rd hcp planes\n(discontinuous and continues curves, respectively) are qu ite\nsimilar throughout the temperature range of the ordered\nphase. They differ significantly from the magnetic moments\non the 2nd hcp plane (continues curves). The down arrows\ncorrespond to the trend of the magnetic moment variations\nwithcstep changes.\n\nσ(1)\nCo\nσ(1)\nGd\nσ(2)\nCo\nσ(2)\nGd\nσ(3)\nCo\n=\nBCo(SCo,T,H(1)\nCo)\nBGd(SGd,T,H(1)\nGd)\nBCo(SCo,T,H(2)\nCo)\nBGd(SGd,T,H(2)\nGd)\nBCo(SCo,T,H(3)\nCo)\n(14)Solving the above equations numerically, we\ncalculate the spin variables σ(1)\nCo,σ(1)\nGd,σ(2)\nCo,σ(2)\nGdand\nσ(3)\nCoas a function of temperature, for any given al-\nloy concentration c. Note that the symmetry of the\nsystem imposes the following equalities for the spin\nvariables\n/parenleftigg\nσ(1)\nCo\nσ(1)\nGd/parenrightigg\n≡/parenleftigg\nσ(6)\nCo\nσ(6)\nGd/parenrightigg\n,/parenleftigg\nσ(2)\nCo\nσ(2)\nGd/parenrightigg\n≡/parenleftigg\nσ(5)\nCo\nσ(5)\nGd/parenrightigg\n,\n/parenleftig\nσ(3)\nCo/parenrightig\n≡/parenleftig\nσ(4)\nCo/parenrightig\n(15)\nThe calculated spin variables σ(n′)\nCo,σ(n′)\nGdfor the\nnominal n′= 1 andn′= 2 hcp basal planes are pre-\nsented inFig.2, as afunctionof temperatureandfor\neutectic concentrations c= [0.1,0.5] in steps of 0.1.\nFurther, Eq.(6), and g(n′)\nCo≡gCo= 2.2,g(n′)\nGd≡\ngGd= 2 for all n′, yield the magnetic moments per\nsite on the 1st and 2nd hcp basal planes as a func-\ntion of temperature. These are presented for com-\nparison in Fig.3, where a small interesting differ-\nence is observed at the high temperature kTend\nof the ordered phase. Compensation temperatures\nkTcomp<21 meV, are observed for the ferrimag-\nnetic hcp planes for eutectic stable concentrations\nin the range 0 .23< c <0.5.\n3.2.Alloy layered\n[Co1−cGdc]3[Co]3[Co1−cGdc]3\nferrimagnetic nanostructure\nbetween cobalt leads\nThe layered [3′33′] nanostructures under considera-\ntion are symmetric about the origin here taken as\nthe hcp plane n= 0. In this system the symmetry\nproperties to be used are\n/parenleftigg\nσ(1)\nCo\nσ(1)\nGd/parenrightigg\n≡/parenleftigg\nσ(9)\nCo\nσ(9)\nGd/parenrightigg\n,/parenleftigg\nσ(2)\nCo\nσ(2)\nGd/parenrightigg\n≡/parenleftigg\nσ(8)\nCo\nσ(8)\nGd/parenrightigg\n,\n/parenleftigg\nσ(3)\nCo\nσ(3)\nGd/parenrightigg\n≡/parenleftigg\nσ(7)\nCo\nσ(7)\nGd/parenrightigg\n,and/parenleftig\nσ(4)\nCo/parenrightig\n≡/parenleftig\nσ(6)\nCo/parenrightig\n(16)\nSimilarly, as in the previous case [2′22′], the\nequivalent molecular field energy results can be cast\nhere in matrix form as8Vinod Ashokan; A. Khater and M. Abou Ghantous\n\nH(1)\nCo\nH(1)\nGd\nH(2)\nCo\nH(2)\nGd\nH(3)\nCo\nH(3)\nGd\nH(4)\nCo\nH(5)\nCo\n=\nA1\nA2\n0\n0\n0\n0\n0\n0\n\n+\n2βJcc6cJcgβJcc3cJcg0 0 0 0\n2βJcg6cJggβJcg3cJgg0 0 0 0\nβJcc3cJcg2βJcc6cJcgβJcc3cJcg0 0\nβJcg3cJgg2βJcg6cJggβJcg3cJgg0 0\n0 0 βJcc3cJcg2βJcc6cJcg3Jcc0\n0 0 βJcg3cJgg2βJcg6cJgg3Jcg0\n0 0 0 0 βJcc3cJcg6Jcc3Jcc\n0 0 0 0 0 0 6 Jcc6Jcc\n\n×\nσ(1)\nCo\nσ(1)\nGd\nσ(2)\nCo\nσ(2)\nGd\nσ(3)\nCo\nσ(3)\nGd\nσ(4)\nCo\nσ(5)\nCo\n(17)\nwhereA1(kT) = 3MCo(kT)JCo−CoandA2(kT) =\n3MCo(kT)JCo−Gd. This yields the new irreducible\nvariables\n\nσ(1)\nCo\nσ(1)\nGd\nσ(2)\nCo\nσ(2)\nGd\nσ(3)\nCo\nσ(3)\nGd\nσ(4)\nCo\nσ(5)\nCo\n=\nBCo(SCo,T,H(1)\nCo)\nBGd(SGd,T,H(1)\nGd)\nBCo(SCo,T,H(2)\nCo)\nBGd(SGd,T,H(2)\nGd)\nBCo(SCo,T,H(3)\nCo)\nBGd(SGd,T,H(3)\nGd)\nBCo(SCo,T,H(4)\nCo)\nBCo(SCo,T,H(5)\nCo)\n(18)c=0.47\n2|22|3|33|\nmeV(a)\n020406080100120-2.0-1.5-1.0-0.50.00.5\nkTM1(persite) / μB\n3|33|\n2|22|c=0.47 (b)\nmeV\n020406080100120-2.0-1.5-1.0-0.50.00.5\nkTM2(persite) / μB\nFig. 5. Calculated effective magnetic moments per site for\nthe nominal: (a) 1st alloyed, n′= 1, and (b) 2nd alloyed,\nn′= 2, hcp basal planes,\nfor the alloy layered [ Co0.53Gd0.47]2[Co]2[Co0.53Gd0.47]2\nand [Co0.53Gd0.47]3[Co]3[Co0.53Gd0.47]3nanostructures be-\ntween cobalt leads.\nThe above nonlinear equations can be solve nu-\nmerically to obtain the spin variables and magnetic\nmomentspersite,ontheindividualhcpbasalplanes\nof the alloy layered magnetic [3′33′] nanostructure.\nFig.4 presents the calculated results for the mag-\nnetic moments on the 1st, 2nd, and 3rd hcp atomic\nplanes for this system. The magnetic moments per\nsite for the 1st and 3rd hcp planes (discontinuous\nand continues curves, respectively) are quite similar\nthroughout the temperature range of the ordered\nphase. Together, they differ significantly from theSublattice magnetizations of nano-magnetic layered mater ials9\nmagnetic moments per site for the 2nd hcp plane,\nthroughout the temperature range of the ordered\nphase.\nAs defined, the fundamental atomic spins are\nS= 1 forCo, andS= 7/2 forGd, where we con-\nsider the spins to be up for Coand down for Gd,\nin the ferrimagnetic alloy. Note that the Gdfun-\ndamental spin is 3.5 times that of Co. However,\nthe magnetic exchange Jggbetween GdandGdis\nweaker than Jccbetween CoandCo. By studying\nequations7,8,9,and10, onecanseethatthemolec-\nular fields for CoandGdsites vary with tempera-\ntureandcanchangesigns.Thecompetitionbetween\nthese molecular fields along the entire temperature\nrange determines the magnetizations per site, M;\nin the low temperature regime they increase then\nreach a maximum value before losing their magne-\ntized phase. See figures 3 and 4.\nThe computational model is general and can be\nextended to treat individual hcp atomic planes of\nalloy layered [ Co1−cGdc]p[Co]q[Co1−cGdc]rnanos-\ntructures, with r,p,q≥1. This procedure may\nbe generalized to larger layered nanostructures be-\ntween semi-infinite Coleads. It has been observed\nthat while increasing r,p,qthe results for the sub-\nlattice magnetization properties in the core atomic\nplanes tend to limiting solutions.\nFig. 6. Calculated spin variables σ(n′)\nCoand their detailed\nvariations as a function of temperature on the nominal n′\ncobalt hcp basal planes inside the alloy layered ferrimag-\nnetic nanostructures [ Co0.53Gd0.47]2[Co]2[Co0.53Gd0.47]2\nand[Co0.53Gd0.47]3[Co]3[Co0.53Gd0.47]3incomparison with\nthe temperature variation of the spin variable σ(B)\nCoon the\ncobalt leads; see details in the text.3.3.Alloy layered\n[Co0.53Gd0.47]ℓ′[Co]ℓ[Co0.53Gd0.47]ℓ′\nferrimagnetic nanostructure\nbetween cobalt leads\nThis alloy layered magnetic nanostructure between\ncobalt leads, at the characteristic eutectic concen-\ntrationc= 0.47, is particularly interesting since\nCo/Gdmagnetic multilayers at the same composi-\ntion have been reported to be very stable,12. We\nhave appliedhencethe EFT- MFT model approach\nto deduce the spin variables σ(n′)\nCoandσ(n′)\nGd, and\nthe effective magnetic moments per site, for the in-\ndividual hcp basal planes of layered ferrimagnetic\nnanostructures between cobalt leads, as a function\nof temperature, eutectic concentration, and thick-\nnessesℓ= 2 and 3. The integer n′numbers the hcp\nplanes from 1 to 6 for ℓ= 2, and from 1 to 9 for\nℓ= 3.\nThe effective magnetic moments per site calcu-\nlated in the units of Bohr magnetons for the alloyed\nnominal 1st, n′= 1, and 2nd, n′= 2, hcp basal\nplanes for the alloy layered ferrimagnetic nanos-\ntructures [ Co0.53Gd0.47]2[Co]2[Co0.53Gd0.47]2and\n[Co0.53Gd0.47]3[Co]3[Co0.53Gd0.47]3arepresentedin\nFig.5. It is observed that the computed effective\nmagnetic moments per site as a function of tem-\nperature on the nominal n′= 1 hcp basal planes\ndo not vary significantly with increased thickness of\nnanostructure,see Fig.5(a). Thiscan beunderstood\nclearly since the corresponding matrix elements in\nEq.(17) do not change significantly with increasing\nthickness. In contrast, it is observed that the ef-\nfective magnetic moments per site on the nominal\nn′= 2 hcp basal planes do vary significantly with\nincreasing thickness, see Fig.5(b). This is expected\nphysically owing to the changes of the correspond-\ning effective molecular fields for CoandGdsites.\nThe observed variations start at ≈15 meV and per-\nsist for higher temperatures, including room tem-\nperature ≈26 meV.\nIt is also interesting to compute spin vari-\nablesσ(n′)\nCo, and their detailed variations as a func-\ntion of temperature on the nominal n′cobalt\nhcp basal planes inside the alloy layered ferri-\nmagnetic nanostructures, in comparison with the\nspin variable σ(B)\nCoon the cobalt leads as a\nfunction of temperature. This is done for the\n[Co0.53Gd0.47]ℓ′[Co]ℓ[Co0.53Gd0.47]ℓ′layered nanos-\ntructure between cobalt leads, for thicknesses ℓ= 2\nand 3. The calculated results are presented in Fig.6.10Vinod Ashokan; A. Khater and M. Abou Ghantous\nTable 1. Spin variable values σGd=< SGd>for theGdsites on atomic planes 1, 2, .., ℓ, of the\n2′22′and 3′33′nanojunction systems between cobalt leads, are given for st able eutectic compositions\nc≤0.5, at room temperature T=300K, using the theoretical EFT- MFT combined method. The spin\nvariable σCo=< SCo>is≈1 at room temperature for the Cosites throughout the system.\nConcentrations Spin variables σGd\n2′22′3′33′\nc [Co1−cGdc]2,L[Co1−cGdc]2,R[Co1−cGdc]3,L [Co1−cGdc]3,R\n0.1 -1.10 -1.10 -1.10 -1.10 -1.10 -1.08 -1.10 -1.10 -1.08 -1. 10\n0.2 -1.03 -1.03 -1.03 -1.03 -1.03 -1.00 -1.03 -1.03 -1.00 -1. 03\n0.3 -0.97 -0.97 -0.97 -0.97 -0.97 -0.91 -0.97 -0.97 -0.91 -0. 97\n0.4 -0.91 -0.91 -0.91 -0.91 -0.90 -0.81 -0.90 -0.90 -0.81 -0. 90\n0.5 -0.83 -0.83 -0.83 -0.83 -0.82 -0.71 -0.82 -0.82 -0.71 -0. 82\nAs is physically expected, the σ(B)\nCois≥σ(n′)\nCofor all\npureCohcp planes n′inside the layered thicknesses\nℓ, at all temperatures of the ordered ferrimagnetic\nphase. Also as expected, σ(5)\nCois≥σ(4)\nCofor the al-\nloy layered [ Co0.53Gd0.47]3[Co]3[Co0.53Gd0.47]3fer-\nrimagnetic nanostructure between cobalt leads, and\nboth are greater or equal to the σ(3)\nCoof the\n[Co0.53Gd0.47]2[Co]2[Co0.53Gd0.47]2layered nanos-\ntructure. Note that the n′= 3 pure cobalt plane\nfor the [2′22′] layered nanostructure is equivalent\nnominally to the n′= 4 pure cobalt plane for the\n[3′33′] layered nanostructure. The results confirm a\nphysical trend which is expected, and which would\nlead to limiting values with increasing thickness of\nthe layered ferrimagnetic structure between cobalt\nleads.\nWe emphasize that the basic physical variables,\nsuch as the exchange and sublattice magnetizations\nforCoandGdsitesfortheembeddedlayerednanos-\ntructures between cobalt leads, are necessary ele-\nments for the computations of the spin-dynamics\nof magnetic nanojunctions in the field of magnon-\nics, as for the ballistic magnon transport25. In Ta-\nble 1 we present an example for the calculated spin\nvariables < SCo>≈1 throughout the system, and\n< SGd>on theidentified atomic planes1, 2, .., ℓ,of\nthe layered ...Co][2′22′][Co...and...Co][3′33′][Co...\nnanostructures between cobalt leads, at room tem-\nperature T=300K and stable eutectic compositions\nc≤0.5, using the EFT-MFT combined method.\n4. Summary and conclusions\nIn this work, we model the salient sub-lattice mag-\nnetic properties of the alloy layered ferrimagnetic\n[Co1−cGdc]ℓ′[Co]ℓ[Co1−cGdc]ℓ′nanostructures be-\ntween magnetically ordered cobalt leads. In par-\nticular, sublattice magnetizations of the CoandGdsites on the individual hcp (0001) basal planes\nof the alloy layered lamellar nanostructures by\nusing EFT-MFT combined method. The effective\nmagnetic moments per site and sublattice mag-\nnetizations are plotted as a function of tempera-\nture and thicknesses of the lamellar nanostructure.\nThe computational model is general and can repre-\nsents other composite magnetic elements andlamel-\nlar nanostructures. The calculated magnetic ex-\nchange, and spin variables for gadolinium < SGd>\nand cobalt < SCo>, site on the identified hcp\n(0001) basal planes of the alloy layered ferrimag-\nnetic [Co1−cGdc]ℓ′[Co]ℓ[Co1−cGdc]ℓ′nanostructures\nbetween cobalt leads, are very important quanti-\nties for the self-consistent analysis of quantum spin\ndynamics system and the coherent magnon ballis-\ntic transport across such nanostructures. The cal-\nculated results are also important fo the applica-\ntions in the fields of magnonics. The Ising EFT\nmethod serves to determine the magnetic exchange\nconstants for CoandGdsites of purecrystals, char-\nacterized by their fundamental quantum spins, by\ncomparing EFT with the experimental data. By\nseeding the MFT results on the alloy lamellar ferri-\nmagnetic nanostructure by the EFT computations\nof the cobalt leads from the interface inwards, the\nsublattice magnetizations for the CoandGdsites\nin the nanostructure are computed.\nReferences\n1. A.A. Serga, A.V. Chumak, A. Andre, G.A. Melkov,\nA.N. Slavin, S.O. Demokritov, and B. Hillebrands,\nPhys. Rev. Lett. 99, (2007) 227202\n2. T. Schneider, A.A. Serga, B. Leven, B. Hillebrands,\nR.L. Stamps, and M.P. Kostylev, Appl. Phys. Lett.\n92, (2008) 022505\n3. V. V. Kruglyak, S. O. Demokritov and D. Grundler,\nJ. Phys. D: Appl. Phys. 43, (2010) 264001Sublattice magnetizations of nano-magnetic layered mater ials11\n4. K. Lee and S. Kim, J. Appl. Phys. 104, (2008)\n053909\n5. S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. 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Spedding, Phys.\nRev.132, (1963) 1092."    },    {        "title": "2303.00294v1.Covalency__correlations__and_inter_layer_interactions_governing_the_magnetic_and_electronic_structure_of_Mn__3_Si__2_Te__6_.pdf",        "content": "Covalency, correlations, and inter-layer interactions governing the\nmagnetic and electronic structure of Mn 3Si2Te6\nChiara Bigi,1Lei Qiao,2, 3Chao Liu,2, 3Paolo Barone,4Monica Ciomaga Hatnean,5,\u0003\nGesa-R. Siemann,1Barat Achinuq,6Daniel Alexander Mayoh,5Giovanni Vinai,7Vincent\nPolewczyk,7Deepak Dagur,7Federico Mazzola,7Peter Bencok,8Thorsten Hesjedal,6Gerrit\nvan der Laan,8Wei Ren,2Geetha Balakrishnan,5Silvia Picozzi,3,yand Phil D. C. King1,z\n1SUPA, School of Physics and Astronomy,\nUniversity of St Andrews, St Andrews KY16 9SS, UK\n2Physics Department, International Center of Quantum and Molecular Structures,\nMaterials Genome Institute, State Key Laboratory of Advanced Special Steel,\nShanghai Key Laboratory of High Temperature Superconductors,\nShanghai University, Shanghai 200444, China\n3Consiglio Nazionale delle Ricerche (CNR-SPIN),\nUnit\u0012 a di Ricerca presso Terzi c/o Universit\u0012 a \\G. D'Annunzio\", 66100 Chieti, Italy\n4Consiglio Nazionale delle Ricerche CNR-SPIN, Area della Ricerca di Tor Vergata,\nVia del Fosso del Cavaliere 100, I-00133 Rome, Italy\n5Department of Physics, University of Warwick,\nCoventry, CV4 7AL, United Kingdom\n6Department of Physics, Clarendon Laboratory,\nUniversity of Oxford, Oxford, OX1 3PU, United Kingdom\n7Istituto O\u000ecina dei Materiali (IOM)-CNR,\nLaboratorio TASC, Area Science Park,\nS.S.14, Km 163.5, 34149 Trieste, Italy\n8Diamond Light Source, Harwell Science and Innovation Campus,\nDidcot, OX11 0DE, United Kingdom\n(Dated: March 2, 2023)\n1arXiv:2303.00294v1  [cond-mat.mtrl-sci]  1 Mar 2023Abstract\nMn3Si2Te6is a rare example of a layered ferrimagnet. It has recently been shown to host a\ncolossal angular magnetoresistance as the spin orientation is rotated from the in- to out-of-plane\ndirection, proposed to be underpinned by a topological nodal-line degeneracy in its electronic\nstructure. Nonetheless, the origins of its ferrimagnetic structure remain controversial, while its\nexperimental electronic structure, and the role of correlations in shaping this, are little explored\nto date. Here, we combine x-ray and photoemission-based spectroscopies with \frst-principles cal-\nculations, to probe the elemental-selective electronic structure and magnetic order in Mn 3Si2Te6.\nThrough these, we identify a marked Mn-Te hybridisation, which weakens the electronic correla-\ntions and enhances the magnetic anisotropy. We demonstrate how this strengthens the magnetic\nfrustration in Mn 3Si2Te6, which is key to stabilising its ferrimagnetic order, and \fnd a crucial role\nof both exchange interactions extending beyond nearest-neighbours and anti-symmetric exchange\nin dictating its ordering temperature. Together, our results demonstrate a powerful methodology of\nusing experimental electronic structure probes to constrain the parameter space for \frst-principles\ncalculations of magnetic materials, and through this approach, reveal a pivotal role played by\ncovalency in stabilising the ferrimagnetic order in Mn 3Si2Te6.\nI. INTRODUCTION\nWhile three-dimensional magnets are common-place, long range order is strictly forbid-\nden to occur in one-dimensional systems [1]. Layered magnetic materials present a novel\nenvironment in which to study the critical dimensionality between these two extremes, with\n\fnite inter-plane coupling expected to have a strong in\ruence on the development of long-\nrange order, magnetic anisotropy, and the role of \ructuations [2]. In this respect, Mn 3Si2Te6\n(MST) is a particularly intriguing compound. It forms in the P31cspace group (No. 163,\nFig. 1(a,b)) [3, 4], containing two inequivalent Mn sites. The Mn (1)atoms sit at the centre\nof edge-sharing MnTe 6octahedra. The resulting Mn atoms would form a triangular lattice.\nHowever, 1=3 of the sites are occupied by a Si-Si dimer, leaving the Mn (1)atoms in a hon-\neycomb con\fguration (Fig. 1(b)), akin to the Cr sites of the layered van der Waals magnet\n\u0003Current address: Paul Scherrer Institut, Forschungsstrasse 111, 5232 Villigen PSI, Switzerland\nysilvia.picozzi@spin.cnr.it\nzpdk6@st-andrews.ac.uk\n2Cr2Ge2Te6(CGT) [5{7]. Unlike CGT, however, extra magnetic ions (Mn (2)) are situated\nbetween the Mn (1)layers, \flling the van der Waals gap. These self-intercalated Mn (2)sites\nform a triangular lattice, with one Mn (2)atop every second Mn (1)site, providing a bridging\nlink between the Mn (1)layers and establishing a more three-dimensional structure. Nev-\nertheless, the samples cleave easily with a standard top-post method, exposing a \rat and\nuniform surface.\nThe Mn ions have been reported to develop a long-range magnetic order below \u001875 K [8],\nwith the Mn moments aligned ferromagnetically within each Mn layer, but with antiferro-\nmagnetic coupling between neighbouring layers (Fig. 1(a)) [9]. This leads to an overall ferri-\nmagnetic structure [8, 10]. This speaks to a critical role of the bridging Mn (2)sites, leading to\nmarkedly di\u000berent interlayer interaction as compared to ferromagnetic CGT [9, 11]. Indeed,\nthe exchange interactions for the \frst three nearest-neighbours, J1,J2andJ3(Fig. 1(b))\nare known to be antiferromagnetic [9]. This creates competing interactions which, com-\nbined with the geometrical arrangement of the Mn-atoms in the MST lattice (side-view\nof Fig. 1(b)), has been proposed to result in a high degree of magnetic frustration in this\nsystem.\nIt remains poorly understood, however, how the overall ferrimagnetic ground state of MST\nis stabilised. Moreover, while the Mn ions would nominally be expected in a 2+ valence\nstate with a quenched orbital moment, this has recently been questioned, as a huge magnetic\nanisotropy (up to 13 T) has been reported [12{14]. The intra- and inter-layer magnetic\ncouplings in MST thus require further exploration, while the underlying electronic structure\nfrom which the magnetic order emerges, and the degree to which ligand hybridisation vs.\nelectronic correlation e\u000bects dominate the magnetic coupling, remain almost completely\nunexplored to date.\nTo address this, here we present a combined theoretical and experimental study into the\nelectronic structure and magnetic interactions of MST. Our results point to a signi\fcant Mn-\nTe orbital hybridisation. We show how this promotes additional exchange coupling terms,\ninclusion of which is essential to obtain an accurate description of the magnetic anisotropies\nand ordering temperature in MST. Together, our results thus provide key insights on the\ninterplay and in\ruence of dimensionality, anisotropy, covalency, and correlations on the\nmagnetic \ructuations of quasi-layered magnetic materials.\n3II. METHODS\nA. DFT\nWe performed density functional theory (DFT) simulations, using the Vienna ab initio\nSimulation Package (VASP) [15, 16]. The generalized gradient approximation (GGA) based\non the Perdew-Burke-Ernzerhof (PBE) functional [17] was employed to treat the exchange-\ncorrelation interaction. We considered the localized 3 delectron correlation of Mn atoms\nby using the GGA+ Umethod [18, 19], with an e\u000bective Hubbard Uparameter chosen to\nbe 1 eV, and other e\u000bective Uchoices of 0, 2, and 3 eV also tested and compared. The\nprojector-augmented-wave (PAW) potentials [20, 21] were used to describe the electron-ion\ninteraction. The energy cuto\u000b was selected to be 550 eV for the plane-wave basis set. The\nBrillouin zones were sampled using a 6 ×6×4 k-grid mesh in the \u0000-centered scheme. The\nforces convergence threshold on each atom was chosen as 0 :005 eV/ \u0017A and the self-consistent\ncalculations were stopped when the energy di\u000berence was smaller than 10\u00007eV per atom.\nWe optimized ionic positions with ground state ferrimagnetic magnetic con\fguration, and\nexperimental lattice constants were used and \fxed in all calculations.\nB. Monte Carlo Calculations\nA standard Metropolis algorithm has been used for Monte Carlo (MC) simulations, with\n105MC steps for equilibration and 5 \u0002105MC steps for averaging. Starting from the crystal-\nlographic unit cell, comprising four Mn (1)and two Mn (2)sites, we performed calculations on\na 16\u000216\u00028 supercell with Ns= 12288 spins. The transition temperature can be estimated\nfrom the peak appearing in the temperature evolution of the speci\fc heat per spin, evaluated\nas\nCv=kB\f2\nNs\u0002\nhE2i\u0000hEi2\u0003\n(1)\nwhereEis the energy calculated using model Eq. (4), kBis the Boltzmann constant and\n\f= 1=kBT, whileh:::iindicates statistical averages. We also de\fne the ferromagnetic F\n4and ferrimagnetic forder parameters as:\nF=1\n6 X\ni=1;4Si+X\ni=1;2si!\nf=1\n6 X\ni=1;4Si\u0000X\ni=1;2si!\n(2)\nwhere Siandsilabel here the spins on Mn (1)and Mn (2)sites, respectively, within the\nunit cell, with associated (generalized) susceptibilities for the magnetic order parameters\nOP=F;fcalculated as:\n\u001fOP=\fNs\u0002\nhOP2i\u0000hOPi2\u0003\n(3)\nResults of Monte Carlo calculations are summarized in Fig. 5. We used four sets of pa-\nrameters in our simulations: two sets correspond to the full model parametrization sum-\nmarized in Tab. I and Tab. II, while two sets consist in simpli\fed model with isotropic\nJiso\n1;(Jiso\n2+Jiso\n5);Jiso\n3exchange interactions and single-ion anisotropy terms. In all cases,\nwe \fnd a transition to an ordered ferrimagnetic phase characterized by ferromagnetic layers\nof Mn (1)and Mn (2)spins antiferromagnetically aligned and lying in the basal plane. The\ncritical temperature is inversely proportional to the value of U, getting lower as the Hubbard\nparameter is increased, as expected. On the other hand, the full model parametrization en-\ntails a stronger magnetic frustration arising from longer-range antiferromagnetic interactions\nwithin the honeycomb layers, that reduces the transition temperature by about 20% with\nrespect to the simpli\fed model. The range of the estimated critical temperatures settles\nbetween 89 and 118 K for Ubetween 2 and 1, respectively.\nC. Sample growth and characterisation\nMn3Si2Te6single-crystals were grown by the chemical vapour transport method using\nIodine as a transport agent, as described in Ref. [4]. X-ray Di\u000braction (XRD) reported\nin Supplementary Fig. S1 show only (00 l) Bragg peaks of the expected MST structure,\nindicating that the facets of the obtained crystals (see inset in Fig. S1) are parallel to the ab-\nplane. The cell parameters extracted from \ftting the XRD data are a=b= 7:07700(6) \u0017Aand\nc= 14:25081(2) \u0017A. A Quantum Design Magnetic Property Measurement System was used to\nmeasure the magnetization between 1.8 and 300 K in applied \felds up to 5 T. A Quantum\n5Design Physical Property Measurement System was used to measure electrical resistivity\nbetween 20 and 300 K.\nD. X-ray and UV spectroscopies\nFor our spectroscopic measurements, samples were cleaved in ultra-high-vacuum (UHV)\nusing a top-post method. X-ray absorption spectroscopy (XAS) and x-ray magnetic circular\ndichroism (XMCD) measurements were performed using the electromagnet end station of\nthe I10 beamline at Diamond Light Source, UK. The pressure was better than 10\u00009mbar,\nand measurements were performed across the Mn L2;3absorption edge in total electron yield\n(TEY) detection, thus probing \u00186 nm depth from the sample surface. Spectra were mea-\nsured at 10 K with left- (CL) and right-circularly (CR) polarized x-rays in both normal- and\ngrazing-incidence (i.e., at an angle of 20\u000efrom theab-plane) geometries (see Supplementary\nFig. S2). An applied \feld of 1 :4 T collinear with the beam axis was applied to magnetize\nthe sample. The main error source on the XAS/XMCD quantitative data analysis arises\nfrom the step-edge background subtraction. To estimate this, each spectrum was analysed\nusing di\u000berent step-edge background choices (mainly varying the energy range of interest\nas this was found to a\u000bect the background shape the most) and we extracted the standard\nerror deviation from the set of obtained values. The mssum rules have been corrected for\nMnjj-coupling [22] and the temperature dependent XMCD asymmetry has been calculated\nfrom the dichroic signal at the L3absorption peak as\nA=ICR;L3\u0000ICL;L3\nICR;L3+ICL;L3:\nThe error on the asymmetry has been estimated from the XMCD residual value obtained\nfor several measurements performed well above the transition temperature.\nResonant photoemission spectroscopy (resPES) measurements were performed at the\nAPE-HE beamline (Elettra synchrotron, Italy) at a base pressure lower than 10\u000010mbar,\nwith the sample temperature kept at \u0018107 K. XAS across the Mn L2;3edge was performed\nin linear horizontal polarization and in TEY detection to determine the relevant energies for\nresPES. Measurements were acquired by a Scienta Omicron R3000 hemispherical electron\nenergy analyser. The binding and photon energies were calibrated with the Fermi edge of a\ngold reference sample, assuming a work function of 4 :4 eV. For better comparison with the\n6experiment, our P-DOS DFT calculations were convolved with a Voigt function to emulate\nlifetime broadening of 0 :01 eV and an experimental resolution 0 :3 eV.\nThe ARPES data were acquired at the APE-LE beamline (Elettra synchrotron) with a\nScienta DA30 hemispherical electron energy and momentum analyzer, with the samples held\nat a temperature of 77 K. The base pressure in the chamber was better than 10\u000010mbar.\nEnergy and angular resolution were better than 40 meV and 20\u000e, respectively.\nIII. RESULTS AND DISCUSSION\nA. Magnetic order\nConsistent with prior studies [8, 9, 23], our single-crystal samples (see Supplementary\nFig. S1) exhibit \fnite net magnetization onsetting below a transition temperature ( TC) of\n\u001879 K (Fig. 1(d)). Mvs.Hmagnetization curves performed at T= 2 K (Fig. 1(c)) indicate\na large magnetic anisotropy with a magnetic hard axis along the caxis: the magnetization\nfully saturates to 1 :42\u00060:02\u0016B/Mn in the ab-plane (H?c) for less than 2 T magnetic \feld,\nwhile 5 T is insu\u000ecient to saturate the moment in the Hkccon\fguration. Consistent\nwith this large anisotropy, Fig. 1(d) shows that the magnetic molar susceptibility is more\nthan a factor of two larger for the magnetic \feld aligned perpendicular vs.parallel to the\nc-axis.\nThe simplest picture in which to describe this magnetic order is to consider fully localised\nMn spins, with Mn in a 2+ oxidation state, and hence nominal 3 d5con\fguration [23, 24].\nMeanwhile, Te2\u0000would be expected to contribute a \flled p6valence band, leading to semi-\nconducting transport (Fig. 1(e), Refs. [8, 9, 12, 25]). Nonetheless, a pronounced kink in\nthe measured resistivity at the magnetic ordering temperature points to a non-negligible\ncoupling between the magnetism and charge carriers in the system, beyond the simple pic-\nture outlined above. To investigate this further, we measured x-ray spectroscopy at the Mn\nL2;3edge, as shown in Fig.2. The x-ray absorption measurements (yellow curve in Fig. 2)\nshows a spectrum broadly reminiscent of those observed in other nominally Mn2+com-\npounds [26, 27]. The branching ratio L3/(L3+L2) exceeds the expected 2 =3 statistical value\n(i.e., 0:768\u00060:004), consistent with a nominal Mn high-spin state and with the observed\nsemiconducting transport behavior [28, 29]. We stress, however, that a picture of fully lo-\n7calised Mn moments is an oversimpli\fcation. Hints of this are already detectable in its XAS\nfeatures, where the characteristic multiplet structure is smeared out and less pronounced\nthan in typical ionic Mn2+compounds [30]. This provides a \frst spectroscopic indication\nthat in fact there is a non-negligible hybridisation of the Mn with the chalcogen ions, a point\nwe return to below.\nOur x-ray absorption data show a clear circular dichroism at low temperature, con\frming\nthat the magnetism of MST primarily proceeds via ordering of the Mn spins. The x-ray\nmagnetic circular dichroism (XMCD) signal obtained at 10 K in a grazing incidence geometry\n(with a 1:4 T \feld applied at 20\u000eto theab-plane) is approximately twice as large as for a\nnormal incidence geometry where the \feld is applied along the c-axis, in good agreement\nwith the magnetic susceptibility measured by SQUID (Fig. 1.(d)). Temperature-dependent\nXMCD spectra measured in remanence, after the sample was \feld-cooled to 10 K in 1 :4 T\nplanar \feld (inset of Fig. 2), yield an XMCD asymmetry which closely follows the bulk\ndemagnetization (Fig. 1(d), reproduced in green in the inset of Fig. 2). In good agreement\nwith the bulk TC, the XMCD asymmetry vanishes at T\u001975 K becoming comparable to\nour measurement error (see Fig. S2 in Supplementary Material for the full temperature-\ndependent XMCD spectra).\nHaving established that our spectroscopic measurements probe bulk-like magnetic prop-\nerties in MST, we turn to the insights on the magnetic and electronic structure which they\nadvance. The main spectrum arises from the multiplet structure of the Mn d5states that\nare split by the crystal-\feld interaction. In addition, we observe a distinct pre-peak feature\nof (\u0000;+) shape, marked by a black arrow in the bottom graph of Fig.2. Such a pre-edge\nfeature in other Mn-based compounds has been ascribed to transitions from the Mn 2 pcore\nlevel to unoccupied p-dhybridized valence states [31], and thus indicates hybridization with\nthe Te 5pvalence states here. Indeed, the shape of our measured XMCD is similar to that\nfound previously for Ga 1\u0000xMnxAs, where the ground state of Mn is found as a hybridised\nstate with 16% d4, 58%d5, and 26%d6character, yielding a d-count of\u00185:1 electrons/Mn\natom.[27]. To consolidate these \fndings, we applied quantitative sum-rule analysis to our\nXMCD data-set, to extract the atom-speci\fc orbital and spin magnetic moments ( mland\nms, respectively) [22, 32, 33]. Our results indicate a smaller than expected ordered spin\nmoment of ms= 0:546\u00060:004\u0016B=Mn - likely due to a \fnite out of plane component of\nthe applied \feld for our experimental geometry, thus not reaching saturation for our avail-\n8able \felds. Nonetheless, constitent with prior calculations [9], we \fnd a small but positive\norbital moment of ml= 0:022\u00060:004\u0016B=Mn (in agreememt with the DFT value of 0.027\n\u0016Bper Mn), with ml=ms= 0:040\u00060:005, further con\frming the deviation from a purely\nhigh-spin localised picture of the Mn atoms in this system. The positive sign of this ratio\npoints directly to a dshell which is more than half \flled, entirely consistent with our above\nqualitative assessment of the spectral lineshapes observed. Indeed, an orbital to spin ratio of\n0.04 tallies with a d-count of\u00185.1 electrons/Mn, with the small excess of electrons compared\nto Mn2+implying the presence of holes in the Te 5 pband.\nB. Electronic structure\nTo address the role and extent of this implied covalency, we combine resonant photoe-\nmission spectroscopy (resPES) measurements of the valence band electronic structure with\ncalculations from density-functional theory. Fig. 3(a) shows the valence band photoemission\nmeasured while scanning the photon energy across the Mn L3absorption edge. The data\nshow a prominent Mn-derived state located at \u00183:6 eV binding energy, whose photoemit-\nted intensity closely follows the L3XAS signal. This can thus be attributed as a nominally\nlocalised Mn component, which carries the majority of the spin moment in this system.\nNonetheless, we \fnd that there are increases in spectral weight through the Mn resonance\nnot just for this peak, but more broadly for the entire valence band region. This implies\nthat Mn becomes hybridised throughout the valence bands, as is evident from the Mn partial\ndensity of states (P-DOS) extracted as the di\u000berence of the PES measurements performed\non- and o\u000b- resonance shown in Fig. 3(b).\nThis directly points to a strong hybridisation between Mn and Te orbitals in this system,\nwhich in turn can be expected to weaken the electronic correlations in MST. Such correlations\nare expected to play a crucial role in the exchange-coupling among the Mn spins [9, 12,\n13, 34], and a quanti\fcation of the extent of electronic correlation vs.covalency in MST\nis strongly required. To this end, we compare the experimental Mn and Te P-DOS (the\nlater extracted from o\u000b-resonant measurements) with the results of DFT+ Ucalculations in\nFig.3(b). While the Te P-DOS only weakly changes with increasing Hubbard- Uparameter\nused in the calculation, the intense Mn-derived peak shifts rapidly away from the Fermi\nlevel, while its shape also becomes modi\fed. The prominent peak in the on-resonance\n9resPES data provides a robust experimental feedback, allowing best matching the Hubbard-\nUparameter for MST from comparison with our Mn P-DOS calculations. Consistency\nbetween the experimental measurements and theoretical calculations is found only for Uin\nthe range of 1\u00002 eV. This is substantially lower than typical U-values of about 4 eV found\nfor Mn2+oxides [35] and even other chalcogenides such as MnSe 2[36] and MnTe 2[37], and\nlower than previously utilised for calculations of MST [9, 25]. A deep physical interpretation\nof theUvalue within the DFT+ Uapproach is not straightforward, due to the lack of formal\ncorrespondence between DFT+ Uand the full many-body approach characteristic of the\nHubbard model. Nonetheless, the small value of the Hubbard parameter here points to a\nsigni\fcant reduction of correlations in MST, due to pronounced ligand hybridisation.\nIn the occupied states, our calculations ( U= 1 eV, Fig. 4(a-d)) indicate a set of \rat\nbands visible at an energy of \u00183\u00004 eV below the valence band top. These match well the\npeak in the Mn P-DOS visible in our res-PES measurements discussed above, as well as a\ncorresponding non-dispersive feature visible in our measured dispersions from angle-resolved\nphotoemission spectroscopy (ARPES, Fig. 4(e),(f)). Our orbitally-projected calculations\nindicate that these derive dominantly from the Mn states. However, consistent with our x-\nray spectroscopic measurements, there is a strong hybridisation of the Mn states throughout\nthe valence region. Interestingly, the two inequivalent Mn sites hybridise di\u000berently with\nthe ligand valence states. Mn (1)(Fig. 4(a)), which sits within the MnSiTe 3layer (Fig. 1(a)),\nhas signi\fcant weight throughout the entire valence band bandwidth, while Mn (2), which\nsits in the interstitial sites between the layers, has a strongly localised Mn peak, exhibiting\nless intermixing with the Te p-orbitals across the rest of the valence band.\nThe Te states, meanwhile, contribute rather dispersive hole-like bands across a broad\nbandwidth of\u00185 eV (Fig. 4(c)). Our orbitally-resolved calculations indicate that the\nvalence band maximum (VBM) is mostly contributed by in-plane Te pxy-states and Mn (1),\nwith wavefunctions thus mainly localised within the honeycomb layer. On the other hand,\nTepzstates provide a channel for inter-layer hybridization between the Mn (1)and Mn (2)\nions. While many states are observed in our calculations due to the large unit cell of MST,\nthe photoemission spectral weight enhances two such dispersive states (Fig. 4(e-g)), with the\nhybridized hole-like dispersion appearing rather broad in the ARPES measurements. This\nis consistent with them having a three-dimensional character, while the surface sensitivity\nintrinsic to ARPES, combined with the large c-axis lattice constant (see supplementary\n10Fig.S1), leads to substantial kzbroadening. This is also re\rected in the surface-projected\nDFT calculations shown in Fig. 4(h). Aside from the surface-projected band features, broad\nbackground intensity is seen across the whole valence band energy range (up to \u00006 eV),\nbroadly consistent with our experimental results.\nFinally, we note that there is a strong demarcation in the contribution of the Mn (1)and\nMn(2)states to the conduction band electronic structure as can be readily seen in Fig.4(a,b).\nThe lowest energy states are contributed almost entirely from the Mn (1)site, with some\nintermixing of the Te, while the Mn (2)states are located approximately 0 :70\u00000:85 eV\nhigher in energy. The di\u000berent chemical environments of Mn (1)and Mn (2)may again account\nfor their di\u000berent contributions to the conduction bands, with the lowest conduction band\nminimum found to be located almost entirely in the honeycomb layer.\nC. Exchange interactions\nThe above comparison between DFT and our spectroscopic results strongly constrains\nthe relevant Uparameter to relatively small values in MST. This, in turn, has a sizeable\nin\ruence on the magnetic interactions and ordering tendencies in this system. To explore\nthis, we have developed a magnetic model, where the exchange coupling constants have been\nestimated by mapping the ab initio total energies onto a classical Heisenberg model, that\ncan be generally expressed as:\nH=1\n2X\nijSi\u0001Jij\u0001Sj+X\niSi\u0001Ai\u0001Si (4)\nExchange interactions between classical spins at sites i;jare described here by the tensor\nJij, including anisotropic e\u000bects, while the second term accounts for the magnetic single-\nion anisotropy (SIA). The full exchange tensor can be decomposed into its isotropic part\nJiso\nij=1\n3TrJij, an antisymmetric term JA\nij=1\n2(Jij\u0000JTij) and a symmetric term JS\nij=\n1\n2(Jij+JTij)\u0000Jiso\nijI.\nIn order to account for the covalency in MST, we have considered up to \ffth nearest-\nneighbor isotropic exchange interactions in our model. Consistent with previous studies, we\nlabel the \frst and second Mn (1)-Mn (2)nearest-neighbour coupling as Jiso\n1andJiso\n3, while we\nconsider two additional in-plane Mn (1)-Mn (1)exchange interactions: Jiso\n4;Jiso\n5(see Fig. 1(b)),\nas well as the next nearest-neighbour Jiso\n2[9, 13]. The exchange parameters have been ex-\n11(meV)Jiso\n1Jiso\n2Jiso\n3Jiso\n4Jiso\n5AMn(1)\ncAMn(2)\nc\nU= 126.06 3.87 9.25 0.84 2.57 0.33 1.14\nU= 220.68 2.33 6.39 0.52 1.61 0.13 1.02\nTABLE I. Estimated isotropic exchange interactions and single-ion anisotropies of MST using two\ndi\u000berent values of U. Within our de\fnition of the Heisenberg model, a positive energy corresponds\nto an antiferromagnetic interaction. Ful\flling the symmetry properties of the system, SIA can be\nparametrized by a unique coupling constant Ac=Azz\u0000A xx. All values are in units of meV,\nassuming classical spins of length jSj= 1.\ntracted using the four-state energy mapping method[38, 39]: this is a supercell approach\nthat allows extracting the full exchange tensor describing the coupling between a selected\npair of magnetic sites at a given distance, while the interaction with all other magnetic\nsites is canceled out by a tailored choice of four magnetic con\fgurations (details can be\nfound in the appendices of Ref. [39]). The four-state method also allows us to extract\nthe single-ion anisotropy, i.e., a site-dependent local quantity, instead of simply the total\nmagnetic anisotropy energy. This is crucial for MST, comprising two inequivalent Mn sites\nthat may in principle display di\u000berent SIA. The anisotropic part of the exchange tensor of\nnearest-neighbor Mn (1)-Mn (2)(J1) and Mn (1)-Mn (1)(J2) were also considered: noticeably,\na non-negligible Dzyaloshinksii-Moriya interaction is found between Mn (1)-Mn (2)magnetic\nmoments, with the Dzyaloshinskii vector parallel to the c-axis (Dz), while the Mn (1)-Mn (1)\nexchange tensor only displays a symmetric anisotropic part. We emphasize that the inclusion\nofi) the exchange coupling in tensorial form, ii) di\u000berent single-ion anisotropies for the two\nMn atomic species, and iii) longer-ranged J4andJ5exchange interactions, not considered\nin the literature, thereby improves our description with respect to previous models.\nIn tables Tab. I and Tab. II we list all the magnetic parameters estimated for U= 1\neV andU= 2 eV. Consistent with previous studies[9, 13], all (isotropic) interactions are\nfound to be antiferromagnetic, denoting a non-negligible magnetic frustration. On the other\nhand, our calculations also reveal sizeable longer-range interactions between Mn (1)magnetic\nmoments within the honeycomb layers, with the third-nearest-neighbor exchange Jiso\n5being\nof the same order of magnitude as Jiso\n2. This is likely due to both the presence of Si-Si dimers,\nwhich are located at the center of the Mn (1)hexagons where they can e\u000eciently mediate the\n12(meV)JS\n1xxJS\n1zzD1zJS\n2xxJS\n2yyJS\n2zzJS\n2xz\nU= 1-0.02 0.04 0.66 -0.05 0.01 0.04 0.04\nU= 2-0.03 0.06 0.49 -0.05 0.02 0.04 0.04\nTABLE II. Estimated anisotropic exchange interactions of MST using two di\u000berent values of U.\nSymmetry imposes J1xx=J1yy, while the cartesian components of tensor JS\n2are given in a local\nreference frame with the yaxis perpendicular to the Mn (1)-Mn (1)bond and the zaxis parallel to the\ncrystallographic cvector. The Dzyaloshinskii vector component is de\fned as D1z= (J1xy\u0000J1yx)=2.\nAll values are in units of meV, assuming classical spins of length jSj= 1.\nmagnetic interactions, as well as to ligand contributions [40]. While the (antiferromagnetic)\nMn(1)-Mn (1)interactions within the honeycomb layer do lead to magnetic frustration of the\nsystem, they are much smaller than the (antiferromagnetic) Mn (1)-Mn (2)exchange coupling,\nso that a ferrimagnetic con\fguration with antiparallel Mn (1)and Mn (2)is expected to be the\nlowest energy state.\nTheJiso\n4;Jiso\n5exchange interactions had not previously been estimated in Refs. [9, 13],\nwhere total energies of di\u000berent magnetic con\fgurations de\fned within the crystallographic\nunit cell were mapped onto a classical Heisenberg model. We con\frmed that such an ap-\nproach cannot provide information on Jiso\n4, whileJiso\n5interactions give rise to a spurious con-\ntribution to the nearest-neighbor Mn (1)-Mn (1)exchange that, within this total-energy map-\nping procedure, would correspond to Jiso\n2+Jiso\n5. Taking into account the di\u000berences between\nthe four-state and the total-energy mapping procedures, our results for Jiso\n1;(Jiso\n2+Jiso\n5);Jiso\n3\nare in excellent agreement with previously reported estimates[13]. On the other hand, both\nSIA and anisotropic exchange interactions support the experimentally observed easy-plane\nmagnetic anisotropy here.\nThe magnetic anisotropy energy per Mn ion, de\fned as the energy di\u000berence EMAE =\nE?\u0000Ekof a ferrimagnetic con\fguration perpendicular or parallel to the honeycomb ( ab)\nlayer, can be expressed from our model in Eq. 4 as:\nEMAE=1\n3\u0010\n2AMn(1)\nc +AMn(2)\nc\u0011\n+2\n3(J1zz\u0000J1xx)\n+\u0014\nJ2zz\u00001\n2(J2xx+J2yy)\u0015\n: (5)\nUsing the parameters listed in Tables I and II, we \fnd a MAE of \u00180:70 and\u00180:54 meV/Mn\nforU= 1 eV and U= 2 eV, respectively. The dominant contribution to the MAE is\n13provided by the Mn single-ion anisotropies. Here, the inequivalency of the Mn atoms is\nclearly re\rected via distinct SIAs: both display an easy-plane character with the hard axis\nparallel to the crystal caxis, but the SIA of Mn (1)is one order of magnitude smaller than\nthat of Mn (2). Interestingly, we found that roughly 15% of the magnetic anisotropy energy is\ncontributed by anisotropic exchange interactions. In agreement with previous studies [9, 13],\nwe \fnd that the dominant Mn (1)-Mn (2)antiferromagnetic interactions favour a ferrimagnetic\nground state comprising ferromagnetic Mn (1)and Mn (2)layers antiferromagnetically aligned.\nWe estimate TCfrom our calculated exchange couplings using Monte Carlo simulations\n(Fig. 5). The transition temperature extracted from the temperature dependence of the\ncalculated speci\fc heat per spin, Cv, is estimated to be Tc'89 K andTc'118 K from\nour calculations taking U= 2 eV and U= 1 eV, respectively. These are in good qualitative\nagreement with the experimentally determined TC= 75\u000079 K, further supporting the\nidenti\fed range of U= 1\u00002 eV as appropriate for MST, and highlighting the crucial\nrole of magnetic anisotropy, which leads to a signi\fcantly reduced TCin our calculations in\ncomparison to an isotropic model. Our \fndings from analysis of the calculated speci\fc heat\nare further validated by direct extraction of the ferrimagnetic order parameter (Fig. 5(b),\nsee Methods) which, together with the abrupt change of its relative susceptibility, shows the\nferrimagnetic state onset. Finally, we note that the dependence of the ferromagnetic order\nparameter with temperature (Fig. 5(c), see Methods) indicates the magnetic easy-plane\nnature of the system, further validating our discussions above.\nIV. CONCLUSIONS\nOur results demonstrate an integrated approach to predicting and understanding the\ninteractions governing long-range magnetic order in the layered ferrimagnet Mn 3Si2Te6. Our\napproach allows characterising the importance of covalency from ligand-metal hybridisation,\nwhich we \fnd to be signi\fcant in this system, weakening the electronic correlations. Using\nthese spectroscopic results to constrain our calculations, we demonstrate how combined \frst-\nprinciples and Monte Carlo methods can be used to accurately predict exchange interactions\nand magnetic anisotropies, elucidating their role in stabilising MST's novel ferrimagnetic\norder and predicting its ordering temperature within \u001815 K of the experimental value. Our\nresults reveal a key role of covalency in ruling these properties in MST, which we expect to\n14be key to understanding magnetism across the emerging class of 2D and layered magnets.\nACKNOWLEDGMENTS:\nWe gratefully acknowledge support from The Leverhulme Trust via Grant No. RL-2016-\n006 and the European Research Council (through the QUESTDO project, 714193). P.B. and\nS.P. acknowledge \fnancial support from the Italian Ministry for Research and Education\nthrough PRIN-2017 projects `Tuning and understanding Quantum phases in 2D materi-\nals|Quantum 2D' (IT-MIUR grant No. 2017Z8TS5B) and `TWEET: Towards ferroelec-\ntricity in two dimensions' (IT-MIUR grant No. 2017YCTB59), respectively. MCH, DM and\nGB acknowledge \fnancial support by the UK Engineering and Physical Sciences Research\nCouncil through grant EP/T005963/1. We thank the Elettra synchrotron for access to the\nAPE-HE beamline under proposal number 20195300. 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Valent\u0013 \u0010, Microscopic origin of magnetism in monolayer 3 d\ntransition metal dihalides, Physical Review B 106, 035156 (2022).\n18FIG. 1. Ferrimagnetism in Mn 3Si2Te6. (a) Crystal structure of MST, indicating the ferrimagnetic\nalignments of Mn moments within the (001) easy ab-plane. (b) Side- and Top-views showing the\natomic arrangements together with the exchange interactions Jibetween the Mn atoms considered\nin this work. (c) Mvs.Hmagnetization curves measured at T= 2 K for HkcandH?c\ngeometries. (d) Temperature-dependent dc magnetic susceptibility ( \u001f) as measured in zero-\feld\ncooled (ZFC) warming mode in a small \feld of 10 mT applied along the crystal directions parallel\nand perpendicular to the c-axis. (e) Planar resistivity ( \u001axx) measured in zero \feld.\n19XMCD (arb.u.)660 655 650 645 640\nhv (eV)pre-peak\n0.8\n0.6\n0.4\n0.2\n0\nXAS intensity (arb.u.)\nSUM XAS\nCR\nCL\n300\n250\n200\n150\n100\n50\n0\nχ(emu/mol)\n80\n 60\n 40\n 20\nTemp. (K)\n8\n6\n4\n2\n0\nA @ L 3peak (10-2)\nTC= 79KFIG. 2. Spectroscopic investigation of the magnetic order in grazing incidence geometry. Top\ngraph: MnL2;3XAS measured at 10 K with an applied \feld of 1 :4 T for the two opposite x-ray\nhelicities: circular right (CR, red) and circular left (CL, blue). The yellow curve shows the Mn\nL2;3XAS summed over the CR and CL spectra after the step-edge background (grey) subtraction.\nBottom graph: x-ray magnetic circular dichroism (XMCD) obtained from the di\u000berence between\nthe two absorption spectra. The black arrow denotes the pre-peak feature, which is a \fngerprint for\nMn hybridisation with the ligand. Inset: Temperature-dependent XMCD asymmetry measured at\nremanence (i.e., in zero applied \feld as a function of increasing temperature) after \frst \feld-cooling\nthe sample. The magnetic susceptibility curve obtained from bulk measurements (green curve) are\nalso shown. The dashed-dotted vertical line marks the magnetic ordering temperature.\n20FIG. 3. Orbital hybridisation in the Mn 3Si2Te6valence band. (a) Angle integrated, resonant\nphotoemission (ResPES) of the valence band measured scanning the excitation energy across the\nMnL3absorption edge. Coloured circles on the XAS spectra ( left panel ) mark the photon energies\nat which the valence band photoemission measurements were performed ( right panel ). The energy\nis referenced to the top of the valence band (E VB) and the VB spectra are vertically o\u000bset for\nclarity. (b) The Mn-( left) and Te-( right ) partial density of states (P-DOS) as determined by\nDFT calculations as a function of Uand compared to the experimental data. The Mn partial\ndensity of states was experimentally evaluated as the di\u000berence between the on- and o\u000b-resonance\nphotoemission spectra (measured at h\u0017= 641:4 eV andh\u0017= 635:5 eV, respectively) while the\nTe partial DOS was taken as the VB measured just before the Mn absorption edge (i.e., at h\u0017=\n635:5 eV). Best agreement between the experimental and calculated data (see, e.g., dashed line as\na guide to the eye) is found for the Mn partial DOS calculated at U= 1\u00002 eV.\n21FIG. 4. Electronic structure of Mn 3Si2Te6. Calculated band dispersions along high symmetry\ndirections, projected on the (a) Mn (1)and (b) Mn (2)d-orbitals and (c) the Te p-orbitals. (d) shows\nthe corresponding partial DOS. Calculations were performed with U= 1 eV. ARPES spectra\nacquired along the \u0016\u0000 -\u0016M at 58:5 eV photon energy for both (e) horizontal (LH) and (f) vertical\n(LV) linearly polarized light, which show a strong dichroism particularly for the Te-derived states.\n(g) A magni\fed view of (e) with enhanced contrast, better highlighting the upper Te-bands. (h)\nBulk states projected on the (001) surface Brillouin zone plane. (i) Bulk and surface-projected\nBrillouin zone marking the relevant high-symmetry points.\n22FIG. 5. Monte Carlo simulations. (a) Speci\fc heat of the full model (dark symbols) with\nparameters obtained from DFT+ Ucalculations for U= 1;2, compared with the speci\fc heat\ncalculated on a simpli\fed model with isotropic Jiso\n1;(Jiso\n2+Jiso\n5);Jiso\n3(light symbols). The stronger\nmagnetic frustration of the full model reduces the critical temperature TC, signalled by the peak\ninCv, by roughly 20% with respect to the simpli\fed model. (b) Temperature evolution of the\nferrimagnetic order parameter and its associated susceptibility which displays a sharp peak at\nthe transition (only U= 2 eV results are shown; the same trend was observed with the other\nset of parameters) and signalling the transition to a ferrimagnetic state. (c) The in-plane Fk=\np\n(F2\nx+F2\ny) and out-of-plane F?=jFzjcomponents of the ferromagnetic order parameter are shown\nas a function of temperature, con\frming the easy-plane character of the magnetically ordered phase.\n23V. SUPPLEMENTARY INFORMATION:\nSupplementary Figure S6. X-ray Di\u000braction pattern of Mn 3Si2Te6single crystal. The pattern\nshows the (00 l) re\rections. The inset shows typical Mn 3Si2Te6single crystals.\n24Supplementary Figure S7. (a) Experimental geometries for X-ray Magnetic Circular Dichroism\n(XMCD) measurements. In grazing incidence geometry ( left) circularly polarized light impinges\non the sample at 20\u000efrom the sample surface (i.e., the ab-plane), while the normal incidence\ngeometry is achieved when the photon beam is aligned perpendicularly to the ab-plane. In both\ngeometries, the magnetic \feld is collinear with the incident photon beam and the signal is acquired\nin total electron yield (TEY) detection by measuring the drain current of the sample. (b) XMCD\nasymmetry signal measured in the two di\u000berent geometries with \u00001:4 T applied \feld and the\nsample temperature kept at 10 K. The XMCD signal in grazing incidence being approximately\ntwice larger than for the normal geometry identi\fes ab-plane as the easy magnetization plane and\nit is in good agreement with the bulk magnetic properties measured by SQUID (Fig.1(c), (d) in\nthe main text). (c) Remanent XMCD spectra series of the Mn L2;3absorption edge measured in\nthe Mn 3Si2Te6ab-plane at di\u000berent temperatures across the ferrimagnetic transition (i.e., between\n10 K and 90 K) showing the onsetting of the long range magnetic order below 75 K. Spectra are\no\u000bset along the vertical axis for clarity.\n25"    },    {        "title": "2205.07541v1.Ferrimagnetism_in_stable_non_metal_covalent_organic_framework.pdf",        "content": "Ferrimagnetism in stable non -metal covalent organic framework  \nDongge Ma1,*, Yuhang Qian1, Mingyang Ji1, Jiani Li1, Jundan Li1, Anan Liu3,  \nYaohui Zhu2,* \n1College of Chemistry and Materials Engineering, Beijing Technology  and Business \nUniversity, Beijing, China  \n2Physics Department, School of Artificial Intelligence, Beijing Technology and \nBusiness University, Beijing, China  \n3Basic Experimental Centre for Natural Science, University  of Science and Technology \nBeijin g, Beijing, China  \n*Corresponding author:  madongge@btbu.edu.cn , yaohuizhu@gmail.com  \nAbstract:  We synthesized a pure organic non -metal crystalline  covalent organic \nframework TAPA -BTD -COF  by bottom -up Schiff base chemical reaction. And this \nimine -based COF is stable in aerobic condition and room -temperature. We discovered \nthat this TAPA -BTD -COF exhibited strong magneticity in 300 K generat ing magnetic \nhysteresis  loop in M -H characterization  and giant χ mol up to 0.028 . And we further \nconducted  zero-field cooling and field -cooling measurement of M -T curves . The as -\nsynthesized materials showed a large  χmol up to  0.028 in 300 K and increasing to 0.037 \nin 4.0 K with 200 Oe measurement field. The TAPA -BTD -COF 1/χ mol ~ T curve supported  its ferrimagnetism, with an intrinsic Δ temperature as -33.03 K by \nextrapolating the 1/χ mol ~ T curve . From the continuously increas ing slope of 1/χmol ~ T, \nwe consider that this TAPA -BTD -COF belongs to ferrimagnetic other than \nantiferromagnetic materials. And the large χmol value 0.028 at 300 K and  0.037 at 4.0 K  \nalso supported this, since common antiferromagnetic materials possess χ mol in the range \nof 10-5 to 10-3 as weak magnetics other than strong magnetic materials such as \nferrimagnetics and ferromagnetics. Since this material  is purely non -metal organic \npolymer, the possibility of d -block and f -block metal with unpaired -electron induced \nmagnetism can be excluded. Besides , since th e COF does not involve free -radical \nmonomer  in the processes of synthesis , we can  also exclude the origin of  free-radical \ninduced  magnetism. According to recent emerging flat-band  strong correlated exotic \nelectron property , this unconventional phenomenon may relate to n -type doping on the \nflat-band locat ing in the CBM  (conduction band minimum) , thus generating highly -\nlocalized electron with infinite  effective mass and  exhibiting st rong correlation , which \naccounts for th is non -trivial  strong and stable ferrimagneticity  at room -temperature and \naerobic atmospheric conditions .  \nI. Introduction  \nRecently, as the emerging of twisted bilayer magic -angle graphene  (TWBMG) (1, \n2), the exotic physics phenomena associated with flat -band induced strong electron \ncorrelation (3), such as superconductivity (4, 5), ferromagnet ism(6), topological states (7) \nand spin Hall  effect (8) have been  investig ated in  low temperature (9) (mK level ) and high magnetic field . Considering the difficulty to obtain a low -temperature condition to \nmK, it is  important to level up the T c of such flat -band materials. Inorganic  materials  \nsuch as TWBMG (10) are emerging as potential flat -band materials. However, due to \ntheir n arrow band -gap, and the considerable challenges to  precisely  tune and control \nthe twisting angle to several degrees, it is increasingly important to develop  tunable  \norganic flat -band materials with exotic electronic property such as magnetism.  \nAs a kind of novel emerging star materials, covalent organic framework (COF) is a \ntype of pure organic polymeric material connected  by strong covalent bond in two or \nthree -dimension other than metal -organic framework (MOF ), which is connected by \nweak coordinative bond (11). Besides, COF is crystalline polymer similar as MOF , \nmostly with crystalline domain as large as  several hundred nanometers to several \nmicrometers , even in certain structures can  be prepa red into  50-100 μm large  single \ncrystal (12). Apart from these , because of its covalent bond connection, COFs possess \nmuch higher chemical stability other than MOFs in aerobic humid, aqueous and organic \nsolvents condition (13). Moreover, due to its crystallinity, the COFs structure can  be \ncertainly confirmed via the assignment of XRD data with Pawley or Rietveld \nrefinement accompanying  theoretical modelling and calculation (14). Their  clear and \nunique structure facilitate s the demonstration of QSAR ( Quantitative structure –activity \nrelationship ) and guide s the design and synthesis of the materials with the optimal \nproperty as desired (15). Since COFs are synthesized via bottom -up organic synthetic \nroutes from organic small molecul e monomers, the main -chain, branch -chain  and pore \nstructures can all be precisely  tuned by choosing different monomers with diversifie d electronic and ster ic configurations. Besides, different dynamic chemical reactions such \nas boronate (16), Schiff -base(17), ketoenamine (18) and C=C (19, 20) formation  \nreactions  can be chosen to optimize COFs with best chemical, physical or biological \napplication performances.  \n  Since in 2005, Yaghi et al. discovered the first COF materials COF -1 and COF -5(16), \nCOFs materials have been utilized in various fields and applications, such as gas \nadsorption /separation (21), catalysi s(22), energy storage (23), proton conduction (24), \nfluorescence sensing (25) and optoelectronics (26). However, using COFs in magnetism \nas magnetic ordered materials is still rare. Jiang et al. reported in 2017 that an iodine -\ndoped C=C linked sp2c-COF can generate χmol up to 0.025 in 8 K (19). The authors \ndemonstrated  that it is a paramagnetic material other than ferromagnetic or \nferrimagnetic because of the absence of  a M-H hysterias loop  and spontaneous \nmagnetization phenomena , and this material exhibited paramagnetic property with \nmeasurable T c curie temperature . Wang (27), Cui (28),  Liu(29) and Br édas(30) \nindependently predicted the potential magneti sm in Lieb -lattice by theoretical \nmodelling and calculation. However, the macroscopic experimental realization of \nstrong magnetism such as ferrimagnetism in pure organic non -metal materials is still \nnot achieved. Herein, we demonstrated that a novel TAPA -BTD -COF, which is \nprepared via a Schiff-base condensation, exhibits large χ mol up to 0. 028 in 300 K and \n0.037 in 4.0 K in aerobic  atmospheric condition with 200 O e measurement field . \nMoreover, this COF exhibits measurably considerable large hysterias loop in M-H \ncurve in 300 K . By extrapolating and analyzing the 1/χ mol ~ T curve, we conclude that the TAPA -BTD -COF is a ferrimagnetics material with χmol ranging from 0.0 28 to 0.037 \nfrom  300 K cooling down  to 4.0 K with 200 Oe measurement field. This is the first \nroom -temperature strong magnetic organic materials exhibiting ferrimagnetism with \ngiant χmol = 0.028 at 300 K in aerobic atmospheric condition  which also possesses M -\nH hysteresis loop in room -temperature and large χ mol = 0.037 in 4.0 K, while previous \nmodel organometallic magnetic material  V(TCNE) xγ(CH2Cl2)  possesses a χmol equal \nto 8*10-6 in 4.2 K  and decompose in aerobic condition with short lifetime (31).  \nII. Experimental  and Methods Details  \nTAPA -BTD -COF was prepared  with a mixture of tris(4 -aminophenyl)amine (TAPA) \n(29.0 mg, 0.10 mmol), 4,4 -(benzothiadiazole -4,7-diyl)dibenzaldehyde  (BTD)  (51.4 mg, \n0.15mmol), n-butanol  (2.0 mL), and 1,4-dioxane  (1.0 mL), and acetonitrile (0.05 mL) \ncharging in a cylindrical glass tube (20 cm of length, ϕin = 0.8 cm, ϕout = 1.0 cm) and \nsonicated for 30 min to get a homogeneous dispersion solution. Then 0. 1 mL of 2 M \naqueous acetic acid was dropwise added to the solution, along with the color change to \ndark red. After it was degassed by the typical three freeze −pump −thaw cycles with \nliquid nitrogen, the tube was sealed and then heated at 60 °C for 3 days. The dark re d \nprecipitate was collected by  Soxhlet extraction and  washed continuously with acetone  \nand methanol . Final ly, solids were washed and immersed in acetone  for 24 h. After \nfiltration, samples were dried in vacuum oven under  120 °C overnight.  Yield:  90 %. Powd er X -ray diffraction data were collected using a Panalytical  Empyrean \ndiffractometer in parallel beam geometry employing Cu K α line focused radiation (λ \n=1.5405 Å) at 1600 W (40 kV , 40 mA) power . Sample powders were placing on glass \nsubstrate  recorded from  2θ =1.5° up to 30° with 0.02° increment.  \nSolid -state nuclear magnetic resonance (SS -NMR) spectra were collected on a Bruker \nA V ANCE III 400 NMR spectrometer using a standard Bruker magic angle -spinning \n(MAS)  probe with 4 -mm zirconia rotors. The magic angle  was adjusted by maximizing \nthe number and amplitudes of the signals of the rotational echoes observed in the 79Br \nMASFID signal from KBr. The transmitter frequency of 13C NMR is 100. 39MHz. The \nsolid -state 13C NMR spectra were acquired using cross -polariza tion (CP) MAS \ntechnique wit h the ninety -degree pulse of 1H with 4μs pulse width. The CP contact time \nwas 3 ms. High power two -pulse phase modulation (TPPM -15) 1H decoupling was \napplied during data acquisition. The decoupling frequency corresponded to 62.5 kHz. \nThe MAS sample spinning rates were 8 kHz. Recycle delays between scans were varied \nfrom 2 to 2.5 s. The 13C chemical shifts are given relative to neat tetramethylsilane as \nzero ppm, calibrated using the methylene carbon signal of adamantane assigned t o \n38.48 ppm a s secondary reference.  \nFourier -transform infrared (FTIR) spectra of starting monomer materials and COF \nsamples were recorded from 400 to 4000 cm–1 by using KBr pellets on a Bruker Tenso r-\n27 Fourier -transform infrared spectrometer  Solid -state UV -Vis electron absorption spectra  of starting materials  monomers  and COF \nsample were recorded on a Perkin -Elmer L amda  diffuse -reflectance -spectrometer.  \nScanning electron microscopy and its mapping -mode characterization of COF samples \nwere measured by dispersing the materials onto silica wafers attached to a flat \naluminum sample holder, which were further coated with platinum. Samples were \nanalyzed on a Zeiss  Sigma  300 field-emission SEM operating at 10 kV .  \nElement al analys es were carried out on a Thermo Flash Smart CHNOS organic element \nanalyzer.  \nElectron -paramagnetic -resonance (EPR)  measurements were carried out using an X -\nband spectrometer, Bruker E500  in 300 K under aerobic atmospheric conditions .  \nMagnetization measurements were  performed using a commercial  superconducting \nquantum interference device (SQUID) magnetometer  (MPMS -3, Quantum Design). All \nthe magnetization data of  TAPA -BTD -COF powder crystals were corrected by \nsubtracting diamagnetic sus ceptibility  of empty sample holder in atmospheric air \ncondition . \nModeling of crystal structures for target COFs were performed using the Vesta \n(Visualization and Electronic Structural Analysis) Version 3.5.2 software (K. Momma \nand F. Izumi, \"VESTA 3 for th ree-dimensional visualization of crystal, volumetric and \nmorphology data,\" J. Appl. Crystallogr., 1272 -1276 (2011) (32)). DFT calculation s were carried out within the  framework of the Perdew –Burke –Ernzerhof generalized gradient \napproximation  (PBE -GGA), as embedded in the Vienna ab initio simulation package \ncode (33). All the calculat ions were performed with a plane -wave cutoff energy of 500 \neV . For the  TAPA -BTD -COF, we adopted the experimental lattice constants and \ntransferred to the  primitive cell with a= 45.05700  Å; b= 45.05700 Å; c=3.65000  Å, α= \nβ= 90°, γ=120 ° [full structural relaxation would only change the lattice constants \nnegligibly  <0.1%)]. TAPA -BTD -COF  were  optimized to have AA stacking which will \nbe used for our study of  both bulk and bilayer systems. The geometric optimizations \nwere performed  without any constraint unti l the force on each atom is <0.01 eV ∙Å−1 \nand the  change of total energy is smaller than 10−4 eV per unit cell. The Γ centered \nBrillouin  zone k -point sampling was set with a spacing of 0.03 × 2π∙Å−1, which \ncorresponds  to 3 × 3 × 11  k-point meshes for bulk  unit cell .  \nIII．Results and discussions  \nWe condensed  the two  monomers t ris(4-aminophenyl)amine  (TAPA)  and 4,4' -\n(benzo (1,2,5 )thiadiazole -4,7-diyl)dibenzaldehyd e (BTD) to form imine -connected  \nTAPA -BTD -COF via a Schiff -base reaction. To improve its crystallinity and the \nreaction yield, w e optimized the preparation  conditions by screening multiple \nparameters. After tedious exploration, the optimum preparation conditions  were \nidentified, and we obtained the dark red crystalline powder with 90% yield ( the \nsynthetic route and structure details are shown in  Fig. 1). The successful preparation  of \nthe proposed COF was confirmed by a series of characterization techniques including powder X -ray diffraction (PXRD), solid -state-cross -polarization magic -angle -spinning  \n13C-NMR (SS -CP-MAS -13C-NMR), Fourier -transform infrared spectrometry (FT -IR), \nelement analysis  (EA) , scanning electron microscopy (SEM)  and element -mapping \n(See Fig . 2 and 3 ). The COF ’s structure was first characterized by PXRD comb ining  \nwith theoretical modelling and simulations  (see Fig 2A).  TAPA-BTD-COF exhibited \nsix diffraction peaks in PXRD spectrum, at 2θ = 2.20o, 3.86o, 4.46o, 5.93o, 7.82o and \n24.73o. With the proposed structure and the PXRD pattern at hand, we conducted the \ntheoretical calculation to simulate  the minimum -energy  geometry configuration via the  \nDFT methods  by modelling with VESTA 64  programs  and calculat ing with V ASP  \nsoftware package (see experimental  and methods  details part) . The theoretical \ncalculations indicated that the AA -stacking  structure possessing the following \ndiffraction peaks at 2 .26o, 3.92o, 4.53o, 5.99o, 7.84o and 24.36o well matched  the \nexperimental spectrum.  Then, the full -profile Rietveld refinement using GSAS \nsoftware was conducted. As sho wn in Fig.2 A, the refined AA -stacking structure \nprovided satisfactory  matching results between calculated curves and refined \nexperimental spectr a with small  Rwp=2.47% and Rp=1.80%. From the experimental \ncrystal lographic  results, t he TAPA -BTD -COF lattice parameters were determined as a= \n46.70 Å, b= 4 5.06 Å, c= 3.6 5 Å, α=β=  90.000°, γ= 120.000°. Furthermore,  the COF \nchemical structure  was i nvestigated by SS -CP-MAS -13C-NMR.  There appear four \nprominent  13C signals peaking at 153.7 8, 145.26, 136.74 and  129.37 ppm  of TAPA -\nBTD -COF  (see Figure 2B). Based on the previous reporte  TAPA and BTD -containing \nCOFs structure s, the corresponding carbons in different chemical environment s were assigned.  The peak at the most downfield (153.78 ppm) is belonged to the imine C=N \nand benzothiadiazole C=N carbons (marked as a and b in Figure 2 B inbox). The peak \nat 145.26 ppm origin ates from the quaternary carbons in the TAPA and phenyl rings. \nThe peak center ed at 136.74 ppm corresponds to the quaternary and tertiary carbon s of \nthe phenyl rings between T APA and BTD rings. The peak at the upper -most  field \n(129.37 ppm) corresponded to the ternary carbons in the T APA rings, benzothiadiazole \ncarbons and the phenyl g -position carbons signals. Moreover,  thorough disappearance \nof signals at 196.77 ppm  of C=O indicates that  the aldehy de is  totally  consumed up. \nWith the  new emerg ing signal at 153.78 ppm of imine C=N carbon together , the \nformation of the imine -linked TAPA -BTD -COF was further confirmed. The FTIR \nspectrometry was conduct ed to characterize  the formation of the imine functional group \nand the consummation of the amine and carbonyl  functional groups. As shown in Fig. \n2C, after condensation  reaction , the TAPA amine N -H stretching vibration signals at \n3336 and 3406 cm-1 wavenumbers  disappeared . Meanwhile, both the aldehyde H -C=O \nstretching vibration at 2720 and 2820 cm-1 and C=O stretching vibration at 1687 cm-1 \nwavelength of BTD precursor disappear  in newly formed  COF  FTIR spectrum . Besides , \nthe C=N imine stretching vibr ation at 1617 cm-1 appeared  in the COF  spectrum  whereas \nboth T APA and BTD precursors spectra lacked t his signal. These results further \nevidenced  the formation of imine COF from TAPA and BTD . The electronic absorption \nspectra of the three COFs studied in  this work were characterized via diffuse reflectance \nspectrometry. As shown in Fig. 2D, the obtained TAPA -BTD -COF exhibited red -shift \nin their absorption in comparison with the TAPA  and BTD monomers. This result demonstrated that the formation of imine-based COFs structures brought about the \nhigher degree of conjugation, which widened their UV -Vis absorption range. To obtain \nthe information of their microscopic morphology, the SEM  and element mapping  were \nundertaken (see Fig 3). From the SEM images, we obse rved that the TAPA -BTD -COF \nexhibited ordered spherical nanoparticle morphology with 500 -800 nm diameter  as \nshown in Fig  3A. The element -mapping images by SEM demonstrate s that the C, N and \nS elements distributed homogeneously in the whole COF nanocrystals  (see Figure 3B -\nF). Furthermore,  the chemical compositions of the as -synthesized T APA-BTD -COF,  \nwas confirmed by the element analysis (EA) technique . The obtained EA experimental \nresults well  agreed with the theoretical calculated value based on the proposed COF \nstructures (see Table 1). The experimental weight composition of  TAPA -BTD -COF is \nC (75.07%), H (4.39%), N (12.19%), and S (5.78%) , while the  infinite two-dimensional \nlayer model is (calculated for a single cell : C96H60N14S3) C 76.59%, H 3.99%, N 13.03%, \nand S 6.38%).    Figure 1. The schematic representation of the chemical structure s of TAPA-BTD -COF   \nFigure 2. Structural characterization of TAPA -BTD -COF (A)  X-ray-diffraction  \n(XRD) analysis of TAPA -BTD -COF with the experimentally observed pattern in \nblack, the Rietveld full -profile refined  curves in red, and the difference plot ted in blue. \n(B) Solid -state 13C CP -MAS NMR spectr um of TAPA -BTD -COF. (C) F ourier \ntransform -infrared  spectr um of (C) TAPA -BTD -COF, blue; TAPA monomer , red; \nBTD monomer, black. (D) Solid -state UV -Vis electron -absorption  spectrum of \nTAPA-BTD -COF, blue; TAPA monomer red; BTD monomer , black;  \n \n \n \n \nFigure 3.  Scanning electron microscopy of TAPA -BTD -COF (A) SEM image of TAPA -\nBTD -COF  (B) C, N and S element total mapping in SEM -mapping mode. (C) Separate \nC element distribution in SEM -mapping. (D) S element distribution in SEM -mapping. \n(E) S element distribution in SEM -mapping. (F) The energy spectrum in SEM -mapping.  \nTable 1.  Element analysis results of T APA-BTD -COF  \nTheoretical value  C (76.59%) H (3.9 9%),  N (13.03%)  S (6.38%)  \nExperimental value  C (75.07%)  H (4.39%)  N (12.19%)  S (5.78%)  \nFrom the deficiency of sulfur element as shown in element analysis results \nbetween experimentally obtained TAPA -BTD -COF material and theoretically modelled \nperfect  TAPA -BTD -COF crystal structure  (see Table 1) , we considered that th is S \ndeficiency also represents the deficiency in BTD unit compared with TAPA unit. As \nwell known, sulfur -contain ing BTD (benzothiadiazole) unit is extremely electron -poor, \nwhile TAPA (tris -4-aminophenyl -amine)  is an electron -rich group. The deficiency in \nBTD indicates that the COF  possesses excess electron s compared with its intrinsic state.          \n \n \n \n \n \n \nFigure 4. Electron -paramagnetic -resonance (ESR) spectr um of TAPA -BTD -COF  in \n300 K under aerobic atmospheric conditions.  \nBased on such consideration, we conducted the band -structure calculation and \nanalysis. To our delight, in the intrinsic TAPA -BTD -COF band structure, there appear \na set of 3 bands in including the conduction band bottom (CBM) just above the Fermi \nlevel. The  three bands include two Dirac bands with a crossing point and a flat band \n(conduction band) under these two Dirac bands (see Figure 5). The VBM bands  contain \nRight below the Fermi level is a set of two Dirac bands with a crossing point (Dirac \ncone). The a ppearance of non -dispersive flat band means the existence of infinite heavy \nelectron with zero -velocity. These  largely localized zero -velocity electrons is evidenced \nto belong to strong -correlated electron system. The strong correlation between electrons \nin flat band would generate various exotic non -trivial phenomena such as \nferromagneticity, ferrimagneticity, antiferromagneticity, superconductivity, topological \nstates, etc.  \n \n \n \n \n \nFigure  5. DFT calculated electronic band structure for k z =0 plane and the orbital -\nresolved  projected band.  And the projected density of states (PDOS) with blue , grey \nand yellow lines corresponding to nitrogen, carbon and sulfur element  respectively.  \n(Upper) The full -profile  and (bottom) the magnified band  structure s of VBM and CBM.  \n  \n \n \n \n \n \n \n \n \n \nFigure 6 . Hysteresis loop  in M-H curves  of TAPA -BTD -COF nanocrystal measured  by \nSQUID MPMS -3 at 300 K in aerobic atmospheric conditions. (Upper) the full -profile \nM-H spectrum (bottom) the magnified spectrum at low external field.  \nIn our system, because of the deficiency of electron -poor BTD unit, the whole \nCOF structure is electron excessive, which was prove n by the electron -paramagnetic -\nresonance  (EPR)  characterization in room temperature (300 K) under atmospheric \naerobic conditio n (see Fig ure 4). There appears  a strong signal of electron  with intensity \nabout 2000  in EPR spectrum. To investigate its magnetic property, we conducted the \nM-H magnetization experiments in 300 K under  aerobic  condition . To our surprise, the \nas-synthesize d TAPA -BTD -COF  exhibits pronounced magnetic hysteresis loop during \nM-H measurement  (See Fig ure 6). To further confirm this result, we conducted parallel \nmeasurements of different batches of our TAPA -BTD -COF materials synthesized and \nstored in different tim e. To our delight, the M -H hysteresis loop remained almost \nunchanged despite the choice of different batches COF s material s. To further explore \nthe intrinsic magnetic property of this COF material, the  M-T measurements under \nzero-field cooling and field co oling conditions were undertaken  (see Figure 7) . The M -\nT curves were transformed  to χmol ~ T curve  according  to the density and molar mass \nof TAPA -BTD -COF. As shown in Figure 7 , the χmol of TAPA -BTD -COF exhibits a very \nlarge value about 0.028  in 300 K. This value is considerably larger than common \nparamagnetic material with χmol about 10-3 to 10-5 level. With the continuously cooling, \nthe χ mol slowly reduces from 0.028 to 0.025. However, the curve demonstrates a steep \njump upwards at about 76. 4 K from 0.025 to 0.030 . The reason of this upwards jumping \nis not totally clarified . After the jumping to χ mol = 0.0 30 at 76. 4 K, the COF exhibits the \nsame trend as befor e the jumping. The χmol ~ T slope did not show  apparent change . \nAnd the χ mol remains near 0.030 from 76.7 K until cooling down to approximately 27.4 \nK. Further cooling from 27.4 K to 4. 0 K, the χ mol ~ T slope  value continues to increase  \nfrom 0.030 to 0.037 . And the χ mol value increased almost vertically from 8 K to 4.25 K , \nincreasing from 0.033 to 0. 037. This large χ mol is about the level of ferrimagnetic materials, which belong to strong magnetic materials similar as ferromagne tics with \nspontaneous magnetization  and coercive force  but magnetic domain antiparallel aligned \nwith different mag netic moment, which posses ses χmol in the range of 100 to 102. And \nthe 1/χ mol ~ T curve was also plotted to further analyze its magnetic property. From the \n1/χmol ~ T curve, we did not observe the T N (Néel temperature), which is critical to \nassign an antiferromagnetic materi al. Moreover, from the continuously  increasing \n1/χmol ~ T slope value, we can also ass ign it as a ferrimagnetic material other than \nantiferromagnetics . The relatively lower χ mol and coercive force differs it with \nferromagnetic materials . From the extrapolating  of 1/χmol ~ T curve to cross with x axis \n(y axis value equals zero), we obtained the critical temper ature Δ  equal ing to -33.03 K. \nThis negative value also differ entiates  it with  typical  ferromagnetic materials. Due to \nthese phenomena, we consider that our TAPA -BTD -COF belongs to strong magnetic \nferrimagnetic material.  This is the first stable pure organic non -metal material with \nconsiderable M -H hysteresis loop , coercive force  and giant χ mol up to  0.028 (previously \naerobic non -stable organometallic magnet mainly in the range of 10-6 to 10-4) at room \ntemperature in aerobic condition, and possess a large r χmol up to 0.037 at 4.0 K . \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \nFigure 7. (Upper ) Zero -Field cooled  (ZFC) and field cooled (FC)  magnetization curves  \nχmol ~ T of TAPA -BTD -COF nanocrystals measured at 200 Oe field.  (Bottom) Zero -\nField cooled  (ZFC) and field cooled (FC) curves of 1/χ mol ~T of TAPA -BTD -COF \nnanocrystals measured at 200 Oe field.  \nIV． Conclus ion \nIn summary, we reported the discovery of the first  non-metal organic ferrimagnetic \nmaterial TAPA -BTD -COF , which exhibited unconventional strong ferrimagnetism  and \neven keep its strong and stable magneticity in 300 K and aerobic atmosphere with \nconsiderable M-H hysteresis , coercive force and  χmol up to 0.028 . The synthesis, \ncharacterization and magnetic performance s were provided in detail. We discussed the \norigin of this non -trivial physics phenomena and attributed it to the population of excess \nelectrons  locating and localized  in CBM flat -band. The electron -popula ted flat band \nhosted the location for strong electron correlation. The strongly correlated electron  \nsystem s lead to  this unprecedented ferrimagneticity for our  non-metal and non -free-\nradical organic polymeric material. 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Physical \nReview B  47, 558 -561 (1993); published online Epub01/01/ \n(10.1103/PhysRevB.47.558).  \n "    },    {        "title": "1804.01724v1.Stochastic_ferrimagnetic_Landau_Lifshitz_Bloch_equation_for_finite_magnetic_structures.pdf",        "content": "Stochastic ferrimagnetic Landau-Lifshitz-Bloch equation\nfor finite magnetic structures\nChristoph Vogler\u0003\nFaculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria\nClaas Abert, Florian Bruckner, and Dieter Suess\nChristian Doppler Laboratory for Advanced Magnetic Sensing and Materials,\nFaculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria\nPrecise modeling of the magnetization dynamics of nanoparticles with finite size effects at fast\nvarying temperatures is a computationally challenging task. Based on the Landau-Lifshitz-Bloch\n(LLB) equation we derive a coarse grained model for disordered ferrimagnets, which is both fast\nand accurate. First, we incorporate stochastic fluctuations to the existing ferrimagnetic LLB equa-\ntion. Further, we derive a thermodynamic expression for the temperature dependent susceptibilities,\nwhich is essential to model finite size effects. Together with the zero field equilibrium magnetization\nthe susceptibilities are used in the stochastic ferrimagnetic LLB to simulate a 5\u000210nm2ferri-\nmagnetic GdFeCo particle with 70% FeCo and 30% Gd under various external applied fields and\nheat pulses. The obtained trajectories agree well with those of an atomistic model, which solves\nthe stochastic Landau-Lifshitz-Gilbert equation for each atom. Additionally, we derive an expres-\nsion for the intergrain exchange field which couple the ferromagnetic sublattices of a ferrimagnet.\nA comparison of the magnetization dynamics obtained from this simpler model with those of the\nferrimagnetic LLB equation shows a perfect agreement.\nI. INTRODUCTION\nThe calculation of the magnetization dynamics of\nlarge systems under the influence of fast varying tem-\nperatures is of great interest from both the scientific\nand the technological perspective. Heat-assisted mag-\nnetic recording (HAMR) [1–5] should be mentioned\nfirst and foremost here. Despite the computing power\nof modern supercomputers, coarse-grained models are\nneeded to manage the computational effort created\nby such complex systems. The development of the\nLandau-Lifshitz-Bloch (LLB) equation for pure ferro-\nmagnets by Garanin [6] and the subsequent improve-\nments [7–9] paved the way to make concrete design\nproposals for real HAMR devices [10–14].\nSimilar to the derivation of the Landau-Lifshitz-\nBloch (LLB) equation for pure ferromagnets by\nGaranin [6] (see Appendix A), Atxitia et al. [15] have\nrecently shown how the LLB equation can be adapted\nfor disordered ferrimagnets with two sublattices. Be-\nfore going into detail and presenting extensions to fer-\nrimagnetic LLB equation, we would like to briefly re-\nview the results of Ref. [15]. The temporal evolution\nof the reduced magnetization mA=MA=MA;0(with\nMA;0beingthezerotemperaturesublatticesaturation\nmagnetization) of sublattice Acan be calculated per\n@mA\n@t=\u0000\u00160\r0\nA(mA\u0002He\u000b;A)\n+\u00160\r0\nA\u000bk\nA\nm2\nA(mA\u0001He\u000b;A)mA\n\u0000\u00160\r0\nA\u000b?\nA\nm2\nA[mA\u0002(mA\u0002He\u000b;A)];(1)\n\u0003christoph.vogler@univie.ac.atwhere\u000b?\nAand\u000bk\nAare the perpendicular and the par-\nallel dimensionless damping constants, respectively.\n\r0\nAis the reduced electron gyromagnetic ratio \r0\nA=\n\re=(1 +\u00152\nA), which is defined via the coupling param-\neter\u0015Aof sublattice A to the heat bath. It is not\nsurprising that Eq. 1 is of the same form as the fer-\nroLLB equation, because within each sublattice the\nmagnetizations and the field terms are treated with\nthe mean field approximation usually used for ferro-\nmagnets.\nThe effective field He\u000b;Aof each sublattice is de-\nfined per [15]\n\u00160He\u000b;A=\u00160Hext+2dA\n\u0016Amz;Aez\n\u0000J0;AB\n\u0016Am2\nA[mA\u0002(mA\u0002mB)]\n+\u00031\"\n1\u0000\u0012mA\u0001mB\nme;A\u0001me;B\u00132m2\ne;A\nm2\nA#\nmA\n\u0000\u00032 \n1\u0000m2\nA\nm2\ne;A!\nmA; (2)\nwith\n\u00031=jme;A\u0001me;Bj\n2m2\ne;AjJ0;ABj\n\u0016A: (3)\nand\n\u00032=1\n2~\u001fk\nA\u0012\n1 +jJ0;ABj\n\u0016A~\u001fk\nB\u0013\n: (4)\nHere,\u0016Ais the magnetic moment of each spin in sub-\nlattice A,dAis the uniaxial anisotropy energy per\nspin,me;Ais the equilibrium magnetization and ~\u001fk\nA\nis the longitudinal susceptibility of the sublattice. InarXiv:1804.01724v1  [cond-mat.mtrl-sci]  5 Apr 20182\nsublattice Bthe same quantities are defined. In the\ncase of two sublattices with atoms A and B there exist\nthree exchange energies, JA\u0000A,JB\u0000BandJA\u0000B. The\nexchange energies in the LLB model depend on the\nnumber of nearest neighbors zand on the concentra-\ntionsxAof the atoms. Hence, the exchange energies\nbecomeJ0;AA=zxAJA\u0000AandJ0;BA=zxAJA\u0000B.\nThe described formalism was successfully applied in\nthe past [16–20]. Most of these works investigate fast\nrelaxation processes in ferrimagnets and use a simpli-\nfied or a linearized version of the ferrimagnetic LLB.\nDue to the deterministic nature of Eq. 1 all results\ncan be interpreted as ensemble averages. We are in-\nterested in the full dynamical response of ferrimagnets\nwith finite size under arbitrary external conditions.\nIn the presence of temperature, this response has a\nstochastic nature.\nNote, all equations are identical for sublattice B if\nsubscript A is replaced by subscript B. For the sake\nof clarity we will call the LLB equation for pure fer-\nromagnets ferroLLB (see Appendix A) and the LLB\nequation for ferrimagnets ferriLLB equation in the fol-\nlowing.\nII. EXTENSIONS TO THE\nFERRIMAGNETIC LLB EQUATION\nA. stochastic form\nTo account for stochastic fluctuations due to tem-\nperature we follow the derivations of Evans et al. [8]\nfor the ferroLLB equation, which lead to a Boltzmann\ndistribution of the magnetization in equilibrium. The\nbasic assumption is that thermal fluctuations can be\nintroduced to the LLB via thermal fields. These fields\nare uncorrelated in time and space, which means that\nitscomponentsconsistofwhitenoiserandomnumbers\nwith zero mean and a variance\n\n\u0018\u0011\n\u0014;i(t;r)\u0018\u0011\n\u0014;j(t0;r0)\u000b\n= 2D\u0011\n\u0014\u000eij\u000e(r\u0000r0)\u000e(t\u0000t0);(5)\nwherei;jare the Cartesian components of the ther-\nmal field,\u0014is a placeholder for the sublattice type (Aor B) and\u0011is a placeholder for parallel and perpen-\ndicular field components. The four diffusion constants\nD\u0011\n\u0014are to be determined for the specific problem. To\nachieve this there exist two strategies, one by means\nof the fluctuation dissipation theorem and one via the\nFokker-Planck equation. We will use the latter ap-\nproach in the following.\nIn its most general form the LLB equation can be\nwritten as a multivariate Langevin equation:\ndmi\ndt=ai(m;t) +X\nk\u0011b\u0011\nik(m;t)\u0018\u0011\nk(t):(6)\nIf the vector ai(m;t)and the tensor bik(m;t)are\nknown the corresponding Fokker-Planck (FP) equa-\ntion can be directly constructed per\n@\u001a\n@t=\u0000X\ni@\n@mi\" \nai\u0000X\n\u0011D\u0011X\nkb\u0011\nikX\nj@b\u0011\njk\n@mj\n\u0000X\n\u0011D\u0011X\njkb\u0011\nikb\u0011\njk@\n@mj!\n\u001a#\n;(7)\nThis equation describes the temporal evolution of the\nprobability density \u001a(m;t)of finding a magnetic con-\nfiguration with magnetization mat timet. In accor-\ndancewiththeferromagneticcasewedefine aA;i(m;t)\nandbA;ik(m;t)for sublattice A as follows\naA;i(mA;t) =\u0000\r0\nA\u00160(mA\u0002He\u000b)\n\u0000\u000b?\nA\r0\nA\u00160\nm2\nA[mA\u0002(mA\u0002He\u000b)]\n+\u000bk\nA\r0\nA\u00160\nm2\nAmA(mA\u0001He\u000b);(8)\nand\nbk\nA;ik(mA;t) =\u000eik\nb?\nA;ik(mA;t) =\u000b?\nA\r0\nA\u00160\u0012\n\u000eik\u0000mA;imA;k\nm2\nA\u0013\n:(9)\nInserting Eqs. 8 and 9 into Eq. 7 yields the FP equa-\ntion for the sublattice\n@\u001aA\n@t=\u0000@\n@mA\u0001\u001a\u0014\n\u0000\r0\nA\u00160(mA\u0002He\u000b)\u0000\u000b?\nA\r0\nA\u00160\nm2\nAmA\u0002(mA\u0002He\u000b) (10)\n+\u000bk\nA\r0\nA\u00160\nm2\nAmA(mA\u0001He\u000b) +D?\nA(\u000b?\nA\r0\nA\u00160)2\nm2\nAmA\u0002\u0012\nmA\u0002@\n@mA\u0013\n\u0000Dk\nA@\n@mA\u0015\n\u001aA\u001b\n:\nAs already mentioned, the main objective is to deter-\nmine the coefficients D\u0011\n\u0017, which are a measure for the\nmagnitude of thermal fluctuations. To compute these\ncoefficients we assume that in equilibrium the proba-\nbility density of each sublattice magnetization followsa Boltzmann distribution per:\n\u001aA=\u001aA;0exp(\u0000E(mA)=kBT);(11)3\n@\u001aA\n@mA=\u001aA\u00160MA;0V\nkBTHe\u000b=\u001aA\u00160natxA\u0016AV\nl3\natkBTHe\u000b:\n(12)\nThis equation holds for a discrete system with dis-\ncretization volume V. In the last term of Eq. 12 we\nidentified the total magnetic moment of sublattice A\nwith atomistic quantities. Here, xAis the concentra-\ntion of atoms A, latis the lattice constant and nat\nis the number of atoms per unit cell. Using the ex-\npression of Eq. 12 in the FP equation and demanding\nthat@\u001aA=@t= 0is valid in equilibrium, the diffusion\nconstants of sublattice A can be computed to\nD?\nA=\u0010\n\u000b?\nA\u0000\u000bk\nA\u0011\nl3\natkBT\n(\u000b?\nA)2\r0\nA\u00162\n0natxA\u0016AV(13)\nDk\nA=\u000bk\nA\r0\nAl3\natkBT\nnatxA\u0016AV: (14)\nFinally, the corresponding stochastic LLB equationfor ferrimagnets can be obtained by using Eqs. 8 and\n9 together with Eqs. 13, 14 and 5 in the Langevin\nequation (Eq. 6) per\n@mA\n@t=\u0000\u00160\r0\nA(mA\u0002He\u000b;A)\n+\u00160\r0\nA\u000bk\nA\nm2\nA(mA\u0001He\u000b;A)mA+\u0018k\nA(15)\n\u0000\u00160\r0\nA\u000b?\nA\nm2\nAn\nmA\u0002h\nmA\u0002\u0010\nHe\u000b;A+\u0018?\nA\u0011io\n:\nB. finite system susceptibilities\nTo integrate the LLB equation detailed knowledge\nof the longitudinal susceptibilities ~\u001fk\nAand ~\u001fk\nBare re-\nquired. In the original work of Atxitia et al. [15] a\nmean field approach was derived\n~\u001fk\nA;mean =\u0016BL0\nA(\u0010A)jJ0;ABjL0\nB(\u0010B) +\u0016AL0\nA(\u0010A)[kBT\u0000J0;BBL0\nB(\u0010B)]\n[kBT\u0000J0;AAL0\nA(\u0010A)][kBT\u0000J0;BBL0\nB(\u0010B)]\u0000jJ0;BAjL0\nA(\u0010A)jJ0;ABjL0\nB(\u0010B): (16)\nIn this equation LAis the Langevin function with\nargument\u0010A= (J0;AAmA+jJ0;ABjmB)=(kBT)and\nL0\nAis the corresponding derivative with respect to \u0010A.\nEquation 16 is, strictly speaking, correct only for in-\nfinite systems. Hence, to properly model a magnet\nwith finite size other strategies are needed. This dis-\ncrepancy was already extensively discussed in the case\nof pure ferromagnets [7, 9, 21]. Additionally, the im-\nportance of modeling the temperature dependence of\nthe anisotropy field was shown. By means of the per-\npendicular susceptibility the anisotropy field in each\nsublattice can be defined as\nHani;A=1\n~\u001f?\nA(mx;Aex+my;Aey):(17)\nHere, the temperature dependence is included in ~\u001f?\nA.\nA benefit is that both parallel and perpendicular sus-\nceptibility can be computed from thermodynamics.\nSpin fluctuations at zero field along and perpendic-\nular to the anisotropy axis can be used to derive an\nexpression for the response function. How this is done\nfor a ferromagnet is briefly reviewed in the following.\nThe result will help to understand the response of sus-\nceptibilities of sublattices in a ferrimagnet.\nThecanonicalpartitionfunction Zofmagnetization\nMiin microstate i, which is subject to a field B, can\nbe expressed per:\nZ=X\nie\u0000\f(Ei\u0000VMi\u0001B): (18)The expectation value of the magnetization can be\nwritten as\nhMi=1\nZX\niMie\u0000\f(Ei\u0000VMi\u0001B)\n=1\nZ1\n\fV@Z\n@B: (19)\nA similar expression for the expectation value of the\nsquared magnetization can be easily found per\nhM2i=1\nZ1\n\f2V2@2Z\n@B2: (20)\nBased on the definition of the susceptibility\n\u001f=\u0012@hMi\n@H\u0013\nT=\u00160\u0012@hMi\n@B\u0013\nT;(21)\nEqs. 19 and 20 can be used to calculate \u001fper\n\u001f=\u00160\fV\u0002\nhM2i\u0000hMi2\u0003\n: (22)\nObviously, the same expressions hold for the compo-\nnents of the susceptibility\n\u001f\u0011=\u00160\fV\u0002\nhM2\n\u0011i\u0000hM\u0011i2\u0003\n: (23)\nWe now assume that the ferromagnet is split into two\nsublattices with concentrations xAandxB, withxA+\nxB= 1. Hence, the partition function can be written4\nas\nZ=X\nie\u0000\f[Ei\u0000(xA+xB)VMi\u0001B]:(24)\nThe same procedure as shown above can now be ap-\nplied to obtain the susceptibility\n\u001f\u0011=\u00160\f(xA+xB)V\u0002\nhM2\n\u0011i\u0000hM\u0011i2\u0003\n:(25)\nObviously,\u001f\u0011can be divided into two expressions for\nthe corresponding sublattices. Without loss of gener-\nality we further analyze the longitudinal susceptibility\nof sublattice A\n\u001fk\nA=\u00160\fxAV\u0002\nhM2\nzi\u0000hMzi2\u0003\n=\u00160\f\nxAV\u0002\nh(xAVMz)2i\u0000hxAVMzi2\u0003\n:(26)\nThe expression xAVMz=PNA\niez\u0001\u0016ican be identi-\nfied with the total magnetic moment of sublattice A\nin z direction resulting in\n\u001fk\nA=\u00160\f(NA\u0016A)2\nxAVh\nm2\nA\u000b\n\u0000hmAi2i\n;(27)\nwith the normalized magnetization of the sublattice\nmA=PNA\niez\u0001\u0016i\nNA\u0016A: (28)\nIn Eq. 4 we are interested in the quantity ~\u001fk\nA=\n\u001fk\nA=(\u00160MA;0). Hence, the final expression takes the\nfollowing form\n~\u001fk\nA=\f(NA\u0016A)2\nxAVl3\nat\nnatxA\u0016Ah\nm2\nA\u000b\n\u0000hmAi2i\n=NA\u0016A\nkBT1\nxAh\nm2\nA\u000b\n\u0000hmAi2i\n: (29)\nIn contrast to ~\u001fkof the whole system a factor x\u00001\nA\nappears in the susceptibility of the ferromagnetic sub-\nlattice ~\u001fk\nA, which is an important but non-obvious re-\nsult. Since a ferrimagnet consists of two ferromag-\nnetic sublattices we need Eq. 29 to correctly extract\nthe sublattice susceptibilities from spin fluctuations.\nC. material function scaling\nThe susceptibilities obtained must be adjusted be-\nfore being entered in the LLB equation via the ef-\nfective field. Typically, functions from a mean field\nmodel are fitted for this purpose. To show how this\nprocedure works for ferrimagnets we would like to\nrely on an example. For better comparability with\nRef.[15]weuseacylindricalnanoparticleconsistingof\nGdFeCoassamplesystem. Thegeometryandthema-\nterial parameters of the particle are shown in Tab. I.\nIn order to be able to quantitatively and quali-A (FeCo) B (Gd)\nd[J] 8:07251x10\u0000248:07251x10\u000024\n\u0016[\u0016Bohr] 2.217 7.63\nx 0.7 0.3\nJ\u0014\u0000\u0014[J] 4:5x10\u0000211:26x10\u000021\nJ\u0014\u0000\u0017[J]\u00001:09x10\u000021\nnat 4\nr[nm] 5.0\nh[nm] 10.0\nTC[K] 697\nTcomp[K] 313\nTABLE I. Geometry and material parameters of both sub-\nlattices A and B in GdFeCo (taken from Ref. [15]). dis\nthe anisotropy energy per atom, \u0016is the magnetic moment\nin units of Bohr magnetons, xis the concentration, J\u0014\u0000\u0014\ndenotes the exchange energy per atom link between equal\natoms,J\u0014\u0000\u0017denotes the exchange energy per atom link\nbetween different atoms, natis the number of atoms per\nunit cell,latis the lattice parameter and randhare the\nradius and the height of the particle. Curie temperature\nand compensation point are denoted with TCandTcomp,\nrespectively.\ntatively validate the results of the proposed coarse\ngrained LLB model we use a finite difference model\nwithatomisticdiscretizationasreference. Themagne-\ntization dynamics of this reference model are assumed\nto be correct in a sense that we aim to reproduce them\nwiththepresentedcoarsegrainedferriLLBmodel. We\nuse the atomistic code VAMPIRE [22] solving the\nstochastic Landau-Lifshitz-Gilbert equation for each\nspin. VAMPIRE is also used to compute the tem-\nperature dependent average magnetization and the\ntemperature dependent spin fluctuations in order to\ndetermine the needed input functions (magnetization\nandsusceptibilities)fortheintegrationoftheferriLLB\nequation. For this purpose system trajectories with\n107time steps (after 2x104equilibration steps) with\nan integration time step of 10\u000015s for each temper-\nature value in the range of 0\u0000950K are simulated\nby means of a stochastic Heun integration schema.\nFirgure 1 displays the resulting equilibrium magneti-\nzation at zero field for both sublattices. To use the\ndata in Eq. 2 we first fit the FeCo curve me;Awith\nthe mean field expression\nme(T) =c1\u0012\n1\u0000T\nc2\u0013c3\n; (30)\nwith fit parameters c1;c2andc3. Here, the Curie tem-\nperatureTC=c2= 697K of the ferrimagnet is deter-\nmined. In the fit procedure of the second sublattice\nthis Curie temperature is fixed and just the other two\nparameters are adjusted. The resulting fit functions\nare plotted in Fig. 1 with black solid lines.\nThe same trajectories from which the equilibrium\nmagnetizations were determined can also be used to5\n0 200 400 600 8000.00.20.40.60.81.0\nT[K]meFeCo\nGd\nfit\nFIG.1. (coloronline)Zerofieldequilibriummagnetization\nmeversustemperature, computedwithanatomisticmodel\nof GdFeCo (parameters are given in Tab. I). The black\nsolid lines show fits, representing an infinite system.\ncalculate the fluctuations of the magnetization paral-\nlel and perpendicular to the anisotropy axis by means\nof Eq. 29. ~\u001f?for both sublattices is shown in Fig. 2.\nWith the expression (Eq. 29) derived in Sec. IIB the\n0 200 400 600 8000.02.04.0\nµA\n2dAµB\n2dB\nT[K]˜χ⊥[1/T]FeCo\nGd\nfit\nFIG. 2. (color online) Perpendicular susceptibility ~\u001f?,\ncomputedwithanatomisticmodelofGdFeCo(parameters\nare given in Tab. I) from magnetization fluctuations. The\nblack solid lines show fits, representing an infinite system.\nsusceptibilities agree well with the inverse anisotropy\nfield at zero temperature, which is also displayed as\ndashed line in Fig. 2 for both sublattices. Note, that\nthe susceptibilities change considerably with temper-\nature. This fact suggests that it is very important\nto correctly model the temperature dependence of ~\u001f?\nand not only to use the zero temperature value for\nthe whole temperature range. A detailed comparison\nwill be presented in Sec. III. To extract the suscepti-\nbilities for the usage in Eq. 2 we use the same fitting\nprocedure as proposed in Ref. [9] for ferromagnets per\ne\u001f?(T) =(\nc4mc5eT <<T C\nc6\nT\u0000TCT >T C:(31)\nParallel susceptibilities are presented in Fig. 3. As\n0 200 400 600 8000.00.20.4\nT[K]˜χ/bardbl[1/T]FeCo\nGd\nFeComzfluctuations\nfitFIG. 3. (color online) Longitudinal susceptibility ~\u001fk, com-\nputed with an atomistic model of GdFeCo (parameters\nare given in Tab. I) from magnetization fluctuations. The\nblack solid lines show fits, representing an infinite system.\nsuggested and explained in detail in Ref. [9] we use\nthefluctuationsofthemagnitudeofthemagnetization\nto determine the parallel susceptibilities. Since both\nsublattices are soft magnetic the fluctuations of mz\nare too noisy near the Curie temperature to be able\nto extract the true parallel susceptibilities from them,\nas pointed out in Fig. 3 for the FeCo sublattice. As\nfit function for ~\u001fkwe use the mean field expression of\nEq. 16 with two fit parameters c7andc8as follows\n~\u001fk(T) =c7~\u001fk\nmean(c8J0;AA;c8J0;BB;c8J0;AB;c8J0;BA;T):\n(32)\nThis means that each exchange energy appearing in\nEq.16isscaledbythefitparameter c8, whichisequiv-\nalent to a scaling of the Curie temperature. To under-\nstand this behavior the denominator of Eq. 16 can be\nanalyzed. Since the susceptibility diverges at TCthe\ndenominator becomes zero. With this condition the\nmean field Curie temperature can be determined to\nTC;mean =1\n6kB\u0010q\n(J0;AA\u0000J0;BB)2\u00004J0;ABJ0;BA\n+J0;AA+J0;BB\u0011\n: (33)\nFrom Eq. 33 it becomes clear that a scaling of all ex-\nchange energies is equivalent to a scaling of TC. But,\njust shifting the Curie temperature is not enough to\nadapt the susceptibilities to the correct finite size be-\nhavior. Scaling of the whole susceptibility function is\nadditionally required via fit parameter c7of Eq. 32.\nAnother issue that needs to be clarified is the mean-\ningoftheexpression \u00031atTCinEq.4. Sincebothsus-\nceptibilities diverge, the limit of the quotient ~\u001fk\nB=~\u001fk\nA\nmust be determined. Nieves et al. [23] derived a com-\npact form of \u00031atTCper\n\u00031=3kBTC\u0000c8J0;AA\n\u0016A: (34)\nNote, in this equation the scaling parameter c8is\nagainneededtoensurethattheexchangeenergy J0;AA6\n300400500600\nT[K]0.0 0.2 0.4 0.6−1.0−0.50.00.51.0\nµ0Hext=−0.8T,T= 500 K(a)\ntime [ns]mzVAMPIRE\nLLB\n0.0 0.2 0.4 0.6−1.0−0.50.00.51.0(b)\ntime [ns]mz\nFIG. 4. (color online) Temporal evolution of the z compo-\nnent of the normalized magnetization of both sublattices\nof GdFeCo computed with the proposed coarse grained\nferriLLB model and the atomistic code VAMPIRE. (a) A\nconstant magnetic field with \u00160Hext=\u00000:8T and an an-\ngle of 6\u000ewith the z direction is applied. (b) A Gaussian\nshaped heat pulse is applied (blue solid line, right y axis).\nyields the finite size Curie temperature. Near TC\nEq. 34 (instead of Eq. 4) is used in Eq. 2 in the coarse\ngrained ferriLLB model.\nIII. RESULTS\nIn order to confirm the validity of the proposed\ncoarse grained model numerical tests for the presented\nGdFeCo system (see Tab. I) are performed in the\nfollowing. First, the dynamics of single magnetiza-\ntion trajectories under the influence of heat and mag-\nnetic field are compared with corresponding trajec-\ntories computed with the atomistic code VAMPIRE.\nIn Fig. 4a) a constant temperature of 500K and a\nconstant magnetic field of \u00000:8T are applied to the\nferrimagnet. Field and easy axis of the grain (along z\ndirection) enclose an angle of 6\u000e. 500K is well above\nthe compensation point and the ferrimagnet is FeCo\ndominated. The simulations are started with an ini-\ntial magnetization of the FeCo sublattice in the pos-\nitive z direction and the Gd sublattice magnetization\npointing in the negative z direction. Unless otherwise\nstated, this initial configuration is used for all sub-\nsequent simulations. Figure 4a) illustrates that the\ntemporal evolution of mzof both sublattices obtained\nby the proposed coarse grained model agrees very well\n−2.00.02.0−1.00.01.0(a)\nm·ˆHextVAMPIRE LLB\n−2.00.02.0−1.00.01.0(b)\n−2.00.02.0−1.00.01.0(c)\nµ0H[T]m·ˆHext\n−2.00.02.0−1.00.01.0(d)\nµ0H[T]FIG. 5. (color online) Hysteresis loops of GdFeCo with a\nfield rate of 1T/ns calculated with the proposed coarse\ngrained ferriLLB model and the atomistic code VAM-\nPIRE. (a) Easy axis loop at a constant temperature of\n(a) 100K and (b) 500K. Hysteresis loop with the applied\nfield tilted 45\u000eagainst the z direction at a constant tem-\nperature of (c) 100K and (d) 500K.\nwith the resulting VAMPIRE trajectories.\nIn a second test we investigate the magnetization\ndynamics under a heat pulse, without an external\nfield. A Gaussian shaped heat pulse is used\nT(t) =Tmin+ (Tmax\u0000Tmin)e(t\u0000t0)2\n\u001c2;(35)\nwithTmin= 300K,Tmax= 600K,t0= 0:3ns and\n\u001c= 0:1ns. The temperature pulse starts slightly be-\nlow the compensation point and heats the ferrimagnet\nnearTC, before the system cools down again. Tem-\nperature pulse and mzof both sublattices are shown\nin Fig. 4b). The results of our coarse grained model\nand VAMPIRE again agree perfectly.\nIn a next step hysteresis loops at constant temper-\natures are compared. We analyze easy axis loops and\nloops with a field angle of 45\u000ewith respect to the\neasy axis of the ferrimagnet. The loops start with a\nsaturating field with a magnitude of 3T, which is de-\ncreased with a rate of 1T per nanosecond until \u00003T\nis reached. After that the field is again increased\nto 3T. The choice of the fast field rate results from\nthehighcomputationaleffortofatomisticsimulations.\nAll loops are calculated at two different temperatures,\n100K and 500K. Figure 5 displays the calculated hys-\nteresis loops of the total normalized magnetization of\nthe ferrimagnet for the four cases. Again, the coarse\ngrained ferriLLB model is in good agreement with\natomistic VAMPIRE simulations.\nIn a last validation step switching probabilities of\nGdFeCo under the influence of various Gaussian heat\npulses and a constant external field are analyzed.7\nAgain, a field with a magnitude of -0.8T and a field\nangle of 6\u000ewith the z direction tries to align the total\nmagnetization of the ferrimagnet along the negative\nz direction. Additionally, a heat pulse, according to\nEq. 35, with Tmin= 300Kand various Tmaxis ap-\nplied to the ferrimagnetic particle. For each Tmax,\nfrom 300K to 680K with \u0001Tmax= 20K, 128 trajec-\ntories are computed. The switching probability then\ncorresponds to the proportion of successfully aligned\nparticles compared to the total number of all started\nsimulations. The comparison of the switching proba-\n300 400 500 6000.00.51.0\nTmax[K]switching probabilityVAMPIRE\nferriLLB\nferriLLB const Hani\nFIG. 6. (color online) Switching probabilities of a GdFeCo\nparticle computed from 128 switching trajectories at each\nTmax. In each simulation a constant field with \u00160Hext=\n\u00000:8T and a Gaussian shaped heat pulse according to\nEq. 35 with Tmin= 300K and\u001c= 0:1ns are applied.\nbilities obtained by the coarse grained ferriLLB model\nand VAMPIRE simulations in Fig. 6 confirms the de-\nsired perfect agreement of the ferriLLB model. To\ncheck the influence of the temperature dependence\nof the perpendicular susceptibility in the ferriLLB\nmodel, which was introduced in Sec. IIC, the prob-\nabilities are recomputed with the same setup, with\nthe only difference that a constant anisotropy field\nHani;A= 2dA=\u0016A, is used. The resulting probabili-\nties, as illustrated in Fig. 6, show a completely dif-\nferent behavior. This fact strengthens the conclusion\nthat it is important to consider the temperature de-\npendence of the anisotropy field in the coarse grained\nferriLLB model.\nA. Equivalence of the ferromagnetic LLB\nequation\nAs already mentioned the ferriLLB equation for\neach sublattice (Eq. 15) has the same form as the fer-\nroLLB equation (Eq. A1). At first glance they differ\nonly in the effective field. In this section we derive an\nexpressionfortheintergrainexchangefieldfortwofer-\nromagnetic sublattices, which couples their ferroLLB\nequations. Furtherweshowthattheresultingeffective\nfield is very similar to Eq. 2 and that using the same\ninput functions (zero field magnetization and suscep-\ntibilities) the magnetization dynamics of a ferrimag-\nnet can be computed equally with both, the ferriLLBequation as well as the ferroLLB together with the\nderived intergrain exchange field.\nTo compute the intergrain exchange field between\ntwo ferromagnetic sublattices we refer to Ref. [9],\nwhere the desired intergrain exchange field was de-\nduced by determining the number of interacting spins\nbetween two coupled ferromagnetic layers on the\nboundary surface. Here, we follow the same strat-\negy by computing the mean number of interacting\nspins between the two ferromagnetic sublattices. The\nHeisenberg Hamiltonian serves as a starting point\nH=\u00001\n2X\nnnJklsk\u0001sl; (36)\nwherelandkarelattice sites and sk=\u0016k=\u0016kdenotes\nthe unit vector of the magnetic moment on lattice site\nk. To obtain the intergrain exchange energy the sum\njustgoesoverallneighboringlatticesitesnnwhichare\noccupied with atoms of different sublattices. We as-\nsume that the exchange integrals Jklare independent\nfrom the lattice site\nH=\u00001\n2JX\nnnsk\u0001sl: (37)\nIfzis the number of nearest neighbors in the lattice\neach atom A has on average zxBneighbors B and each\natom B has on average zxAneighbors A. Hence, the\nsum can be rewritten over all interacting pairs\nH=\u00001\n2Jzx BNAX\ni=1si;A\u0001si;B\u00001\n2Jzx ANBX\nj=1sj;B\u0001sj;A:\n(38)\nAs explained in Ref. [9] the next step is the transition\nfrom the atomistic to the LLB description, where each\nsublattice is represented by one magnetization vector\nH=\u00001\n2Jz(NAxB+NBxA)mA\nmA\u0001mB\nmB:(39)\nSince,N\u0017=x\u0017Vnat=l3\nat, Eq. 39 becomes\nH=\u0000Jzn atxAxBV\nl3\nat\u0012mA\nmA\u0001mB\nmB\u0013\n:(40)\nFinally, the intergrain exchange field of sublattice A\nis computed per\n\u00160Hiex;A=\u00001\nVM A;0@\n@mAH:(41)\nKeeping in mind that the absolute value mAcan\nbe written aspmA\u0001mAand with the definitions\nMA;0=natxA\u0016A=l3\natandJ0;AB=zxBJ, the inter-8\ngrain exchange field yields\n\u00160Hiex;A=J0;AB\n\u0016Aq\nm\u000b\ne;A(T)m\f\ne;B(T)\nmAmB\n\u0001\u0012\nmB\u0000mA\u0001mB\nm2\nAmA\u0013\n:(42)\nThe factorq\nm\u000b\ne;A(T)m\f\ne;B(T)was introduced in\nRef. [9] to account for the temperature dependence\nof the exchange constant. \u000band\fare power law\nexponents describing the temperature dependence of\nthe bulk exchange constant in the sublattices. For\na generic soft magnetic ferromagnet the exponent is\n1:66.\nThe intergrain exchange field of Eq. 42 together\nwith the ferroLLB equation in each sublattice is now\nused to compute the same switching probabilities as\nin Sec. III. Note, we use the temperature dependent\nfunctionsme(T),~\u001fk(T)and ~\u001f?(T)determined for\nthe sublattices of the ferrimagnet in Sec. IIC, which\nwere also used in the ferriLLBequation. The resulting\n300 400 500 6000.00.51.0\nTmax[K]switching probabilityVAMPIRE\nferriLLB\nferroLLB\nFIG. 7. (color online) Switching probabilities of a GdFeCo\nparticle computed for the same setup as shown in Fig. 6.\nHere the results of the atomistic code VAMPIRE, the pro-\nposed coarse grained ferriLLB model are again illustrated.\nAdditionally, each sublattice is computed with the coarse\ngrained ferroLLB model (see Appendix A) and coupled via\nthe derived intergrain exchange field of Eq. 42.\nswitching probabilities in Fig. 7 display that using the\nproper intergrain exchange field the ferroLLB equa-\ntion yields the same agreement with VAMPIRE simu-\nlations as the more complex ferriLLB equation. This\nagreementmightbeabitsurprisingatfirstglance, but\nif the effective fields of ferroLLB and ferriLLB equa-\ntionareexaminedmorecloselythesimilaritiesbecome\nobvious.\nFirst of all, in the proposed form the anisotropy\nfieldsHani;A(compare Eq. 17 and Eq. A4) are identi-\ncal. Further, thethirdterminEq.2isavectornormal\ntomA, and thus is only not vanishing if it enters into\nthe terms of the ferriLLB that change the direction of\nthe magnetization. If the identity of the double crossproduct is used we obtain\n\u0000J0;AB\n\u0016Am2\nA[mA\u0002(mA\u0002mB)]\n=J0;AB\n\u0016AmB\u0000J0;AB\n\u0016Am2\nA(mA\u0001mB)mA:(43)\nThe second term in Eq. 43 does not influence the mag-\nnetization dynamics. If the ferrimagnet is near equi-\nlibrium, which is a good assumption for the majority\nofthesimulationtimeduetotherapidlongitudinalre-\nlaxation of the LLB equation, the first term in Eq. 42\ncorresponds to the remaining term of Eq. 43.\nThe effective field of both formulations has a term\nwhichquicklyrelaxesthemagnitudeofthemagnetiza-\ntion to its equilibrium value. In the ferriLLB equation\nthe corresponding field term consists of two contribu-\ntions as can be seen in Eq. 4. Only the first term has\nits counterpart in the ferroLLB equation (Eq. A5).\nNevertheless, the second term\n~\u001fk\nB\n2~\u001fk\nAjJ0;ABj\n\u0016A(44)\nis only dominating very close to TC, where the suscep-\ntibilities diverge, while the quotient ~\u001fk\nB=~\u001fk\nAremains fi-\nnite. In this small range the ferriLLB equation shows\na faster relaxation of the sublattice magnetizations to-\nwardsitsequilibriumvalue, comparedtotheferroLLB\nequation. Obviously, this faster relaxation has not a\nlarge influence up to the simulated temperatures.\nAdditionally, the fourth term of Eq. 2 controls the\nangle between the magnetization of both sublattices.\nUnder the assumption that the magnetizations are\nnear equilibrium we can expand (mA\u0001mB)around\n(me;A\u0001me;B), yielding\n1\u0000\u0012mA\u0001mB\nme;A\u0001me;B\u00132m2\ne;A\nm2\nA\n\u0019\u00002\f\f\f\fmA\u0001mB\nme;A\u0001me;B\f\f\f\fm2\ne;A\nm2\nA+ 1 +m2\nB\nm2\ne;B:(45)\nTogetherwiththeprefactorthefirsttermofthisequa-\ntion becomes\n\u0000jJ0;ABj\n\u0016AjmA\u0001mBj\nm2\nAmA: (46)\nNear equilibrium this expression corresponds well to\nthe second term of the derived intergrain exchange\nfield in Eq. 42.\nInanutshell,wehaveshownthatalmosteveryeffec-\ntive field term of the ferriLLB equation has its coun-\nterpart in the ferroLLB equation if the derived inter-\ngrain exchange field of Eq. 42 is used to couple the\nferromagnetic sublattices. As a consequence the good\nagreement of simulated switching probabilities with\nboth equations in Fig. 7 can be well understood.9\nIV. CONCLUSION\nIn this work we developed a coarse grained model\nof disordered ferrimagnets based on the ferrimagnetic\nLandau-Lifshitz-Bloch (ferriLLB) equation [15]. In a\nfirst step, stochastic fields were incorporated into the\nferriLLBequationinordertoaccountforthermalfluc-\ntuations of individual system trajectories. In a sec-\nond step, an expression for the susceptibilities of finite\nsized ferrimagnets was derived from thermodynamics.\nAswiththeLLBequationofferromagnets(ferroLLB),\nmodeling these temperature-dependent material func-\ntions, including the zero field equilibrium magneti-\nzation, is the key to accurately describing the mag-\nnetization dynamics of ferrimagnets with high com-\nputational efficiency. We have shown that the pre-\nsentedcoarsegrainedmodelagreeswellwithatomistic\nsimulation, in which the stochastic Landau-Lifshitz-\nGilbert equation is solved for each atom of a parti-\ncle. The agreement was proven for simulations of a\nsmall GdFeCo ferrimagnetic particle with 70% FeCo\nand 30% Gd with a diameter of 5nm and a length of\n10nm subject to various external applied fields and\nheat pulses.\nIn the last part of the work we investigated the\ndifference between the ferriLLB equation and a more\nstraightforward model of a ferrimagnet, in which the\nferromagnetic sublattices are described with the fer-\nroLLB and coupled with an intergrain exchange field.\nWe derived this intergrain exchange field based on the\nHeisenberg Hamiltonian of the ferrimagnet under the\nassumption that the exchange is an interface exchange\nbetween the sublattices, with the interface extending\nacross all atoms. The fact that both models produced\nidentical results seemed surprising at first glance. But\nafter comparing the individual field terms it turned\nout that almost every field term of the ferriLLB equa-\ntion has a counterpart in the exchange coupled fer-\nroLLBequations. Forthisreason, thegoodagreement\ncan be well understood.V. ACKNOWLEDGEMENTS\nThe authors would like to thank the Vienna\nScience and Technology Fund (WWTF) under grant\nNo. MA14-044, the Advanced Storage Technology\nConsortium (ASTC), and the Austrian Science Fund\n(FWF) under grant No. I2214-N20 for financial\nsupport. The computational results presented have\nbeen achieved using the Vienna Scientific Cluster\n(VSC).\nAppendix A: ferromagnetic LLB equation\nTheferromagneticLLBequationreadsasfollows[9]\ndm\ndt=\u0000\u00160\r0(m\u0002He\u000b)\n\u0000\u000b?\u00160\r0\nm2fm\u0002[m\u0002(He\u000b+\u0018?)]g\n+\u000bk\u00160\r0\nm2m(m\u0001He\u000b) +\u0018k; (A1)\nwith\r0=j\rej=(1 +\u00152). Longitudinal and perpendic-\nular thermal field components consist of white noise\nrandom numbers with zero mean and variance\nh\u0018\u0011;i(t;r)\u0018\u0011;j(t0;r0)i= 2D\u0011\u000eij\u000e(r\u0000r0)\u000e(t\u0000t0):(A2)\nDiffusion constants D\u0011can be computed per:\nD?=\u0000\n\u000b?\u0000\u000bk\u0001\nkBT\n\r0\u00162\n0M0V\u000b2\n?\nDk=\u000bk\r0kBT\nM0V: (A3)\nThe effective magnetic field consists of external field\nHext, anisotropy field\n\u00160Hani=1\ne\u001f?(mxex+myey);(A4)\nand internal exchange field\n\u00160HJ=1\n2e\u001fk\u0012\n1\u0000m2\nm2e\u0013\nmforT<\u0018TC:(A5)\nAdditionally, the intergrain exchange field of Eq. 42,\nas derived in Sec. IIIA adds to the effective magnetic\nfield.\n[1] L. Mayer, J. Appl. Phys. 29, 1003 (1958).\n[2] C. Mee and G. Fan, IEEE Trans. Magn. 3, 72 (1967).\n[3] G. W. Lewicki and J. E. Guisinger, “Thermomagnetic\nrecording and magneto-optic playback system,” (10\nMarch 1969), patent No. US3626114.\n[4] H. Kobayashi, M. Tanaka, H. Machida, T. Yano, and\nU. M. Hwang, “Thermomagnetic recording,” (5 Jan-\nuary 1984), patent No. JPS57113402.[5] R. Rottmayer, S. Batra, D. Buechel, W. Challener,\nJ. Hohlfeld, Y. Kubota, L. Li, B. Lu, C. Mihalcea,\nK. Mountfield, K. Pelhos, C. Peng, T. Rausch, M. A.\nSeigler, D. Weller, and X. Yang, IEEE Trans. Magn.\n42, 2417 (2006).\n[6] D. A. Garanin, Phys. Rev. B 55, 3050 (1997).\n[7] N. Kazantseva, D. Hinzke, U. Nowak, R. W.\nChantrell, U. Atxitia, and O. Chubykalo-Fesenko,\nPhys. Rev. B 77, 184428 (2008).10\n[8] R. F. L. Evans, D. Hinzke, U. Atxitia, U. Nowak,\nR. W. Chantrell, and O. Chubykalo-Fesenko, Phys.\nRev. B85, 014433 (2012).\n[9] C.Vogler, C.Abert, F.Bruckner, andD.Suess,Phys.\nRev. B90, 214431 (2014).\n[10] J.-G. Zhu and H. Li, Magnetics, IEEE Transactions\non49, 765 (2013).\n[11] C. Vogler, C. Abert, F. Bruckner, D. Suess, and\nD. Praetorius, Appl. Phys. Lett. 108, 102406 (2016).\n[12] J.-G. Zhu and H. Li, IEEE Transactions on Magnetics\n53, 1 (2017).\n[13] C.Vogler, C.Abert, F.Bruckner, andD.Suess,Appl.\nPhys. Lett. 110, 182406 (2017).\n[14] C.Vogler, C.Abert, F.Bruckner, andD.Suess,Phys.\nRev. Applied 8, 054021 (2017).\n[15] U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko,\nPhysical Review B 86, 104414 (2012).\n[16] U.Atxitia, T.Ostler, J.Barker, R.F.L.Evans, R.W.\nChantrell, and O. Chubykalo-Fesenko, Physical Re-\nview B87, 224417 (2013).[17] U. Atxitia, J. Barker, R. W. Chantrell, and\nO. Chubykalo-Fesenko, Physical Review B 89, 224421\n(2014).\n[18] O. J. Suarez, P. Nieves, D. Laroze, D. Altbir, and\nO. Chubykalo-Fesenko, Physical Review B 92, 144425\n(2015).\n[19] D. Hinzke, U. Atxitia, K. Carva, P. Nieves,\nO. Chubykalo-Fesenko, P. M. Oppeneer, and\nU. Nowak, Physical Review B 92, 054412 (2015).\n[20] U. Atxitia, D. Hinzke, and U. Nowak, Journal of\nPhysics D: Applied Physics 50, 033003 (2017).\n[21] C. Vogler, C. Abert, F. Bruckner, D. Suess, and\nD.Praetorius,JournalofAppliedPhysics 120,223903\n(2016).\n[22] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A.\nOstler, M. O. A. Ellis, and R. W. Chantrell, J. Phys.:\nCondens. Matter 26, 103202 (2014).\n[23] P. Nieves, U. Atxitia, R. W. Chantrell, and\nO. Chubykalo-Fesenko, Low Temperature Physics 41,\n739 (2015)."    },    {        "title": "2208.08148v2.Polarization_selective_magneto_optical_modulation.pdf",        "content": "arXiv:2208.08148v2  [physics.optics]  29 Sep 2022Polarization-selective magneto-optical modulation\nBanoj Kumar Nayak and Eyal Buks\nAndrew and Erna Viterbi Department of Electrical Engineeri ng, Technion, Haifa 32000 Israel\n(Dated: September 30, 2022)\nWe study magneto-optical coupling in a ferrimagnetic spher e resonator made of Yttrium iron\ngarnet. We find that the resonator can be operated in the telec om band as a polarization-selective\noptical modulator. Intermodulation gain can be employed in the nonlinear regime for amplification.\nI. INTRODUCTION\nInformation is commonly transmitted by modulating a\nmonochromatic carrier wave. The method of single side-\nband modulation (SSM) allows reducing both transmis-\nsion power and bandwidth, in comparison with simpler\nmethods such as amplitude, frequency and phase mod-\nulation [1]. In the radio frequency band SSM can be\nimplemented using electronic circuits, however, SSM im-\nplementation in the optical band is challenging, since it\nrequires that different out of phase modulation methods\nare simultaneously applied [2, 3].\nMagneto-optical(MO)coupling[4–10] in ferrimagnetic\nsphere resonators (FSR) can be used for optical modu-\nlation of signals in the microwave band. Such a modu-\nlation has been demonstrated before [11–21] by exciting\nindividual whispering gallery FSR optical modes using\neither a tapered optical fiber or a prism. Here we em-\nployed a modified experimental setup, in which light in\nthe telecom band is transmitted through the FSR bulk.\nDriving the FSR near its resonance generates sidebands\nin the transmitted optical spectrum. We find that the\nFSR can be used as a polarization-selective SSM. The\npolarization selectivity is attributed to angular momen-\ntum conservation in photon-magnon scattering [22–27].\nWe demonstrate that intermodulation (IMD) gain can\nbe exploited in the nonlinear regime for amplification.\nII. EXPERIMENTAL SETUP\nThe experimental setup is schematically shown in\nFig.1. Optical components and fibers are red colored,\nwhereasblue coloris usedto label microwave(MW) com-\nponents and coaxial cables. A MW cavity made of a\nloop gap resonator (LGR) allows achieving a relatively\nlarge coupling between magnons and MW photons [29–\n32]. The LGR is fabricated from a hollow concentric\naluminium tube. A sapphire (S) strip of 260 µm thick-\nness is inserted into the gap in order to increase its ca-\npacitance, which in turn reduces the frequency fcof the\nLGR fundamental mode. An FSR made of Yttrium iron\ngarnet (YIG) having radius of Rs= 125µm is held by\ntwo ceramic ferrules (CF) inside the LGR. The two CFs,\nwhich are held by a concentric sleeve, provide transverse\nalignment for both input and output single mode opti-\ncal fibers. Fiber longitudinal alignment is performed by\nmaximizing optical transmission.\nFigure 1: Experimental setup. Optical fibers are installed o n\nboth sides of the FSR for transmission of light through the\nsphere. Optical components [TL (tunable laser), Att (opti-\ncal attenuator), PC (polarization controller) and PBS (pol ar-\nization beam splitter)] and fibers are red colored, and MW\ncomponents [MWA (microwave loop antenna), S (splitter), C\n(circulator), VNA (vector network analyzer), SA (spectrum\nanalyzer) and SG (signal generator)] and coaxial cables are\nblue colored. TL2 together with two PBSs (labelled as PBS1\nand PBS2) and two differential photo detectors (labelled as\nDPD1 and DPD2) operate as a polarization-selective optical\nspectrum analyzer (OSA) [28]. A power amplifier is serially\nconnected to the SG. The MWA is weakly coupled to the\nFSR-LGR system.\nThe angular frequency of the Kittel mode ωmis ap-\nproximately given by ωm=µ0γeHs, where Hsis the\nstatic magnetic field, µ0is the free space permeability,\nandγe/2π= 28GHzT−1is the gyromagnetic ratio [10].\nThe applied static magnetic field Hsis controlled by ad-\njusting the relative position of a magnetized Neodymium\nusing a motorized stage. The static magnetic field is nor-\nmal to the light propagation direction k, and the mag-\nnetic field of MW drive is nearly parallel to k. The\nLGR-FSR coupled system is encapsulated inside a metal-\nlic rectangular shield made of aluminum. The LGR is\nweakly coupled to a microwave loop antenna (MWA).\nThe plot in Fig. 2exhibits a vector network analyzer\n(VNA) reflectivity measurement of the LGR-FSR cou-\npled system. The static applied magnetic field Hsin this\nmeasurement is varied near the value corresponding to\navoided-crossing between the FSR and LGR resonances.2\nFigure 2: VNA reflectivity in dB units as a function of mag-\nnetic field Hsat applied microwave power of −30 dBm.\n1538.85 1538.9 1538.95−80−60−40−20\nλ2 [nm]IT [dBm]\nFigure 3: The transmitted optical spectrum. For this mea-\nsurement the TL1 is set at optical power of 31mW and wave-\nlengthλLof 1538.887nm, and the driving microwave is set at\nfrequency ωp/(2π) of 3.79GHz and power of Ppof 20 dBm.\nIII. OPTICAL SIDE BANDS\nOptical side bands are observed in the transmission\nspectrumwhenthedrivingmicrowavefrequency ωp/(2π)\nistunedclosetotheFSRresonanceat ωm/(2π). Theplot\nshown in Fig. 3exhibits the measured total optical inten-\nsityIT=IDPD1+IDPD2as a function of the wavelength\nλ2of TL2, where IDPD1andIDPD2are the intensities\nmeasured by the two differential photodetectors (labelled\nas DPD1 and DPD2 in Fig. 1). The side band wave-\nlengths are given by λL±λSB, whereλSB≃λ2\nLωp/(2πc),\nandλLis the TL1 wavelength, which is related to the\nTL1 frequency ωL/(2π) byωL= 2πc/λL, wherecis the\nspeed of light in vacuum. The value of λSB= 30.0pm is\nobtained for TL1 wavelength of λL= 1539nm and FSR\ndriving frequency of ωp/(2π) = 3.79GHz.\nBoth motorized polarization controllers (labelled as\nPC1 and PC2 in Fig. 1) have three optomechanical com-\nponents (paddles), which act as either quarter or half\nwave plates. The paddles’ angles of PC1 (PC2) are de-\nnoted by θ1A,θ1Bandθ1C(θ2A,θ2Bandθ2C). The in-cident light state of polarization (SOP) can be manip-\nulated using PC1. We observe that intensity of lower\nwavelength λL−λSBanti-Stokes sideband and higher\nwavelength λL+λSBStokes side band depend on the in-\nput SOP. SSM in the transmission spectrum, with either\nsingle anti-Stokes side band, or with single Stokes side\nband, canbe obtained byadjusting PC1. The plot shown\nin Fig.4(a) exhibits the measured anti-Stokes side band\nintensity as a function of microwave driving frequency\nfp=ωp/(2π) and PC1 angle θ1Cnear the avoided-\ncrossing region. The plot shown in Fig. 4(c) exhibits si-\nmultaneously measured Stokes side band intensity in the\nsame region. We clearly observe appreciable anti-Stokes\nand Stokes intensity in Fig. 4(a) and (c), respectively,\nwhen driving frequency ωp/(2π) becomes close to FSR\nresonance ωm/(2π). However, they are asymmetric. For\na certain range of PC1 position, SSM is obtained, i.e.\nonly one side band, either anti-Stokes or Stokes, is ob-\nserved. Contrary to other experimental setups, in which\nthe FSR is optically coupled by either a tapered optical\nfiber or a prism, for our setup, for which the measured\noptical transmission only weakly depends on the input\nwavelength λL, SSM can be obtained in wide range of\nλL.\nA rotating lambda plate polarimeter is employed to\nmeasure the input SOP. The polarimeter measurements\nrevealthattheinputSOPforthetwoextremecases(SSM\nof either anti-Stokes or Stokes peak) are orthogonal to\neach other (i.e. separated by a diameter on the Poincar´ e\nsphere).\nFigure 4: Side bands in dBm units. (a) anti-Stokes intensity\nas a function of PC1 angle θ1C. (b) anti-Stokes intensity as\na function of magnetic field Hs. (c) Stokes intensity as a\nfunction of θ1C. (d) Stokes intensity as a function of Hs.\nThe magnetic field Hsin (a) and (c) is tuned near avoided-\ncrossing regime. TL1 is set at optical power of 31mW and\nwavelength of λLof 1537.7nm, and the driving microwave\npower is set at Pp= 20 dBm. In (a) and (c), θ1A= 170◦\nandθ1B= 85◦, andθ1Cis varied from 0◦to 170◦, whereas in\n(b) and (d) ( θ1A,θ1B,θ1C) = (170◦,85◦,60◦) (for this setting\nboth Stokes and anti-Stokes peaks are clearly visible near t he\nFSR resonance).3\nFigure 5: Sideband SOP. The measured intensity IDPD1\n(IDPD2) is shown (in dBm units) in the plots labeled by the\nletter ’a’ (’b’). The intensity at wavelengths λL−λSB,λLand\nλL+λSBis shown in the plots labelled by the numbers ’1’,\n’2’ and ’3’, respectively. The TL1 is set at optical power of\n31mW and wavelength of λLof 1538.9nm, the driving mi-\ncrowave is set at power Ppof 20 dBm.\nThe plots shown in Fig. 4(b) and (d) exhibit anti-\nStokes and Stokes intensity, respectively, as a function of\nmicrowave driving frequency fpand static magnetic field\nHs. The FSR resonance changes as we vary the static\nmagnetic field Hs. Accordingly, from Fig. 4(b) and (d),\nwe see that both anti-Stokes and Stokes intensity gets\npronounced when driving frequency ωp/(2π) is within\nthe bandwidth of FSR resonance at ωm/(2π).\nOur experimental setup (see Fig. 1) allows measuring\nthe SOP of both sidebands. While the plots shown in\nFig.4display the total optical intensity IT=IDPD1+\nIDPD2, the intensity IDPD1(IDPD2) is separately dis-\nplayed in the top (bottom) row plots shown in Fig. 5.\nThese two intensities IDPD1andIDPD2represent two or-\nthogonal SOP, which can be set by adjusting PC2 (see\nFig.1). The left (right) column plotsin Fig. 5displaythe\nmeasured intensity of the left anti-Stokes (right Stokes)\nsideband at wavelength λL−λSB(λL+λSB), whereas\nthe intensity at the central wavelength λLis displayed\nby the central column plots in Fig. 5. For the measure-\nments shown in Fig. 5, PC1 is set to a nearly SSM state.\nBy varying the setting of PC2, we find that the cen-\ntral peak at wavelength λLis maximized (minimized)\nin the same region where the sidebands at wavelength\nλL±λSBare minimized (maximized). This observation\nimplies that in the region of SSM, the SOP of the side-\nbands is nearly orthogonal to the SOP of the incident\nlight. This orthogonality can be exploited at the receiver\nend of a data transmission system based on our proposed\nMO modulation, since it allows demodulation by polar-\nization filtering-out of the carrier at wavelength λL.IV. MO COUPLING\nThe MO coupling giving rise to the optical side-\nbands originatesfrom an interactionterm in the system’s\nHamiltonian, which is denoted by VSB. This term VSB\nis commonly derived from the classical energy density\nassociated with the interaction between magnetization\nand optical modes. For the case where only whispering\ngallery FSR optical modes participate in the interaction,\nthe term VSBwas derived in [11–19], whereas for our ex-\nperimental configurationwe consider the case wherelight\npropagates through the FSR bulk.\nConsider an incident I (scattered S) optical field, hav-\ningrightandleft handedcircularpolarizationamplitudes\nEI+andEI−(ES+andES−), respectively. The time-\naveragedenergydensity umassociatedwith MO coupling\nis given by um= (1/4)ReUm, where\nUm=/parenleftbigE∗\nS+E∗\nS−/parenrightbig\nǫm/parenleftbiggEI+\nEI−/parenrightbigg\n, (1)\nand where ǫm=ǫm0+ǫm+m+′+ǫm−m−′is a trans-\nverse permittivity tensor. The static part ǫm0is given\nby Eq. ( A2) of appendix A. The diagonal elements of\nǫm0give rise to the static Faraday effect, whereas the\nstatic Voigt (Cotton-Mouton) effect originates from the\noff-diagonal elements of ǫm0[see Eq. ( A2)]. The terms\nǫm+m+′andǫm−m−′account for the effect of magne-\ntization precession, where ǫm±is given by Eq. ( A3) of\nappendix A, andm±′representamplitudes ofmagnetiza-\ntionprecession. Note thatthe matrix ǫm±isproportional\ntoe±iϕ, whereϕis the azimuthal angle [see Eq. ( A3)].\nThesphericalsymmetryoftheFSR ispartiallybrokenby\nthe two CFs that are employed for holding it (see Fig. 1).\nIn the semiclassical approximation VSBis derived from\num= (1/4)ReUm[seeEq.( 1)]. Considerapairofoptical\nmodeshavingnormalizedscalarspatialwaveforms,which\nin spherical coordinates are expressed as un′(r,θ,ϕ) and\nun′′(r,θ,ϕ), respectively. The contribution of this pair\nto the total interaction term VSB, which is denoted by\nVn′,n′′, is expressed as\nVn′,n′′=a†\nn′an′′/parenleftbig\ngn′,n′′,+b†+gn′,n′′,−b/parenrightbig\n+h.c. ,(2)\nwherean(b) is an annihilation operator for the n’th opti-\ncal mode (magnon mode), and h .c.stands for Hermitian\nconjugate. The coupling coefficients gn′,n′′,±aregivenby\n(recall that in our experiment the static magnetic field is\nnormal to the light propagation direction)\n/planckover2pi1−1gn′,n′′,±≃g0/integraldisplay\ndr′e±iϕun′(r′)u∗\nn′′(r′),(3)\nwhereg0=ωLQs//parenleftBig\n8n2\n0N1/2\ns/parenrightBig\n, andNsis the number of\nFSR spins ( Ns= 3.4×1016for the FSR under study).\nFor YIG in the telecom band (free space wavelength\nλ0≃1550nm), the refractive index is n0= 2.19, and\nthe dimensionless MO coupling coefficient is Qs≃10−44\n[33], and thus g0/(2π) = 2.7Hz. The overlap integral in\nEq. (3) represents a photon-magnon scattering selection\nrule [11, 12].\nThe ratio of side band output optical power to the in-\nput optical power is denoted by ηSB. The largest value\nofηSBis obtained at the triple resonance [11], for which\nthe MW driving is tuned to the FSR resonance ωm, the\nlaser frequency ωLmatches the frequency of one opti-\ncal mode, and the second one has a frequency detuned\nfromωLbyωm. For this case ηSB≃(2n0Rsg0/c)2Nm\n[it is assumed that the overlap integral in Eq. ( 3) is of\norder unity], where Nmis the averaged number of ex-\ncited magnons in steady state. For the case where the\nMWA is nearly critically coupled to the FSR, at reso-\nnanceNm≃Pp/(/planckover2pi1ωmκm), whereκmistheFSRdamping\nrate. The values of Pp= 20 dBm, ωm/(2π) = 3.8GHz\nandκm/(2π) = 1MHz yield ηSB≃10−5. This rough\nestimate agrees with the experimentally observed value\nofηSB[see Fig. 5].\nV. KERR NONLINEARITY\nMagnetic anisotropy gives rise to Kerr nonlinearity in\nthe FSR response [34]. The nonlinearity can be exploited\nfor modulation amplification [35]. Modulation measure-\nments in the nonlinear regime are shown in Fig. 6. The\nresults indicate that the Kerr coefficient is negative (giv-\ningrisetosoftening). Fortheplotsshowninthetop(bot-\ntom) row of Fig. 6, the microwave driving frequency is\nswept upwards (downwards). The dependency on sweep-\ning direction is attributed to nonlinearity-induced bista-\nbility, which, in turn, gives rise to hysteresis.\nVI. SUMMARY\nIn summary, polarization-selective SSM in the telecom\nband is achieved using an FSR strongly coupled to an\nLGR. The modulator can be used in a wide optical band,\nand it is compatible with ultra low temperatures. Future\nstudy will explore potential applications, including quan-\ntum state readout of superconducting circuits.\nThis work was supported by the Israeli science founda-\ntion, the Israeli ministry of science, and by the Technion\nsecurity research foundation.\nAppendix A: Transverse permittivity tensor\nThe evolution of electromagnetic waves propagating\ninside a magnetized medium is governed by a 3 ×3\npermittivity tensor [36–38]. Consider a Cartesian co-\nordinate system ( x,y,z), for which the propagation di-\nrection is parallel to the zdirection. In this system\nthe static magnetic field (magnetization vector) is par-\nallel to a unit vector denoted by ˆh(ˆ m). The angle\nFigure 6: Spectral peaks (in dBm units) in the nonlinear\nregime as a function of MW driving power Pp. The intensity\nof the left (right) sideband at wavelength λL−λSB(λL+λSB)\nis shown in the plots in the left (right) column, whereas the\nplots in the central column show the intensity of the central\noptical peak (at TL1 wavelength λL). For the plots shown\nin the top (bottom) row, the frequency fpis swept upwards\n(downwards).\nbetween ˆh= (hx,hy,hz) = (sin θcosϕ,sinθsinϕ,cosθ)\nandˆ m= (mx,my,mz) is assumed to be small.\nFrom the 3 ×3 permittivity tensor, a 2 ×2 transverse\npermittivity tensor ǫTcan be derived. In a basis of cir-\ncular SOP ǫTis given by ǫT=n2\n0I+ǫm, wheren0is the\nmedium refractive index, Iis the 2×2 identity matrix,\nand the 2 ×2 matrix ǫm(in a basis of circular SOPs) is\ngiven by [39]\nǫm\nn2\n0=/parenleftbigg\nQsmzQ2\nsm2\n−\nQ2\nsm2\n+−Qsmz/parenrightbigg\n, (A1)\nwherem±= (mx±imy)/√\n2. For YIG in the telecom\nband, the refractive index is n0= 2.19, and the dimen-\nsionless MO coupling coefficient is Qs≃10−4[33].\nThe eigenvalues of ǫm/n2\n0(A1) are given by\n±Qs/radicalBig\nm2z+Q2sm2\n−m2\n+. For the Faraday configuration,\nfor which mx=my= 0 and mz= 1, i.e. ˆ mis parallel\nto the propagation direction, the eigenvectors of ǫm/n2\n0\nrepresent circular SOPs, the corresponding eigenvalues\nare±Qs, and MO coupling gives rise to circular bire-\nfringence, whereas for the Voigt (Cotton-Mouton) con-\nfiguration, for which mz= 0 and m2\nx+m2\ny= 1, i.e.\nˆ mis perpendicular to the propagation direction, the\neigenvectors of ǫm/n2\n0represent colinear SOPs, the cor-\nresponding eigenvalues are ±Q2\ns/2 [note that m2\n−m2\n+=/parenleftbig\nm2\nx+m2\ny/parenrightbig2/4], and MO coupling gives rise to colinear5\nbirefringence. Note that for the Faraday configuration,\nthe SOP rotation angle that is accumulated over a trav-\neling distance of a single optical wavelength is 2 πQs.\nTo describe the effect of magnetization precession on\nǫm, it isconvenienttoexpress ˆ m(magnetizationunitvec-\ntor) as a sum of parallel and perpendicular components,\nwithrespectto ˆh(magneticfieldunitvector). InaCarte-\nsian coordinate system ( x′,y′,z′), for which the static\nmagnetic field is parallel to the z′direction, the unit\nvector parallel to the magnetization vector is expressed\nasˆ m′=mx′ˆ x′+my′ˆ y′+mz′ˆ z′=m+′ˆ u′\n++m−′ˆ u′\n−+\nmz′ˆ z′, where ˆ u′\n±= (ˆ x′±iˆ y′)/√\n2, and where m±′=\n(mx′∓imy′)/√\n2. The unit vectors ˆ mandˆ m′are re-\nlatedby ˆ m=R−1\nˆhˆ m′, whereforagivenunit vector ˆ n, the\nrotation matrix Rˆ nis defined by the relation Rˆ nˆ n=ˆ z,\nand thus ˆ m=m+′ˆ v++m−′ˆ v−+mz′R−1\nˆhˆ z′, whereˆ v±=\nR−1\nˆhˆ u′\n±. The matrix elements of the 3 ×3rotationmatrix\nRˆhare given by R11= 1+(cos θ−1)cos2ϕ,R22= 1+\n(cosθ−1)sin2ϕ,R12=R21= (1/2)(cosθ−1)sin(2ϕ),\nR31=−R13= sinθcosϕ,R32=−R23= sinθsinϕ\nandR33= cosθ. The following holds ˆ v±=cos2(θ/2)ˆ u±−e±2iϕsin2(θ/2)ˆ u∓−2−1/2e±iϕ(sinθ)ˆ z,\nhenceˆ m=µ+ˆ u++µ−ˆ u−+µzˆ z+mz′ˆh, where\nµ±=m±′cos2(θ/2)−m∓′e∓2iϕsin2(θ/2) andµz=\n−2−1/2/parenleftbig\nm+′eiϕ+m−′e−iϕ/parenrightbig\nsinθ, and thus m±=µ∓+\n2−1/2mz′e∓iϕsinθ.\nTheassumptionthattheanglebetweenthestaticmag-\nnetic field and the magnetization vector is small implies\nthatmz′≃1 and|m±′| ≪1. To first order in |m±′|,\nǫmcan be expanded as ǫm=ǫm0+ǫm+m+′+ǫm−m−′,\nwhereǫm0, which is given by [compare with Eq. 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Shaw,3 H. T. Nembach,3,5 S. J. Poon,2 and A. D. Kent1  1Center for Quantum Phenomena, Department of Physics, New York University, New York, New York 10003, USA 2Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA 3Quantum Electromagnetics Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA 4Center for Memory and Recording Research, University of California San Diego, La Jolla, CA92093, USA 5JILA, University of Colorado, Boulder, Colorado 80305, USA Abstract  Skyrmions can be stabilized in magnetic systems with broken inversion symmetry and chiral interactions, such as Dzyaloshinskii-Moriya interactions (DMI). Further, compensation of magnetic moments in ferrimagnetic materials can significantly reduce magnetic dipolar interactions, which tend to favor large skyrmions. Tuning DMI is essential to control skyrmion properties, with symmetry breaking at interfaces offering the greatest flexibility. However, in contrast to the ferromagnet case, few studies have investigated interfacial DMI in ferrimagnets. Here we present a systematic study of DMI in ferrimagnetic CoGd films by Brillouin light scattering. We demonstrate the ability to control DMI by the CoGd cap layer composition, the stack symmetry and the ferrimagnetic layer thickness. The DMI thickness dependence confirms its interfacial nature. In addition, magnetic force microscopy reveals the ability to tune DMI in a range that stabilizes sub-100 nm skyrmions at room temperature in zero field. Our work opens new paths for controlling interfacial DMI in ferrimagnets to nucleate and manipulate skyrmions.    Introduction: Magnetic skyrmions due to their non-trivial topology have interesting properties1-3 that make them attractive for spintronic applications, such as racetrack memory and logic devices4-6. A magnetic skyrmion designates a chiral spin texture with a whirling spin configuration7. Skyrmions can be stabilized by broken inversion symmetry and chiral interactions, such as the Dzyaloshinskii-Moriya interactions (DMI)8,9, which is an antisymmetric exchange interaction that favors non-collinear neighboring spins. Ultrathin magnetic materials with interfaces to heavy non-magnetic metals with large spin-orbit coupling exhibit interfacial DMI that stabilizes skyrmions and chiral domain walls10-13. The interfacial DMI and the nucleation of skyrmions have been extensively investigated in ferromagnetic materials10,14-18. Very recently, magnetic skyrmions and chiral domains were reported in ferrimagnetic systems19-21. Nearly compensated thin ferrimagnetic films with interfacial DMI are interesting materials due to their low stray fields, reduced sensitivity to external magnetic fields, and fast spin dynamics, which are predicted to lead to ultrasmall and ultrafast skyrmions19,22. Unlike in ferromagnets where fast current-induced motion of chiral textures is impeded by the Walker breakdown and domain wall pinning13,23-25, high domain wall velocities––reaching 1000 m s-1––have been observed in ferrimagnetic CoGd films near the angular momentum compensation temperature19. Hence, ferrimagnetic thin films are promising candidates for ultrafast skyrmion-based spintronics.  Recently, bulk DMI was reported in an amorphous ferrimagnetic GdFeCo alloy26. However, the significant advantages of interfacial DMI are that it can be controlled by the nature of the interfaces and widely tuned to stabilize skyrmions. Yet, interfacial DMI has not been studied in this class of materials. The DMI competes with the perpendicular magnetic anisotropy and Heisenberg exchange interaction and tuning the DMI in a range that favors small skyrmions can be challenging. Asymmetric domain wall nucleation and motion is commonly used to measure DMI12,27. However, these methods require advanced models of the domain wall dynamics28,29. A more direct method is Brillouin light scattering (BLS) in the Damon-Eshbach geometry, which relies on the asymmetric spin-wave frequency dispersion in the presence of DMI30; the asymmetry is directly related to the strength of the DMI.  Here we present a systematic study of the interfacial DMI in CoGd thin films by BLS as a function of the capping layer composition and magnetic layer thickness. We aim to understand how the DMI in CoGd films depends on the structural symmetry and magnetic properties. The interfaces are studied by cross-sectional transmission electron microscopy (TEM). We found that as little as 10% of W in the cap layer in Pt/CoGd/Pt1-xWx thin films is sufficient to induce a DMI of about 0.15 mJ m-2, larger than the bulk DMI found in much thicker films26. We also observed that the DMI is inversely proportional to the magnetic thickness in asymmetric CoGd stacks, confirming the interfacial nature of the DMI. In addition, we were able to tune the DMI in a range that stabilizes sub-100 nm skyrmions at room temperature in zero field, as observed by magnetic force microscopy (MFM). Our findings provide insight into the key parameters that control the DMI in ferrimagnetic films toward achieving ultrasmall and ultrafast skyrmions.   Results: Tuning the DMI with the capping layer composition: The ferrimagnetic CoGd thin films were grown by RF magnetron co-sputtering on oxidized silicon wafers in the following sequence: W(3)/Pt(3)/Co78Gd22(t)/Pt1-xWx(3)/Pt(3) (thicknesses in nanometers) [Methods]. The W/Pt seed layer provides good adhesion to the substrate and texture to ensure perpendicular magnetic anisotropy (PMA). The top Pt layer prevents sample oxidation. The DMI of a 5-nm thick CoGd film was studied as a function of the W composition (x) of the cap layer Pt1-xWx. The magnetic properties of the films were measured by vibrating sample magnetometry (VSM) and are summarized in Table I (Methods). Figures 1(a) and 1(b) show an out-of-plane field room-temperature magnetization hysteresis loop and the temperature (T) dependence of the saturation magnetization (MS) for Pt/CoGd(5 nm)/W, respectively. MS (T) is greatly reduced around 150 K, corresponding to the magnetization compensation temperature (TM).  Spin wave spectroscopy using BLS was performed to measure the DMI in the CoGd films (Methods). The DMI leads to an asymmetric frequency dispersion of the counterpropagating spin waves30. The DMI energy D (mJ m-2) is proportional to the frequency shift (∆f DMI) and given by: |#|=\t&'\t()*+,(-./∆12-3,         (1) where g is the spectroscopic splitting factor (we take g = 2), μB the Bohr magneton and h Planck’s constant and k = 16.7 µm-1 is the spin wave vector. Notably, the DMI energy given by BLS measurements is an effective DMI averaged over the film thickness, i.e., a sum of the bottom and top interfacial contributions. An example of BLS spectra is displayed in Fig. 1(c) for Pt/CoGd(5 nm)/W. We fitted the spectra for positive and negative field polarity. The frequency shift was determined for the Stokes and the anti-Stokes peaks separately and then averaged.  The diameter of a skyrmion results from the competition between different energies such as the Heisenberg exchange energy, the magnetic anisotropy energy and the DMI strength. Ultrasmall skyrmions can be nucleated at room temperature only in a narrow range of D31-33. For DMI strength larger than a scale set by the magnetic anisotropy, the formation of stripe domains become energetically favorable32,33. Conversely, a weak DMI cannot stabilize a skyrmion. Theoretical work has predicted that ferrimagnetic materials are better candidates than ferromagnets to host ultrasmall and ultrafast skyrmions due to their low saturation magnetization, which causes only small stray fields22. Indeed, in ferrimagnetic materials, the interfacial DMI can dominate over the dipolar interactions and enable the formation of ultrasmall DMI skyrmions, which is difficult to achieve in ferromagnets. Hence, our goal was to provide a new method for controlling the interfacial DMI in thin ferrimagnetic CoGd films, which could allow one to precisely tune the DMI in a range that would enable skyrmion nucleation. Changing the nature of the CoGd interfaces can be used to engineer the DMI strength. Therefore, the idea is to leave the Pt underlayer at the bottom interface of the CoGd film unchanged and insert a Pt1-xWx alloy at the top interface. Thus, by changing the composition of the Pt1-xWx alloy, the structural symmetry of the film can be gradually broken to induce DMI. Pt is chosen for its strong spin-orbit coupling that gives a large interfacial DMI on Co spins34,35. Theoretical calculations based on Hund’s first rule have shown that, on the contrary, a weaker  DMI arises from interactions between W and Co34 and with the same chirality as Pt and Co. In addition, W, due to its giant spin-Hall angle36,37, would serve as a spin current source to enable skyrmion motion induced by spin-orbit torque (SOT). Figure 2(a) shows the DMI energy as a function of the W composition (x) in Pt/CoGd(5 nm)/Pt1-xWx measured by BLS. A maximum DMI of 0.23 ± 0.02 mJ m-2 is obtained for the asymmetric stack (x = 1). In comparison, a bulk DMI of up to 0.10 mJ m-2 was reported in ferrimagnetic GdFeCo films26. Conversely, the DMI is almost zero for the symmetric film (x = 0). Indeed, in the Pt/CoGd/Pt film, the top and bottom interfaces induce an interfacial DMI of similar amplitude but opposite sign (as the DMI is a chiral interaction), thus, resulting in a near vanishing effective DMI. In Pt/CoGd/W, since W gives rise to a weaker interfacial DMI, the contributions of the two interfaces are not compensated, leading to a larger effective DMI. As seen in Fig. 2(a), as little as 10 % of W introduced in the top layer is enough to significantly break the symmetry and induce a DMI of 0.15 mJ m-2. Yet, for x > 0.1, the DMI is less sensitive to the W content in the top interface. This would indicate that for 0.1 < x < 0.9, the W rather greatly reduces the DMI between the Co spins and the Pt. Indeed, if the W were actively contributing to the interfacial DMI, a stronger dependence of the DMI energy with the alloy composition would have been expected. Additionally, the quality of the interfaces, which has a great impact on the DMI38,39, was assessed by cross-sectional TEM. Figure 2(b) shows a cross-section of the asymmetric Pt/CoGd(5 nm)/W film, while a closer view of the top and bottom interfaces of the CoGd layer is displayed in Fig. 2(c). The CoGd alloy and the W layers are amorphous and the Pt is polycrystalline. Figure 2(c) shows that the Pt/CoGd and the CoGd/W interfaces are smooth.  Thickness-dependence of the DMI: It is necessary to study the dependence of the DMI on CoGd thickness to establish its nature, i.e. to know whether the DMI is arising from interfacial effects. In our Pt/Co78Gd22(t)/Pt1-xWx films, the W composition (x) was fixed either to 0 or 1 to investigate the DMI in a symmetric (x=0) and asymmetric (x=1) stack as a function of the magnetic thickness t. t was increased from 5 nm to 15 nm. The magnetic properties were systematically measured by VSM as a function of thickness. The results are presented in Fig. 3(a) and the DMI is plotted versus the inverse magnetic thickness. In the asymmetric Pt/CoGd/W stack, the DMI is inversely proportional to the magnetic thickness and reaches a minimum of 0.09 ± 0.01 mJ m-2 for 1/t = 0.067 nm-1 (t = 15 nm). The DMI linearly increases with the inverse thickness at a rate of ~ 1 mJ nm-1. This confirms that the strength of the DMI at the interface remains unchanged and underlines its interfacial nature. In Fig. 3(b), the saturation magnetization times the magnetic thickness is plotted versus the CoGd thickness. It has a linear dependence on thickness with an x-axis intercept near zero thickness, which indicates that there is no measurable dead layer in the CoGd film. Notably, in Fig. 3(a), the intercept of the linear fit is non-zero for 1/t = 0 (i.e. an infinitely thick film). This indicates that there is a residual DMI of 0.03 mJ m-2, which may result from a change of the magnetization compensation temperature throughout the thickness as evidenced in another rare-earth transition-metal alloy40 that could induce inversion symmetry breaking. Yet, as the thickness decreases, the interfacial effects become more important and the DMI increases as seen in Fig. 3(a). Thus, the interfacial DMI dominates in the entire thickness range we have studied. On the other hand, for the symmetric Pt/CoGd/Pt film, the DMI increases with the thickness to a value of 0.09 ± 0.02 mJ m-2 as seen in Fig. 3(a) (red data points). This behavior is surprising as the interfacial DMI is expected to be almost zero in symmetric layer structures. This result shows there is a difference in the nature of the top and bottom CoGd interfaces. In order to verify the latter, we performed TEM imaging in the Pt/CoGd(15 nm)/Pt film. The full stack is shown in Fig. 4(a) and a closer view of the top and bottom interfaces in Figs. 4(b) and 4(c), respectively. In Fig. 4(b), a thin layer of intermediate gray contrast (indicated by the white arrows) can be seen at the top CoGd interface and not in the bottom interface. It appears that the Pt from the capping layer has diffused into the amorphous CoGd film. As a result, the bottom and top interfaces have different roughness and intermixing. Hence, the DMI contributions of the top and bottom interfaces are not equal. Thus, due to the chirality of the interaction, they do not cancel out, leading to an increase of the net DMI. Intermixing and roughness effects appear to be more predominant in thicker films as the DMI increases with thickness in Pt/CoGd/Pt as seen in Fig. 3(a). Interestingly, the residual DMI observed for the asymmetric stack is of the same order of magnitude as that for the symmetric Pt/CoGd(5 nm)/Pt structure. Therefore, this suggests that this residual DMI is independent of the CoGd interfaces. Finally, as the skyrmion size depends on the magnetic film thickness22,33, it is thus important to understand how the interfacial DMI scales with the thickness. Evidence of magnetic skyrmions by MFM: Finally, we aimed to verify whether these thin ferrimagnetic alloy films would indeed host skyrmions. We focused on the asymmetric Pt/CoGd/W stacks as they are more promising for skyrmion motion via spin-orbit torque because of the giant spin-Hall angle of W36,37. In fact, in Pt/CoGd/Pt, the spin-orbit torques from the top and bottom interfaces would tend to cancel each other out. The Pt/CoGd/W films were subject to AC in-plane magnetic field demagnetization and imaged by atomic and magnetic force microscopy (AFM and MFM) at room temperature in zero field. Figure 5 shows images for Pt/CoGd(10 nm)/W. The left column is AFM data (Figs. 5(a) and (c)) and right column MFM images (Figs. 5(b) and (d)). The surface roughness was measured and is on the order of 0.2 nm (rms). Magnetic contrast is indicated by dark areas in the MFM images. By comparing the AFM and MFM images, it is clear that this contrast comes from magnetic textures and is not due to topography. Several skyrmion-like textures can be seen in Fig. 5(b). Figure 5(d) corresponds to a smaller MFM scan performed around of one of them marked by a square box in Fig. 5(b). This skyrmion-like texture is on the order of 100 nm. 50 nm skyrmions were observed in Pt/CoGd(8 nm)/W (see supplemental materials41). Arguably, considering the size of these textures, the DMI energy values [see Fig. 3(a)], and the fact that the CoGd films are weakly magnetized (MS ~ 140 – 150 kA m-1, see supplemental materials41), it is unlikely that these textures be magnetic bubbles4,7 stabilized by dipolar interactions. Thus, MFM images would rather indicate the presence of skyrmions. However, accurate estimation of the skyrmion size is difficult. Indeed, the MFM tip is sensitive to the dipolar field emerging from the magnetic texture which is spatially spread out. Furthermore, smaller magnetic features may be present in Fig. 5(b), yet they cannot be clearly distinguished due to the background noise and small magnetic contrast.  Discussion: To summarize, we have demonstrated that by capping the ferrimagnetic CoGd layer with a PtW alloy we could tune the DMI energy over a large range, from almost no DMI to an interfacial DMI energy of 0.23 mJ m-2. The DMI thickness dependence reveals the interfacial nature of the DMI in CoGd thin films. Thus, the DMI strength can be controlled by the interfaces in the thickness range we studied, which is also the range relevant for skyrmion nucleation. Moreover, the DMI was found to be non-zero in thicker symmetric structures emphasizing the role of interface roughness and intermixing. Lastly, we showed evidence that films can have a DMI in a range that allows sub-100 nm skyrmion nucleation at room temperature in zero field. Our experimental results provide insight into the key parameters that control the DMI in ferrimagnetic films toward achieving ultrasmall and ultrafast skyrmion motion for spintronic applications.  Methods: Thin film deposition:  The thin films were prepared by RF magnetron sputtering and deposited onto Si-SiO2 substrates at room temperature with a base pressure of 2.7´10-5 Pa. The Ar deposition pressures of W, Pt, CoGd, and Pt1-xWx were 0.93 Pa, 0.1 Pa, 0.16 Pa, and 0.16 Pa, respectively. CoGd films were obtained by co-sputtering from the Co and Gd targets. The powers of the Co and Gd sources were tuned to obtain CoGd films with approximately 78 at. % of Co. The deposition rates were calibrated using x-ray reflectometry.  Magnetometry: The magnetic properties of the samples were measured by vibrating sample magnetometry. Magnetic hysteresis loops were measured by varying the temperature from 100 K to 300 K with steps of 25 K in order to extract the temperature dependence of the saturation magnetization and the coercive field. Magnetometry was systematically performed prior to BLS experiments.  Brillouin light scattering: Spin wave spectroscopy using BLS is sensitive to interfacial effects and can be used to measure the DMI strength. The spin waves (SWs) inelastically scatter the monochromatic laser beam that is focused onto the sample surface. The frequency of the scattered photons is shifted by the SWs frequency. The SWs frequency is determined by analyzing the backscattered light with a (3 + 3)-pass tandem Fabry-Pérot interferometer. The counterpropagating Damon-Eshbach SWs have a non-reciprocal frequency dispersion characterized by a frequency shift (noted ∆fDMI). The frequency shift is considered here in absolute value. An in-plane bias magnetic field was applied to allow the SW propagate in-plane (Damon- Eshbach geometry). For a λ = 532 nm laser beam with an incidence of θi = π/4, the SW vector, k defined as k = 4πsin(θi)/λ was set to 16.7 μm-1.  Acknowledgments: This work was supported by DARPA grants No. D18AP00009 and R186870004 and by the DOE grant No DE-SC0018237.  Author contributions: A.D.K. and S.J.P. conceived of and supervised the project. Y.Q, H.T N., S.J.P. and A.D.K. planned the experiments. BLS measurements were conducted by G.A.R., J.M.S. and H.T.N. BLS data was analyzed by Y.Q., J.-W.X. A.D.K. and H.T.N. Sample fabrication and magnetometry measurements were performed by C.T M. and W.Z. MFM imaging was done by Y.Q. and J.-W.X. All of the authors participated in the discussion and interpretation of the experimental results. Y.Q. and A.D.K. drafted the manuscript and all of the authors contributed to and commented on it.   x MS (kA m-1) µ0HC (mT) TM (K) 0 160 12.5 125 – 150 0.1 166 11.0 125 0.25 160 10.0 125 0.5 180 12.0 125 – 150 0.75 160 12.0 125 – 150 1 145 11.0 125 – 150 TABLE I. Summary of the magnetic properties of Pt/Co78Gd22(5 nm)/Pt1-xWx films as a function of W composition (x). The room temperature saturation magnetization (MS) and coercive field (µ0HC) are indicated. \n  FIG. 1. (a) Out-of-plane magnetization hysteresis loop (b) and temperature dependence of the saturation magnetization measured by VSM for the Pt/CoGd(5 nm)/W sample. Magnetic compensation of this CoGd composition occurs around 150 K. (c) Spin wave spectroscopy obtained by BLS in Pt/CoGd(5 nm)/W. The shift in the frequency dispersion, ∆fDMI, is proportional to the DMI. The applied in-plane field was 0.460 T. The solid lines are fit to the BLS data obtained for positive (blue curve) and negative (red curve) field polarity.    StokesAnti-Stokes2ΔfDMI = 2.92 GHz-1000100-200-1000100200\n100150200250300050100150\n-30-20-101020300.00.51.0Magnetization (kA m-1)\nMagnetic Field (mT) Pt/CoGd(5 nm)/W(a)(b)\n(c)\nMagnetization (kA m-1)\nTemperature (K) Pt/CoGd(5 nm)/W\n  Positive field polarity  Negative field polarityPhoton counts (a.u.)\nFrequency (GHz) \n  FIG. 2. (a) DMI energy measured by BLS in Pt/CoGd(5 nm)/Pt1-xWx as a function of the W composition (x). The solid black line is a guide to the eye. (b) and (c) Cross-sectional TEM images of the Pt/CoGd(5 nm)/W film. (b) The full stack and (c) a magnified view of the top and bottom interface of the CoGd layer.   \n FIG. 3. (a) Magnetic thickness (t) dependence of the DMI in Pt/CoGd(t)/(W or Pt) with the DMI energy plotted against 1/t. In Pt/CoGd/W, the increase indicates the interfacial nature of the DMI interactions. (b) Room temperature magnetization thickness product versus thickness. The solid blue lines are linear fits to the Pt/CoGd/W data.   \n10 nm\n(b)(c)(a)SiO2WCoGdPtPtW\nWCoGdPtPtW0.00.20.40.60.81.00.000.050.100.150.200.25DMI (mJ m-2)\nTungsten composition (x) Pt/CoGd(5 nm)/Pt1-xWx\n0.00.10.20.000.050.100.150.200.25\n0510150.00.51.01.52.02.5Pt/CoGd(t)/WPt/CoGd(t)/PtDMI (mJ m-2)\n1/CoGd (t) thickness (nm-1)Pt/CoGd(t)/WPt/CoGd(t)/PtMS t (x10-6A)\nCoGd (t) thickness (nm)(b)(a) FIG. 4. Cross-sectional TEM images of the symmetric Pt/CoGd(15 nm)/Pt structure. The full stack is shown in (a) with a magnified view of the top (b) and bottom (c) interface of the CoGd layer.  \n FIG. 5. AFM (a,c) and MFM (b,d) images showing skyrmion-like magnetic textures nucleated in Pt/CoGd(10 nm)/W at room temperature in zero-field. The skyrmion imaged in (d) is indicated by a square box in (b).       \n(a)(b)\n(c)SiO2PtCoGdPtCoGdPt\nCoGdPt\nMFMAFM(a)(b)\n(c)(d)1 µm1 µm\n100 nm100 nmReferences:   1 Mühlbauer, S. et al. Skyrmion Lattice in a Chiral Magnet. Science 323, 915 (2009). 2 Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nature Nanotechnology 8, 899–911, doi:10.1038/nnano.2013.243; (2013). 3 Rohart, S., Miltat, J. & Thiaville, A. Path to collapse for an isolated Neel skyrmion. Physical Review B 93, 214412 (2016). 4 Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nature Nanotechnology 8, 152–156, doi:10.1038/nnano.2013.29; https://www.nature.com/articles/nnano.2013.29#supplementary-information (2013). 5 Koshibae, W. et al. Memory functions of magnetic skyrmions. 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"    },    {        "title": "1901.09524v1.Structural__magnetic__and_electrical_properties_of_collinear_antiferromagnetic_heteroepitaxy_cubic_Mn__3_Ga_thin_films.pdf",        "content": "1 Structural, magnetic, and electrical properties of collinear \nantiferromagnetic heteroepitaxy cubic Mn 3Ga thin films  \n \nHyun -Woo Bang1, Woosuk Yoo1, Chung man Kim1, Sungh un Lee2, Jiyeong Gu3, Yunchang Park4, \nKyujoon Lee5 and Myung -Hwa Jung1,* \n \n1Department of Physics, Sogang University, Seoul 04107 , Republic of Korea  \n2Department of Physics and Astronomy, Sejong University, Seoul 05006, Republic of Korea  \n3Department of Physics and Astronomy, California State University  LongBeach , Long Beach, CA 90840, USA  \n4National nanofab  center, Daejeon 34141 , Republic of Korea  \n5Institute of Physics, Johannes Gutenberg University Mainz, Mainz 55128, Germany  \n \n \nAbstract  \nAlthough a cubic phase of Mn 3Ga with an antiferromagnetic order has been theoretically predicted , it \nhas not been experimentally verifie d in a bulk or film form. Here, w e report the structural, magnetic, \nand electrical properties of antiferromagnetic cubic Mn 3Ga (C-Mn 3Ga) thin films, in compar ison with \nferrimagnetic tetragonal Mn 3Ga (T -Mn 3Ga). The structural analyses reveal that C -Mn 3Ga is hetero -\nepitaxially grown on MgO substrate with the Cu3Au-type cubic structure , which transforms to T-\nMn 3Ga as the  RF sputtering  power increases. The magnetic and magnetotransport da ta show the \nantiferromagnetic transition at TN = 400 K for C -Mn 3Ga and  the ferri magnetic transition at TC = 820 K \nfor T -Mn 3Ga. Furthermore, we find that the antiferromagnetic C -Mn 3Ga exhibits a higher electrical \nresistivity than the ferrimagnetic T -Mn 3Ga, which can be understood by spin -dependent scattering \nmechanism.  \nKeywords: Heusler compound, antifer romagnetic  cubic material, magnetic transition , hetero -epitaxy  \n \n* Corresponding author: Tel.: +82 2 705 8828, E-mail: mhjung@sogang.ac.kr  \n \n  2 Introduction  \nMost of Heusler compounds crystallize in the cubic L2 1 structure, where special attention has \nbeen focused on  half-metallic ferromagnetism to exhibit  a metallic behavior in one spin channel and \nan insulating behavior in the other spin channel, resulting in complete spin polarization of electrons at \nthe Fermi level [1-3]. Among them , Mn -based Heusler compounds have  been received of great \ninterest  due to  the tetragonally distorted structure showing half -metallic ferrimagnetism  [4-13], which \nhas an advantage of low saturation magnetization requisite for spin -transfer -torque based spin devices \n[14-16]. The magnetization of tetragonal Mn 3Ga vanishes over a wide range of temperature because \nthe magnetic moments of two Mn sublattice s are antiferromagnetically aligned with different \nmagnitude  [4,5,13]. However, the tetragonal distortion tends to destroy  the compensated \nferrimagnetism as well as  the half -metallic behavior  [2,13]. In theory, a cubic phase of Mn 3Ga is \npredicted to display half -metallicity with collinear antiferromagnetic order, anticipating  both complete \nspin polarization and zero net magnetic moment  [11-13,17 -22], which are meaningful  to lower energy \nloss in spintronic device applications. However, the cubic phase of Mn 3Ga has not been \nexperimentally verified yet in a bulk  or film  form. There has been only one report on nanostructured \nribbons of cubic Mn 3Ga phase , built with nano -sized particles , which are made  by quenching method \nof arc melting  and melt spinning [ 7]. Unfortunately, the cubic antiferromagnetic phase is not \nthermally stable and undergoes phase transitions to tetragonal ferr imagnetic phase at 600 K and to \nhexagonal antiferromagnetic phase at 800 K.  \nIn the present work , we have successfully fabricated hetero -epitaxial Mn 3Ga films with stable \ncubic phase by using RF magnetron sputtering method. We report the structural, magnetic , and \nelectrical properties of the cubic Mn 3Ga (C-Mn 3Ga), in comparison with the tetragonal phase (T -\nMn 3Ga). The structural analyses reveal that C -Mn 3Ga deposited  with low RF power crystallizes in the \ndisordered Cu 3Au-type structure, and it transforms to T -Mn 3Ga as the power  increases . We find the \nantiferromagnetic transition at TN = 400 K for C -Mn 3Ga, while the ferri magnetic transition at TC = \n800 K for T -Mn 3Ga. Furthermore, the electrical resistivity is higher in the antiferromagnetic phase of \nC-Mn 3Ga than that in the ferrimagnetic  phase of  T-Mn 3Ga. 3   \nExperimental Details  \nThe films of Mn 3Ga were deposited  on MgO(001)  substrate  using RF magnetron sputtering with \na base pressure of 1.0 × 10-6 Torr. The RF power was varied from 10 W to 55 W with a constant \nsubstrate temperature of 400oC and Argon pressure of 2mTorr during the depo sition. The crystal \nstructure of the samples was determined by using X -ray diffraction  (XRD  Bruker AXS D8 Discover \ndiffractometer using Cu Kα radiation) . In addition, high -resolution transmission electron microscopy  \n(HR-TEM  FEI Tecnai G2 F30 S -TWIN) and transmission electron diffraction  (TED) were used for \ndetailed structural investigation of Mn 3Ga with MgO substrate.  The surface morphology and relative \nMn composition were measured using the scanning electron microscope  (SEM) and electron  \ndispersive x -ray spectroscopy  (EDX  JEOL JSM -6700F) . The magnetic properties  and electron -\ntransport properties  were measured using a superconducting quantum interference device -vibrating \nsample magnetometer  (SQUID -VSM  Quantum Dsign MPMS ) where  the magnetic  field was swept  \nfrom -7 T to 7 T  and temperature range  2 ~ 300 K . The temperature dependence of magnetization  \nwith high temperature  was measured from 300 K to 800  K by using a physical propert y measurement \nsystem (VSM PPMS) .  \n \nResults a nd Discussion  \nThe structural evolution with varying the deposition conditions has been investigated by the X -\nray diffraction (XRD) measurements. Figure 1( a) shows  the XRD patterns of Mn 3Ga films  grown \nwith various deposition power s of the RF  magnetron  sputtering system . As the RF power decreases \nfrom 50 to 25 W, the  D0 22 tetragonal phase of Mn 3Ga (T -Mn 3Ga in Fig. 1(b)) is slowly transformed to \nthe cubic Mn 3Ga phase (C -Mn 3Ga in Fig. 1(c)) with the  disordered Cu 3Au-type (L1 2) structure. \nBesides the peak s from MgO  substrate, the XRD patterns mainly show three different peaks from the \nsamples, which are two (002) and (004) tetragonal peaks and one (002) cubic peak. For the films \ndeposited with high powers ( P > 43 W), there are two dominant peaks at 24.99 and 51.28  which \ncoincide with the (002) and (004) peaks of the  D0 22 tetragonal structure, respectively. For the films 4 grown with low powers ( P < 41 W), we observe a peak at 48.27 which  is matched with the (002) \npeak of the disordered L12 cubic structure [ 7,23,24]. In the intermediate RF powers (4 1 W ≤ P ≤ 43 \nW), a mixture of both tetragonal and cubic phase s is found  even though the diffraction peaks become \nbroader than those of single phase of T -Mn 3Ga or C -Mn 3Ga. It is clear that the structural phase \nchange from the tetragonal structure to the cubic structure occurs as the RF power decreases . The \nlattice parameters obtained from the XRD analyses for T -Mn 3Ga are c = 7.11 Å  and a = 3.89 Å by \nsetting c/a = 1.83, which  are the same value s reported by other literatures  [4-6,12]. For C -Mn 3Ga, we \nestimate the lattice parameter of a = c = 3.76 Å, which is consistent with that in the nanostructured \nribbons of Mn 3Ga proposed to have the Cu 3Au-type cubic structure [ 7]. Here, it should be pointed out \nthat the lattice mismatch of C-Mn 3Ga with the MgO substrate  (a = c = 4.21 Å) is larger than  that of T-\nMn 3Ga, and the C -Mn 3Ga phase is not a stable  phase in nature . Nevertheless, we obtain epitaxial \nfilms of both T -Mn 3Ga and C -Mn 3Ga, which are demonstrated by the in -plane  phi scan s. The \nrepresentative  plots are shown in  Figs. 1(d) and (e) , where the peaks are observed at 90 degree  \ninterval s indicating  four-fold symmetry. The results suggest that both T - and C - Mn 3Ga films on MgO \nsubstrate are epitaxially grown with high quality.   \nTo elucidate the epitaxy of C -Mn 3Ga, we have performed the transmission electron microscopy \n(TEM) and transmission electron diffraction (TED) measurements. Fig. 2(a) show s the TEM image of \nC-Mn 3Ga deposited  with the RF power of 25 W. The t wo different layers of MgO (100) substrate  and \nC-Mn 3Ga sample  are clearly distinguished  in the TEM image . The thickness of C -Mn 3Ga is about 10 \nnm. In Fig. 2(b), we observe two distinct  TED patterns corresponding to (200) orientation  of C -\nMn 3Ga and MgO  with four -fold symmetry, in consistent  with the result of XRD pi scan experiments.  \nThe lattice parameter s from the TED results are  estimated to be a = c = 3.78 Å  and 4.21 Å  for C -\nMn 3Ga and MgO, respectively, which also agree  well with the value s from the XRD results . The \nlattice mismatch between C-Mn 3Ga and MgO  is about 10.2% , which is too large to consider the \nepitaxial growth of C-Mn 3Ga phase. In order to  investigate  the microstructure at  the interface between \nC-Mn 3Ga and the MgO , we have taken a magnified image at the interface . Fig ure 2(c) shows the \nmagnified TEM image of the area in the red box of  Fig. 2(a). The atoms of C-Mn 3Ga are marked  with 5 red circles in the upper part and the atoms of MgO are marked with yellow circles in the lower part. It \nis clearly seen that t here is  an atomic  stacking ratio of 10:9 between C-Mn 3Ga and MgO  at the \ninterface  with small  dislocation of a few atomic layers . This kind of growth me chanism is well known \nin hetero -epitaxial  thin films with a  large lattice mismatch of  more than 9%  [25-27]. In the hetero -\nepitaxial growth, the films are gro wn by domain -matching epitaxy . In the scanning electron \nmicroscopy  (SEM)  images of the surface morphology  shown i n Figs. 2(d) and ( e), the domain  \nboundaries are observed in C -Mn 3Ga, compared with the flat surface in T-Mn 3Ga. This difference in \ndomain  structure  would be a natural feature when considering the  domain -matching epitax ial growth . \nThe most prominent change in the crystal structure is clearly seen in the magnetism. We probe \nthe magnetic transition temperatures of two T -Mn 3Ga and C -Mn 3Ga phases by measu ring high -\ntemperature magnetization up to 820 K. Figure 3(a) shows the temperature dependence of remanent \nmagnetization for T -Mn 3Ga, measured when the external magnetic field is removed after field cooling. \nThe magnetization abruptly increases below TC = 800 K corresponding to the ferr imagnetic transition \ntemperature of T -Mn 3Ga. For C -Mn 3Ga, on the other hand, we have measured the temperature \ndependent  magnetization in an applied magnetic  field of 1 kOe  because of no remanence . In Figure \n3(b), the magnetization exhibits a sharp peak at TN = 400 K, which is close to the temperature \nproposed as an antiferromagnetic transition temperature in the cubic phase of  Mn 3Ga [7,8,28-30]. \nThese magnetic data recorded in thin films are different from those taken with nano -ribbons [8], \nwhere the antiferromagnetic cubic Mn 3Ga undergoes multiple magnetic and structural transitions to \nferrimagnetic tetragonal phase at 600 K and to antiferroma gnetical hexagonal phase at 800 K, and \nthey are thermally irreversible. The irreversibility has been explained by the unstable cubic phase of \nMn 3Ga in nature because the cubic phase is obtained only by a nonequilibrium synthesis process such \nas rapid quenc hing from a very high temperature . However, in our case of a thin -film form, we obtain \na quite stable cubic phase of Mn 3Ga, which may be  related to  a strain  effect of the MgO substrate. \nNotab ly, we obtain two different stable phases of ferrimagnetic T -Mn 3Ga and antiferromagnetic C -\nMn 3Ga simply by changing the RF deposition power. As aforementioned, the lattice mismatch \nbetween Mn 3Ga sample  and MgO substrate is large  (~ 10.2%) . When such materials are deposited on 6 the substrate with a large lattice mismatch,  higher kinetic energy is necessary to overcome the energy \nbarrier of metastable state and achieve the stable state . In the present case, the metastable state is \ncubic phase of Mn 3Ga and the stable state is the tetragonal phase of Mn 3Ga, and the  higher deposition \npower means higher kinetic energy giving rise to the deposition of stable T -Mn 3Ga phase.  On the \nother hand, the lower deposition power could result in the growth of metastable C -Mn 3Ga phase  [31-\n34].  \nFigs. 3(c)-(e) show the magnetization  M(H) curves  at room temperature for the three typical \nphases of T-Mn 3Ga, M -Mn 3Ga, and C -Mn 3Ga. The magnetic fields are applied perpendicular and \nparallel to the film plane , and the background signals from the diamagnetic substrate  are subtracted. In \nFig. 3( c), T-Mn 3Ga exhibits clear hysteresis loop in the out -of-plane configuration, indicating the \nperpendicular magnetic anisotropy  found in a tetragonal system [ 4,10]. The saturation magnetization \nand anisotropy constant values are e xtracted to be MS = 220 emu/cc and Keff = 0.97 ×106 J/m3, which \nare consistent with previous results [ 4,10, 12,13 ].  In Fig. 3( d) for M -Mn 3Ga deposited with an  \nintermediate power  of 43 W , which is a mixture of the cubic and tetragonal phases,  the saturation \nmagnetization is approximately three times lower than that of T -Mn 3Ga. The low saturation \nmagnetization  is due to the appearance of the cubic phase of Mn 3Ga. In other words, the total  volume \nof ferromagnetic component decreases compared to the pure T -Mn 3Ga phase. In Fig. 3( e), C-Mn 3Ga \nshows no hysteresis behavior  but only a linear field dependence , demonstrating the antiferromagnetic \norder . Note that all the samples show abrupt change  at low magnetic fields, which may come from \nsmall misalignment from the c axis or small misorientation in lattice.  \nFigures 3( f)-(h) represent the Hall resistivity xy(H) curves of  T-Mn 3Ga, M -Mn 3Ga, and C -\nMn 3Ga obtain ed at room temperature  for the field along the c axis. The results are in good agreement \nwith the M(H) curves. We observe clear hysteresis loops for T-Mn 3Ga and M-Mn 3Ga, whereas no \nhysteresis loop is found in C-Mn 3Ga. Here it is noteworthy that there is a slight shift of the hysteresis \nloop in M -Mn 3Ga, which is an indication of exch ange bias effect. If the ferrimagnetic states of T -\nMn 3Ga coexist with the antiferromagnetic states of C -Mn 3Ga, the exchange bias effect can be \nexpected. The shift of hysteresis  loop is quite small  because the magnetic field and temperature  7 required for the conventional exchange bias effect  are too low enough to affect the exchange bias.  \nFrom the high -field data  with linear dependence , we calculate the carrier density of n = 1.0  1020, 1.3 \n 1020, and 1.9  1020 cm-3 for T -Mn 3Ga, M -Mn 3Ga, and C -Mn 3Ga, respectively. These values lie in \npoor metallic regime, which is necessary for later discussion on the electrical  transport.  \nWe investigate  the temperature  dependence of electrical resistivity (T) for C-, M-, and T -Mn 3Ga. \nThe results are displayed in Fig. 4(a). Since M-Mn 3Ga can have dominant contributions from the \ndifferent volume of mixed C - and T -Mn 3Ga phases, we select two different M -Mn 3Ga films deposited \nwith the RF powers of 4 3 and 4 1 W, which correspond to tetragonal - and cubic -phase dominant \nsamples , respectively. As shown in Fig. 4(a) , the electrical resistivity of T -Mn 3Ga is distinct from that \nof C-Mn 3Ga, i.e., they display very different behavior not only in temperature dependence but also in \nmagnitude. T-Mn 3Ga displays metallic behavior , C-Mn 3Ga exhibits  semiconducting behavior, and M -\nMn 3Ga shows the intermediate behavior  depending on the dominant phase; (T) of the tetragonal -\nphase dominant M -Mn 3Ga (43 W)  is close to that of T -Mn 3Ga and  (T) of the cubic -phase dominant \nM-Mn 3Ga (41 W)  is close to that of C -Mn 3Ga. The res istivity values are also changed sequentially \ndepending on the structural change. According to the carrier density estimated from the Hall \nmeasurements, C -Mn 3Ga has more carriers than T -Mn 3Ga, so that (T) of C -Mn 3Ga must be lower \nthan that of T -Mn 3Ga. However, we observe the opposite behavior in experiment . The carrier mobility \nestimated from the carrier density and resistivity value is 1,400, 950, and 510 cm2/Vs for T -Mn 3Ga, \nM-Mn 3Ga, and C -Mn 3Ga, respectively,  suggesting that the electrical resisti vity is governed mostly by \nthe carrier mobility. One possible explanation for the difference between T -Mn 3Ga and C -Mn 3Ga is \nthe effect of grain boundary scattering on the electron  transport. As shown in the SEM images in Fig. \n2(d) and (e), more grain boundaries exist in C -Mn 3Ga, resulting in the reduced carrier mobility and \nthe increased electrical resistivity. However, this grain boundary effect cannot explain the \nintermediate beh avior of M -Mn 3Ga. Another explanation  can be the spin-dependent scattering \nmechanism, which is normally discussed in giant magnetoresistance effect [ 35-38]. The electrical \nresistance is larger for the collinear antiferromagnetic spin configuration.  8 Since t he electrical transport is strongly affected by the magnetic order  in magnetic materials, it \nis useful to compare magnetoresistance with magnetization.  As displayed i n Fig. 4(b) and (c), clear \ntwo peaks in the magnetoresistance of T -Mn 3Ga coincides with the sharp peaks of differential \nmagnetization data at HC = 15 kOe. On the other hand, no anomaly is found in C -Mn 3Ga, where the \nmagnetoresistance changes by the order of 0.1% of the total resistance.  \n \nConclusions  \nOur results show that the cubic phase of Mn 3Ga can be stabilized and manipulated by reducing \nthe deposition power in RF magnetron sputtering. Notably, d epending on the crystal structure of \nMn 3Ga, two distinct magnetic phases hav e been observed experimentally; cubic Mn 3Ga (C -Mn 3Ga) \nand tetragonal Mn 3Ga ( T-Mn 3Ga). The XRD and TEM analyses show that C -Mn 3Ga is hetero -\nepitaxially grown on MgO substrate in spite of large lattice mismatch. From the magnetic field and \ntemperature depend ent magnetization measurements, we confirm C -Mn 3Ga to be antiferromagnetic \nwith TN = 400 K and T -Mn 3Ga to be ferrimagnetic with TC = 800 K. The electrical transport data \nprovide poor metallicity in C -Mn 3Ga, which can be understood by spin -dependent scatter ing in \ncollinear antiferromagnetic spin structure. These results enlarge the family of Heusler compounds and \npave a new  way to the engineering of new antiferromagnetic material for future spintronic device \napplications.  \n  \nACKNOWLEDGEMENT  \nThis work was supported by the National Research Foundation of Korea (NRF) grant funded by \nthe Korea government (No. 2016M3A7B4910400, 2017R1A2B3007918).  \n \n  9  Reference s \n1) T. Graf , C. Felser, and S. S. P. Parkin, Prog. Solid State Chem. 39, 1 (2011).  \n2) L. Wollmann, A. K. Nayak, S. 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McCarron, Thin Solid Films 155, 53 (1987) . \n33) R. F. C. Farrow, Mater. Res. Soc. Symp. Proc.  37, 275 (1985).  \n34) A. Chaoumead, Y. -M. Sung, and D. -J. Kwak, Adv. Condens. Matter Phys. 2012 , 1 (2012).  \n35) S. S. P. Parkin, Phys. Rev. Lett.  71, 1641 (1993).  \n36) W.H. Butler, X. -G. Zhang, D. M C. Nichol son, and J. M. MacLaren, J. Magn. Magn. Mater. 151, \n354 (1995).  \n37) J. F. Gregg, W. Allen, K. Ounadjela, M. Viret, M. Hehn, S. M. Thompson, and J. M. D. Coey, \nPhys. Rev. Lett.  77, 1580 (1996).  \n38) F. G. Aliev,  R. Schad,  A. Volodin,  K. Temst,  C. Van Haesend onck,  Y. Bruynseraede,  I. Vavra,  V. \nK. Dugaev  and R. Villar, Europhys. Lett., 63, 888 (2003).  \n  11 Figure captions  \n \nFigure 1.  (a) X -ray diffraction patterns of Mn 3Ga films,  where the line colors represent the \ndiffraction patterns for different power s of 25, 30, 35, 41, 42, 43, 40, 45, and 50  W. Crystal structure s \nof (b) tetragonal and (c) cubic Mn 3Ga. The red , green, and blue  spheres indicate  Ga, Mn I, and Mn II \natoms, respectively. In-plane phi scans of (d) tetragonal and (e) cubic Mn 3Ga. \n \nFigure 2.  (a) Transmission electron microscope image , (b) transmission electron diffraction patterns, \nand (c) the magnified image of cubic Mn 3Ga film and MgO su bstrate. The red and yellow colors \nrepresent the cubic Mn 3Ga film and the MgO substrate, respectively. Scanning electron microscope \nimages of (d) tetragonal and (e) cubic Mn 3Ga. \n \nFigure 3.  Temperature dependence of magnetization for (a) tetragonal and (b) cubic Mn 3Ga. \nMagnetic field dependence of (c -e) magnetization and (f -h) Hall resistivity data for tetragonal, mixed, \nand cubic Mn 3Ga, respectively.  \n \nFigure 4.  (a) Temperature dependence of electrical resistivity for cubic, (cubic - and tetragonal -phase \ndominant ) mixed, and tetragonal Mn 3Ga. (b) Magnetoresist ance data for  tetragonal  and cubic Mn 3Ga, \ncompared with (c) the first derivative of magnetization data for tetragonal Mn 3Ga. \n  12  \n \n \n \n \n \n \n \nFigure 1.  Bang et al.  \n \n \n \n \n \n \n \n \n13  \n \n \n \n \nFigure 2.  Bang et al.  \n \n \n \n \n \n14  \n \n \n \nFigure 3.  Bang et al.  \n \n \n \n \n \n \n15  \n \n \n \nFigure 4.  Bang et al.  \n \n"    },    {        "title": "1911.05393v1.Unveiling_domain_wall_dynamics_of_ferrimagnets_in_thermal_magnon_currents__competition_of_angular_momentum_transfer_and_entropic_torque.pdf",        "content": "Unveiling domain wall dynamics of ferrimagnets in thermal magnon currents:\ncompetition of angular momentum transfer and entropic torque\nAndreas Donges,1Niklas Grimm,1Florian Jakobs,1, 2Severin\nSelzer,1Ulrike Ritzmann,1, 2Unai Atxitia,1, 2and Ulrich Nowak1\n1Fachbereich Physik, Universit¨ at Konstanz, Universit¨ atsstraße 10, DE-78457 Konstanz, Germany\n2Dahlem Center for Complex Quantum Systems and Fachbereich Physik,\nFreie Universit¨ at Berlin, Arnimallee 14, DE-14195 Berlin, Germany\n(Dated: November 14, 2019)\nControl of magnetic domain wall motion holds promise for efficient manipulation and transfer of\nmagnetically stored information. Thermal magnon currents, generated by temperature gradients,\ncan be used to move magnetic textures, from domain walls, to magnetic vortices and skyrmions. In\nthe last years, theoretical studies have centered in ferro- and antiferromagnetic spin structures, where\ndomain walls always move towards the hotter end of the thermal gradient. Here we perform numerical\nstudies using atomistic spin dynamics simulations and complementary analytical calculations to\nderive an equation of motion for the domain wall velocity. We demonstrate that in ferrimagnets,\ndomain wall motion under thermal magnon currents shows a much richer dynamics. Below the\nWalker breakdown, we find that the temperature gradient always pulls the domain wall towards\nthe hot end by minimizating its free energy, in agreement with the observations for ferro- and\nantiferromagnets in the same regime. Above Walker breakdown, the ferrimagnetic domain wall can\nshow the opposite, counterintuitive behavior of moving towards the cold end. We show that in\nthis case, the motion to the hotter or the colder ends is driven by angular momentum transfer and\ntherefore strongly related to the angular momentum compensation temperature, a unique property\nof ferrimagnets where the intrinsic angular momentum of the ferrimagnet is zero while the sublattice\nangular momentum remains finite. In particular, we find that below the compensation temperature\nthe wall moves towards the cold end, whereas above it, towards the hot end. Moreover, we find that\nfor ferrimagnets, there is a torque compensation temperature at which the domain wall dynamics\nshows similar characteristics to antiferromagnets, that is, quasi-inertia-free motion and the absence of\nWalker breakdown. This finding opens the door for fast control of magnetic domains as given by the\nantiferromagnetic character while conserving the advantage of ferromagnets in terms of measuring\nand control by conventional means such as magnetic fields.\nI. INTRODUCTION\nFundamental interest in the understanding of the inter-\naction of thermal stimuli and magnetic domains has been\npropelled by its potential to impact recording and pro-\ncessing technologies for magnetically stored information\n[1,2]. Control of magnetic states by thermally generated\nstimuli is hence a growing field of research. Prominent\nexamples include the fields of spin caloritronics [ 3,4], e.g.\ndomain wall ( DW) motion by temperature gradients [ 5–12],\nand the field of ultrafast spin dynamics, e.g. thermally-\ninduced magnetic toggle-switching by ultrafast heat load\nin ferrimagnets ( FIs) [13–22]. Other coherent means of ma-\nnipulating magnetic textures, such as heat-assisted mag-\nnetic recording ( HAMR ) [23,24] and helicity-dependent\nall-optical switching ( HD-AOS ) for instance [ 25–30]—albeit\nnot primarily induced thermally—are facilitated tremen-\ndously by additional application of an ultrashort thermal\nexcitation und subsequent demagnetization [31–36].\nA key ingredient for the theoretical description of\nthe aforementioned magnetothermal effects, especially\nthermally-induced DWmotion, lies in the understanding\nof transport processes for energy and angular momentum.\nWhile in metals thermal spin currents are also transported\nby electrons, in insulators magnons, low energy magnetic\nexcitations, are responsible for the transport of angularmomentum via the spin Seebeck effect ( SSE) [37]. No-\ntably, thermal magnons can be used to move magnetic\ntextures, such as DWs, vortices, and skyrmions [ 38–40]. In\nprevious works the DWmotion of ferromagnets ( FMs) and\nantiferromagnets ( AFM s) induced by temperature gradi-\nents has been investigated thoroughly [ 7–11]. For instance,\nboth, experimental [ 5,41] and theoretical [ 7,8,10,12]\nstudies on FMs, have shown that a DWin a temperature\ngradient moves towards the hotter end of the sample.\nOn a microscopic level, the hot sample region acts as\na magnon source. Since ferromagnetic magnons carry\nspin, angular momentum conservation dictates that a\nmagnon which is transmitted through a DWexerts an\nadiabatic spin transfer torque ( STT) onto the wall. As a\nconsequence, the DWmoves in opposite direction to the\nmagnon propagation direction, i.e. towards the source\n[7,42,43]. Differently to the mechanism based on angu-\nlar momentum conservation, an alternative explanation\nbased on thermodynamic arguments has been suggested.\nSince the DW-free energy decreases as the temperature\nincreases [ 44], the so-called non-adiabatic entropic torque\nacts on the magnetization pulling the magnetic texture to-\nwards the hotter region of the sample, thereby maximizing\nthe entropy and minimizing the free energy [ 8,10]. The\ngenerality of the latter picture makes it also applicable to\nDWsinAFM s, in which thermal magnons do on averagearXiv:1911.05393v1  [cond-mat.mtrl-sci]  13 Nov 20192\nnot carry angular momentum [ 9,11], but also to more\ncomplex systems such as spin-spirals and skyrmions [ 39].\nDomain wall motion by thermal gradients in AFM sof-\nfers complementary properties to the motion in FMs. On\nthe one hand, AFM DW motion can be faster due to the\nalmost complete lack of inertia and the missing Walker\nbreakdown, which limits the maximum velocity. On the\nother hand, a disadvantage of AFM DW sis the difficulty to\nmanipulate, control and measure by conventional means,\nsuch as external magnetic fields. This kind of conven-\ntional magnetization control is only possible in a subclass\nofAFM s, so-called weak ferromagnets ( WFM s) such as\nthe rare earth ( RE)-orthoferrites for instance, in which\nthe Dzyaloshinskii-Moriya interaction ( DMI) induces a\nsmall net-magnetic moment, perpendicular to the N´ eel\norder parameter [ 45,46]. So-called “pure” AFM s, such as\nNiO for instance, in which there is no net-magnetization\nin bulk, require more sophisticated means of excitation\n[47,48].FIscan be seen as a generalization of both sys-\ntems, FMsand AFM s, since one may selectively tune the\nrelevant magnetic properties by modifying for instance\nthe sample temperature or composition [ 49,50]. This\nallows for an enhanced control of the ferromagnetic- or\nantiferromagnetic-like character of the spin dynamics and\nenables to potentially exploit the characteristically fast\nspin-dynamics of an AFM [49–51], while at the same time\none can easily manipulate them by using magnetic fields,\nand measure it by conventional detection methods such as\nthe magneto-optical Kerr effect ( MOKE ) or x-ray magnetic\ncircular dichroism ( XMCD ).\nNaturally, the larger parameter space of the FI, which\nemerges from the (at least) two non-equivalent magnetic\nsublattices, also implies that its magnetization dynamics\nbecomes more complex to understand, i.e. the properties\nof thermal magnon currents strongly depend on the un-\nderlying microscopic spin structure [ 52,53]. Thus, DW\nmotion in FIdriven by temperature gradients has been\nscarcely investigated so far [ 6] and previous works on DW\nmotion in FIs(and synthetic AFM s) were focused on more\ncontrollable stimuli, such as electric currents [ 51,54] and\nmagnetic fields [49].\nIn this work we study DW-dynamics in FIsdriven by\nthermal magnon currents in constant temperature gra-\ndients [ 53,55]. We use an atomistic spin model based\non the stochastic Landau-Lifshitz-Gilbert ( LLG) equation,\nto simulate ferrimagnetic DWsin a temperature gradient.\nOur simulation results will be compared to the previously\ndeveloped theory for DWmotion in FMs[56–58], based on\nthe collective coordinates approach. Depending on the\nstrength of the thermal gradient and the base temper-\nature, we find similarities in the DWdynamics to both,\nthe FMand AFM. For instance we can find a Walker\nbreakdown as observed for FMs[7,8], but we also find the\nquasi-inertia-free motion observed in AFMs [9]. However,\nin addition we find a completely new feature that is unique\nto the FIand has so far neither been reported for the FM\nnor the AFM: a motion towards the cold sample region in\nthe case of a FIbelow angular momentum compensationandabove Walker breakdown. Using a theoretical model\nbased on linear spin-wave theory ( LSWT ) we show that\nthis peculiar motion is due to angular momentum transfer\nand not linear momentum transfer.\nII. METHODS\nA. Atomistic spin model\nWe model the most simple kind of FI, that is a two-\nsublattice FIwith a rock salt structure (G-type magnetic\nordering) as depicted in Fig. 1. Our atomistic spin model\nis based on an extended Heisenberg Hamiltonian\nH=−/summationdisplay\ni<jJijST\niSj−/summationdisplay\niST\niKiSi (1)\nfor normalized magnetic moments Si=µi/µi.Jijde-\nnotes the isotropic Heisenberg exchange coupling and\nKi=dz\niˆ zˆ zT+dy\niˆ yˆ yTis the biaxial on-site anisotropy with\neasyz-axis and hard x-axis (0≤dy\ni< dz\ni=0.5 meV ).\nWe use the following exchange parameters for the interac-\ntion between spins located on the same sublattice A/B,\nJAA=16 meV ,JBB=0.5 meV , and between spins on\ndifferent sublattices, JAB=−6 meV . These values are\ncharacteristic for ferrimagnetic RE-transition metal ( TM)\nalloys [ 59], which are testbed materials in the field of ultra-\nfast spin dynamics [ 13–17], and are receiving an increasing\nattention in the field of spintronics [54].\nThe time evolution of the spins is computed with the\nstochastic LLGequation [60]\n(1 +α2\nG)∂Si\n∂t=−γi\nµiSi×(Hi+αGSi×Hi) (2)\nwith the effective field Hi=−∂H/∂Si+ζi, containing\nboth the deterministic field from the spin Hamiltonian H,\nEq.(1), and the stochastic field ζiin the form of Gaußian\nwhite noise, with\n/angbracketleftζi/angbracketright= 0 and/angbracketleftζi(0)ζT\nj(t)/angbracketright=2αGµikBTi\nγiδijδ(t).(3)\nFor the atomic magnetic moments we use µA= 4µBohr\nandµB= 5µBohr. Here, for simplicity, we assume that\nthe Gilbert damping and gyromagnetic ratios are the\nsame for both sublattices. The gyromagnetic ratios are\nγi= 2µBohr/~=1.76×1011s−1T−1, and the Gilbert\ndamping is set to αG= 0.01. By numerical integration\nof Eq. (2), for a range of temperatures we calculate the\nthermal average of the sublattice-specific as well as the\nnet angular momentum (Fig. 1). The angular momen-\ntum compensation temperature, at which the net angular\nmomentum is zero, is found at TA=107 K (Fig. 1). More-\nover, our numerical calculations allow us to determine\nthe Curie temperature of the system, TC=616 K , in the\nrange of ferrimagnetic RE-TMalloys [ 61,62]. We assume\na lattice constant of a=250 pm . Similar models have3\n0 100 200 300 400 500 600 700\ntemperature (K)0.00.51.01.52.02.5angular momentum ( )\nlA\nlB\nlfu\na2a\nTA(M)\nFigure 1. Sublattice-specific and net thermal average angular\nmomentum as a function of temperature (points). Solid lines\nare fits to the simulation data. At the angular momentum\ncompensation temperature, TA, the net angular momentum\nvanishes. Note that TA=TM, due to our choice of γA=\nγB. The sketch shows the G-type magnetic ordering of the\nunderlying atomic spin model of the ferrimagnet.\nbeen already used in the literature to model the spin\ndynamics of ferrimagnetic systems, the most prominent\nexample being GdFeCo alloys used for ultrafast toggle\nswitching [ 13–16,18–20] and HD-AOS [25,29,30]. Despite\ntheir potential key role on such a switching process, the\nstudy of DWmotion under thermal gradients of such kind\nof materials [6] has gained far less attention.\nB. Computation of domain wall dynamics\nIn our simulations a DWis placed in a constant temper-\nature gradient and the magnetization is relaxed to a base\ntemperature of T0. The base temperature determines the\nremanent angular-momentum and thus enables us to tune\nthe magnetic properties of the FI. During this relaxation\nphase (t<0) we setαG= 1 which efficiently suppresses\nany DWdynamics. At t= 0 we set αGto 0.01, which\nreleases the DWinstantaneously. The wall coordinates,\ni.e. angles Φ νand positions Zν(ν=A,B), are tracked\nby fitting the wall profiles\nm⊥\nν(z) =mν(z) exp(iΦν)\ncosh((z−Zν)/∆ν),and (4)\nmz\nν(z) =mν(z) tanh((z−Zν)/∆ν) (5)\nto the simulation data. ∆ νis hereby the wall width\nandmν(z) is the saturation magnetization for which we\nassume a linear correction in zto improve the fitting\naccuracy, compared to a spatially constant saturation\nmagnetization. Absorbing boundaries in the form of\nenhanced Gilbert damping are applied in the longitudinal\ndirection, whereas the transverse boundaries are periodic\nin order to have bulk-like properties. Due to the sizable\ninter-sublattice coupling JAB, the deviation of the DW\n6080100120T (K)\nT0=92 KdT/dz=0.82 K/nm\nZDW(t0)\nZDW(tmax)\n1.0\n0.5\n0.00.51.0m(t0)\n020406080100120\nposition (nm)1.0\n0.5\n0.00.51.0m(tmax)\nmA\nx\nmB\nxmA\ny\nmB\nymA\nz\nmB\nzFigure 2. Top panel shows the temperature profile used in\nthe simulation setup, together with the initial (solid circle)\nand final (open circle) DWpositions;T0indicates the base\ntemperature at the DWcenter at the start of the simulation\nt0. Central and bottom panels show the wall profile of the\nnormalized A (circles) and B (crosses) sublattice magnetiza-\ntions. Black lines correspond to the fits according to Eqs. (4)\nand (5). The data shown corresponds to Fig. 3 (b).\ncoordinates of the two sublattices A and B from each\nother are relatively small, such that for the tracking of\ntheDWcoordinates it is not necessary to distinguish the\nDWvariablesZ,Φ,∆ for the two sublattices A and B. A\nsimulation setup of a typical DWprofile in a temperature\ngradient is shown in Fig. 2.\nWe use a comparably large grid cross section of 96 ×192\nspins, with a length of 480spins in the direction of the\ntemperature gradient, to reduce thermal fluctuations of\nthe data [ 63]. To handle the large computational effort of\nalmost 107spins in total with simulation times of several\nhundreds of picoseconds, we use a previously developed,\nhighly efficient GPU accelerated atomistic spin dynamics\nsimulation routine based on the Nvidia CUDA C-API[64].\nIII. THEORY OF DOMAIN WALL MOTION\nA. Adiabatic and non-adiabatic spin transfer\ntorques\nThe theory of DWmotion driven by spin-polarized\nelectric currents in FMsis well established. Initial works4\nhave suggested that thermal magnon currents can be\nviewed as spin currents, whose amplitude is proportional\nto the temperature gradient [ 8]. For FIs, the question is\nto what extend a similar picture holds and how theory\nhas to be modified in order to account for the particular\nproperties of FIs.\nIn a temperature gradient, the spatial variation of the\nstochastic noise ζi(Ti), Eq. (3), in the effective field of the\nLLGequation (2)can be interpreted as sources of thermal\nmagnons. This thermal magnon current acts on a DWin a\nsimilar way as the STTused to describe DWmotion under\nspin-polarized electric currents in micromagnetic models\n[56,57]. In those models, the LLGequation is augmented\nby two additional torque terms to take into account the\ninteraction of a spin-polarized electron current on the\nmagnetization. The so-called adiabatic torque\nTad=−(u·∇)m∝−(∇T·∇)m (6)\nand the non-adiabatic torque\nTnad=m×(βeffu·∇)m∝m×(∇T·∇)m. (7)\nThe parameter βeffis the dimensionless non-adiabaticity\nand by definition specifies the ratio between the two STTs.\nThe adiabatic torque can be related to angular mo-\nmentum conservation [ 7,42,43]: when a magnon current\n(or spin-polarized electron current) passes through the\nwall, their polarization is continuously rotated by 180◦\nthereby changing the magnons’ angular momentum. To\nobey angular momentum conservation in the combined\ndomain+magnon system, one domain has to grow in size,\ni.e. the DWhas to move—the direction depending on the\nrelative polarization of the particle current and magne-\ntization with respect to the DW. The adiabatic torque\namplitude in this case is simply given by (cf. [43])\nu=Ja3/lfu (8)\nwhereJ∝∂T/∂z (see also Ref. [ 65] and Appendix A and\nB) is the spin current density and lfu= (lA−lB)/2 the\nangular momentum per unit volume ( lν=µνmν/γν), see\nFig. 1. The difficulty here is to find an expression for the\nspin current Jfor a FI.\nOn the other hand, Schlickeiser et al. [8]introduced the\nconcept of non-adiabatic entropic torque due to the spa-\ntially varying exchange stiffness ∇Aeff= (∂Aeff/∂T)∇T\nin a FM. Here, we adapt their model for the FMto the FI,\nand we find the following expression for the non-adiabatic\nentropic torque strength (see Appendix C for the temper-\nature dependence of the exchange stiffness Aeff)\nβeffu=−2a3\n|lfu|dAeff\ndT∂T\n∂z. (9)\nOne of the main result of the present work is the demon-\nstration of the validity of these relations, Eqs. (8)and\n(9), for FI, by comprehensive comparison to atomistic\nspin dynamics simulations of the DWmotion under a\ntemperature gradient.B. Dynamics of the domain wall\nThe conceptual idea behind Eqs. (8)and(9)is that\nthe dynamics of a FIcan be viewed as an effective FM\nwith angular momentum given by lfu—a model which has\nrecently been employed in a similar fashion by Kim et al.\n[49]for field-driven FI-DWmotion. Such a model should\nbe valid for the low wall velocities in thermal gradients;\nalthough some deviations might occur close to TAsince\nwe do not take into account inertial effects proportional\nto¨Z, and ¨Φ [48, 66].\nThe dynamics of a rigid, ferromagnetic DWcan then be\ndescribed by the two collective coordinates that are the\nwall position ZDWand tilting angle Φ DW[57,58]. We can\nadapt the corresponding equations of motion and rewrite\nthem for the ferrimagnetic DW, leading to\n˙ZDW=βeffu\nα⊥\neff−∆/prime\nDW˙ΦDW\nα⊥\neff(10)\n˙ΦDW=α⊥\neff\n1 + (α⊥\neff)2/bracketleftbiggβeff−α⊥\neff\n∆/prime\nDWα⊥\neffu+K⊥\n|lfu|sin 2Φ DW/bracketrightbigg\n(11)\nwhere K⊥=Kyy−Kxxis the in-plane anisotropy, α⊥\neffis\nthe transverse Gilbert damping parameter, known from\nmagnetic resonance for instance [ 67,68], and ∆/prime\nDW=\n−sign(lfu)∆DWis the signed wall width [ 69]. The coupled\nequations (10) and(11) have two kinds of steady state\nsolution.\nBelow the Walker breakdown “current”\nuW=K⊥\n|lfu|∆DWα⊥\neff\n|βeff−α⊥\neff|(12)\nthe driving stimulus is insufficient for Φ DWto overcome\nthe potential barrier, scaling with K⊥, hence the wall\nangle becomes stationary and consequently the motion\nlinear. In this case, the DWvelocity reads\nVDW=βeffu\nα⊥\neff=−4a3\nαG|lA+lB|dAeff\ndT∂T\n∂z, (13)\nwhere in the second equality we used the relation [70]\nα⊥\neff≈αG|lA+lB|\n|lA−lB|, (14)\ncommonly derived under magnetic resonance conditions\nand related to its linewidth in FIs. We note that Eq.\n(13) is a generalization of the already known relations\nfor the velocity of the DWsforFMs[8] and AFM s[9].\nImportantly, the difference to those cases is that the\neffective damping in FIshas a non-monotonous behavior.\nThe relatively simple expression in Eq. (14) holds if the\nmagnetization is not too close to the compensation point,\nat which an apparent divergence occurs, and where more\nsophisticated damping models would be necessary [ 67,68].\nThus, very close to this point we expect some deviations\nin the wall velocity and precession which we derive from5\nthe ferromagnetic model here in this section. Though\nqualitatively, this singular behavior accelerates the spin\ndynamics around TAand is the reason why FIswith\nangular momentum compensation points are becoming so\nrelevant for applications and functionalities related to the\nspeed of the spin dynamics [ 49,51,54]. Additionally, while\nthe micromagnetic exchange stiffness and its temperature\ndependence for FMsand AFM sis somehow known [ 71], the\nspecifics ofAeffinFIsremains an open problem—especially\nat elevated temperatures.\nAccordingly, above the Walker breakdown, the wall\nangle Φ DWprecesses continuously as the repelling force\nis limited by K⊥/|lfu|. The corresponding wall velocity\nthen reads [58]\nVDW=βeffu\nα⊥\neff±|βeff−α⊥\neff|/radicalbig\nu2−u2\nW\nα⊥\neff(1 +α⊥2\neff)(15)\nwhere the positive sign is for the case ( βeff−α⊥\neff)u<0.\nIn the limit of a vanishing Walker threshold uW(∼K⊥)\n[Eq. (12)] or high driving currents u/greatermuchuW, this solution\nconverges to\nlim\nuW→0VDW= lim\nu→∞VDW=1 +βeffα⊥\neff\n1 +α⊥2\neffu. (16)\nInterestingly, the velocity in this limit and in the small\ndamping regime α⊥\neff/lessmuch1 can be approximated by the sim-\nple relation, VDW≈u=Ja3/lfu. We note that the small\ndamping condition, αG/lessmuch|lA−lB|/|lA+lB|holds in a\nwide range of temperature, when the system temperature\nis not too close to the compensation temperature.\nTo evaluate the equations presented in this section, we\nwant to refer to Appendix A-C. First of all, in Appendix\nA we derive the dispersion relation of the FI, from the\nLLG-equation. In Appendix B this dispersion relation is\nused to calculate the thermal spin current density J, from\nwhich we can then derive the adiabatic STTparameteru.\nFinally, in Appendix C we compute the non-adiabatic STT\nfor this system using the effective exchange stiffness of the\nFI. Altogether this allows us to compute the effective DW\nvelocity and precession in a self-consistent way from the\nLLG-equation, (2), and the spin Hamiltonian, (1), without\nthe need of additional parameters.\nIV. ATOMISTIC SPIN DYNAMICS\nSIMULATIONS\nA. Overview of the domain wall dynamics\nFigure 3 shows the wall velocity VDWwe obtained from\nour atomistic spin dynamics simulations, as a function\nof the temperature gradient ∂T/∂z , i.e. the amplitude\nuof the driving STT(the calculation of the steady-state\nwall velocity is described in detail in Appendix D). We\nstudy a range of temperature gradients such that we can\ninvestigate the dynamics below and above the Walker\n100\n0100VDW(m/s)\nT= 58+6\n16K\n100\n0100200VDW(m/s)\nT= 92+12\n11K\n0100200VDW(m/s)\nT= 127+15\n0K\n0.0 0.2 0.4 0.6 0.8 1.0\ntemperature gradient (K/nm)0100VDW(m/s)\nT= 174+21\n0K\nbiaxial ferrimagnetFigure 3. DWvelocity as a function of temperature gradient\nfor various temperatures. Filled (open) symbols denote walls\nbelow (above) Walker breakdown. Error margins indicate the\nmaximum thermal drift of the wall towards the hot (plus) and\ncold (minus) sample regions throughout the simulation time,\ni.e. the maximum deviation from the starting temperature.\nThe intermediate anisotropy is dy\nA=12.5µeV. Black lines are\nfits to Eqs. (13) and (15).\nbreakdown. Additionally, we consider four different base\ntemperatures T0, ranging from below to above the com-\npensation temperature TA=107 K that will allow us\nto assess the role of the net angular momentum in the\ndynamics of the DW.\nBelow the Walker breakdown (filled symbols) we find\nthat the wall velocity VDWscales linearly with the ther-\nmal torque amplitude uas it is expected according to\nEq.(13). The positive sign of the velocity hereby indicates\na motion towards the hotter end as previously predicted\nforFMsand AFM s[7–10]. Above the Walker breakdown\nthe situation is vastly different: here we find that below\nthe compensation point the wall moves towards the cold\nregion (VDW<0) in the limit u/greatermuchuW, and above the\ncompensation point to the hot one ( VDW>0) for any\nvalue ofu. Nevertheless, the theoretical wall velocity\npredicted by Eqs. (13) and(15) nicely traces our simula-\ntions results for all four temperatures displayed in Fig. 3.\nThus, different propagation directions of the DWmotion\nare found for temperatures below and above the angular\nmomentum compensation temperature TA. This implies6\nthat around TA(i) the adiabatic torque parameter u(T)\nchanges sign due to the sign change of J/lfuin Eq. (8)\nand (ii) the non-adiabaticity βeff(T) changes sign likewise\nsince the product βeffu, Eq. (9), is strictly positive (for\n∂T/∂z> 0).\nAnother intriguing observation is the apparent increase\nin the Walker threshold uWfor the upper three pan-\nels (a)-(c), where in (c) the threshold was actually too\nhigh to be determined by our simulations. This is highly\ncounterintuitive since the critical current in a biaxial\nmagnet is determined by the in-plane anisotropy K⊥\n[see Eq. (11)] which decreases quickly with temperature\n/angbracketleftKν/angbracketright(T)∼Kν(0)m3\nν(T) [72]. Thus, the Walker threshold\nuW(T) is expected to decrease monotonically with Tlike-\nwise which we observe only at even higher temperatures\nshown in panel (d). However, in the FI, the expression\nuW∝K⊥/|lfu|in Eq. (12)increases, since the net-angular\nmomentum lfu(T) decreases faster than the individual\nsublattice order parameters mν(T) and hence faster than\n/angbracketleftK⊥/angbracketright(T). These insights about the Walker threshold could\nhave impact in the design of ferrimagnetic devices with\nimproved functionalities, as the temperature dependence\nof the individual order parameters can be readily tuned\nby material engineering techniques, e.g. by modification\nof sample composition.\nB. Diversity of temperature dependence of the\ndomain wall dynamics\nThe DWdynamics below the Walker breakdown ( u<\nuW) behave as one would expect from previous works in\nFMbut with effective parameters accounting for the fact\nthat the FIis composed of two antiferromagnetically cou-\npled sublattices. Thus, it is worth to further investigate\nthe range of validity of this idea. As shown above, the\nWalker threshold can be controlled via the perpendicular\nanisotropy parameter, dy\nA. Therefore, in order to investi-\ngate the regime below Walker breakdown ( u<u W) we\nhave to consider systems with biaxial anisotropy. On the\nother hand, one of the main results of this work is the\ndemonstration that a DWmotion towards the cold end\nof the sample is possible above the Walker breakdown\n(u/greatermuchuW). Since for a uniaxial FI(dy\nν= 0) the wall mo-\ntion is always above the Walker breakdown ( uW= 0), we\ncan investigate the validity of Eq. (16)for the wall velocity\nwithout mixing effects coming from the presence of per-\npendicular anisotropy. To study the DWdynamics below\nand above the Walker breakdown more thoroughly, we\ncomputed the temperature dependence of the steady state\nDWvelocity for the biaxial ( dy\nA=25µeV) and uniaxial\n(dy\nA= 0) FI, shown in Fig. 4 (a,b), for a fixed temperature\ngradient of ∂T/∂z = 260 K µm−1.1. Domain wall velocity in biaxial systems\nAs we could already expect from the data in Fig. 3,\nthe wall velocity below the Walker breakdown is only\nweakly sensitive to the base temperature, as can be seen\nin the panel (a) of Fig. 4. The wall velocity decreases\nonly slightly at higher temperatures. However, this ef-\nfect is already known from previous studies and can be\nrelated to the changing equilibrium magnetic properties\n[11,71]. We can estimate the expected DWvelocity by\nevaluating Eq. (13) (see also Appendix C). For the sim-\nulation parameters described in Sec. II A, this yields an\nSTTofβeffu=8.6 m s−1and an expected DWvelocity of\nabout 95 m s−1(black cross) and is in good agreement\nwith the simulation data (colored squares), especially\nwhen we consider the rough approximations we applied.\nNote that although the entropic torque (9)alone is pro-\nportional to 1 /|lfu|and hence expected to diverge at the\nangular momentum compensation point (white marker\ncolor) where lfu→0, the DWvelocity Eq. (13) remains\nfinite since the damping coefficient diverges in the same\nfashion:α⊥\neff∝1/|lfu|.\n2. Domain wall velocity in uniaxial systems\nMore fascinating dynamics can be found for the uniax-\nialFI, where predominantly the adiabatic STTdrives the\nDW, see Fig. 4 (b). Here we can make two clear obser-\nvations: (i) the direction of motion of the wall changes\nsign very close to the compensation temperature (white\nmarker color). Below the compensation point the wall\nmoves to the colder sample regions, i.e. co-propagates\nwith the magnon current, whereas above the compensa-\ntion point the regular motion towards the thermal source\nis obtained. (ii) The absolute value of the wall veloc-\nity drastically increases towards the compensation point,\nwhich is supported by Eq. (15) for a magnon current\ndensityJwhich depends only weakly on temperature.\nThis assumption, in particular that Jdoes not change\nsign atTA, is hereby motivated by its derivation from\nthe dispersion relation (see Appendix A), which does not\ndepend on temperature in a qualitative way, as long as\nT/lessmuchTC[73].\nHowever, to fully understand the DWdynamics for the\nfreely precessing wall, far above the Walker breakdown,\nwe first need a better understanding of the origin of the\nspin current density J. Until now, it is not even clear\nwhat the sign of the spin current density Jis in the FI\n[52]. For a single sublattice FMthe magnetic moment\nof the magnon is given by the reduction of the Szspin\ncomponent with respect to the saturation value ( mz=±1)\nand is hence antiparallel to the ground state magnetization\n[7,42]. In the FIthe net-momentum of the magnon will be\ndetermined by the ratio of the two components µASz\nA/γA\nandµBSz\nB/γBrelative to each other.\nThe classical spin-wave amplitudes for the uniaxial\nFIatT=0 Kcan be calculated from LSWT , follow-7\nTT(b) velocity\n(d) precession\nfrequencyuniaxial ferrimagnet\n(c) tilting angle(a) velocitybiaxial ferrimagnet\nTT(e)\nTT\nFigure 4. Top panels show the DWvelocityVDWfor a biaxial (a) and a uniaxial (b) FIas a function of temperature for a\nconstant thermal gradient of ∂T/∂z =260 K µm−1. Bottom panels show the corresponding tilting angle of the wall (c) for the\nbiaxial FIbelow Walker breakdown and the DWprecession frequency for the uniaxial magnet (d), respectively. The inset (e)\nshows the non-adiabaticity parameter βeff(markers) calculated from Eq. (18) in comparison to the effective damping coefficient\nα⊥\neff(dash-dotted line), corresponding to the magnetic resonance linewidth, Eq. (14). The black crosses in (a) and (e) correspond\nto the entropic torque (9). Solid black lines in (b) and (d) correspond to Eqs. (11) and(16), using the spin current Eq. (17)\nas input and a constant non-adiabaticity parameter |βeff|=0.55 (see text). Dashed vertical lines mark the so-called torque\ncompensation temperature TT(see text). The marker color indicates the net-angular momentum lfu; horizontal error bars\nindicate the wall drift.\ning Refs. [ 55,74] (see Appendix A). We find that the\nlow-frequency branch ( σ=−1) in the dispersion rela-\ntion carries momentum parallel to mz\nA= +1 and the\nhigh-frequency branch ( σ= +1) parallel to mz\nB=−1.\nThe question now is which of these two branches dom-\ninates the net spin current? Lower frequency implies\nhigher thermal population—in the classical model that\nis a population according to a Rayleigh-Jeans distribu-\ntionn0\nk,σ=kBT/~ωk,σ—and at the same time longer life\ntimesτk,σ. However, the high-frequency branch has a\nmuch steeper dispersion relation and hence much higher\ngroup velocities vk,σ=∂ωk,σ/∂kand propagation lengths\nξk,σ=|vk,σ|τk,σ.\nTo answer this question, we calculate the thermal\nmagnon current density Jquantitatively, by solving the\nfollowing k-space integral (see Appendix A and B for the\nderivation):\nJ=/summationdisplay\nσ=±1kB∂T\n∂zσ/integraldisplay\nvz\nk>0d3k\n(2π)3∂lnωk,σ\n∂kzξk,σcosϑk.(17)\nThe dispersion relation ωk,σand the magnon propagation\nlengthsξk,σcan be written in closed form expression,\nandϑkis simply the angle between the k-vector and the\nz-direction. Thus the numerical solution of Eq. (17)poses\nonly minimal computational effort, and we find that the\nhigh-frequency branch of the dispersion clearly dominates\nthe net-magnon current density J. Thus, we have the\npeculiar situation in which below the compensation point\nTA, the net-magnon current has a polarization parallel\nto the ground state angular momentum lfu—a situationopposite to the case of a simple FM—leading to the op-\nposite direction of DWmotion, that is, towards the cold\nsample regions for T <T A(see Fig. 5 left). We can use\nthe spin current Eq. (17) to compute the adiabatic STT\nquantitatively, yielding u=−15.7 m s−1atT= 0. More-\nover, we can now calculate the non-adiabaticity parameter\nβeffusing our previously determined non-adiabatic torque\nparameter βeffu= +8.6 m s−1yieldingβeff=−0.55. In\ncombination with Eq. (16) we can now predict a wall\nvelocity of about VDW(T= 0) = −14.8 m s−1for the\n260 K µm−1thermal gradient at T= 0, which as already\nexpected, can be well approximated by VDW≈u. This\nvalue matches quite well with the results from our atom-\nistic spin dynamics simulation, although we are unable\nto simulate T= 0 exactly.\nNow that we have gained some insight on the role of\nthe adiabatic STTon the DWmotion at low temperatures,\nwe can address the domain wall velocity in the uniaxial\nFIat finite temperature. In particular, the DWvelocity\nin Fig. 4 (b) presents an apparent asymmetry close to\nthe angular momentum compensation point TA. As we\ndiscussed above, the magnon current density Jabove and\nbelowTAshould not change drastically as it is derived\ndirectly from the dispersion relation, cf. [ 73]. Thus, we\nargue that the temperature dependence of u=Ja3/lfu\nis mostly due to lfu(T). Consequently, we expect the\nadiabatic STT to act approximately antisymmetrically\non the DWin the vicinity of TA, implying min[VDW]≈\n−max[VDW], as indicated in Fig. 5. However, the wall\nvelocity from our numerical simulations clearly does not\nfollow this symmetry—we find min[VDW] =−54 m s−18\nZDW(t)\nZDW(t+Δt)lA\nlB\nhot coldthermal\nmagnonsJ+\nJ–T < TA\nZDW(t+Δt)lA\nlB\nhot coldthermal\nmagnonsT > TA\nZDW(t)J+\nJ–\nFigure 5. Schematics of DWmotion in a thermal gradient due to a magnon current Jσfor base temperatures T <T A(left) and\nT >T A(right). The magnon current density Jσof the two branches ( σ=±1) of the dispersion relation carry opposite angular\nmomentum. We find that for both base temperatures T, the high frequency branch σ= +1 (light blue) dominates over the low\nfrequency branch σ=−1 (orange), see Appendix A. This leads to a polarization of the net-magnon current density J=J++J−\nparallel to the lBsublattice angular momentum (in the source region), i.e. |J+|>|J−|. Due to the change in the magnon\npolarization when passing through the wall, to satisfy angular-momentum conservation in the combined domain+magnon system,\nthe domain with the net-angular momentum lfuin the “down” direction has to grow in size. Below TAthis is the domain on the\nhot side (left), whereas above TAit is the domain on the cold magnon side (right). Hence, for the same spin current density J,\nwe obtain different DWpropagation directions above and below TA.\nand max[VDW] = +142 m s−1.\nWe can trace back the origin of such asymmetry by\nconsidering Eq. (16)for the domain velocity in the temper-\nature regime close to TA. We know that both the adiabatic\n(8)and non-adiabatic STTs(9)scale via∼1/|lfu|, thus,\nthe temperature dependence of βeffshould be mostly due\nto the softening of the exchange stiffness Aeff(T), i.e. weak\nforT/lessmuchTC(cf. Appendix C). This allows us to assume\n|βeff|∼const and only include a temperature dependence\nin the form of the necessary sign change at TA. The tem-\nperature dependence of VDW(T) in Eq. (16)is thus due to\nα⊥\neff(T) [Eq. (14)] andu(T) =Ja3/lfu(T) [Eq. (8)]—the\nformer being only relevant close to TA. Using these as-\nsumptions we can calculate VDW(T) from our theoretical\nmodel, shown as the black line in Fig. 4 (b). This model\nexcellently describes the temperature dependence of the\nwall velocity over the full temperature range, including\nits asymmetry.\nWe conclude that below TA, the entropic torque and\nthe angular momentum transfer work against each other,\nwhereas above TAthey act in the same direction. Natu-\nrally, the angular momentum transfer becomes less impor-\ntant if angular momentum conservation is broken, that\nis, whenα⊥\neffbecomes large in the vicinity of TA. At the\nsame time, the contribution of the non-adiabatic term\n∼βeffα⊥\neff/(1 +α⊥2\neff) increases. These findings demon-\nstrate for the first time that a ferrimagnetic DWcan be\npushed away from a thermal magnon source by angular\nmomentum transfer—an effect which in FMsand AFM s\ncan be achieved only by a less efficient linear momentumtransfer, i.e. magnon reflection [12, 75–77].\nC. Emergence of torque compensation temperature\nAside from the DWvelocity, Fig. 4 (c) and (d) also show\nthe temperature dependence of the dynamics of both the\nwall tilting (biaxial) and precession (uniaxial). For the\nbiaxial FIthe steady state tilting angle Φ DWgradually\ndecreases with temperature until it reaches zero at a tem-\nperature of about 125 K after which it increases again\nwith opposite sense of rotation. A similar behavior is\nfound for the wall precession Ω DW=˙ΦDWin the uniaxial\nFI. The wall precession changes sign at the very same\ntemperature of 125 K , suggesting that this phenomenon\nis independent of the in-plane anisotropy K⊥. In the\nfollowing we define this point of completely suppressed\nDWtilting and precession the torque compensation tem-\nperatureTT. However, unlike in the biaxial FI, in the\nuniaxial FIthere is an additional rapid increase of the\nwall precession frequency Ω DWat the angular momentum\ncompensation point. Such an increased wall precession\nclose toTAwas recently predicted for field-driven DWmo-\ntion in ferrimagnetic GdFeCo by Kim et al. [49], however,\nin their case, the sign change of the wall precession coin-\ncides with the angular momentum compensation point.\nFor the case of thermal magnon current driven DWmotion\nTTdiffers from TAimplying that field driven DWmotion\nis fundamentally different from thermally induced motion.\nWe can understand the existence of the torque com-9\n(a)\n(b)\n(c)(d)\n(e)\n(f)(g)\n(h)\n(i)(j)\n(k)\n(l)38 m/s 31 m/s31 m/s\n28 m/s108 m/s92 m/s\n-45 m/s\n187 m/s\n71 m/s9 m/s10 m/s36 m/s\nFigure 6. DWdisplacement ∆ ZDWfor the biaxial FIat different temperatures [(a)-(i); dy\nA=12.5µeV] in comparison to the FM\n[(j)-(l);dy\nA=62.5µeV]. Note the much faster acceleration of the ferrimagnetic walls (left and center panels), compared to the\nferromagnetic ones (right panels). Labels indicate the steady state velocities.\npensation by comparison to the AFM. In the AFM, the\nsymmetry of the non-adiabatic STTcan only lead to prop-\nagationVDWof the wall along the temperature gradient.\nA rotation of the wall angle Φ DWon the other hand does\nnot occur, since both sublattices try to rotate in opposite\ndirections, canting the sublattice magnetizations instead\nof tilting the DWangle [ 9]. In the FIthis is also the case,\nbut unlike in the AFM the torques from the two sublattices\nwill in general not have equal magnitude and are thus not\nfully compensated.\nAnother explanation of these results is provided by the\ncoupled equations of motions for the collective coordinates\nZDWand Φ DW, Eqs. (10) and(11). Crucial is hereby the\nrole of the non-adiabaticity parameter βeff(T), shown in\nFig. 4 (e). In the previous section we already determined\na crude estimate of βeff≈0.55 by calculating the non-\nadiabaticity at low temperature from Eq. (9). We can use\nthis number again, to solve Eq. (11) for the steady-state\nprecession frequency, shown in Fig. 4 (d), where we find\nexcellent agreement with the simulation results.\nA more rigorous approach to compute the non-\nadiabaticity βeff(T), including its temperature de-\npendence, can be computed from the ratio R=\nVbiaxial\nDW/Vuniaxial\nDW of the wall velocity Vbiaxial\nDW below\nWalker breakdown, Fig. 4 (a), and the one for the freely\nprecessing wall Vuniaxial\nDW Fig. 4 (b). By dividing Eqs. (13)\nand(16) we can then solve for βeff(T) usingR(T) fromthe numerical simulations:\nβeff=α⊥\neffR\n1 +α⊥2\neff(1−R). (18)\nThe magnonic torque udrives the wall precession via\nΩDW∝(βeff−α⊥\neff)u. Thus, the DWprecession is expected\nto cease for βeff(T) =α⊥\neff(T), or, in other words, the crit-\nical gradient uW, Eq. (12), diverges for βeff(T)→α⊥\neff(T).\nThe intersection point βeff(T) =α⊥\neff(T), shown in Fig. 4\n(e), is in very good agreement with the torque compensa-\ntion point shown in panels (c) and (d) which have been\ndetermined directly from wall tilting and precession, re-\nspectively. Note that by definition, this intersection point\nalso marks the temperature at which the wall velocities\nabove and below Walker breakdown, depicted in Figs. 4\n(a) and (b), coincide. For our parameters that is a steady-\nstate velocity at TT=125 K of about 90 m s−1in both\npanels.\nFurthermore, we are now able to generalize one of\nour findings, namely that the torque compensation point\nTTis found above the angular momentum compensation\ntemperature TA: if the adiabatic STTmediated by the\nthermal magnon current uacts repulsive on the DW, the\nnon-adiabaticity βeffhas to be negative, to ensure that the\nnon-adiabatic STT(9), i.e. the product βeffu, remains pos-\nitive (for∂T/∂z> 0) [8–10]. Thus, the term ( βeff−α⊥\neff)u\nin Eq. (11) can only be zero for an attractive adiabatic\nSTT, sinceα⊥\neffis strictly positive.10\nD. Domain wall motion in time domain\nFor discussing the DWmotion in the time domain it is\nhelpful to compare our results on the FI’s dynamics to the\nprevious works on FMsand AFM s[7–9,11,57,58]. For the\nFMwe can do this even in a quantitative manner by simply\nswitching the sign of the inter-sublattice coupling JAB\nto get a ferromagnetic exchange between the sublattices\nA and B. The steady-state wall velocity VDWbelow the\nWalker breakdown appears to be more or less unaffected\nby the sign change of JAB, as can be seen in the top panel\nof Fig. 6. This is indeed expected from Eq. (13) where\nonly the sum of the sublattice angular momenta enters\nand the effective exchange stiffness Aeff(T) of the system\nshould be equal in the FM,AFM, and FI.\nHowever, the time to reach this steady-state velocity\nin the FMis greatly extended with acceleration times on\nthe order of nanoseconds [Fig. 6 (j)-(l)], whereas in the FI\ntheDWcan reach its steady state velocity on time scales\nof several tens of picoseconds [Fig. 6 (a)-(c) and (g)-(i)].\nIn fact, close to the torque compensation point [Fig. 6\n(d)-(f)] the acceleration is even faster, though, the exact\ntime constant there is difficult to determine due to strong\nfluctuations of the DWmotion even for a grid cross section\nof 96×192 spins. This is especially problematic for very\nlow gradients as for instance in Fig. 6 (a), (d), and (g).\nThe steady-state below the Walker breakdown is char-\nacterized by a constant tilting angle Φ DW, where the\ntorques of the non-adiabatic STT are balanced by the\nanisotropy torques, see Eqs. (10) and(11). During the\ninitial rotation of the DWup to this angle the velocity\nincreases to its steady-state value, and hence one can\ninterpret it as an inertial mass of the wall [ 78]. As men-\ntioned before, these torques are partially compensated\nin the FIgreatly reducing the tilting angle and therefore\nalso the effective inertia of the DW. For the same rea-\nson the Walker breakdown uW, at which the wall starts\nto rotate continuously, is shifted to much higher critical\ngradients. At T=58 K we find a threshold gradient in\ntheFIof aboutkB|∂T/∂z|FI\nW/dy\nA,FI≈1.8 nm−1compared\ntokB|∂T/∂z|FM\nW/dy\nA,FM≈0.15 nm−1in the FM. At the\ntorque compensation point the FIresembles an AFM, for\nwhich there is no tilting and hence the wall can move\nquasi-inertia-free, i.e. without a relevant acceleration time\n[9]. Ultrafast DWacceleration in the FIis not only found\nat exactly the torque compensation point TT, but also\nslightly below, due to the diverging wall precession Ω DW\nclose to the angular momentum compensation point TA\n[see Fig. 4 (d)]. Thus, even though the wall has to tilt\nby a finite angle, the steady-state angle is reached on\nultrashort time scales of only few picoseconds.\nIt should be noted though that there are other effects\nin an AFM that can be attributed to a mass of the DW\n[48,66]. However, these effects are much smaller and\nproportional to the velocity of the wall, which is here\nrestricted by feasible temperature gradients.V. CONCLUSION\nTo summarize our results, we calculated the DWdy-\nnamics of a FIin a thermal gradient using both, large\nscale atomistic spin dynamics simulations based on the\nstochastic LLG-equation and analytical calculations based\nonLSWT . Our simulation results are in good agreement\nwith our theoretical findings that we derived from LSWT .\nWhereas in the thoroughly studied ferromagnetic systems\nthe adiabatic and non-adiabatic STTlead qualitatively to\nthe same result [ 7,8,10]—a motion to the hotter sample\nregion—a ferrimagnetic DWreacts differently to these two\nkinds of torques. The non-adiabatic torque leads to a\nconsistent motion towards the hotter end, as it is the\ncase for the FMand AFM and can be explained by the\nfree energy minimization via an entropic torque [ 8,9].\nOn the other hand the adiabatic STT can either push\nor pull the ferrimagnetic DWaway from or towards the\nspin-wave source, depending on whether the temperature\nis below or above the angular momentum compensation\npoint. In the FIthe copropagation of the DWwith the\nmagnon current at low temperature is not due to lin-\nearmomentum transfer resulting from magnon reflection\n[12,75–77], but due to the angular momentum transfer\nfrom the transmitted magnons. Moreover, the FIshows\nanother distinct characteristic point, besides the angular\nmomentum and magnetic compensation points, that is a\ntorque compensation point at which we find a reversal of\ntheDWrotation. Consequently, at the torque compensa-\ntion point the Walker breakdown is strongly suppressed\nwhich suggests that high DWvelocities and ultrafast DW\nacceleration should be achievable at this point.\nFinally, we want to mention that first experimental ev-\nidence on copropagation of a DWwith a thermal magnon\ncurrent, induced by ultrashort laser pulses, has been re-\nported recently by Shokr et al. [6]. In their work it is\nreported that DWsin a ferrimagnetic GdFeCo alloy will\nmove away from the laser spot center, i.e. against the\nthermal gradient and towards the cold region, corrobo-\nrating our findings for the DWmotion above the Walker\nbreakdown.\nACKNOWLEDGMENTS\nThe authors would like to thank Philipp Graus and\nJohannes Boneberg for fruitful discussions. This work was\nfinancially supported by the Deutsche Forschungsgemein-\nschaft through the SFB 767 “Controlled Nanosystems”.\nAt the FU Berlin support by the Deutsche Forschungsge-\nmeinschaft through SFB/TRR 227 “Ultrafast Spin Dy-\nnamics”, Project A08 is gratefully acknowledged. Fur-\nthermore, U.R. acknowledges funding from the Deutsche\nForschungsgemeinschaft via the project RI 2891/2-1.11\nAppendix A: Dispersion relation of the rocksalt-type\nferrimagnet\nWe start the derivation of the spin-wave dispersion by\nintroducing the complex vector S= [Sx\nA+iSy\nA,Sx\nB+iSy\nB].\nThis ansatz implies an x-y-symmetry and thus vanishing\nin-plane anisotropy K⊥=Kyy−Kxx= 0 in order to avoid\ndealing with squeezed magnon states [ 79]. We assume a\ngroundstate magnetization of mz\nA= +1 andmz\nB=−1.\nFollowing Refs. [ 55,74] one can deduce the linearized\nLLG-equation in k-space in analogy to the FMand AFM\ncase:\n∂Sk\n∂t=−Ωk·Sk (A1)\nwhere the frequency matrix on the right hand side is given\nby\nΩk=/bracketleftbigg\n(+i−αG)Ωk\nAA(+i−αG)Ωk\nAB\n(−i−αG)Ωk\nBA(−i−αG)Ωk\nBB/bracketrightbigg\n. (A2)\nThe matrix elements of Eq. (A2) are\nΩk\nAA=γA\nµA/parenleftBig\n6JAB−2dz\nA−2JAAC(2)\nk/parenrightBig\n(A3)\nΩk\nAB=γA\nµA2JABC(1)\nk(A4)\nΩk\nBB=γB\nµB/parenleftBig\n6JAB−2dz\nB−2JBBC(2)\nk/parenrightBig\n(A5)\nΩk\nBA=γB\nµB2JABC(1)\nk. (A6)\nThe structure factors C(n)\nkare related to the neighbor\npositions of shell nand can be expressed as\nC(1)\nk=/summationdisplay\nνcos(kνaν) (A7)\nC(2)\nk=/summationdisplay\nν,κ\nν/negationslash=κ[1−cos(kνaν) cos(kκaκ)] ; (A8)\n2aνare hereby the lattice constants of the face centered\northorhombic unit cell (see Fig. 1), in the following we\nassumeaν=afor simplicity.\nThe solution of Eq. (A1) is given by the eigenvalues\nof Eq. (A2) and can be computed in closed form for the\n2×2-matrix:\nΛk\n±=−αG−i\n2Ωk\nAA−αG+i\n2Ωk\nBB (A9)\n∓1\n2/bracketleftBig\n4(1 +α2\nG)/parenleftbig\nΩk\nABΩk\nBA−Ωk\nAAΩk\nBB/parenrightbig\n+/parenleftbig\n(αG−i)Ωk\nAA+ (αG+i)Ωk\nBB/parenrightbig2/bracketrightBig1/2\n.\nWe can further simplify this expression by assuming\nαG/lessmuch1, leading to\nωk\n±= +Ωk\nAA−Ωk\nBB\n2∓1\n2˜Ωk (A10)\nλk\n±\nαG=−Ωk\nAA+ Ωk\nBB\n2±(Ωk\nAA)2−(Ωk\nBB)2\n2˜Ωk(A11)\nXW L\nK XU\nk\n020406080100120140k,and k,/  G (meV)\nk, +\nk,\nk, +\nk,\nFigure 7. Dispersion relation of the FIalong the high symme-\ntry path of the magnetic BZ(ΓX=π/a). Solid lines are the\nmagnon energies and dashed lines their linewidth (normalized\ntoαG), respectively. Shown are the absolute values of the\nfrequency, neglecting the rotation sense.\nFigure 8. Sublattice-resolved magnon amplitudes, along\nthe high-symmetry lines in the BZ. From the ratio of the\namplitudes one can deduce in which direction the net-angular\nmomentum of the magnon points. The top/bottom sketch\nqualitatively show a low-/high-frequency magnon excitation\namplitude far away from the Γ-point. At the Γ-point (not\nshown as sketch) mixed excitations can occur in which both\nsublattices are excited.\nwhere the frequencies ωk\n±=/Ifractur{Λk\n±}and damping rates\nλk\n±=/Rfractur{Λk\n±}are the imaginary and real parts of the\ncomplex eigenvalues Λk\n±, respectively. The frequency ˜Ωk\nsimply reads\n˜Ωk=/radicalBig\n(Ωk\nAA+ Ωk\nBB)2−4Ωk\nABΩk\nBA. (A12)\nNote that for k= 0 Eqs. (A10) and(A11) coincide with\nthe results of Kamra et al. [68, Eq. (16) and (17) ]for the\nmagnetic resonance mode of a FI.\nNext we want to derive the “amplitude” of the magnon.\nAlthough the absolute value of a magnon is not well de-\nfined in our semiclassical picture, it is sufficient to compute12\nthe relative amplitudes between the A and B sublattices,\nas the absolute values of the amplitudes will cancel for a\nthermal magnon distribution. These relative amplitudes\nare related to the (non-normalized) eigenvectors\nSk\n±=/bracketleftbigg\n−Ωk\nAA+ Ωk\nBB∓˜Ωk\n2Ωk\nBA,1/bracketrightbigg\n, (A13)\nto the eigenvalues Λk\n±of Eq. (A2). The classical equivalent\nto the magnon amplitude µk\n±/lessmuchµA,Bfollows from simple\ngeometrical considerations as\nµk\n±=−S2\n2||Sk\n±||2/parenleftBig\nµA/parenleftbig\nSk\n±,A/parenrightbig2−µB/parenleftbig\nSk\n±,B/parenrightbig2/parenrightBig\n,(A14)\nor\nlk\n±=−S2\n2||Sk\n±||2/parenleftbiggµA\nγA/parenleftbig\nSk\n±,A/parenrightbig2−µB\nγB/parenleftbig\nSk\n±,B/parenrightbig2/parenrightbigg\n,(A15)\nwhereSis a scaling parameter which quantifies the classi-\ncal spin-wave amplitudes. The sublattice-resolved magnon\namplitudes as defined inside the brackets of Eq. (A15)\nare shown in Fig. 8. One can clearly see that the sign\nofµk\nσdoes not depend on k, but only on σ, since for a\ngiven branch σ, one sublattice is always excited much\nmore strongly. In fact, apart from the modes close to the\nΓ-point, the magnon amplitude can be approximated by\nan excitation of only one of the two sublattices: for the\nlow frequency branch (orange), that is the B-sublattice\n(top), whereas for the high frequency branch (blue) it is\nthe A-sublattice.\nAppendix B: Linear spin wave theory for thermally\ninduced domain wall motion\n1. Temperature step\nFirst we suppose a system of an extended nanostrip\nwith a temperature profile in the form of a step function\nT(z) =T0+∆TΘ(−z). The system is then isotropic along\nthex-/y-directions and we only expect a net-spin current\npropagating along z-direction. Since the thermal magnon\noccupation in the classical limit follows a Rayleigh-Jeans\ndistribution n0\nk,σ=kBT/~ωk,σ, one can drop the base\ntemperature T0as long as it is low enough that it does\nnot affect the effective magnetic parameters, i.e. as long\nas the dispersion relation (A10) and (A11) is still valid.\nNote, that unlike previous works that studied the\naction of spin-waves on DWs[43,75], an effective 1-\ndimensional model is not sufficient here, since thermally\nexcited magnons with off-axis wave vector k/negationslash=kˆ zare\nrelevant and due to the large grid cross-section included in\nthe numerical simulations. The macroscopic spin current\ndensity follows from integrating over all thermally excited\nmodes in the BZ. The two sublattices of the checkerboard\nAFM (Fig. 1) are two fcc-lattices with magnetic lattice\nconstant of 2 a, respectively. Thus, the BZis a truncated\nVDW\nm⊥/meqT/ΔTFigure 9. DWvelocity for a uniaxial FIas a function of\ndistancezfrom a temperature step of kBT=1 meV ; compari-\nson between LLG-simulations (points) and LSWT (line). The\ntheory line was obtained by integrating Eq. (B5) numerically\nvia a Monte-Carlo method with about 5×105k-points in\nthevz\nk,σ>0 half of the BZ. For the LLG simulation a grid\nof 96×96×480 and a simulation time of 192 ps was taken\n(t/greatermuchz/vz\nk,σfor the majority of the BZ). Errorbars indicate\nthe initial and final position of the DW. The DWcopropagates\nwith the magnon current, i.e. moves to the colder sample\npart. The inset shows the schmematics of the angular momen-\ntum transfer due to high (blue) and low (yellow) frequency\nmagnons.\noctahedron with qX=π/a, [80]. For a DWat a position\nz>0 away from the temperature step, we can restrict the\nk-space integral to the half space vz\nk>0 to only include\nforward propagating magnons:\nJ(z) =/summationdisplay\nσ=±1/integraldisplay\nvz\nk>0d3k\n(2π)3jz\nk,σ(z) (B1)\nEach mode k,σcontributes with\njk,σ=−σ~nk,σ∂ωk,σ\n∂kzexp/parenleftbigg−2z\nξk,σcosϑk/parenrightbigg\n(B2)\nto the net-current J. Here∂ωk,σ/∂k=vk,σis the group\nvelocity of the mode, nk,σis the magnon occupation num-\nber at the source, and the exponential factor accounts\nfor the absorption of the current with propagation length\nξk,σ=|vk,σ|τk,σ. The factor of two in the exponential\naccounts for the conversion of the spin-wave amplitude to\nthe magnon number, proportional to the squared ampli-\ntude.ϑkdenotes the angle between kand thez-direction\nand is needed to compute the actual propagated distance\nrk=z/cosϑk.\nIgnoring depletion effects at the interface, i.e. at the\nmagnon source, we can assume that the magnon occu-\npation at the source is given by the thermal population13\nnk,σ≈n0\nk,σand hence we get\njk,σ=−σ~kB∆T\n~ωk,σ∂ωk,σ\n∂kzexp/parenleftbigg−2z\nξk,σcosϑk/parenrightbigg\n(B3)\n=−σkB∆T∂lnωk,σ\n∂kzexp/parenleftbigg−2z\nξk,σcosϑk/parenrightbigg\n(B4)\nand we arrive at our preliminary result for the magnon\ncurrent density due to a temperature step\nJ=−/summationdisplay\nσ=±1kB∆Tσ/integraldisplay\nvz\nk>0d3k\n(2π)3∂lnωk,σ\n∂kzexp/parenleftbigg−2z\nξk,σcosϑk/parenrightbigg\n.\n(B5)\nWe should note that we assumed a fixed magnon am-\nplitude of ~in the derivation, which for the quantum\nmechanical case is a reasonable assumption for the FI[79],\nbut seems arbitrary for the classical case [see Eq. (A14) ].\nThis is however not further relevant, since the magnon\namplitude eventually cancels when we put in the thermal\noccupation n0\nk,σ∝1/~ωk,σ.\nIn Fig 9 we compare the DWvelocity calculated accord-\ning to Eq. (B5) with our numerical simulations results.\nSince the base temperature is set to zero, the system is\nbelow the compensation temperature. The DWvelocity\nis plotted as a function of distance zfrom a 1 meV tem-\nperature step. We find excellent quantitative agreement\nbetween numerical simulations and the LSWT . Further-\nmore, as for the thermal gradients, we also find the motion\nof the wall to be away from the magnon source.\n2. Temperature gradient\nThe solution of the previous section B 1 is easily applica-\nble to temperature gradients, by simply summing up over\nseveral temperature steps d T(z) =∂T/∂z dz. Once more\nwe will use the fact that the Rayleigh-Jeans distribution\nis linear in T(z). Therefore, in a constant temperature\ngradient, we have the same amount of magnons flowing\nfrom the right to the left (carrying spin −σ~), than we\nhave “magnon-holes” flowing from right to left (carrying\nspin +σ~). This means we can again restrict the k-space\nintegral over half of the BZand multiplying the result\nby two, such that the final result for the spin current is\nthe sum of equal contributions of magnon current and\n“magnon-hole” current.\ndJ= 2/summationdisplay\nσ=±1kBσ/integraldisplayd3k\n(2π)3∂lnωk,σ\n∂kz\n×exp/parenleftbigg\n−2z\nξk,σcosϑk/parenrightbigg∂T\n∂zdz. (B6)\nWe obtain the final result Eq. (17) by performing the\nz-integration\nJ=/summationdisplay\nσ=±1kB∂T\n∂zσ/integraldisplayd3k\n(2π)3∂lnωk,σ\n∂kzξk,σcosϑk.(B7)\n0 10 20 30 40 50 60 70 80\nz (nm)103\n102\n101\n100101VDW (m/s)BE 5 meV\nBE 10 meV\nBE 25 meV\nRJFigure 10. Comparison of the semiclassical spin current\nemitted from a kB∆T=1 meV temperature step, derived\nvia the Rayleigh-Jeans distribution (RJ; red line) and the\nquantum statistical derivation based on the Bose-Einstein\ndistribution (BE) for different base temperatures T0. At very\nlow temperature the spin current is dominated by the low-\nfrequency branch, since the high-frequency magnons are still\nfrozen out. Thus the sign of the net-spin current is reversed\nin the vicinity of the source, indicated by the dashed line.\n3. Quantum effects\nOne can further compute the spin current in the quan-\ntized form by replacing the Rayleigh-Jeans distribution\nn0\nk,σwith the Bose-Einstein distribution nBE\nk,σ. For simplic-\nity we restrict this discussion to the case of a temperature\nstep as the findings are expected to be qualitatively similar\nfor the thermal gradient.\nIn quantum statistics, the magnon occupation number\nis no longer linear in the temperature, hence, the base\ntemperature will be relevant. The resulting spin current\n(B2) emitted from our temperature step should thus be\nproportional to\nnk,σ=nBE\nk,σ[kB(T0+ ∆T)]−nBE\nk,σ[kBT0]. (B8)\nFigure 10 shows the DWvelocity corresponding to a\nthermal spin current calculated with the correct quan-\ntum statistics for a set of base temperatures kBT0(the\nCurie temperature for the given exchange constants is\nTC=616 K ). At very low temperature only the lowest\nmagnon energies will be occupied, i.e. the low-frequency\nbranch of the dispersion will dominate the spin transport,\ndespite the low propagation length. Moreover it is implied\nthat real systems can exhibit a sign change of the net-spin\ncurrentJat very low temperature. Thus, for uW→0, the\nDWvelocityVDW=Ja3/lfu[Eqs. (8)and(15)] not only\nchanges sign at the compensation point where lfuchanges\nsign, but also a second time when the temperature is suf-\nficiently high, to populate the long-range, high-frequency\nmagnons of the upper branch—the ones that carry nega-\ntive momentum (see Fig. 8). The overall magnon current14\nand magnon accumulation at low temperature is greatly\nreduced with respect to the classical case. However, for\nhigher temperatures and in particular at room tempera-\nture, we qualitatively retain the semiclassical magnon cur-\nrent derived with the Rayleigh-Jeans distribution. From\nthis we can conclude that the semiclassical treatment\nis sufficiently accurate for describing most experiments,\nwhich are usually carried out near room temperature with\nmagnetic materials of similar ordering temperature [61].\nAppendix C: Entropic torque in the ferrimagnet\nThe entropic or magnetothermal torque in the FIcan\nbe defined in analogy to the FMcase [ 8]. The effective\nexchange stiffness for our cubical FIis composed by the\nthree exchange contributions Aeff=AAA+AAB+ABB. In\nthe molecular field approximation, for a magnetic texture\nalong the (001) direction, these can be written as [71]\nAAA=2JAA\nam2\nA, ABB=2JBB\nam2\nB,\nandAAB=−JAB\n2amAmB. (C1)\nThe different numerical factors come from the symmetry\nof the shells which is fcc for the ferromagnetic exchanges\nAAAandABB(eight neighbors with ∆ z=±1), and sim-\nple cubic for the antiferromagnetic exchange AAB(two\nneighbors with ∆ z=±1), see Fig. 1. Their temperature\ndependence is hereby assumed to be well approximated\nby the mean field expressions Aij/Aij(0) =mimj. It\nshould be noted that although we chose exchange param-\neters withJAA/greatermuchJBBand thusJBBis not significantly\naffecting the magnetic ordering ( TCfor instance), it does\nadd a non-negligible contribution to the entropic torque\ndue to the faster demagnetization of mBcompared to\nmA. The temperature dependence of the equilibrium\nmagnetizations mi(T) is taken from the data in Fig.1.\nWe find∂mA/∂T≈−4.87×10−4K−1for the strongly\ncoupled sublattice A and ∂mB/∂T≈−2.15×10−3K−1\nfor the weakly coupled lattice B. The temperature deriva-\ntives which we obtained for the three exchange stiffness\ncontributions are summarized in Tb. I.\nTable I. Low-temperature exchange stiffnesses and their tem-\nperature derivatives in 10−11J/m and 10−14J/(m K), respec-\ntively.\nAA BB AB sum\nAij 2.05 0.064 0.192 2.31\n∂Aij/∂T −2.00 −0.275 −0.507 −2.78\nThe effective magnetocrystalline anisotropy density is\nKzz\neff= (dz\nA+dz\nB)/2a3=5.13×106J m−3. This value ischosen rather high in order to (i) keep the DWwidth and\nhence the required computation grid small and (ii) to\nreduce the characteristic time scale of the DWaccelera-\ntion which is proportional to the wall width ∆ DW[see\nEq.(11)]. In our simulations we observe a DWwidth\nof about 1.6 nm to 2.2 nm (depending on temperature)\nwhich is in good agreement with the theoretical predic-\ntion of ∆ DW=/radicalbig\nAeff/2Kzz\neff= 1.50 nm\nAppendix D: Computation of steady state domain\nwall dynamics\nDue to the different time scales involved in the ferrimag-\nnetic DWdynamics, determining the steady-state velocity,\nprecession, and tilting is challenging. On the one hand,\nsimulation time should be as short as possible, in order to\nminimize the thermal drift, i.e. the error margins of the\ntemperature, but at the same time one has to assure the\nsimulation time is sufficiently long for the wall to reach\nits steady-state motion.\nFor the data in Fig. 3 the steady-state dynamics were\ndetermined as follows: below Walker breakdown, we simu-\nlated a fixed amount of time of 320 ps ,128 ps ,128 ps , and\n384 ps for the panels (a)-(d), respectively. These numbers\nreflect the acceleration time scales at the different base\ntemperatures. The first 25 % of this simulation time was\nhereby discarded in order to reach the steady-state veloc-\nity (and tilting), the other 75 % were used for computing\nthe time average of VDWdisplayed in Fig. 3. The Walker\nbreakdown was defined by the wall angle tilting by more\nthan 45◦plus a five degree error margin, in order to ac-\ncount for the diverging wall precession time at exactly the\nWalker threshold. Above the torque compensation point,\npanel (d), or very close to the Walker thresholds, the DW\nprecession is slow and the precession period can be several\nhundreds of picoseconds. In this case, the steady-state\nvelocity was time-averaged over only one 180◦-rotation\nto keep thermal drift as low as possible and ensure a\nwell-defined temperature. 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Cass Avenue Argonne, IL  60439, USA  \n3Neutron Scattering Science Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831, USA \n4 Institut Laue-Langevin  6, rue Jules Horowitz  - BP 156 F- 38042 Grenoble Cedex 9, France \n \n \nAbstract \n \n \nThe single crystal study of CaBaCo 4O7, a non collinear ferrimagnet (T C=64K), with a \npolar orthorhombic space group (Pbn2 1) between 4 K and 293 K, shows the appearance below \nTC of a large electric  polarization along its c\n axis, reaching 17 mC.m-2 at 10K. At 62.5 K, a \nmagnetic field driven giant variation of polarization, P(9T) - P(0T) = 8 mC/m2, is observed. \nMoreover, the present magnetoelectric measurem ents are fully consistent with the m’m2’ \nmagnetic point group, strongly supporting that this oxide is also ferrotoroidic.  This \nferrimagnetic oxide, which belongs to the “114” structural fam ily, opens an avenue for the \nsearch of new magnetoelectrics.    \n    \n* Antoine Maignan \nLaboratoire CRISMAT, ENSICAEN/CNRS,  6 boulevard du Maréchal Juin, 14050 \n Caen cedex 4 - France \nantoine.maignan@ensicaen.fr\n \nTel: 02.31.45.26.04 Fax: 02.31.95.16.00  Numerous investigations of multiferroics ha ve shown that two physical characteristics \nof these materials are of great importance in  view of technological applications, the \nmagnetoelectric coupling and the electric polarization, which should be as high as possible [1-\n2]. Based on these two prerequisites, improper fe rroelectrics, where ferroelectricity originates \nfrom a particular magnetic order, are cha llenging for discovering new performances and \nunderstanding multiferroism. Significant coupl ing between magnetism and ferroelectricity \nwas observed in a rather large number of improper ferroelectrics, but the coefficients of the tensor characterizing the linear magnetoelectric effect α\nij were rarely measured. The highest \nvalue of α that has been reported to da te is close to 20000 ps/m for \nBa0.5Sr1.5Zn2(Fe 0.92Al0.08)12O22 [3]. Unfortunately, it is also observed that the magnetically \ninduced polarization of these magnetoelectric fe rroelectrics remains rather low, generally \nsmaller than 100 μCm-2, as shown for example for TbMnO 3 [4], MnWO 4 [5], TbMn 2O5 [6], \nNi3V2O8 [7]. Recently, the multiferroic GdMn 2O5 was shown to exhibit giant ferroelectricity \n[8-9], with ~3600 μ C/m2, the largest observed value fo r improper ferroelectrics. \nThe “114” CaBaCo 4O7 cobaltite [10-12] exhibits a pur e tetrahedral framework [10-11] \nwhere the CoO 4 tetrahedra are three-dimensionally in terconnected and form a geometrically \nfrustrated network (Fig. 1). The magnetic struct ure of this oxide shows that, similarly to \nseveral improper ferroelectrics which are antiferromagnets, the cobalt spins are non collinear \nbut differently from the latter, CaBaCo 4O7 is ferrimagnetic, below T C~64K, with as easy \naxis. In contrast to m\nany impr oper ferroelectrics, it crystallizes  in a noncentrosymmetric space \ngroup Pbn2 1, in the whole temperature rang e, from 4 K to 293 K, with c as polar axis. \nStudies on polycrystalline samples have s hown that below 64 K th e magnetic ordering \ninduces an additional polarization [12], suggesti ng the existence of improper ferroelectricity. \nIn order to conclusively determine the nature  of the magnetoelectri c coupling and of the \nelectric polarization in this pha se, a single crystal study was necessary. Such a study is also \nmotivated by the fact that th e magnetic order in th is oxide reduces the point group symmetry \nto m’m2’, similarly to several magnetoelectri c boracites [13-17], so that the existence of \nferrotoroidicity can be predicted [12]. Here, we show that CaBaCo 4O7 exhibits a high \nmagnetoelectric coupling factor le ading to a magnetic field driv en gigantic change in the \npolarization near T C. Neutron diffraction data reveal an  abrupt structur al change at T C, and it \nis proposed that the electrical polariz ation has a magnetost rictive origin.   b\n\nMillim\neter size crystals were grown usi ng the floating zone technique in a mirror \nfurnace under air at 3.5 bars. Laue diffraction patte rns showed that the si ngle crystals exhibit the same characteristics as the polycr ystalline samples with the space group Pbn2 1. Laue \npattern were collected to obtain the geomet ric relation between crystal faces and the \ncrystallographic axes. Finally pl atelet-like crystals were cut with the thinnest dimension \n(1mm) corresponding to the  axis and with the largest faces ( xy planes) reach ing 5x5 mm. \nFigure 2a shows the field cooled magne tization measured along the easy-axis ( b) upon \nwarming at 10-2T. A sharp transition is evidenced with T C=64 K, in good agreement with data \npreviously reported for the polycrystalline precu rsor [10]. The temperature dependence of the \ndielectric permittivity ( ε’) was measured along the cc\n\n and b\n directions (Fig. 2b and inset). \nAlong , a clear peak is observed at T C, supporting the possibility  of a magnetoelectric \ncoupling along cas predicted by the symmetry analysis [see ref. 12]. A second peak, with no \ncorresponding anomaly on the M(T) curve wa s also observed at 69 K, implying a non-\nmagnetic origin. In contrast, along c\n\nb\n(easy-axis for magnetization) only a change of slope is \nobserved at T C with a sharp drop of ε’ at T C. To test the origin of the anomaly above T C, \nspecific heat measurements were  made (on cooling), without a nd with an applied external \nmagnetic field of 2 T, using a larger  crystal (Fig. 2c). The peak at T C with 0H=0 becomes \nbroader and shifts towards higher T within 0H=2 T. It corresponds to the ferrimagnetic \nordering. For H=0, a second peak is detected at 69 K, which does not change in fields \nmeasured up to 2 T (Fig. 2c, inset). This strongly supports the lack of magn etic origin for this \nsmall, high temperature peak on the ε’(T) curve (Fig. 2b). \nThe presence of a well-defined dielectric peak along c at T C motivated polarization \nmeasurements. For that purpose, a thin platelet with contacts on the largest xy faces was cut \nfrom the crystal used for the M(T) in or der to apply a small electric field along c (E=1.1 \nkV/cm) during the cooling from 80 K to 8 K. At 8 K, E was removed and P was measured \nupon warming at 0.5K/min (Fig. 2d). A gigantic variation of the polar ization is evidenced, \nbetween 10K and 80 K. Also, the sharp transition at T C towards P=0 demonstrates the \nimproper origin of the electric polarization in the magnetically  ordered phase. However this \npolarization cannot be reversed completely by changing the polarity of a poling electric field \n(E= 14kV/cm). Moreover it should be pointed out th at an electric polar ization is observed \nbelow T C even in a zero poling electric field. Above T C, the polarization remains constant \n(i.e., the pyroelectric current is nu ll) but its variation cannot be measured above ~150K due to \nthe high value of th e dielectric losses. \nTo dem\nonstrate the existence of  a magnetoelectric (ME) coupling, P measurements \nwere performed under magnetic field.  It must be mentioned that w ith our experimental set-up, no polarization measurements under both magnetic and electric field could be made. So, the \ncoupling terms xyz, linear in H and E, and allowed in the ordered magnetic state [12], could \nnot be determined. Moreover, considering th e anisotropic shape of the crystals, the P \nmeasurements were made only al ong the thinnest  direction, c, which is also the polar axis. In \nthat geometry, upon application of an external  magnetic field H, the induced polarization \nalong  axis is given by P z=32Hy + (311Hx2+322Hy2+333Hz2)/2 (eq.1), which ij is the \ncoefficient of the linear magnetoelectric tensor and ijk is the coefficient of the bilinear \nmagnetoelectric susceptibility tensor. Applying H along bc\n\n, the formula reduces to: P z=32Hy \n+ 322Hy2/2 (eq.2), from which the two coefficients 32 and  322 can be extracted below T C \nwhile above T C, only 322 is allowed. As aforementioned, H was applied along the easy-axis \n in order to be perpendicular to E (c b  axis). The induced polarization Pz was recorded from \n0 to 9 T at several temperatures between  10 K and 75 K, i.e. below and above T C. An increase \nof the polarization with the applied field is obs erved whatever the temperature in this range \n(Fig.3). The values of 32 and  322 coefficients as a function of T, obtained by fitting the \nPz(Hy) curves (inset of Fig. 4) between -3 T and 3T with eq.2, are given in Fig. 4.  \nFrom these curves, it is clear th at the temperature dependence of 32 goes through a \nmaximum just below T C, as observed for the Ni-Cl or Co-I boracites [13, 15] . At 60 K, the \nvalue of the coefficient of the tensor 32 for the linear ME coupling, obtained by fitting the P z \n(Hy)T=60K  curve as shown in Fig. 4, reaches 32=764 ps/m value in SI uni t. This comp ares with \nthe value xy=730 ps/m reported at 1.5 K for a TbPO 4 single crystal [18]. As expected for the \nparamagnetic state, 32 is found to be close to zero for T>T C. The bilinear coefficient  322 is \nnegative below T C and positive above. Near the magnetic transition 322 decreases abruptly \ntowards negative values, changes sign at the tr ansition and decreases again with temperature \n(Fig.4).  A similar behavior has been previously measured in Ni-Cl boracites [13]. Additional \nmeasurements were also made to verify the predictions coming from the m’m2’ point group. \nUnder application of H along c (H z), the induced polarization P z should only depend on 333 \nwith P z =  333Hz2/2, since the linear magnetoelectric coefficient 33 is expected to be equal to \nzero by symmetry. At 10K, the P z(Hz) curve leads to 333=18.4(2) as/A and 33=0.07(11) \nps/m, i.e. 330, as expected for the point group m’m2’. Thus, the present ME H measurements \nwith H along  and  confirm the magnetic point group m’m2’ for CaBaCo 4O7. As \ntoroidization is allowed in this point group [12], CaBaCo 4O7 may also be ferrotoroidic. In that \nrespect, the divergence of 32(T) near T C is an indirect method to  probe the existence of a \nb\nctoroidal moment [19] . The shape of the 32(T) curve for CaBaCo 4O7 is consistent with the \ntheory (for a review see the re ferences in [18] and [19]). In  addition, at T=10 K, a butterfly \nloop in the P(H) curve is observed (inset of Fig. 3) with a ch aracteristic symmetric minimum, \ncorresponding with the coercive magnetic field ( 0.6 T)  and consistent with a spontaneous \nmagnetization as for LiCoPO 4 [20] and Ni-I boracite [21]. \nAs shown in Fig. 3, the largest ME effects are achieved close to T C.  This motivated \nthe measurements of H-dependent M, ’ and P at 65K (Fig. 5). A la rge magnetodielectric \neffect of 80% is found in only 1T ( ε’(H), Fig. 5b) together with a large magnetoelectric \nresponse P(H) (Fig. 3). This can be compared to the derivative of th e magnetization with \nrespect to magnetic field (inset  of Fig. 5a), showing a maximum at ~1-1.5 T. A metamagnetic \ntransition occurs from a paramagnetic state below 0H~1T towards an ordered magnetic state \nabove that value. This transition is reflected by the derivative curve of  P(H) (right inset, fig. \n5b) and confirmed by the value of magnetoelectric coefficients (linear and bilinear), \ncalculated from the P(H) curve. The  322 coefficient is positive below 1 T as in the \nparamagnetic state with a value 2.0(3) fs/A. In contrast the 322 coefficient is negative above 1 \nT as is observed in the ferrimagnetic state below T C. It should be pointed out that the sign \nchange of the bilinear coefficient, i.e. the metamagnetic transition, is also observed at 68 K \n(fig.3) but under a higher field, around 7 T. \nAlthough the present measurements of pyroel ectric current indicate that exceedingly \nhigh values of induced polarization are achieved in this non collinear ferrimagnet, the lack of \nevidence for P switching argues that this oxide is not a ferroelectric below T C. Nevertheless, \nits magnetoelectric effects are remarkable. Th e variation of the pol arization under magnetic \nfield reaches values as high as ~ 8 mC/m2 around T C between 62.5 and 65 K, which is nearly \ntwice the largest variation among the known magnetoelectric compounds [8]. The present \nresults indicate also that exceedingly high values of induced P in the magnetic state can be \nachieved in non collinear ferrim agnets. The values for CaBaCo 4O7 are higher by almost three \norders of magnitude than that recorded  at 300K for the Z-type hexaferrite, Sr 3Co2Fe24O41, \n25C/m2 [22] and by a factor of five compared to CaMn 7O12 [9] or GdMn 2O5 [8].  \nWe now consider possible orig ins for the high values of ΔP and magnetoelectric \ncoupling coefficients. Different mechanisms  have been invoked to explain improper \nferroelectricity in centrosymmetric materials:  spin spiral magnetic order which breaks the \ninversion symmetry [23], and asymmetric exchange  as in a chain of a lternating magnetic ions \nwith antiferromagnetic and ferromagnetic exch anges [24]. The case of some boracites and CaBaCo 4O7 differs from these examples as the latter compounds are polar by symmetry with \na possible electric polarization in their paramagne tic states. They exhibit an extra polarization \nin their ordered magnetic state and a magnetoelectric coupling resulting from the m’m2’ point \ngroup. However, ΔP and   values are much higher in the case of CaBaCo 4O7 as compared to \nmagnetoelectric boracites. Moreover, since the vari ation of the polarization and the values of \nthe magnetoelectric coefficients are maximum near T C, the magnetostricti on may indeed play \na major role. Our measurements by neut ron diffraction on polycrystalline CaBaCo 4O7, using \nthe POWGEN diffractometer at Oak Ridge Nati onal Laboratory, revealed small but abrupt \nvariations of the unit cell para meters to be detected at T C. As shown in Fig.6, the variation of \nthe cell parameter below T C follows the variation of both the polarization and the \nmagnetization. This suggests that the increase of the polarization below T C is strongly linked \nto the magnetostriction. However the structure of CaBaCo 4O7 is quite complex and the \nrelatively large values of the estimated standa rd deviations on positi onal parameters preclude \na quantitatively reliable calculation of the induced polarization. The sensitivity of the \nstructural and electrical properties to the spin ordering is also reflec ted by the effect of an \nexternal magnetic field.  As shown in Fig. 5a, at 65 K (i.e. just 1K above T C) and for H>1 T, a \nmetamagnetic transition from a paramagnetic to a ferrimagnetic st ate is induced. Thus, \nmagnetic field application could control the change from the pol ar paramagnetic phase to the \npolar ferrimagnetic phase with a giga ntic variation of the polarization. \nThis study of a CaBaCo 4O7 single crystal demonstrates that its ferrimagnetic ordering \ninduces a gigantic variation of its electric polar ization near T C, five times larger than the \nhighest value reported for GdMn 2O5 [8]. Moreover, its linear magnetoelectric effect is also \none of the highest that has been reported up to now. \nRemarkably, this phase which belongs to th e “114” structural family exhibits a pure \ntetrahedral coordination of cobalt, differen tly from most of ma gnoelectrics where the \nmagnetic cations are either in octahedral or pyramidal coordi nation. Several oxides of this \n“114” series such as YbBaCo 4O7 [25], TmBaCo 4O7 [cited in 25] and YBaCo 4O7 [26-27] have \nbeen shown to exhibit, like CaBaCo 4O7, a k =0 magnetic propagation vector which is a \nsymmetry condition for the appearance of  spontaneous magnetization and linear \nmagnetoelectric effect. Thus, the “114” family of fers a potential new route toward design of \nperforming magnetoelectrics, and a challe nge for understanding the role of the \nmagnetostriction in the appearance of induc ed polarization and strong magnetoelectric \ncoupling. \n  \n  \nAcknowledgment : Work in the Materials Science Di vision of Argonne Na tional Laboratory \n(single crystal growth, heat capacity measuremen ts) is sponsored by the U.S. DOE, Office of \nScience, Office of Basic Energy Sciences, Mate rials Science and Engi neering Division under \ncontract No. DE-AC02-06CH211357. The Aut hors thank Laurence Hervé (CRISMAT) for \ncrystal growth. \n \n5 current address: UGC-DAE Consortium for Scientific Research, University Campus, Khandwa \nRoad, Indore, 452017, India References: \n1. M. Fiebig, J. Phys. D. Appl. Phys. 38, R 123 (2005) \n2. S.-W. Cheong and M. Mostovoy Nat. Mater.  6, 13 (2007) \n3. S. H. Chun, Y. S. Chai, Y. S. Oh,  D. Jaiswal-Nagar, S. Y. Haam, I. Kim,B. Lee, D. H. \nNam, K.-T. Ko, J.-H. Park, J.-H. Chung, and K. H. Kim, Phys. Rev. Lett.  104, 037204 \n(2010) \n4. T. Kimma, T. Goto, H. Shinta ni, T. Arima and Y. Tokura, Nature 426, 55 (2003) \n5. K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa and T. Arima, Phys. Rev. Lett  97, \n097203 (2006) \n6. N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha and S.-W. Cheong, Nature 429, 392 \n(2004) \n7. G. Laws, A. B. Harris, T. Kimura, N. R ogado, R. J. Cava, A. Ahrarong, O. Entin-\nWohlman, T. Yildrim, M. Kenzelma nn, C. Broholm and A. P. Ramirez Phys. Rev. \nLett. 95, 087205 (2005) \n8. N. Lee, C. 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B  74, 172401 \n(2006)  \nFigure captions: \nFig 1: Crystal structure of CaBaCo 4O7. \nFig.2: a) FC Magnetization along the b axis; Cooling and magnetic measurement were carried \nout under 100 Oe; b) Dielectric permittivity along the c axis at 100 kHz, and along the b axis \n(inset); c) Specific heat; Inse t of c shows enlargement around T C of specific heat measured in \n0 T (blue dots), 1T (green down triangles)  an d 2 T (red up triangles); d) Variation of the \npolarization along the c axis after poling under E c=1.1 kV/cm. All the measurements were \nperformed upon heating except the specific heat measurement.  \nFig.3: Magnetic field dependence of the polarization along the c axis at various temperatures. \nMagnetic field is applied along the b axis. Inset: polarization hysteresis loop measured at 10 \nK. \nFig. 4: Linear magnetoelectric coefficient 32 (dots) and bilinear magnetoelectric coefficient \n322 (squares) versus T in SI units (Doted lines are guided for eyes). Inset: Least-squares fit of \nPz(Hy) above (bottom)  and below (top) T C. For clarity the figure s hows only the fit between 0 \nand 3T. At T=65K, the fit of P z(Hy) is performed between to -1T and +1T. \nFig. 5: Isothermal curves collected at 65 K a) Magnetization versus field with H//b; b) \nMagnetodielectric effect at 100 kHz versus field with E//c and H//b; Inset enlargements for \n0T0H3T of (a) M/dH = f(H), (b-left) ’(H) and (b-right) P/dH = f(H). \nFig. 6: From top to bottom: a, b and c cell parameters of polycrystalline CaBaCo 4O7 versus T \nrefined from neutron powder diffraction data.  \n Figure 1: \n \n \n \n \n Figure 2:  \n Figure 3: \n \nFigure 4: \n Figure 5: \n Figure 6: \n \n "    },    {        "title": "1509.00901v1.Effect_of_Chemical_Pressure_on_High_Temperature_Ferrimagnetic_Double_Perovskites_Sr2CrOsO6_and_Ca2CrOsO6.pdf",        "content": "Effect of Chemical Pressure  on High Temperature Ferrimagnet ic \nDouble Perovskites Sr 2CrOsO 6 and Ca 2CrOsO 6 \nRyan Morrow,1 Jennifer R. Soliz,1,2 Adam J. Hauser,3,4 James C. Gallag her,3 Michael A. \nSusner,5,6 Michael D. Sumption,5 Adam  A. Aczel,7 Jiaqiang Yan,6,8 Fengy uan Yang,3 Patrick M. \nWoodward1 \n1 Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210 -1185, USA  \n2 Edgewood Chemical Biological Center, 5183 Blackhawk Road, Aberdeen Proving Ground, Maryland 21010, USA  \n3 Departme nt of Physics, The Ohio State University, Columbus, Ohio 43210 -1185, USA  \n4 California Nanosystems Institute, University of California, Santa Barbara, California 93106, USA  \n5 Department of Materials Science and Engineering, The Ohio State University, Columb us, Ohio 43210 -1185, USA  \n6 Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA  \n7 Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA  \n8 Department of Materials Science and  Engineering, The University of Tennessee, Knoxville , TN 37996 , USA  \n \nAbstract  \nThe o rdered double perovskite s Sr2CrOsO 6 and Ca 2CrOsO 6 have been synthesized and \ncharacterized with neutron powder diffraction, electrical transport measurements, and high field \nmagnetization experiments.  Sr 2CrOsO 6 and Ca 2CrOsO 6 crystallize with \n3R and P21/n space \ngroup  symmetry , respectively.  Both materials are found to be ferrimagnetic insulators with \nsaturation magnetizations near 0.2 μB.  Sr2CrOsO 6 orders at 660 K, showing non -monotonic \nmagnetization temperature dependence, while Ca 2CrOsO 6 orders at 490 K and does not show \nnon-monotonic behavior.  Evidence for a theoretically predicted canted magnetic structure in \nSr2CrOsO 6 is sought a nd not found.  \n \nIntroduction  \nTransition metal oxides with the p erovskite  structure  are one of the most heavily studied class es \nof materials due to their ability to incorporate elements from throughout  the periodic table which \nresults in  properties that span  the gamut of modern functional materials .  Recently, ordered \ndouble perovskite s with formula A2BB'O6 and consisting of a network of corner sharing BO6 and \nB'O6 octahedra have been the focus of intense research [1] due to the occurrence of highly spin -\npolarized electrical transport  [2-4], unusual sequences of magnetic phase transitions  [5-7], and \ngeometric frustration  [8-10].  One material, a ferrimagnetic insulator Sr 2CrOsO 6, has the \ndistinction  of possessing the highest  known  Curie temperature  (TC) amongs t double perov skites \nat 725 K  [11].  However, despite the fact that it has been the subject of  numerous theoretical \nstudies  [12-14], including one that proposes  a canted magnetic structure to explain its unusual \nnon-monotonic magnetic susceptibility temper ature dependence  [15], Sr 2CrOsO 6 has not been \nexperimentally revisited since the initial publication  [11]. Here in, we revisit the magnetic properties of Sr 2CrOsO 6 with neutron diffraction , electrical \ntransport measurements, and high field magnetization exp eriments.  Additionally, we synthesize \nand characterize the  crystal structure and magnetic properties  of Ca 2CrOsO 6. This compound for \nwas prepared by Sleight and Ward in 1962, but its properties have not previously been studied  \n[16].  The results shed ligh t on the way that  chemical pressure affects  the magnetic interactions \nand the resulting magnetic ground state s of these two compounds .  The ir behavior is  compared \nand contrasted with A2FeOsO 6 and A2CoOsO 6 (A = Sr, Ca) osmate double perovskites where \nchemic al pressure drives a crossover from an antiferromagnetic to a ferrimagnetic ground state \n[17-19]. \nExperimental  \n1.6 g powder samples were prepared by solid state reaction of stoichiometric quantities of SrO \n(99.9% pure, Sigma Aldrich), CaO (99.9% pure, Sigm a Aldrich), CrO 3 (99.99% pure, Sigma \nAldrich), and Os metal (99.98% pure, Alfa Aesar).  Reactants were ground in a n argon  glove box  \nand then placed in a high density alumina tube that was  sealed in a silica ampoule (40 mL \nvolume, 3 mm wall thickness)  along  with a separate vessel containing PbO 2 that decompose d \ninto PbO and acted as an in-situ source of O 2 gas.  One quarter mole excess O2 gas was produced \nin this way in order to ensure full oxidation of the reactants.  The sealed tubes were heated to  \n1000  °C for Sr 2CrOsO 6 and 950 °C for Ca 2CrOsO 6 and held  for 48 hours in a box furnace \nlocated within a fume hood as an additional precaution.  Great caution must be used as heating \nosmium in the presence of oxygen  can produce toxic OsO 4 gas.  Larger batch sizes o r ampoules \nwith thinner walls  may produce conditions resulting in rupture of the sealed tub e at elevated \ntemperatures .  The following equations were used for these reactions.  \n4 SrO + 2 CrO 3 + 2 Os + (3 PbO 2)  2 Sr 2CrOsO 6 + (3 PbO) + 1/2 O 2 \n4 CaO + 2 CrO 3 + 2 Os + (3 PbO 2)  2 Ca 2CrOsO 6 + (3 PbO) + 1/2 O 2 \nLaboratory  X-ray diffraction measurements were conducted on a Bruker D8 Advance instrument \nequipped with a Ge(111) monochromator in order to verify composition and cation order ing.  \nTime of flight n eutron powder diffraction  (NPD)  data were collected on the POWGEN [20] \nbeamline at Oak Ridge National Laboratory  (ORNL) . Sr 2CrOsO 6 was measured using the JANIS \ncryofurnace with high and low d spacing frames of 0.2760 –3.0906 Å and 2.2076 –10.3019 Å at 5  \nK and 500  K as well as single intermediate d spacing frames of 1.1038 –6.1811 Å at 50  K \nintervals .  Ca 2CrOsO 6 was measured in the Powgen Automatic Changer  (PAC) environment at \n10 K and 300  K with frames of 0.2760 –4.6064 Å and 2.2076 –15.3548 Å at each temperature.  \nRietveld refinements were conducted using the GSAS EXPGUI software package  [21, 22] .  \nAdditional neutron powder diffraction data was collected on Sr 2CrOsO 6 using  ORNL’s HB -1A \ninstrument with a constant wavelength of λ = 2.367 Å .  The HB -1A sample of approximately 5 g \nwas comprised of three combined batches of material. Diffraction patterns were collected at T = \n4 and 300 K using a closed cycle refrigerator, and th e instrument collimation was 40' -40'-40'-80'.  Sr2CrOsO 6 and Ca 2CrOsO 6 powders were pressed and  sintered  at 1000  °C and 950 °C, \nrespectively,  in evacuated sealed ampoules in order to produce bar shaped polycrystalline pellets  \nfor electrical measurements . Silver paint was applied to attach copper leads to the pellet s. DC \nresistivity  data was collected  over a temperature range of 140 to 400 K for Sr 2CrOsO 6 and 70 to \n335 K for Ca 2CrOsO 6 using the four-point probe method in a Quantum Design model 6000 \nphysical property measurement system (PPMS).   The samples were too resistive for accurate \nmeasurements at low er temperatures.   No magnetic field was applied during the measurements.  \nAdditionally, n o corrections were made for porosity.  \nThe field dependence of the magnetization of  Sr2CrOsO 6 and Ca 2CrOsO 6 pellet s was measured \nusing the Vibrating Sample Magnetometer (VSM) option of the PPMS with the maximum field \nof 14 0 kOe at 5, 300 , and 400 K  for Sr 2CrOsO 6 and at 5 and 300 K for Ca 2CrOsO 6. Higher \nmagnetic fields (± 350 kOe) were used to measure the hysteresis loops  of Sr 2CrOsO 6 at 4.5 K \nand 295 K using the VSM at the National High Magnetic Field Laboratory (NHMFL).  \nMagnetic susceptibility of a pelletized  Sr2CrOsO 6 sample was obtained using a LakeShore VSM. \nThe sample  was heated to 800 K and then cooled in a 15 kOe field to field cool the sample. The \nfield cooled susceptibility curve was then measured in field strength of 1 kOe in the temperature \nrange from 300 to 800 K. An analogous set of measurements was conducted f or Ca 2CrOsO 6 \nwithin the temperature range 300 to 600  K. \nResults  \nSr2CrOsO 6 crystallize s in the \n3R space group as reported previously  [11], while Ca 2CrOsO 6 \ncrystallize s in the monoclinic P21/n space group common to double perovskites w ith a tolerance \nfactor smaller than approximately 0.97  [23].  Cr/Os cation ordering for both compounds was \ndetermined from laboratory  XRD data in order to take advantage of the substantial X -ray \nscattering contrast between Cr and Os . The order parameter , which is  defined as η = 2 θ – 1 \nwhere θ is the occupancy of the cation on its assigned site (i.e. the Cr occupancy on the Cr -rich \noctahedral site) , was found to be 80.2(4)% for Sr 2CrOsO 6 and 76.2(5)% for Ca 2CrOsO 6.  \nThe results from neutron powder diffractio n are given in Tables 1 and 2 for Sr 2CrOsO 6 and \nCa2CrOsO 6 respectively , while the refined NPD patterns are shown in Figures 1 and 2. The \ncalculated bond valence sums [24] for chromium are 3. 27 (5 K) and 3. 38 (500 K) for Sr2CrOsO 6, \nwhile they are  3.21 (10 K ) and 3. 22 (300 K) for Ca 2CrOsO 6.  These results support the \nassignment of Cr3+ in these materials, indicating that Os is in the 5+ oxidation state.  These \nconclusions are supported by direct comparison to the M−O bond lengths in  related perovskites \nwith these oxidation states  [11, 17, 25, 26] .  Other than the change in symmetry driven by the \nchange in octahedral tilting , the most significant variance between the two is the reduction in the \nCr−O−Os bond angle fro m 170.8° in Sr 2CrOsO 6 to an average of 153.2° in Ca 2CrOsO 6 at low \ntemperatures . An increased bending of the B−O−B′ bonds is the expected response to increasing \nchemical pressure  when the stiffness of the B/B′−O bonds is higher than that of the A−O bonds .   It is interesting to note that the continuous evolution of the Cr−O−Os bond angle in Sr 2CrOsO 6 \nwith temperature, as shown in Figure 3, typical of rhombohedral perovskites approaching the \ncubic phase transition  [27].  The symmetry of Sr 2CrOsO 6 is rhombohed ral at 500 K , which is \npresumably quite close to a structural phase transition , given the earlier report of cubic symmetry \nat 540 K  [11].   \nThe electrical transport of Sr 2CrOsO 6 and Ca 2CrOsO 6 is given in Figure 4 showing linear \nbehavior s on a T−1/4 scale  which is consistent with a variable range hopping transport model as \nhas been reported in related double perovskite osmates  [28, 29] .  The resistivity  of Sr 2CrOsO 6 \nincreases from 8.6×102 Ω·cm at 300 K to 2.5×106 Ω·cm at 140 K , below  which the resistance \nexceeds the maximum value that can be measured with our instrument , clearly demonstrating \ninsulating behavior .  A room temperature resistivity value of 10 Ω·cm had been reported \npreviously , but the temperature dependence has not been reported  previously  [11].  Ca2CrOsO 6 \nalso demonstrates insulating behavior, albeit with resistivity which is significantly reduced from \nSr2CrOsO 6. \nThe temperature depend ence of the magnetic susceptibilit ies of Sr 2CrOsO 6 and Ca 2CrOsO 6 are \nplotted  in Figure 5.  The observed TC of 66 0 K for Sr 2CrOsO 6 is smaller than the value of 725 K \nreported previously  [11] but still larger than any other known double perovskite.  The difference \nin ordering temperature is unclear but can likely be attributed to differences  in cation order, as \nKrocke nberger et al. indicate full Cr/Os cation order  [11].  Reductions in cation order have been \nshown to result in reduction in magnetic ordering temperature in other double perovskites  [30]. \nThe magnetic susceptibility is shown to increase non -monotonically f rom 300 K to \napproximately 500 K  where it peaks .  The temperature dependence of the magnetic susceptibility \nof Ca 2CrOsO 6 was measured between 300 and 600 K as shown in Figure 5b revealing a TC of \napproximately 490 K.  Unlike Sr 2CrOsO 6, Ca2CrOsO 6 does not e xhibit a non -monotonic \ntemperature dependence of the field cooled susceptibility.   \nIn rare earth ( RE) chromate perovskites,  RECrO 3, a reduction in Cr−O−Cr bond angles (i.e. \nincreased octahedral tilting) is driven  by progressively smaller RE cations .  This  weakens the \nsuperexchange interactions leading to a reduction in the Neél temperature  (TN): LaCrO 3 TN = 282 \nK, NdCrO 3 TN = 224 K, TbCrO 3 TN = 158 K, YCrO 3 TN = 141 K, LuCrO 3 TN = 112 K  [31]. We \nobserve analogous behavior in Sr 2CrOsO 6 and Ca 2CrOsO 6.  This implies  that the nearest \nneighbor antiferromagnetic coupling is the dominant exchange interaction in  A2CrOsO 6 double \nperovskites just as it is in RECrO 3 perovskites . The higher ordering temperatures seen in the \nA2CrOsO 6 double perovskites imply stronger su perexchange coupling  than in RECrO 3 \nperovskites,  clear evidence that superexchange coupling involving the t2g orbitals of 3d and 5d \nions can be very strong.  \nThe field dependence of the magnetization  of Sr 2CrOsO 6 is shown in Figure 6.  A saturation \nmagnetiz ation (Msat) of 0.22 μB per formula unit (f.u.) is measured at 4.5 and 295 K under \nmaximum applied fields of 35 0 kOe, while  lesser fields of the available PPMS were insufficient to saturate the magnetization  [11], as shown in Figure 6b .  The field dependence of the \nmagnetizati on of Ca 2CrOsO 6 at 5 K and 300 K under maximum applied fields of 140 kOe is \ngiven in Figure 6c.  The magnetization clearly does not saturate under these conditions , and \nfurther high field measurements would be necessary to reach saturation .  However, as th e \nmagnetization  of Ca 2CrOsO 6 approaches similar values as Sr 2CrOsO 6, it can safely be assumed \nthat both compounds are ferrimagnetic  with similar saturation magnetization values .   \nThe magnetic scattering intensity observed in the neutron powder diffraction  patterns could be  \nmodeled  using a collinear  ferrimagnetic model.  Unfortunately  strong correlations between the \nmoments on the Cr and Os sites make it difficult to independently refine the moments of the two \nmagnetic ions. I t is possible to produce simila r quality fits to the data with a continuum of Cr/Os \nmoment s, provided the difference between the Cr and Os moments stays roughly constant, as has \nbeen reported in prior  studies of ferrimagnetic double perovskites  [17, 32] . \nThe data can be modeled by refin ing the moment on chromium and neglecting any contribution \nfrom osmium. In the case of Sr 2CrOsO 6 this approach  results in chromium moments  of 2.24(6) \nand 4.01(8) μB at 500 and 5 K respectively , the latter value is physically unreasonable for a d3 ion \nlike Cr3+.  In order to systematically refine the magnetic structure with moments on both ions the \nmoment on Cr was fixed to 2.5 μB, in line with values recorded in earlier  studies of RECrO 3 \nperovskites [31], while  the Os moment was allowed to refine.  This res ulted in a n Os  moment of \n0.82(9) μB at 5 K .  This value is reduced from the values obtained for Os5+ in other studies of \nosmate perovskites [33 -35], possibly due to Cr/Os antisite disorder.  Figure 7 shows the \ntemperature dependence of the refined Os momen t using this approach .  The refined moment is \nfairly constant in the temperature range 5 to 350 K, above which it drops substantially at 400 and \n450 K.  Finally at  500 K  the Os moment seems to go to zero and the  Cr moment reduces to  \n2.24(6)  μB.  This is ne ar the temperature where a maximum is seen in the magnetization \nmeasurement.  It should be noted that while this  refinement approach could also be consistent \nwith both Cr and Os moments decreasing with temperature at a similar rate, that scenario would \nnot lead to the increase in magnetization seen in Figure 5a for Sr 2CrOsO 6 between 300 and 500 \nK. \nA similar approach was employed in the refinement of the magnetic moments in Ca 2CrOsO 6.  \nRefining only Cr moments resulted in physically unreasonable values of 4. 07(6) and 4.34(5) μB \nat 300 and 10 K, respectively.  Fixing the Cr moment at 2.5 μB antiparallel to Os resulted in  \nrefined  Os moments of 1.26(6)  and 0.55(6)  μB at 10 and 300 K respectively .   \nAdditional purely magnetic reflections were not observed  in neut ron powder diffraction  for \neither Ca 2CrOsO 6 or Sr 2CrOsO 6 precluding the consideration of canted magnetic models.  As \nshown in Figure 8, Sr 2CrOsO 6 was further scrutinized with neutron scattering at Oak Ridge \nNational Laboratory’s HFIR  facility  on the HB -1A instrument to search for possible weak \nmagnetic reflections in accordance with the theoretical canted magnetic structure mode l [15].  \nHB-1A is the ideal instrument for this particular investigation because of its excellent signal -to-noise ratio, arising fr om the combined use of a double -bounce monochromator and an analyzer.  \nHowever, all weak features in the data were accounted for in powder X -ray diffraction data \nindicating that they are not of magnetic origin.  Specifically, the  strong  magnetic reflection  \npredicted to occur at Q = 1.14 Å-1, corresponding to a (110) reflection  on a cubic double \nperovskite lattice (used in the theoretical model [15]),  is not observed as highlighted in Figure \n8c.  Thus, within the limitations of a powder neutron experiment, t here is no evidence of  the \ntheoretically predicted  spin canting in the magnetic structure of Sr 2CrOsO 6.  \nPerovskites containing the d3 electronic configuration are known for having high magnetic \ntransition temperatures . Examples include the  antiferromagnet s NaOsO 3 (TN = 410  K [35]), \nSrRu 2O6 (TN = 565 K [36]), and SrTcO 3 (TN = 1000  K [37]).  While Sr 2CrOsO 6 has a similar \nelectronic configuration (3d3−5d3), it is ferrimagnetic rather than antiferromagnetic due to a \nsmall but non -negligible orbital contribution  [33] that reduces the moment on  Os preventing  \nfully compensated magnetism between Cr and Os sublattices.   Recent work suggests that  spin-\norbit c oupling  (SOC ) is allowed in the 5d3 case due to an electronic state which is intermediate \nbetween L -S and j-j coupling schemes , rendering the assumption that SOC is completely \nquenched invalid [ 38].  While no evidence was found for a canted magnetic struct ure, an \nalternative cause of net magnetization  as proposed by  Meetei et al. [15],  for the non -monotonic \nmagnetic susceptibility of Sr 2CrOsO 6 is confirmed.  The decrease in magnetic scattering , \nobserved upon warming above 350 K and  modeled here as a decreas e in the Os moment,  is \ncoincident with the non -monotonic portion of the magnetization temperature dependence and  \nlends support to the idea that the magnetization of the Os and Cr sublattices have differing \ntemperature dependence s. While the neutron data ha ve been modeled in accordance with this \nidea, conclusive evidence will require an elemental probe such  as X-ray magnetic circular \ndichroism  (XMCD ). Unfortunately such an  experiment would be very  challenging due to the \nwide temperature range of the measurem ent and the exceptionally  high magnetic  fields needed  to \nsaturate the magnetization of the material.  \nConclusion  \nSr2CrOsO 6 and Ca 2CrOsO 6 have been synthesized and characterized  by a number of techniques .  \nThe compounds form partially ordered double perovski tes with \n3R  (Sr2CrOsO 6) and P21/n \n(Ca 2CrOsO 6) space group symmetries.  Sr 2CrOsO 6 is an insulating  ferrimagnet with TC = 660  K \nand Msat = 0.22  μB/f.u. while Ca 2CrOsO 6 is also a n insulating  ferrimagnet with a reduced TC of \n490 K and an Msat of approximately 0.2  μB/f.u.  The temperature dependence of the magnetic \nsusceptibility of Sr 2CrOsO 6 is non -monotonic, an unusual feature for a ferrimagnetic d ouble \nperovskite. This behavior is attributed to  different temperature dependences in the magnetization \nof the chromium and osmium sublattices.  Interestingly, similar behavior is not observed for  \nCa2CrOsO 6.  \n \nAuthor Information  Corresponding Author  \nwoodwa rd@chemistry.ohio -state.edu  \nNotes  \nThe authors declare no competing financial interest.  \nAcknowledgements  \nSupport for this research was provided by the Center for Emergent Materials an NSF Materials \nResearch Science and Engineering Center (DMR -1420451 ). Add itional support was provided by \nthe U.S. Department of Energy, Office of High Energy Physics under Grant Number DE -FG02 -\n95ER40900  and DE -SC0001304 (magnetic characterization) . A portion of this work was \nperformed at the National High Magnetic Field Laborat ory, which is supported by National \nScience Foundation Cooperative Agreement No. DMR -1157490 and the State of Florida.  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Jpn. 82, 073703 (2013).  \n \n \n  \nTemperature (K)  5 500 \nSpace Group  R    R    \na (Å) 5.5200(1)  5.5300(1)  \nc (Å) 13.4403(3)  13.5242(6)  \nV (Å)3  354.662)  358.2(1)  \nRwp 2.74%  3.60%  \n    \nCr−O (×6 , Å) 1.964(2)  1.952(1)  \nOs−O (×6 , Å) 1.944(2)  1.958(1)  \n∠Cr−O−Os  () 170.82(5)  176.5(1)  \n    \nSr z 0.2500(3)  0.2502(7)  \nO x 0.3350(6)  0.3323(8)  \nO y 0.1958(3)  0.1770(6)  \nO z 0.4162(2)  0.4166(4)  \n \nTABLE 1. Neutron powder diffraction parameters obtained from Rietveld refinement for \nSr2CrOsO 6 at 5 and 500K.  \n   \nTemperature (K)  10 300 \nSpace Group  P21/n P21/n \na (Å) 5.3513(1)  5.36287(9)  \nb (Å) 5.4561(1)  5.4541 (1)  \nc (Å) 7.6204(1)  7.6321(1)  \nV (Å)3  222.50(1)  223.24 (1)  \nβ (°) 90.092(2)  90.074(2)  \nRwp 3.98%  3.89%  \n   \nCr−O1 (×2 , Å) 1.973(3)  1.971(2)  \nCr−O2 (×2 , Å) 1.973(2)  1.970(2)  \nCr−O3 (×2 , Å) 1.967(3)  1.972(3)  \n   \nOs−O1 (×2 , Å) 1.955(3)  1.959(2)  \nOs−O2 (×2 , Å) 1.960(2)  1.962(2)  \nOs−O3 (×2 , Å) 1.946(3)  1.943(3)  \n   \n∠Cr−O1−Os  () 153.3(2)  153.4(2)  \n∠Cr−O2−Os  () 152.6(2)  153.1(2)  \n∠Cr−O3−Os  () 153.8(1)  154.20(8)  \n   \nCa x −0.0096(3)  −0.0086(4)  \nCa y 0.0458(2)  0.0434(2)  \nCa z 0.2499(4)  0.2505(5)  \nO1 x 0.2044(4)  0.2050(4)  \nO1 y 0.2087(4)  0.2081(5)  \nO1 z −0.0407(3)  −0.040 3(4) \nO2 x 0.2067(3)  0.2069(4)  \nO2 y 0.2064(4)  0.2073(5)  \nO2 z 0.5428(3)  0.5417(4)  \nO3 x 0.4196(2)  0.4211(3)  \nO3 y −0.0200(2)  −0.0199(3)  \nO3 z 0.2514(3)  0.2519(4)  \n \nTABLE 2 : Neutron powder diffraction parameters obtained from Rietveld refinement for \nCa2CrOsO6 at 10 and 300 K.  \n \n \n  \nFIG. 1 (color online).  Refined neutron powder diffraction pattern of Sr 2CrOsO 6 at a) 500 K and \nb) 5 K.  Black symbols, red curve s, and blue curve s correspond to observed data, the calculated \npattern s, and the difference  curve , respectively .  The black hashes correspond to both the nuclear \nand magnetic peak positions of the compound.  \n \nFIG. 2 (color online).   Refined neutron powder diffraction pattern of Ca 2CrOsO 6 at a) 300 K and \nb) 10 K.  Black symbols, red curve s, and blue curve s correspond to observed data, the calculated \npattern s, and the difference  curves , respectively .  Black hashes correspond to both the nuclear \nand magnetic peak positions of the compound. There is a trace amount of CaO which is \nunrefined in the pattern s with a single  peak identified by the black arrow.  \n \n \n \nFIG. 3 (color online).  The temperature dependence of the refined Cr−O−Os bond angle in \nSr2CrOsO 6. \n \n \n \nFIG. 4 (color online).  The temperature dependence of the electrical resistivity of Sr 2CrOsO 6 \n(black) and Ca 2CrOsO 6 (red).  \n \n \nFIG. 5 (color online).  The field cooled (15 k Oe) temperature dependence of the magnetization  of \na) Ca 2CrOsO 6 and b) Sr2CrOsO 6 measured under an applied field of 1 k Oe. \n \n \nFIG. 6 (color online).  The field dependence of the magnetization of a) Sr2CrOsO 6 at 4.5 and \n295K under a m aximum applied field of 350 kOe as well as at b) 5, 300, and 400  K under  \nmaximum applied field s of 140 kOe .  The field dependence of the magnetization of Ca 2CrOsO 6 \nat 5 and 300 K under a maximum applied field of 140 kOe is given in c).  \n \n \nFIG. 7. The temperature dependence of the refined Os moment in Sr 2CrOsO 6 with a fixed Cr \nmoment at 2.5 μ B as described in the text.  At 500 K, the Os moment is fixed to 0, and the Cr \nmoment is refined to 2. 24(6) μB. \n \nFIG. 8 (color online).  a) Powder X -ray diffraction pattern of Sr 2CrOsO 6 sample used for HB -1A \nmeasurements, b) neutron scatteri ng data from Sr 2CrOsO 6 powder collected at 4 and 300 K, and \nc) comparison of the data set s in log scale.  The XRD pattern has been offset for clarity.  All \nweak features seen in neutron scattering can be accounted for in the X -ray data indicating that \nthey are not of magnetic origin and are likely associated with trace unknown impurities.  The \nposition of a predicted magnetic reflection as described in the text is indicated with an arrow.  \n"    },    {        "title": "1702.02554v1.Self_Focusing_Skyrmion_Racetracks_in_Ferrimagnets.pdf",        "content": "Self-Focusing Skyrmion Racetracks in Ferrimagnets\nSe Kwon Kim,1Kyung-Jin Lee,2, 3and Yaroslav Tserkovnyak1\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Materials Science and Engineering, Korea University, Seoul 02841, South Korea\n3KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, South Korea\n(Dated: February 9, 2017)\nWe theoretically study the dynamics of ferrimagnetic skyrmions in inhomogeneous metallic \flms\nclose to the angular momentum compensation point. In particular, it is shown that the line of the\nvanishing angular momentum can be utilized as a self-focusing racetrack for skyrmions. To that\nend, we begin by deriving the equations of motion for the dynamics of collinear ferrimagnets in the\npresence of a charge current. The obtained equations of motion reduce to those of ferromagnets and\nantiferromagnets at two special limits. In the collective coordinate approach, a skyrmion behaves\nas a massive charged particle moving in a viscous medium subjected to a magnetic \feld. Analogous\nto the snake orbits of electrons in a nonuniform magnetic \feld, we show that a ferrimagnet with the\nnonuniform angular momentum density can exhibit snake trajectories of skyrmions, which can be\nutilized as racetracks for skyrmions.\nIntroduction. |A free particle with the magnetic mo-\nment precesses at the frequency proportional to the ap-\nplied magnetic \feld and its gyromagnetic ratio, which is\nthe ratio of its magnetic moment to its angular momen-\ntum. When a magnet is composed of equivalent atoms,\nits net magnetization and net angular momentum density\nare collinear with the proportionality given by the gyro-\nmagnetic ratio of constituent atoms. Due to the linear\nrelationship between them, the magnetization and the\nangular momentum density represent the same degrees\nof freedom, and thus are interchangeable in describing\nthe magnetization dynamics. One-sublattice ferromag-\nnets and two-sublattice antiferromagnets are examples\nof such magnets.\nWhen a magnet consists of inequivalent atoms, how-\never, its net magnetization and net angular momentum\ndensity can be independent degrees of freedom [1]. One\nclass of such magnets is rare-earth transition-metal (RE-\nTM) ferrimagnetic alloys [2], in which the moments of\nTM elements and RE elements tend to be antiparallel\ndue to the exchange interaction. Because of di\u000berent gy-\nromagnetic ratios between RE and TM elements, one can\nreach the angular momentum compensation point and\nthe magnetization compensation point, by varying the\nrelative concentrations of the two species or changing the\ntemperature. These compensation points are absent in\nthe ferromagnets and antiferromagnets, which have been\nmainstream materials in spintronics [3], and thus may\nbring a novel phenomenon to the \feld. In particular,\nwe would like to focus on the dynamics of ferrimagnets\naround the angular momentum compensation point in\nthis Letter for the following reason. Away from the com-\npensation point, the dynamics of ferrimagnets is close to\nthat of ferromagnets [4]. At the compensation point, its\ndynamics is antiferromagnetic [2, 5]. Therefore, the ideal\nplace to look for the unique aspects of the dynamics of\nferrimagnets would be close to, but not exactly at, the\nangular momentum compensation point.Topological solitons in magnets [6] have been serving as\nactive units in spintronics. For example, a domain wall,\nwhich is a topological soliton in quasi-one-dimensional\nmagnets with easy-axis anisotropy, can function as a\nmemory unit, as demonstrated in the magnetic domain-\nwall racetrack memory [7]. Two-dimensional magnets\nwith certain spin-orbit coupling can also stabilize an-\nother particle-like topological soliton, which is referred\nto as a skyrmion. Skyrmions have been gaining atten-\ntion in spintronics as information carriers, alternative to\ndomain walls, because of fundamental interest as well as\ntheir practical advantages such as a low depinning electric\ncurrent [8]. Several RE-TM thin \flms such as GdFeCo\nand CoTb have been reported to possess the perpendic-\nular magnetic anisotropy and the bulk Dzyaloshinskii-\nMoriya interaction [9, 10], and thus are expected to be\nable to host skyrmions under appropriate conditions.\nIn this Letter, we study the dynamics of skyrmions\nin metallic collinear ferrimagnets, with a speci\fc goal to\nunderstand and utilize the dynamics of skyrmions close\nto the angular momentum compensation point in RE-\nTM alloys. To that end, we \frst derive the equations of\nmotion for the dynamics of general collinear magnets in\nthe presence of an electric current. The resultant equa-\ntions of motion reduce to those of ferromagnets and an-\ntiferromagnets at two limiting cases. The dynamics of\na skyrmion is then derived within the collective coordi-\nnate approach [11]. Generally, it behaves as a massive\ncharged particle in a magnetic \feld moving in a viscous\nmedium. When there is a line in the sample across which\nthe net angular momentum density reverses its direction,\nthe emergent magnetic \feld acting on skyrmions also\nchanges its sign across it. Motivated by the existence of a\nnarrow channel in two-dimensional electron gas localized\non the line across which the perpendicular magnetic \feld\nchanges its direction [12], we show that, under suitable\nconditions, the line of the vanishing angular momentum\nin RE-TM alloys can serve as a self-focusing racetrack forarXiv:1702.02554v1  [cond-mat.mes-hall]  8 Feb 20172\nskyrmions [13] as a result of combined e\u000bects of the e\u000bec-\ntive Lorentz force and the viscous force. We envision that\nferrimagnets with the tunable spin density can serve as a\nnatural platform to engineer an inhomogeneous emergent\nmagnetic \feld for skyrmions, which would provide us a\nuseful knob to control them.\nMain results. |The system of interest to us is a two-\ndimensional collinear ferrimagnet. Although the angu-\nlar momentum can be rooted in either the spin or the\norbital degrees of freedom, we will use the term, spin,\nas a synonym of angular momentum throughout for the\nsake of brevity. For temperatures much below than the\nmagnetic ordering temperature, T\u001cTc, the low-energy\ndynamics of the collinear ferrimagnet can be described\nby the dynamics of a single three-dimensional unit vec-\ntorn, which determines the collinear structure of the\nmagnet [4]. Our \frst main result, which will be derived\nlater within the Lagrangian formalism taken by Andreev\nand Marchenko [4] for the magnetic dynamics in conjunc-\ntion with the phenomenological treatment of the charge-\ninduced torques [14], is the equations of motion for the\ndynamics of nin the presence of a charge current den-\nsityJand an external \feld hto the linear order in the\nout-of-equilibrium deviations _n,J, and h:\ns_n+s\u000bn\u0002_n+\u001an\u0002n=n\u0002fn+\u0018(J\u0001r)n\n+\u0010n\u0002(J\u0001r)n;(1)\nwheresis the net spin density along the direction of\nn,s\u000band\u001aparametrize the dissipation power density\nP=s\u000b_n2and the inertia associated with the dynam-\nics of n, respectively, and fn\u0011\u0000\u000eU=\u000enis the e\u000bective\n\feld conjugate to nwithU[n] the potential energy [15].\nHere,\u0018and\u0010are the phenomenological parameters for\nthe adiabatic and nonadiabatic torques due to the cur-\nrent, respectively. It is instructive to interpret \u0018Jas the\nproduct of the dimensionless factor ~\u0018\u0011\u0018=(~=2e) and the\nspin current density corresponding to the charge current\ndensity, Js\u0011(~=2e)J, wheree<0 is the electric charge\nof conducting electrons. Hereafter, the symbols with the\ntilde will denote the dimensionless quantities.\nWhen the inertia vanishes, \u001a= 0, the obtained\nequations of motion is reduced to the Landau-Lifshitz-\nGilbert equation for ferromagnets augmented by the\nspin-transfer torques [16, 17], in which s\u000b=sand ~\u0018can\nbe identi\fed as the Gilbert damping constant and the\nspin polarization rate of conducting electrons, respec-\ntively. When the net spin density vanishes, s= 0, it\ncorresponds to the equations of motion for antiferromag-\nnets [14]. The equations of motion for the dynamics of\na two-sublattice ferrimagnet in the absence of an electric\ncurrent and dissipation, s\u000b= 0 and J= 0, has been\nobtained by lvanov and Sukstanskii [18].\nThe low-energy dynamics of rigid magnetic solitons in\ntwo-dimensional collinear magnets can be derived from\nEq. (1) within the collective coordinate approach [11],\n⦿(a)(b)xyz⦿xyzs>0s<0Q=\u00001Q=1\nFQ=\u00001Q=1\nFFIG. 1. Schematic illustrations of a steady-state skyrmion\nmotion [Eq. (5)] in the presence of a current-induced force\nF=F^x. Four possible types are classi\fed by its skyrmion\ncharge Qand the sign of the net spin density s. See the main\ntext for the discussions.\nwhere the dynamics of the order parameter is encoded\nin the time evolution of the soliton position, n(r;t) =\nn0[r\u0000R(t)]. The resultant equations of motion for the\nposition of a circularly symmetric soliton, which are ob-\ntained by integrating Eq. (1) multiplied by n0\u0002@Rn0\nover the space, are our second main result:\nMR=Q_R\u0002B\u0000D_R+FU+FJ; (2)\nwhereM\u0011\u001aR\ndxdy (@xn0)2is the soliton mass [19],\nD\u0011s\u000bR\ndxdy (@xn0)2is the viscous coe\u000ecient, FU\u0011\n\u0000dU=dRis the internal force, ( FJ)i\u0011R\ndxdy [\u0018n\u0001(J\u0001\nr)n\u0002@in\u0000\u0010@in\u0001(J\u0001r)n] is the force due to the charge\ncurrent. The \frst term on the right-hand side is the ef-\nfective Lorentz force on the soliton, which is proportional\nto its topological charge\nQ=1\n4\u0019Z\ndxdyn0\u0001(@xn0\u0002@yn0); (3)\nwhich measures how many times the unit vector n0(r)\nwraps the unit sphere as rspatially varies [20], and the\n\fctitious magnetic \feld\nB\u0011B^z=\u00004\u0019s^z: (4)\nAccording to the equations of motion, a skyrmion in\nchiral ferrimagnets, which is characterized by its topolog-\nical chargeQ=\u00061, behaves as a massive charged particle\nin a magnetic \feld moving in a viscous medium. The \fc-\ntitious magnetic \feld is proportional to the net spin den-\nsitysalong the direction of the order parameter n, which\nleads us to consider collinear magnets with tunable sto\nlook for a possibly interesting dynamics of a skyrmion.\nThe RE-TM ferrimagnetic alloys [2] are such materials.\nFor example, Co 1\u0000xTbxhas been shown to exhibit the\nvanishing angular momentum s\u00190 atx\u001917% at room\ntemperature [10] by varying the chemical composition.\nAs another example, the angular momentum compensa-\ntion temperature of Gd 22%Fe75%Co3%has been reported\nasT\u0019220K [21].\nA skyrmion can be driven by an electric current as can\nbe seen in Eq. (2). In the presence of the corresponding3\n0 20 40 60 80 100-505\n-505255050⌧0 25 500.2.55.\n0.2.55.˜Vx\n05\u00005(a)\n(b)˜y024˜y\n00.1\u00000.1s/s↵\n00.1\u00000.1s/s↵\n0˜x204060800˜x0 20 40 60 80 100-4-2024\n-4-2024\n204060800˜Vx\n0 10 2008\n0881020˜t\u00002\u00004˜Y(0) = 2˜Y(0) =\u00003\nFIG. 2. Trajectories of skyrmions with the topological charge\nQ= 1 in the presence of a current-induced force F=F^x,\nwhich are obtained by numerically solving the dimension-\nless equations of motion for the dynamics of skyrmions in\nEq. (6). (a) Two trajectories for the monotonic net angular\nmomentum density s. The inset shows the convergence of the\nskyrmion velocities. (b) Multiple trajectories for the periodic\nnet angular momentum density s. See the main text for the\ndetailed discussions.\ncurrent-induced force FJ\u0011F^x, the direction of which is\nde\fned as the xaxis, the steady state of a skyrmion is\ngiven by\n_R!V=F\nB2+D2(D^x\u0000QB^y): (5)\nSee Fig. 1 for illustrations of a steady-state skyrmion\nmotion for F > 0. The skyrmion with the topological\nchargeQ= 1 moves down for s <0 and up for s >0,\nwhile moving to the right regardless of the sign of s. If\nthe ferrimagnet is prepared in such a way that s<0 for\ny > 0 ands > 0 fory < 0, the skyrmion with Q= 1\nwill move along the horizontal line y= 0 after certain\nrelaxation time because it is constantly pushed back to\nthe line via the e\u000bective Lorentz force. Note that the\nskyrmion experiences no Lorentz force on the angular\nmomentum compensation line, and thus will move as an\nantiferromagnetic skyrmion along it [22].\nTo corroborate the qualitative prediction, we numeri-\ncally solve the equations of motion [Eq. (2)] in its dimen-\nsionless form:\nId2~R\nd~t2+4\u0019sQ\ns\u000bd~R\nd~t\u0002^z+Id~R\nd~t=~F^x; (6)\nin which time, length, and energy are measured in units\nof the relaxation time \u001c\u0011\u001a=s\u000b, the characteristic length\nscale for the skyrmion size l[23], and\u000f\u0011s2\n\u000bl2=\u001a, respec-\ntively, where I=R\ndxdy (@xn0)2is a dimensionless num-ber determined by the skyrmion structure. Figure 2(a)\nshows the two trajectories of skyrmions of the charge Q=\n1 departing from ( ~X;~Y) = (0;2) and ( ~X;~Y) = (0;\u00003)\nwith the zero initial velocity under the following con\fg-\nurations:I=\u0019=2,~F= 4\u0019, ands=s\u000b=\u00000:1 tanh(~y).\nWe refer the paths as skyrmion snake trajectories due to\ntheir shapes, analogous to the electronic snake orbits in\nan inhomogeneous magnetic \feld [12]. The inset shows\nthat the skyrmion speed converges as ~Vy!~F=I after\nsu\u000eciently long time, ~t\u001d1. Figure 2(b) depicts multi-\nple trajectories of skyrmions when the net spin density\nis spatially periodic, s=s\u000b=\u00000:1 sin(2\u0019~y=5). Skyrmions\nare attracted to the angular momentum compensation\nlines and their velocities converge to the \fnite value. This\nleads us to state our third main result: self-focusing nar-\nrow guides for skyrmions can be realized in certain fer-\nrimagnets such as the RE-TM alloys along the lines of\nthe angular momentum compensation points, which can\nbe useful in using skyrmions for information processing\nby, e.g., providing multiple parallel skyrmion racetracks\nin one sample [24].\nThe dynamics of collinear magnets. |The derivation\nof the equations of motion for the dynamics of collinear\nmagnets in [Eq. (1)] is given below, which follows the\nphenomenological approach taken for antiferromagnets\nby Andreev and Marchenko [4]. Within the exchange\napproximation that the Lagrangian is assumed invariant\nunder the global spin rotations, we can write the La-\ngrangian density for the dynamics of the directional order\nparameter nin the absence of an external \feld as\nL=\u0000sa[n]\u0001_n+\u001a_n2\n2\u0000U[n]; (7)\nto the quadratic order in the time derivative, where a[n]\nis the vector potential for the magnetic monopole, rn\u0002\na=n[25]. The \frst term accounts for the spin Berry\nphase associated with the net spin density along n; The\nsecond term accounts for the inertia for the dynamics of\nn, which can arise due to, e.g., the relative canting of the\nsublattice spins [15].\nNext, the e\u000bects of an external \feld can be taken into\naccount as follows. The conserved Noether charge asso-\nciated with the symmetry of the Lagrangian under the\nglobal spin rotations is the net spin density, and it is\ngiven by s=sn+\u001an\u0002_n. The magnetization in the\npresence of an external \feld Hcan be then written as\nM=glsn+gt\u001an\u0002_n+\u001fH, whereglandgtare the\ngyromagnetic ratios for the longitudinal and transverse\ncomponents of the spin density with respect to the direc-\ntionn, respectively, and \u001fis the magnetic susceptibility\ntensor. The relation, M=@L=@H[4], requires the sus-\nceptibility to be \u001fij=\u001ag2\nt(1\u0000ninj), with which the\nLagrangian is extended to\nL=\u0000sa[n]\u0001_n+\u001a(_n\u0000gtn\u0002H)2\n2\u0000U[n];(8)4\nwhereU[n] includes the Zeeman term, \u0000glsn\u0001H. Finally,\nthe dissipation can be accounted for by the Rayleigh dis-\nsipation function, R=s\u000b_n2=2, which is the half of the\ndissipation rate of the energy density, P= 2R. The\nequations of motion obtained from the Lagrangian and\nthe Rayleigh dissipation function are given by Eq. (1)\nwithout the current-induced torques.\nCurrent-induced torques. |To derive the torque terms\ndue to an electric current, it is convenient to begin by\nphenomenologically constructing the expression for the\ncharge current density Jpumpinduced by the magnetic\ndynamics, and subsequently to invoke the Onsager's reci-\nprocity to obtain the torque terms as done for antiferro-\nmagnets in Ref. [14]. To the lowest order of the space-\ntime gradients and to the \frst order in the deviations\nfrom the equilibrium, we can write two pumping terms\nthat satisfy the appropriate spatial and spin-rotational\nsymmetries: _n\u0001@inandn\u0001(_n\u0002@in). The resultant\nexpression for the induced current density is given by\nJpump\ni=\u001b=\u0010_n\u0001@in+\u0018n\u0001(@in\u0002_n); (9)\nwhere\u001bis the conductivity.\nTo invoke the Onsager reciprocity that is formulated\nin the linear order in the time derivative of the dynamic\nvariables, we turn to the Hamiltonian formalism instead\nof the Lagrangian formalism. We shall restrict ourselves\nhere to the case of a vanishing external \feld for simplicity,\nbut it can be easily generalized to the case of a \fnite\nexternal \feld. The canonical conjugate momenta of n\nis given by p\u0011@L=@_n=\u001a(_n\u0000gtn\u0002h)\u0000sa. The\nHamiltonian density is then given by\nH[n;p] =p\u0001_n\u0000L=(p+sa)2\n2\u001a+U; (10)\nwhich resembles the Hamiltonian for a charged particle\nsubjected to an external magnetic \feld [26]. The Hamil-\nton equations are given by\n_n=@H\n@p\u0011\u0000hp; (11)\n_p=\u0000@H\n@n\u0000@R\n@_n\u0011hn\u0000s\u000b_n=hn+s\u000bhp;(12)\nwhere hpandhnare conjugate \felds to pandn, re-\nspectively. In terms of the conjugate \felds, the pumped\ncharge current is given by Jpump=\u0000\u0010@in\u0001hp\u0000\u0018(n\u0002\n@in)\u0001hp. By using the Onsager reciprocity and Ohm's\nlaw for the current J=\u001bE, we can obtain the torque\nterms in Eq. (1).\nDiscussion. |Let us discuss approximations that have\nbeen used in the Letter. First, we have developed the\ntheory for the dynamics of collinear magnets within the\nexchange approximation [4], in which the total energy\nis invariant under the simultaneous rotation of the con-\nstituent spins. The relativistic interactions including the\nmagnetic anisotropy, which weakly break the exchangesymmetry of the magnet, are added phenomenologically\nto the potential energy. Secondly, when studying the\ndynamics of skyrmions in inhomogeneous ferrimagnetic\n\flms, we have considered the nonuniform spin density s,\nwhile neglecting possible spatial variations of the other\nparameters such as the inertia \u001aor the damping s\u000bbe-\ncause we do not expect those variations to change the\nresults qualitatively. As long as skyrmions are attracted\nto the line of vanishing angular momentum due to the\ncombined e\u000bects of the e\u000bective Lorentz force, the vis-\ncous force, and the current-induced force, the line should\nbe able to convey skyrmions along with it.\nFerrimagnetic RE-TM alloys have not only the angular\nmomentum compensation point, which we have focused\non in this Letter, but also the magnetic moment compen-\nsation point. Motivated by the attraction of skyrmions\ntoward the angular momentum compensation lines that\nwe have discussed, it would be worth looking for an inter-\nesting phenomenon that can occur on the magnetic mo-\nment compensation line. For example, since the magnetic\nmoment governs the magnetostatic energy, there may be\nunusual magnetostatic spin-wave modes [27] localized at\nthe line. In addition, we have considered the dynamics\nof a soliton in two-dimensional ferrimagnets driven by an\nelectric current. In general, the dynamics of a soliton can\nbe induced by other stimuli such as an external magnetic\n\feld [28] and a spin-wave excitation [29{31], which may\nexhibit peculiar features of ferrimagnets that are absent\nin ferromagnets and antiferromagnets.\nThis work was supported by the Army Research Of-\n\fce under Contract No. W911NF-14-1-0016 (S.K.K. and\nY.T.) and by the National Research Foundation of Korea\n(NRF) grant funded by the Korea government (MSIP)\n(2015M3D1A1070465) (K.-J.L.).\n[1] R. K. Wangsness, Phys. Rev. 91, 1085 (1953); Phys. Rev.\n95, 339 (1954); Am. J. Phys. 24, 60 (1956).\n[2] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rep. Prog.\nPhys. 76, 026501 (2013), and references therein.\n[3] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004); T. Jungwirth, X. Marti, P. Wadley, and\nJ. Wunderlich, Nat. Nano. 11, 231 (2016).\n[4] A. F. Andreev and V. I. Marchenko, Sov. Phys. Usp. 23,\n21 (1980).\n[5] K.-J. Kim, S. K. Kim, T. Tono, S.-H. 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Therefore, under-\nstanding the dynamics of general solitons would require\nus to consider the mass Mas a parameter that can be\ndi\u000berent from the given expression.\n[20] A. A. Belavin and A. M. Polyakov, JETP Lett. 22, 245\n(1975).\n[21] C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsukamoto,\nA. Itoh, A. Kirilyuk, and T. Rasing, Phys. Rev. B 73,\n220402 (2006).\n[22] J. Barker and O. A. Tretiakov, Phys. Rev. Lett. 116,\n147203 (2016); X. Zhang, Y. Zhou, and M. Ezawa, Sci.\nRep.6, 24795 EP (2016).\n[23] For example, the energy density, U=A(rn)2=2\u0000\nKn2\nz=2 +Dn\u0001(r\u0002n), yields the characteristic length\nscale for the skyrmion radius, l=D=K [33].\n[24] R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Car-\npentieri, and G. Finocchio, Sci. Rep. 4, 6784 (2014).\n[25] B. Ivanov and A. Sukstanskii, Solid State Commun. 50,\n523 (1984); D. Loss, D. P. DiVincenzo, and G. Grin-\nstein, Phys. Rev. Lett. 69, 3232 (1992).\n[26] H. Goldstein, C. Poole, and J. Safko, Classical Mechan-\nics, 3rd ed. (Addison Wesley, 2002).\n[27] R. Damon and J. Eshbach, J. Phys. Chem. Solids 19,\n308 (1961); R. W. Damon and H. V. D. Vaart, J. Appl.\nPhys. 36, 3453 (1965).\n[28] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n[29] D. Hinzke and U. Nowak, Phys. Rev. Lett. 107, 027205\n(2011); P. Yan, X. S. Wang, and X. R. Wang, Phys.\nRev. Lett. 107, 177207 (2011); A. A. Kovalev and\nY. Tserkovnyak, Europhys. Lett. 97, 67002 (2012).\n[30] E. G. Tveten, A. Qaiumzadeh, and A. Brataas, Phys.\nRev. Lett. 112, 147204 (2014).\n[31] S. K. Kim, Y. Tserkovnyak, and O. Tchernyshyov, Phys.\nRev. B 90, 104406 (2014).\n[32] I. Makhfudz, B. Kr uger, and O. Tchernyshyov, Phys.Rev. Lett. 109, 217201 (2012).\n[33] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP\n68, 101 (1989).6\nSupplemental Material\nIn this supplemental material, we derive the equations\nof motion for a two-sublattice ferrimagnet by following\nthe approach taken in Ref. [14] with the explicit treat-\nment of two sublattices.\nThe model system is a two-dimensional collinear mag-\nnet that consists of two inequivalent sublattices. The\nlocal spin densities of the two sublattices are denoted by\ns1\u0011s1n1ands2\u0011s2n2, where n1andn2are slowly\nvarying unit vectors. We allow the two scalar spin densi-\nties,s1ands2, to be either positive and negative, which\nis useful to construct a general theory for collinear mag-\nnets as will be shown below. In equilibrium, the two spin\ndensities are collinear, which we represent by n1=n2.\nTo describe the dynamics of the magnet, it is convenient\nto use the new vectors, n\u0011(n1+n2)=2 and m\u0011n1\u0000n2,\ninstead of n1andn2, and the new scalars, s=s1+s2and\ns\u000e= (s1\u0000s2)=2, instead of s1ands2. Here, nserves as\nthe order parameter, which captures the collinear struc-\nture in equilibrium; mcorresponds to the relative cant-\ning of the two sublattices, which vanishes in equilibrium;\nsands\u000eare the net and the staggered spin densities\nin equilibrium, respectively. The cases where the two\nsublattices are coupled by a ferromagnetic exchange can\nbe represented by s1;s2>0, for which nis the direc-\ntion of the net spin density in equilibrium. The cases\nof an antiferromagnetic exchange can be represented by\ns1>0> s 2, for which nis the direction of the stag-\ngered spin density in equilibrium. From the de\fnitions,\nwe obtain n\u0001m= 0, and, for small deviations from the\nequilibrium, we can impose the constraints jnj= 1 and\njmj\u001c1 [14]. Without loss of generality, we can assume\ns\u000e\u00150. See Fig. S1 for illustrations of possible types of\ncollinear structures.\nLet us \frst derive the equations of motion in the ab-\nsence of an electric current within the Lagrangian for-\nmalism. The spin Berry phase contribution to the La-\ngrangian density, which governs the magnetic dynamics,\nis given by\nLB=\u0000s1a(n1)\u0001_n1\u0000s2a(n2)\u0001_n2; (S1)\nwhere ais a vector potential for magnetic monopoles,\nwhich satis\fes rn\u0002a(n) =n. By expanding the spin\nBerry phaseLBto the second order in mand _nas done\nin Ref. [31], we obtain\nLB=\u0000sa(n)\u0001_n+s\u000en\u0001(_n\u0002m)\u0000s_m\u0001(n\u0002m)=8:(S2)\nThe \frst term comes from the net spin Berry phase,\nwhich is in the Lagrangian for ferromagnets; The second\nterm comes from the cancelation of the spin Berry phases\nof the two sublattices, which is in the Lagrangian for an-\ntiferromagnets; the third term shall be ignored over the\n\frst term for the slow dynamics. The total Lagrangian\ndensity is given by L=LB\u0000U[n;m]. The dissipation\n(a)(b)(c)nms1s2n1n2nms2n1n2s1nms2n1n2s1nms2n1n2s1nms2n1n2s1\n(d)(e)s>s\u0000>0s>s\u0000=0s\u0000>s>0s\u0000>s=0s\u0000>0>sFIG. S1. Schematic illustrations of possible con\fgurations of\nthe spin densities s1\u0011s1n1ands2\u0011s2n2of the two sublat-\ntices in collinear magnets, which are classi\fed by the relative\nmagnitude and the sign of the net scalar spin density s=\ns1+s2and the staggered scalar spin density s\u000e= (s1\u0000s2)=2.\n(a) and (d) correspond to a one-sublattice ferromagnet and a\ntwo-sublattice antiferromagnet, respectively; (b) corresponds\nto a ferrimagnet, in which the two inequivalent sublattices are\ncoupled by a ferromagnetic exchange; (c) and (e) correspond\nto ferrimagnets, in which the two inequivalent sublattices are\ncoupled by an antiferromagnetic exchange.\ncan be accounted for by the Rayleigh dissipation func-\ntion,R=s\u000b_n2=2, which is the half of the dissipation\nrate of the energy density, P= 2R. Here, we consider\nthe dissipation associated with the dynamics of the order\nparameter n, while neglecting the contribution from the\ndynamics of massumingj_mj\u001cj _nj. The equations of\nmotion for the \felds nandmcan be obtained from the\nLagrangian and the Rayleigh dissipation function [26]:\ns\u000e_n=n\u0002fm; (S3)\ns\u000e_m=n\u0002(fn\u0000s\u000b_n)\u0000(s=s\u000e)n\u0002fm; (S4)\nwhere fn=\u0000\u000eU=\u000enandfm=\u0000\u000eU=\u000emare the e\u000bective\n\felds conjugate to nandm, respectively.\nBy using the Onsager reciprocity as done in the main\ntext, we can obtain the torque terms in the equations of\nmotion:\ns\u000e_n=n\u0002fm; (S5)\ns\u000e_m=n\u0002(fn\u0000s\u000b_n)\u0000(s=s\u000e)n\u0002fm (S6)\n+\u0010n\u0002(J\u0001r)n+\u0018(J\u0001r)n:\nWithin the exchange approximation that the energy is\ninvariant under the global spin rotations, the free energy\nexpanded to the second order in the gradients and the\nrelative canting mis given by U[n;m] =R\ndV[m2=2\u001f+\nA(@in\u0001@in)=2\u0000h\u0001n\u0000g\u0001m], where\u001frepresents the\nmagnetic susceptibility, Ais the sti\u000bness associated with\nthe spatial change of n,h= (M1+M2)H,g= (M1\u0000\nM2)H, and His a static external magnetic \feld. Here,\nM1=\r1s1andM2=\r2s2are the magnetizations of the\ntwo sublattices, where \r1and\r2are their gyromagnetic\nratios. Using fm=\u0000m=\u001f+n\u0002(g\u0002n), we can remove\nmfrom the equations of motion, which results in Eq. (1)\nwith\u001a=s2\n\u000e\u001f."    },    {        "title": "0808.3955v1.Magnetoelectric_coupling_in_the_cubic_ferrimagnet_Cu2OSeO3.pdf",        "content": " 1 Magnetoelectric coupling in the cubic ferrimagnet C u 2OSeO 3 \nJan-Willem G. Bos 1,*, Claire V. Colin 2 and Thomas T.M. Palstra 2 \nSchool of Chemistry and Centre for Science at Extre me Conditions, University of Edinburgh, \nEdinburgh, EH9 3JJ, United Kingdom. \nSolid State Chemistry Laboratory, Zernike Institute  for Advanced Materials, University of \nGroningen, 9747 AG Groningen, The Netherlands.  \n \nWe have investigated the magnetoelectric coupling i n the lone pair containing piezoelectric \nferrimagnet Cu 2OSeO 3. Significant magnetocapacitance develops in the ma gnetically ordered state \n(T C = 60 K). We find critical behavior near T C and a divergence near the metamagnetic transition at \n500 Oe. High-resolution X-ray and neutron powder di ffraction measurements show that Cu 2OSeO 3 \nis metrically cubic down to 10 K but that the ferri magnetic ordering reduces the symmetry to \nrhombohedral R3. The metric cubic lattice dimension s exclude a magnetoelectric coupling \nmechanism involving spontaneous lattice strain, and  this is unique among magnetoelectric and \nmultiferroic materials. \n \nIntroduction \nMagnetoelectrics are materials in which an applied electric field can induce a magnetization or \nconversely where the application of a magnetic fiel d leads to an induced electric polarization. 1, 2  The \nmagnetodielectric effect (changes in the dielectric  constant at the magnetic ordering temperature or \nin a magnetic field) is often large close to a ferr oic transition, which leads to large non-linear \nmagnetoelectric (ME) couplings. Magnetoelectrics ar e usually materials that are magnetically \nordered but not polar and show comparative modest l inear ME coupling. Magnetoelectrics attracted \nsignificant interest in the 1970s and more recently  with the enormous attention for multiferroic \nmaterials. 1-4 The earlier studies include work on BaMnF 4,5-8 Cr 2BeO 4,9 and Gd 2(MoO 4)3,10  while  2 recently studied materials include SeCuO 3,11  BiMnO 3,12  EuTiO 3,13  and CoCr 2O4.14, 15  Among these, \nferromagnetic materials are of special interest as a large spontaneous magnetization M is considered \nto be favorable for large ME effects. 16  In spite of the significant theoretical and experi mental \ninterest there is no generic model to describe the observed dependence of the dielectric constant on \nspin structure and applied magnetic fields. 17  In modern multiferroics, two mechanisms describing  \nthe coupling between electric polarization and magn etic order have come to the fore: the spin-\ncurrent model for spiral magnets and the exchange s triction model for the RMn 2O5 phases. 4 \nHowever, neither of these models makes quantitative  predictions about the dielectric response. \nOther microscopic mechanisms include the coupling b etween long wavelength polar phonon modes \nand spin structure, as proposed for BaMnO 4,7 and single ion effects. 18  Recent work on multiferroic \nmaterials has shown that anomalies in the dielectri c constant that occur at the onset or with changes \nin the magnetic order are generally also associated  with spontaneous lattice distortions. 19-23  This \nstrongly suggests that the ME coupling proceeds via  the lattice (atomic displacements). \nHere, we present the results of our investigation i nto Cu 2OSeO 3, which shows significant ME \ncoupling but does not have a spontaneous lattice di stortion below T c. Cu 2OSeO 3 is ferrimagnetic \nwith a Curie temperature of 60 K and has a saturati on magnetization of 0.50 µB/Cu. 24  At room \ntemperature, it has the cubic space group P2 13, which allows for piezoelectricity but not for a \nspontaneous polarization.25, 26  Magnetic susceptibility measurements reveal a meta magnetic \ntransition around 500 Oe between two ferrimagnetic states with different saturation magnetizations. \nRietveld analysis of neutron powder diffraction dat a shows that the H = 0 magnetic structure is \ncollinear ferrimagnetic with magnetic space group R 3. This reduction in symmetry is required \nbecause ferrimagnetism is not symmetry allowed for cubic crystal structures.27  However, high \nresolution synchrotron X-ray powder diffraction sho ws that there is no observable rhombohedral \ndistortion below T c. The crystal structure of Cu 2OSeO 3 therefore remains metrically cubic but the \nmagnetic ordering lowers the crystal and magnetic s ymmetry to R3. In contrast to most studied \nmagnetoelectric materials, the dielectric constant of Cu 2OSeO 3 is enhanced directly below the  3 magnetic ordering transition, and shows positive ma gnetocapacitance (MC) near T C. At lower \ntemperatures the dielectric constant starts to decr ease. Below T C a negative MC is observed, upon \nwhich the positive MC originating from the metamagn etic transition is superimposed. \nTo the best of our knowledge this is the first demo nstration of ME coupling in a system where \nspontaneous lattice strain can be excluded, and as such is relevant to understand the microscopic \ncoupling mechanisms in magnetoelectric and multifer roic materials. We use spontaneous strain to \ndistinguish from the situation where strain is indu ced by an applied magnetic or electric field or by \nmagnetic ordering. Induced strain participates in t he ME coupling as both piezoelectric and \npiezomagnetic coupling are symmetry allowed, and ma y well be responsible for the observed \nmagnetodielectric effects. \n \nExperimental \nOlive green polycrystalline samples of Cu 2OSeO 3 were prepared by standard solid state chemistry \nmethods. CuO (99.999%) and SeO 2 (99.999%) were thoroughly mixed in a 2:1 ratio usi ng mortar \nand pestle, and pressed into a pellet. The pellet w as sealed in an evacuated quartz tube and heated to  \n600 °C over the duration of a day. After heating for 12 hours at 600 °C the sample was quenched, \nhomogenized using mortar and pestle, pressed into a  pellet and heated for 3 days at 600 °C with one \nmore intermediate homogenization. Phase purity was confirmed by powder X-ray diffraction \n(PXRD). Variable temperature PXRD data (10 ≤ T ≤ 300 K) were collected using a Huber \ndiffractometer with Mo K α radiation. A 10 K dataset suitable for a full stru cture solution was \ncollected on the ID31 diffractometer at the ESRF in  Grenoble, France. Data were collected between \n3 ≤ 2 θ ≤ 50 ° and binned with a step size of 0.003 °. No impurity phases were observed. The \nwavelength was 0.45620 Å. Neutron powder diffractio n experiments were performed on the D20 \ninstrument at the Institute Laue Langevin in Grenob le, France. Datasets were collected at 10 and 70 \nK using the low resolution high flux mode with a mo nochromator take-off angle of 44 °. The  4 neutron wavelength was 2.42 Å. Data were collected in the 10 ≤ 2 θ ≤ 150 ° range in 0.1 ° increments. \nThe GSAS suite of programs was used for Rietveld fi tting of the powder diffraction data. Zero Field \nCooled (ZFC) and Field Cooled (FC) magnetic suscept ibilities in applied fields of 10 and 250 Oe \nwere collected using a Quantum Design Magnetic Prop erty Measurement System (MPMS). The \nfield dependence of the magnetization was measured using a Quantum Design Physical Property \nMeasurement System (PPMS) fitted with ACMS insert. Additional low field M(H) data (inset in \nFig. 4b) at 5 K field were collected using the MPMS . The capacitance was measured in a \ncommercial system (Quantum Design PPMS) using a hom e-made insert and an Andeen-Hagerling \n2500A capacitance bridge operating at a fixed measu rement frequency of 1kHz. Electrical contacts \nwere painted using Ag-epoxy on a pressed pellet wit h capacitor geometry, typically 1*7*7 mm 3. \n \nResults \nStructure : The crystal structure of Cu 2OSeO 3 is depicted in Fig. 1a-b and is characterized by \ntrigonal bi-pyramidal CuO 5, square pyramidal CuO 5 and tetrahedral SeO 3-lp (lp = lone pair) \ncoordination polyhedra. The CuO 5 polyhedra share edges and corners while the SeO 3-lp polyhedra \nshare corners with the CuO 5 polyhedra. The connectivity of the Cu ions is show n in Fig. 1b as are \nthe local coordination environments. The Cu 2+  ions form a network of distorted tetrahedra whose \ncorners are connected via linear Cu-Cu bridges. The  solid lines indicate edge sharing CuO 5 and the \nopen lines indicate corner sharing CuO 5 polyhedra. The Cu-Cu distances for edge-sharing Cu O 5 are \n0.18 to 0.25 Å shorter than the ones for corner sha ring. The Cu coordination polyhedrons deviate \nsignificantly from ideal square pyramidal and trigo nal bi-pyramidal, respectively (Fig. 1b). Bond \nvalence sum calculations confirm the +2 oxidation s tate for the copper ions [BVS(Cu1)=2.06(2), \nBVS(Cu2)=2.02(2)]. 28   \nThe temperature evolution of the lattice parameter (10 ≤ T ≤ 300 K) and the 10 K crystal structure \nwere studied by PXRD (Fig. 2). All patterns were fi tted using the space group P2 13. 25  No structural  5 phase transitions were observed. The temperature de pendence of the lattice constant is shown in \nFig. 3. The data have been scaled using the ID31 da taset at 10 K. The 300 K cell constant ( a = \n8.9235(2) Å) is in good agreement with the literatu re value ( a = 8.925(1) Å).25  The solid line is a fit \nto a(T) = a 0 + Acoth( θ/T), which is an approximation to the bare thermal expansion, due to thermal \nvibrations, of a solid as derived in Ref. 29 . (θ equals half the Einstein temperature). Deviations from \nthis temperature dependence signal the occurrence o f anomalous lattice strain. No deviations were \nobserved and Cu 2OSeO 3 has a conventional thermal expansion due to lattic e vibrations. A \ncomparison of bond lengths and angles at 300 and 10  K does not reveal any significant changes (the \n10 K crystallographic coordinates are given in Tabl e 1), and further confirms the absence of any \nmagneto-structural coupling or polar structural dis tortions. \nMagnetism : The temperature dependences of the ZFC and FC mag netic susceptibilities, and the \ninverse ZFC susceptibility, collected in H = 250 Oe , are shown in Fig. 4a. The susceptibility \ndiverges just below 60 K. Above 100 K, the suscepti bility follows the Curie-Weiss law, and a fit \n(solid line) gives a Curie constant of 0.23(1) emu mol Cu -1 Oe -1 K -1 and Weiss temperature of \n+69(2) K. The positive Weiss temperature indicates the presence of dominant ferromagnetic \ninteractions, in agreement with the observed diverg ence of the susceptibility. The experimental \neffective moment (1.36 µB/Cu) is lower than the expected spin-only value for  S = ½ of Cu 2+  (1.73 \nµB). This reduction is not unusual for Cu 2+  in metal oxides, e.g. in CuO and La 2CuO 4 the moment is \nreduced to ~50-70% of the spin only value.30, 31  The field dependence of the magnetization is shown  \nin Fig. 4b. The magnetization saturates in small ap plied fields and has a saturation value of 0.50 \nµB/Cu at 5 K. This is exactly half the expected satur ation moment for a S = ½ ferromagnet, and \nsuggests a simple collinear ferrimagnetic alignment  with 3 majority and 1 minority spins. A change \nof slope can be noticed around 500 Oe (Fig. 4b). Th is signals the presence of a metamagnetic \ntransition with a small amount of magnetic hysteres is (insets to Fig. 4b). Extrapolating the low field  \nmagnetization suggests a saturation moment around 0 .25(5) µB/Cu at 2 kOe. The inset to Fig. 4a  6 shows the low temperature ZFC and FC curves in 10 O e, confirming the ferrimagnetic state even in \nsmall applied fields. \nNeutron powder diffraction : Long range magnetic order was confirmed by the ob servation of \nmagnetic intensities on the (110) and (201) reflect ions in the 10 K diffraction pattern (inset to Fig \n5a). The magnetic cell is identical to the crystall ographic one, and magnetic symmetry was used to \nconstruct possible magnetic models. The only possib le magnetic space group (MSG) based on the \ncrystal structure is P2 13. This MSG, however, does not allow for ferromagne tic or ferrimagnetic \nmagnetic ordering. In fact, no cubic MSG allows for  ferromagnetic ordering, and a symmetry \nlowering is therefore required.27  Possible crystallographic subgroups are R3 and P2 12121, and \nferrimagnetic structures are possible in MSGs R3 (m  // 3) and P2 121’2 1’ (m // 2 1). Rietveld \nrefinement revealed that models with anti-parallel sublattices based on the Cu1 and Cu2 site from \nthe P2 13 structure gave the best fits. This corresponds to  a magnetic structure with 12 majority and 4 \nminority spins. The R3 solution is shown in Fig. 5b . The Cu moments refine to m x=m y=m z=0.35(3) \nµB for Cu1 and m x=m y=m z= -0.35(2) for Cu2, yielding a moment of 0.61(5) µB per copper \n(wR p=1.5%, R F2= 8.86). The reduced ordered moment (the expected s pin-only value is 1 µB) is \ncommon in copper oxides, e.g. CuO has m=0.68 µB/Cu. 30  For the P2 121’2 1’ model m=m x=0.61(5) \nµB,  and an identical goodness of fit was obtained. Thi s ferrimagnetic arrangement corresponds to a \nsaturation moment of 0.3 µB/Cu in good agreement with the extrapolated low fie ld magnetization \n(Fig. 4). A comparison of the magnetic structure (F ig. 5b) and the crystal structure (Fig. 1b) is of \nsome interest. The Kanamori-Goodenough rules predic t ferromagnetic exchange interactions for \nedge-sharing CuO 5 polyhedra (solid lines) and antiferromagnetic exch ange for corner-sharing (open \nlines). The experimental magnetic structure is larg ely consistent with this. All exchange interactions  \nare satisfied within the Cu4 tetrahedra but the cou pling between tetrahedra is not as expected based \non the Kanamori-Goodenough rules.  7 Dielectric constant : The temperature dependence of the dielectric cons tant is shown in Fig. 6a. \nImmediately below 60 K, the dielectric constant is enhanced at the emergence of long range order. \nThe enhancement is noteworthy since a reduction in dielectric constant is more common, \nirrespective of the type of magnetic order. For exa mple, BiMnO 3 (FM) SeCuO 3 (FM) and YMnO 3 \n(AFM) all show a reduction below the magnetic order ing transition. 11, 12, 32  Upon further cooling the \ndielectric constant decreases, and below 20 K is lo wer than the extrapolated lattice contribution (see  \nbelow). This could reflect the complex temperature evolution of the magnetic order parameter in \nzero applied field. The lattice contribution to the  dielectric constant was fitted (100 ≤ T ≤ 200 K) \nusing the expression for the lattice thermal expans ion. The Einstein temperature was fixed at 316 K, \nand the fit results are given in Fig. 6a. Subtracti on of the lattice contribution allows for the analy sis \nof the critical behavior in the vicinity of the mag netic ordering transition. The dielectric constant \n(minus the lattice contribution) can be fitted with : 32  \n   αε ε−− = −11)/( )( )(c c s s TTA T T      (1) \nFor T c = 59.5 K, the fit shown in Fig. 6b leads to equal slopes before and after the transition. The \ncritical exponent α can be estimated from the slope (1- α) and α ≈ 0.3. The critical exponent for \nAFM FE YMnO 3 is 0.25. 32  \nThe magnetocapacitance: ( ) ) 0 ( /) 0 ( )( C C HC MC − =  was measured for temperatures between 5 \nand 60 K and for fields between 0 and 8 Tesla (Fig 7a-b). At low temperatures and fields, the \nmagnetocapacitance is dominated by a peak at the me tamagnetic transition. The peak position is \nhysteretic revealing that the metamagnetic transiti on is of first order (inset to Fig. 7b). At higher \nfields the magnetocapacitance decreases gradually w ith field. In contrast, in the critical region, the  \nmagnetocapacitance is smooth, positive and has a co nvex curvature. This curvature is also observed \nfor ferromagnetic SeCuO 3 and BiMnO 3 but in those cases the magnetocapacitance is negat ive.11, 12  \nAntiferromagnetic materials, such as YMnO 3 and TeCuO 3, in contrast, have concave negative  8 magnetocapacitances.11, 33  The magnetocapacitance can be described in a pheno menological manner \nusing: \n   α\ncHHA MC MC − + =0       (2) \nat low temperatures, where α≈-0.49 and H c is the field of the metamagnetic transition, and  \n   βBH MC MC + =0        (3) \nnear the magnetic transition. In the intermediate r egime these two functions can be added to fit the \nmagnetocapacitance, and A and B are a measure of th e weights of the long range ordered magnetic \nand critical contribution, respectively. The temper ature dependences of MC 0, A, B and β are given \nin figure 7c. The critical contribution can be seen  to peak in the vicinity of the magnetic transition , \nwhile the long range magnetic contribution is most significant at lower temperatures, and vanishes \nat T c. The exponent β varies between ~0.3 near T C and approaches ~1 at low temperatures. The field \nindependent term (MC 0) gradually decreases with temperature, and has a l ocal maximum just above \nthe magnetic transition. \n \nDiscussion \nThe coupling between magnetic and polar order param eters in multiferroic materials attracts much \ninterest but little is known about the microscopic origin. From Landau theory, the dominant \nsymmetry unrestricted coupling terms are non-linear  terms in the free energy such as M 2P2 or L 2P2, \nwith L the antiferromagnetic sublattice magnetizati on, M the magnetization and P the electric \npolarization. Multiferroics are expected to show st rong coupling as the non-linear terms are large in \nthe vicinity of a phase transition. However, no exp erimental realization of a material with large P \nand M is known. Materials like BiMnO 3, SeCuO 3 and Cu 2OSeO 3 with large M but no spontaneous \nelectric polarization show nevertheless significant  magnetoelectric coupling. This suggests that the \nlarge ordered magnetic moment is important for the coupling. Here, the coupling may operate via an  9 induced polarization by the applied electric field used in the measurements. Alternatively, it has als o \nbeen proposed that the magnetoelectric coupling pro ceeds via lattice strain. \nCu 2OSeO 3 is an interesting model system. This is mainly due  to the fact that our structural studies \nshow no measurable structural distortion occurs dow n 10 K. This excludes the possibility that the \nmagnetoelectric coupling proceeds via a spontaneous  lattice distortion below the magnetic ordering \ntemperature. The dielectric constant initially incr eases below the magnetic ordering temperature. \nThis is followed by a decrease and below about 20 K  the dielectric constant is smaller than the \nextrapolated lattice contribution. This unusual tem perature dependence is probably related to the \ndifferent temperature dependence of the ferro- and antiferromagnetic order parameters, M(T) and \nL(T) respectively. Critical behavior similar to tha t observed for YMnO 3 is observed near the \ntransition. 32  The initial increase in dielectric constant and po sitive magnetocapacitance near the \nmagnetic transition are unusual. Most studied ferro magnetic and antiferromagnetic materials have \nnegative magnetocapacitance and show a decrease in dielectric constant. Magnetization and \nmagnetocapacitance measurements reveal a metamagnet ic transition around 500 Oe. The \nmetamagnetic transition shows up as a large positiv e peak in the magnetocapacitance \nmeasurements, which becomes stronger at lower tempe ratures, and is superposed on a decreasing \ncapacitance. The high field magnetic state is consi stent with a simple collinear ferrimagnetic \narrangement with 3 majority and 1 minority S=1/2 sp in, leading to a saturation moment of 0.5 \nµB/Cu. The magnetic structure in zero magnetic field was determined from neutron powder \ndiffraction and is also collinear ferrimagnetic but  with a reduced copper moment (0.61(5) µB). \nFerrimagnetic structures are incompatible with cubi c symmetry, and the crystal and magnetic \nstructure below T c are therefore described in R3. \nThe absence of lattice strain indicates that linear  magnetoelectric coupling effects may be important \nas expected for non-multiferroic materials. The mag netic space group R3 allows for piezoelectric \ncoupling, and for both a linear magnetoelectric eff ect and coupling via a piezomagnetic effect. Thus,  10  Cu 2OSeO 3 is a unique example of a metrically cubic material that allows linear magnetoelectric \ncoupling as well as piezoelectric and piezomagnetic  coupling. Further measurements are needed to \nfind out which linear coupling terms of the magneto electric tensor dominate.  11  Acknowlegdements \nJWGB acknowledges the Royal Society of Edinburgh fo r financial support, and EPSRC for \nprovision of the beam time at the ESRF and ILL. Dr.  Andy Fitch, Dr. Paul Henry and Dr. Simon \nKimber are acknowledged for help with data collecti on. CC acknowledges the EU STREP project \nCOMEPHS under contract No. NMPT4-CT-2005-517039. TP  acknowledges stimulating \ndiscussions with Umut Adem, Beatriz Noheda, and Max im Mostovoy.  12  Figure Captions  \nFig. 1 (color online) (a) Polyhedral representation  of the crystal structure of Cu 2OSeO 3 with SeO 3lp \npolyhedrons omitted. (b) Connectivity of the Cu ion s and local coordination environments of Cu1 \nand Cu2. The labeling of atoms is consistent with t hat used in Table 1. \n \nFig. 2 (color online) Observed (crosses), calculate d (solid line) and difference PXRD profiles for \nCu 2OSeO 3 at 160 K (laboratory data) and at 10 K (synchrotro n data). The markers correspond to the \nBragg positions of Cu 2OSeO 3. \n \nFig. 3 Temperature evolution of the cubic lattice c onstant. \n \nFig. 4 (a) The temperature dependences of the ZFC a nd FC magnetic susceptibilities in an applied \nfield of 250 Oe. The Curie-Weiss fit to the inverse  ZFC susceptibility is shown. The inset shows the \nZFC and FC susceptibilities measured in 10 Oe. (b) The field dependence of the magnetization at 5, \n25, 50 and 75 K. The insets illustrate the high fie ld behavior, the metamagnetic transition and the \nassociated small magnetic hysteresis. \n \nFig. 5 (color online) (a) Observed (crosses), calcu lated (full line) and difference neutron powder \ndiffraction Rietveld profiles for Cu 2OSeO 3 at 10 K. The inset shows the 10 K dataset fitted w ith the \nstructural model obtained at 70 K. The magnetic ref lections are indexed. The markers correspond to \nthe Bragg positions. (b) Representation of the R3 m agnetic structure. White circles correspond to \nCu1 sites while blue ones correspond to Cu2. \n \nFig. 6 (a) The temperature dependence of the dielec tric constant. The solid line is a fit. (b) critica l \nbehavior of the magnetic contribution to the dielec tric constant. \n  13  Fig. 7 (color online) (a) Magnetocapacitance in low  applied fields illustrating the anomaly that \noccurs at the metamagnetic transition. The inset sh ows the magnetic H-T phase diagram. (b) Fits to \nthe field dependence of the magnetocapacitance (see  text) at temperatures below (T = 25 K) and at \nthe magnetic transition (T = 60 K). (c) Temperature  evolution of the fitting constants MC 0, A, B and \nβ (see text).  14  Table 1. Atomic parameters for Cu 2OSeO 3 at 10 K obtained from Rietveld fitting of the ID31  data. \nGoodness of fit statistics: χ2 = 6.6, wR p = 10.6%, R p = 6.8%, R F2 = 2.4%. Space group P2 13, a = \n8.91113(1) Å. \n  x y z Uiso  (Å2) \nCu1 4 a 0.88557(6) = x =x 0.00105(5) \nCu2 12 b 0.13479(6) 0.12096(6) -0.12733(6) 0.00105(5) \nO1 4 a 0.0103(3) = x = x 0.0022(4) \nO2 4 a 0.7619(4) = x = x 0.0022(4) \nSe1 4 a 0.45993(5) = x = x 0.00075(5) \nSe2 4 a 0.21223(5) = x = x 0.00075(5) \nO3 12 b 0.2306(3) 0.5159(3) -0.0301(3) 0.0034(4) \nO4 12 b 0.2731(3) 0.1872(3) 0.0331(3) 0.0034(4) \n  15  Fig. 1a \n \nFig. 1b \n \n  16  Fig. 2 \n2 4 6 80100 200 300 2 4 6 80200 400 600 800 \n9 10 05\n  \nQ (Å -1 )Intensity (10 3 counts) \n  \nID31, T = 10 K   Intensity (counts) \n  \nCu 2OSeO 3\nT = 160 K \nHuber \n  \nQ (Å -1 )\n \n  17  Fig. 3. \n0 50 100 150 200 250 300 8.910 8.915 8.920 8.925 \n lab data \n ID31 data Cu 2OSeO 3\na(T) = a 0 + Acoth( θ/T) \na0 = 8.8997(9) Å \nA = 0.011(1) \nθ = 158(8) K \n  \nT (K) a-axis (Å)  \n  18  Fig. 4a \n0 25 50 0.0 0.5 1.0 1.5 2.0 \n0 50 100 150 200 250 300 012\n ZFC H=250 Oe \n FC   H=250 Oe \nT (K) M/H (emu/mol Cu  Oe) Cu 2OSeO 3\n01000 (M/H) -1  (emu/mol Cu  Oe) -1  ZFC H=10 Oe \n FC   H=10 Oe   \nT (K) M/H \n  \n \nFig. 4b \n-2000 -1000 0 1000 2000 -0.4 -0.2 0.0 0.2 0.4 \n-1000 0 1000 -50 0 50 -0.5 0.0 0.5 \n0 250 500 750 0.0 0.2 \n  \nH (Oe) M ( µB/Cu) \nCu 2OSeO 35 K \n25 K \n50 K \n75 K \n  dM/dH 5 K \n50 K  \n  \nH (kOe)  \n MH (Oe) T = 5 K \n  19  Fig 5a \n0.5 1.0 1.5 2.0 2.5 0510 \n1.0 1.5 012Intensity (10 3 counts) \nQ (Å -1 )Cu 2OSeO 3\n10 K \n  \nQ (Å -1 )Int. (10 3 counts) \n  \n(110) (201) \n \nFig. 5b \n \n  20  Fig. 6a \n0 20 40 60 80 100 120 14.28 14.30 14.32 14.34 14.36  \nT=60K \n1kHz \n Dielectric constant εr\nT (K) Cu 2OSeO 3ε(T)= ε0 + Acoth( θ/T) \nε0 = 13.91(1) \nA = 0.38(1) \nθ = 158 K \n \nFig. 6b \n0.01 0.1 1E-3 0.01  abs( εs(T)- εs(T c)) \n  \nabs(T/T c-1) Cu 2OSeO 3\nTc = 59.5 K \n  21  Fig. 7a \n0 1000 2000 3000 4000 -0.04 0.00 0.04 \n0 20 40 60 0250 500 750 \n magnetocapacitance (%) \nH (Oe) Cu 2OSeO 35 K \n25 K 42 K 46 K 50 K 54 K 56 K 58 K 60 K T (K) H (Oe) \n  \n H up \n H down \n \nFig. 7b \n0 20 40 60 80 -0.04 0.00 0.04 0.08 \n0 20 40 60 80 0.00 0.02 0.04 0.06  \n \n H (kOe) magnetocapacitance (%) Cu 2OSeO 3T = 25 K \n  MC (%) \nH (kOe) T = 60 K \n \nFig. 7c \n40 45 50 55 60 65 70 75 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 \n0.00 0.01 0.02 0.03 0.04 \n012345\n0.00 0.25 0.50 0.75 1.00 A\nT (K) Cu 2OSeO 3MC 0Bβ A MC 0\n10 -4  B \n \nβ\n \n  22  References \n1 W. Eerenstein, N. D. Mathur, and J. F. 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Ono8\n1Advanced Institute for Materials Research,\nTohoku University, Sendai 980-8577, Japan\n2Center for Spintronics Research Network,\nTohoku University, Sendai 980-8577, Japan\n3Center for Science and Innovation in Spintronics,\nTohoku University, Sendai 980-8577, Japan\n4Department of Applied Physics, Graduate School of Engineer ing,\nTohoku University, Sendai 980-8579, Japan\n5Department of Materials Science and Engineering,\nKorea University, Seoul 02841, Korea\n6College of Science and Technology,\nNihon University, Funabashi, Chiba 274-8501, Japan\n7KU-KIST Graduate School of Converging Science and Technolo gy,\nKorea University, Seoul 02841, Korea\n8Institute for Chemical Research, Kyoto University,\nGokasho, Uji, Kyoto 611-0011, Japan\n(Dated: August 20, 2018)\n1Abstract\nThe magnetization dynamics for ferrimagnets at the angular momentum compensation tempera-\ntureTAis believed to be analogous to that for antiferromagnets. We investigated the pulsed-laser-\ninduced magnetization dynamics in amorphous rare-earth tr ansition-metal ferrimagnet films with\naTAjust above room temperature. For a low pulse fluence, the magn etization precession frequency\ndecreases as the applied magnetic field increases, whereas for a higher pulse fluence, it increases\nas the applied field increases. The result was well explained by the left-handed and right-handed\nprecession modes of the antiferromagnetic-like resonance at temperatures below and above TA,\nrespectively, and the data were in agreement with the theore tical simulation. The study demon-\nstrated the experimental route to achieving antiferromagn etic resonance in ferrimagnets using a\npulsed laser.\n2The fundamental research on antiferromagnets started with th e classic work of N´ eel,\nand the spin dynamics in antiferromagnets has been extensively stu died in the past [1],\ncontributing to the development of the standard theory of antife rromagnetic resonances [2].\nMost of the antiferromagnetic resonances have been observed f or materials with a lower\nN´ eel temperature using the microwave or infrared technique [1]. T here has been renewed\ninterest in the utilization of antiferromagnets in spintronic devices b eyond the one based on\nferromagnets [3]. Recent advances in ultrashort pulse laser and TH z wave technology have\nenabled further exploration of the antiferromagnetic resonance for various antiferromagnets\n[4–6], which is currently still in the early stage of experimental resea rch.\nHerein, we focus on rare-earth (RE) transition-metal (TM) amor phousferrimagnets .\nAlloys films such as GdFeCo have recently been considered as a proto type of ferrimagnets\nwith a perpendicular magnetic easy axis. They serve as good playgro unds for exploiting\nthe fundamental ultrafast physics [7] as well as spintronic devices [8, 9]. The RE and\nTM magnetic moments can be considered as two sublattice magnetic m oments coupled\nantiferromagnetically, leading to a net magnetization tuneable by th e composition ratio\nof RE to TM elements. The alloys generally have two characteristic te mperatures below\nthe Curie temperature: the magnetization compensation tempera ture,TM, at which the\ntwo sublattice magnetic moments are canceled, and the angular momentum compensation\ntemperature, TA, at which the two sublattice angular momenta are canceled. The existence\nofTAin the alloys provides a route for exploring the antiferromagnetic-lik e spin dynamics,\neven though the alloys are not true antiferromagnets, as discuss ed by Kim et al.in terms of\nthe domain wall dynamics [9]. This means that the antiferromagnetic- like resonance should\nalso be observed in these ferrimagnetic alloys at temperatures TnearTA.\nThe ferromagnetic resonance (FMR) and the exchange modes at Tbelow or around TM\nhave been well discussed in relation to such amorphous RE-TM ferrim agnets and crystalline\nferrimagnetic oxides using all-optical pulse laser methods [10–13]. On the other hand, the\nantiferromagnetic-like resonance at TaroundTAis essentially different from those dynamics\nand has not been observed in these alloys. In this Letter, we repor t the observation of\nantiferromagnetic-like resonance in amorphous GdFeCo ferrimagn ets atTnearTA. The\nobserved behaviors are consistent with the simple physical picture s described herein and\nseveral numerical simulations.\nThe resonance dynamics in ferrimagnetic films with perpendicular mag netic anisotropy\n3/g2033/g2878/g2033/g2879/g2033(a)\n(c) (d)\n/g513/g2033/g2878/g513\n/g2033/g2878/g2033/g2879/g2033\n/g513/g2033/g2878/g513MCoFe MCoFe\nMGd MGdH H (b)\n/g2033/g2878/g2033/g2879\n0 0H H\nFIG. 1. An illustration of the antiferromagnetic-like reso nance for ferrimagnets at a temperature\nTnear the angular momentum compensation temperature TAwhen the applied magnetic field H\nis parallel to the magnetic easy axis, i.e., the film normal in the present study. (a) The left-handed\nmode with the angular frequency ω+and (b) the right-handed mode with ω−.MCoFeandMGd\nare the sublattice magnetization vectors for CoFe and Gd, re spectively. The schematic illustration\nof these mode frequencies vs.the external magnetic field HatTjust below (b) and just above TA\n(c). Dashed lines denote the absolute values of ω+vs.H.\n(PMA) are discussed based on the coupled Landau-Lifshitz equatio ns for the magnetization\nvectors of the sublattices M1(2)[2, 14]. The linearized versions of these equations yield the\nangular frequency ω±for the two modes under an external magnetic field Happlied parallel\nto the film normal and parallel (antiparallel) to M1(2)(1: CoFe and 2: Gd) under thermal\nequilibrium:\nω±=∓µ0/bracketleftBig\nγHeff2+2γHeff·γHex+[δ(γHex)]2/bracketrightBig1\n2(1)\n−µ0[γH+δ(γHk)+δ(γHex)].\nHere,γHeff= [γ1(Hk1+H) +γ2(Hk2−H)]/2,γHex= (γ1Hex1+γ2Hex2)/2,δ(γHex) =\n(γ1Hex1−γ2Hex2)/2,δ(γHk) = (γ1Hk1−γ2Hk2)/2, andγ= (γ1+γ2)/2.γ1(2),Hk1(2), and\nHex1(2)are the absolute values for the gyromagnetic ratio, the effective P MA field, and the\neffective magnetic field of the antiferromagnetic exchange coupling for the magnetization\nM1(2)of sublattice 1(2), respectively. µ0is the permeability in a vacuum. We simplify\nEq. (1) to capture the underlying physics, assuming that Hk1(2)>> Hand that δ(γHk) is\n4negligible. δ(γHex) can be rewritten with the mean field coefficient λ(>0) as\nδ(γHex) =γ1γ2λ(S2−S1)/2, (2)\nso that it is determined by the difference in the angular momentum den sityS1(2)(≡\nM1(2)/γ1(2)) for sublattice 1(2). Since δ(γHex) may be small at TnearTA, Eq. (1) may be\ncrudely approximated as follows:\nω±≈ ∓µ0/bracketleftbig\nγHk(γHk+2γHex)/bracketrightbig1\n2(3)\n−µ0[γH+δ(γHex)],\nwithγHk= (γ1Hk1+γ2Hk2)/2. Equation (3) is a counterpart of the well-known relation of\nthe antiferromagnetic resonance mode in pure antiferromagnets [2]. The ω+andω−modes\nrepresent the left-handed and right-handed precession modes, as schematically shown in\nFigs. 1(a) and 1(b), respectively. The absolute value of ω+(−)increases (decreases) as the\nmagnetic field increases, which results from the opposite gyration m otion being similar to\nthe pure antiferromagnetic resonance. Different from the pure a ntiferromagnetic resonance,\nthe correspondence of the respective high and low frequency mod es either to the ω+andω−\nmodes or to the ω−andω+modes varies at Tsmaller or larger than TA, as schematically\nshown in Figs. 1(c) and 1(d). This phenomenon occurs because δ(γHex) in Eq. (3) behaves\nas a negative or positive offset at Tsmaller or larger than TA. This change in the attribution\nmaybetheunique characteristic oftheantiferromagnetic-likeres onance forferrimagnetsand\nshould be experimentally examined.\nThe sample studied is the 30-nm-thick amorphous thin films of Gd 23Fe67.4Co9.6, which\nwere fabricated on thermally oxidized Si substrates by a magnetro n sputtering method. The\n5-nm-thick SiN layers were deposited as a buffer and capping layer. T he film exhibited a net\nmagnetization of 45 kA/m and a perpendicular magnetic anisotropy fi eld of approximately\n1 T at room temperature, measured by a vibrational sample magnet ometer. The sample\nexhibited TM= 239 K and TA= 321 K, evaluated by the anomalous Hall effect and the\ndomain wall velocity measurement for different temperatures, res pectively [9]. The time-\nresolved magneto-optical Kerr effect was measured under the am bient temperature using\nthe all-optical pump-probe setup with a Ti:Sapphire laser and a regen erative amplifier, the\nsame as that previously reported [15–18]. The duration, the centr al wave length, and the\nrepetition rate for the output laser pulse in this study were ∼120 fs,∼800 nm, and 1\n5050100150-1500-1000-50005001000ΔϕK (a.u.)\nΔt (ps)µ0H (T)\n0.781.171.511.752.01\n0.39\n050100150-400-2000200ΔϕK (a.u.)\nΔt (ps)µ0H (T)\n0.781.171.511.752.01\n0.39\n(a) (b)\n0 1 2050100f (GHz)\nµ0H (T)0.9 1.83.6Fp (mJ/cm2)(c)\nFIG. 2. The change in the Kerr rotation angle ∆ φkas a function of the pump-probe delay time\n∆tfor different magnetic field strengths Hwas measured at pump pulse fluences Fpof 0.9 (a)\nand 3.6 mJ/cm2(b). Both data were collected for a field angle θHof 70◦with respect to the film\nnormal. Solid curves are fitted to the data. (c) The precessio n frequency f, evaluated from the\ntime-resolved data as a function of H, for different Fp. The lines and curve are visual guides.\nkHz, respectively. The angle of incidence of the p-polarized pump and s-polarized probe\nbeams were ∼3◦and∼8◦, respectively, with respect to the film normal. The respective spot\nsizes for the pump and probe beams, which were focused on the film s urface with spatial\noverlapping, were 1.3 and 0.37 mm in diameter. The maximum magnetic fie ld applied was\n2 T, with variable field directions.\nFigures 2(a) and 2(b) show the typical data of the change in the Ke rr rotation angle ∆ φk\nas a function of the pump-probe delay time, measured for different applied magnetic fields\nat pump pulse fluences Fpof 0.9 and 3.6 mJ/cm2, respectively. The magnetic field angle θH\nwas fixed at 70◦from the film normal. The very rapid changes in the Kerr rotation ang le\nobserved at the delay zero mainly result from the sub-ps reduction in the normal component\nof the net magnetization owing to the absorption of the pump pulse. Subsequent damped\noscillations of ∆ φk, corresponding to the magnetization precession, are observed, and their\n6oscillation periods vary with the magnetic field strength. At a fluence of 0.9 mJ/cm2, the\noscillation period becomes longer as the magnetic field strength incre ases [Fig. 2(a)]. The\nopposite trend is observed when the fluence is 3.6 mJ/cm2[Fig. 2(b)]. In addition to the\nprecession clearly visible in the figures, additional precession is obse rved, with much smaller\namplitudes and relatively short precession periods (not shown here ). We attribute this\nmode to a high-frequency branch due to the lift of degeneracy for theω−andω+modes in\nferrimagnets, as discussed earlier. Since it was very hard to simulta neously fit both the high-\nand low-frequency modes, we analyzed only the low precession freq uency mode by fitting the\nexponentially damped sinusoidal function to the time-resolved data , as shown by the solid\ncurves in Figs. 2(a) and 2(b). The evaluated frequencies are plott ed as a function of the\nmagnetic field in Fig. 2(c) for different pump fluences. Approximately linear relationships\nbetween the frequencies and the magnetic fields are found at fluen ces of 0.9 and 3.6 mJ/cm2,\nregarding which the negative and positive slopes are considered to b e those for the ω−and\nω+modes, as depicted in Figs. 1(c) and 1(d), respectively. The mode c hange as a function\nof the fluence may stem from the change in the time-averaged samp le temperature from\nbelow to above TA. This interpretation is very reasonable since the ambient temperat ure is\njust below TA= 321 K and the sample temperature easily exceeds TAat the high fluence.\nAt the intermediate fluence 1.8 mJ/cm2, the frequency falls off near 2 T, which may be\nunderstood as the mode crossing from the ω+mode to the ω−mode. Namely, the frequency\nfor theω−mode becomes lower than that for the ω+mode at such high field.\nBefore proceeding further, note the mechanism of the laser-indu ced magnetization pre-\ncession. The primary mechanism of the excitation of these two mode s can be attributed to\nthe sudden change in PMA. This change in the anisotropy functions a s the effective torque\ntriggering magnetization precession, as discussed in relation to the all-optical FMR [19].\nThe effective torque works only when the magnetization makes an an gle with respect to\nthe magnetic easy direction or plane and reaches its maximum (zero) when the magnetic\nfield is parallel (perpendicular) to the film plane in the present case. T he magnetization\nprecession amplitudes tend to decrease as the magnetic field angle θHdecreases. Hence, it is\ndifficult to observe the dynamics when the applied field is parallel to the film normal in the\ncase depicted in Fig. 1. Note that the sudden change in the magnetic anisotropy may be\ncaused by the ultrafast demagnetization and its relevant process [20]. More details regarding\nsimilar alloys have been discussed for the FMR and exchange modes at Tbelow or near TM\n7(a)\n(b)0306090050100f (GHz)µ0H (T)\n0.63\n2.01Fp=0.8 mJ/cm20.33\n0306090050100f (GHz)\nθH (°)Fp=1.8 mJ/cm22.01\n0.63µ0H (T)\nFIG. 3. Magnetic field angle θHdependence of the precessional frequency ffor low-frequency\nmodes, extracted from data similarly measured for different fi eld strengths Hat a pump laser pulse\nfluenceFpof 0.8 (a) and 1.8 mJ/cm2(b). The curves are visual guides.\n[10, 11]. However, the mechanism at TnearTAis still unclear, though a discussion on this\nmechanism is outside the scope of this study.\nThe magnetic field angle dependence of the precession frequency w as also examined to\ngain insight into the role of PMA in the dynamics observed. Figures 3(a ) and 3(b) show that\nthe precession frequencies increase as the magnetic field angle θHdecreases. This trend is\nsimilar to that observed in ferromagnetic films possessing a large PMA , such as Co/Pt and\nCoFeB/MgO multilayers, and ordered alloys films [15–18]. Thus, this an gular dependence\nmay be due to the influence of PMA on the antiferromagnetic-like res onance. Interestingly,\nthe tendency of the frequency under higher magnetic fields being s maller than that under\nlower magnetic fields is maintained for the different magnetic field angle s considered here\nfor a low pump fluence [Fig. 3(a)]; the opposite case is true for a high p ump fluence [Fig.\n3(b)].\nInstead of Eq. (3) describing the simple physics, hereafter we disc uss the present dynam-\nics, particularly the angular dependence, based ona more realistic m icromagnetic simulation\nusing the coupled Landau-Lifshitz-Gilbert equations for two sublat tice magnetizations with\nPMA under various strengths and directions of the external magn etic field [9]. The mesh\nwas set to 0 .4×20×5 nm3, and the exchange stiffness between the sublattices ACoFe−Gdwas\ntaken as 0.04 pJ/m. To model the dynamics at Tbelow (above) TA, we input the following\n8temperature-dependent sublattice magnetization with temperat ure-independent gyromag-\nnetic ratios for TM and RE: MCoFe= 615 (510) kA/m with γCoFe= 193.6×109rad/T·s; and\nMGd= 568(420)kA/mwith γGd= 176×109rad/T·s, respectively. Thedifferenceinthecor-\nresponding angular momentum density is SGd−SCoFe= 0.506 (−2.48)×10−3J·s/rad·m3.\nNote that the magnetization values at Tbelow (above) TAcorrespond to the values at\nT= 312 (380) K that were evaluated from the experimental magnetiz ation-temperature\ncurves using the method described in Ref. 21. The intrinsic PMA cons tantK= 0.245 (0.17)\nMJ/m3forbothsublatticeswasassumedfor Tbelow(above) TA. Then, theeffectiveuniaxial\nPMA constant Keff\nCoFe(Gd)for the sublattices was given as Keff\nCoFe(Gd)=K−2πM2\nCoFe(Gd). The\ntwo-modeprecessional dynamics wascomputedinthetimedomainfo rvariousfieldstrengths\nand directions, where a sublattice-independent Gilbert damping par ameter of 0.001 was in-\nput. Then, themodefrequencies were evaluatedvia thefastFour ier transform. Notethatwe\nalso employed a theoretical calculation based on the fully analytic for mula for an arbitrary\nmagnetic field and direction, with Eq. (1) being a special case; it yielde d data approximately\nequal to those evaluated by simulations with the parameters listed a bove as the input.\nFigure 4 displays the computed mode frequencies. The data reveal a nearly linear re-\nlationship between the frequency and the strength of the out-of -plane magnetic field [Fig.\n4(a)], in which the high- (low-) frequency mode exhibits a positive (ne gative) slope in the\ncase ofTbelowTA(the opposite is true in the case of TaboveTA), being roughly consistent\nwith Eq. (3) and similar to that shown in Figs. 1(c) and 1(d). The comp utation also shows\nthat the negative and positive slopes of the fvs.Hdata are similarly observed as the low-\nfrequency mode even when the magnetic field angle is 70◦, as experimentally verified [Fig.\n4(b)]. Subsequently, the experimental field-angle variation was ve rified via a computation\nwith different angles and a fixed magnetic field. The calculated data fo rTbelow and above\nTAare shown in Figs. 4(c) and 4(d), which well reproduce the experime ntal tendencies\ndisplayed in Figs. 3(a) and 3(b), respectively. Thus, the simulation s atisfactorily supports\nthe conclusion that the dynamics observed is the antiferromagnet ic-like resonance mode in\nferrimagnets. Note that the frequency range of ∼50–150 GHz in Figs. 4(c)-4(d) is also\nroughly consistent with the one experimentally observed ( ∼30–90 GHz). Additional quan-\ntitative investigations should be carried out in future studies, which will require a precise\nevaluation of the high-frequency mode at TnearTA.\nFinally, wecomment onthedifference between ourworkandthatofS tanciuet al.[10]. In\n9ω-(a)\n(c)\n(d) (b)T> TA\nω-ω-\nω+\nω+T< TA\nω-ω+T< TA\nT> TAT> TAT< TA\nθH= 70/g7092.00.60.3\nµ0H(T)\n2.0\n0.6\n0.3ω+µ0H(T)\n0306090050100150\nθH (°) \n050100050100150\nθH (°) 012050100f (GHz)\nµ0H (T)0120100200300400f (GHz)\nµ0H (T)\nFIG. 4. The calculated data of magnetization precession fre quencyfas a function of the magnetic\nfieldHwith (a) the magnetic field angle θH= 0◦and (b) 70◦. The data of ω+andω−atT < TA\n(T > TA) are indicated by the solid and open diamonds (inverse trian gles), respectively. (b) shows\nonly the modes with lower frequencies. The calculated data o ffvs.θHfor the modes with lower\nfrequencies at µ0H= 0.3 (circle), 0.6 (triangle). and 2.0 T (square) at (c) T < TAand (d)T > TA.\nThe input parameters correspond to the ones at Tjust below or above TA; see the main text. The\ncurves are visual guides.\ntheir paper, they discussed the FMR and exchange modes at TnearTMandTAand stated\nthe following: ”When the temperature of the sample approaches the angular m omentum\ncompensation point, both frequency and the Gilbert damping parameter of the magnetization\nprecession increase significantly. In addition, the high-f requency exchange mode softens and\nbecomes observable.” Meanwhile, the theoretical data shown in Fig. 3 in Ref. 10 indicate\nthat the frequency of the FMR mode becomes infinite at TAowing to the divergence of the\neffective gyromagnetic ratio at this point. Our study, however, pr esents a rather different\nphysical picture at TA. The two modes exhibit similar frequencies at TnearTA, originating\nfromthenatureofanantiferromagnetic-like stateatthispoint inf errimagnets. Theessential\ndifference between the two modes at T=TAis the left-handed or right-handed symmetry,\nwhich was experimentally confirmed with the observation of the oppo site response to the\napplied field. This outcome is a very natural consequence of the bea utiful symmetry in\nantiferromagnets, namely, a time-reversal invariant.\n10In summary, the pulsed-laser-induced magnetization precessiona l dynamics in the Gd-\nCoFe ferrimagnetic film with TAjust above the ambient temperature was reported. An\ninversion of the relation of the gyromagnetic precession frequenc y with respect to the mag-\nnetic field was clearly observed as the pump laser fluence changed. T his inversion was well\nexplained by the change between the right-handed and left-hande d precession modes, being\nattributed to the antiferromagnetic-like resonance modes at Tbelow and above TA, under\nlaser-induced heating. This unique dynamics was also examined for diff erent magnetic field\nangles, with all experimental data being consistent with the microma gnetic simulation. The\nfindings of this study will contribute to development of the physics o f the antiferromagnetic\nspintronics underlying these ferrimagnets.\nS.M. thanks CSRN, and Y.S. thanks GP Spin of Tohoku University. This work was\npartially supported by KAKENHI (16H03846, 26103002, and 26103 004). D.K.L. and K.J.L.\nwere supported by the National Research Foundation of Korea (2 017R1A2B2006119) and\nthe KIST Institutional Program (Project No. 2V05750).\n∗shigemi.mizukami.a7@tohoku.ac.jp\n[1] T. Nagamiya, K. Yosida, and R. Kubo, Adv. Phys. 4,1 (1955).\n[2] C. Kittel, Phys. Rev. 82,565 (1951).\n[3] T. Jungwirth, X. 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Phys.\nLett.108,012404 (2016).\n[19] M. van Kampen, C. Jozsa, J. Kohlhepp, P. LeClair, L. Laga e, W. de Jonge, and B. Koopmans,\nPhys. Rev. Lett. 88,227201 (2002).\n[20] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigo t, Phys. Rev. Lett. 76,4250 (1996).\n[21] Y. Hirata, D.-H. Kim, T. Okuno, T. Nishimura, D.-Y. Kim, Y. Futakawa, H. Yoshikawa, A.\nTsukamoto, K.-J. Kim, S.-B. Choe, and T. Ono, Phys. Rev. B 97,220403(R) (2018).\n12"    },    {        "title": "1706.06379v1.Magnetic_signatures_of_quantum_critical_points_of_the_ferrimagnetic_mixed_spin__1_2__S__Heisenberg_chains_at_finite_temperatures.pdf",        "content": "arXiv:1706.06379v1  [cond-mat.stat-mech]  20 Jun 2017Journal of Low Temperature Physics manuscript No.\n(will be inserted by the editor)\nMagnetic signatures of quantum critical points of\nthe ferrimagnetic mixed spin-(1/2, S) Heisenberg\nchains at finite temperatures\nJozef Streˇ cka ·Taras Verkholyak\nReceived: date / Accepted: date\nAbstract Magneticpropertiesoftheferrimagneticmixedspin-(1/2, S)Heisen-\nberg chains are examined using quantum Monte Carlo simulations for t wo dif-\nferent quantum spin numbers S= 1 and 3/2. The calculated magnetization\ncurves at finite temperatures are confronted with zero-temper ature magne-\ntization data obtained within density-matrix renormalization group m ethod,\nwhich imply an existence of two quantum critical points determining a b reak-\ndown of the gapped Lieb-Mattis ferrimagnetic phase and Tomonaga -Luttinger\nspin-liquid phase, respectively. While a square-root behavior of the magneti-\nzation accompanying each quantum critical point is gradually smooth ed upon\nrising temperature, the susceptibility and isothermal entropy cha nge data pro-\nvide a stronger evidence of the quantum critical points at finite tem peratures\nthrough marked local maxima and minima, respectively.\nKeywords ferrimagnetic Heisenberg chains ·quantum critical point ·\nquantum Monte Carlo\nPACS75.10.Pq ·75.10.Kt ·75.30.Kz ·75.40.Cx ·75.60.Ej\nThis work was financially supported by the grant of The Minist ry of Education, Science,\nResearch and Sport of the Slovak Republic under the contract No. VEGA 1/0043/16, as\nwell as, by grants of the Slovak Research and Development Age ncy provided under Contract\nNos. APVV-0097-12 and APVV-14-0073.\nJozef Streˇ cka\nInstitute of Physics, Faculty of Science of P. J. ˇSaf´ arik University,\nPark Angelinum 9, 040 01 Koˇ sice, Slovak Republic\nE-mail: jozef.strecka@upjs.sk\nTaras Verkholyak\nInstitute for Condensed Matter Physics, NASU,\n1 Svientsitskii Street, 790 11 L’viv-11, Ukraine\nE-mail: werch@icmp.lviv.ua2 Jozef Streˇ cka, Taras Verkholyak\n1 Introduction\nQuantum phase transitions traditionally attract a great deal of at tention, be-\ncause they are often accompanied with several remarkable signat ures exper-\nimentally accessible at nonzero temperatures [1]. One-dimensional q uantum\nspin chains provide notable examples of condensed matter systems , which\nbring a deeper understanding into exotic forms of magnetism nearb y quan-\ntum critical points based on rigorous calculations [2]. Apart from con ven-\ntional solid-state compounds affording experimental realizations o f the quan-\ntum spin chains [3], one may simulate a quantum phase transition in spin\nchains through ultracold atoms confined in an optical lattice [4]. Frac tional\nmagnetizationplateaux[5],magnetizationcuspsingularities[6]andsp in-liquid\nground states [5,6,7,8] can be regarded as the most profound m anifestations\nof zero-temperature magnetization curves of the quantum spin c hains.\nThe ferrimagnetic mixed spin-( s,S) Heisenberg chains [9,10,11,12,13,14,\n15,16,17,18] with regularly alternating spins s= 1/2 andS >1/2 display\nthe intermediate magnetization plateau inherent to the gapped Lieb -Mattis\nferrimagnetic ground state, as well as, the gapless Tomonaga-Lu ttinger spin-\nliquid phase. It has been argued [18] on the grounds of Lieb-Mattis t heorem\n[19] and Oshikawa-Yamanaka-Affleck rule [20] that the ferrimagnet ic mixed\nspin-(1/2, S)Heisenberg chains should exhibit just one plateau at the following\nvalue of the total magnetization m/ms= (2S−1)/(2S+1) normalized with\nrespect to its saturation. The gapped Lieb-Mattis ferrimagnetic g round state\nbreaks down at a quantum phase transition towards the gapless To monaga-\nLuttinger spin-liquid phase, which is accompanied with a singular squar e-root\nbehaviorof the magnetization. The same type of the magnetization singularity\nalsoappearsjustbelowthesaturationfield,whichdeterminesaqua ntumphase\ntransition between the Tomonaga-Luttinger spin liquid and the fully p olarized\nphase.Inthepresentworkwewillexaminemagneticsignaturesoft hequantum\ncritical points of the ferrimagnetic mixed spin-(1/2, S) Heisenberg chains at\nfinite temperatures by making use of quantum Monte Carlo simulation s.\n2 Model and method\nLetusconsidertheferrimagneticmixedspin-(1/2, S)Heisenbergchainsdefined\nthrough the Hamiltonian\nˆH=JL/summationdisplay\nj=1ˆSj·(ˆsj+ˆsj+1)−hL/summationdisplay\nj=1(ˆSz\nj+ ˆsz\nj), (1)\nwhereˆsj≡(ˆsx\nj,ˆsy\nj,ˆsz\nj) andˆSj≡(ˆSx\nj,ˆSy\nj,ˆSz\nj) denote the standard spin-1/2\nand spin- Soperators, respectively. The first term in the Hamiltonian (1) take s\ninto account the antiferromagnetic Heisenberg interaction J >0 between the\nnearest-neighborspins and the second term h=gµBHincorporatingthe equalMagnetic signatures of quantum critical points of the Heise nberg chains 3\nLand´ e g-factors gs=gS=gand Bohr magneton µBaccounts for the Zee-\nmann’s energy of individual magnetic moments in an external magnet ic field.\nSince the elementary unit contains two spins an overall chain length is 2L,\nwhereas a translational invariance is ensured by the choice of perio dic bound-\nary conditions ˆsL+1≡ˆs1.\nTo explore magnetic properties of the ferrimagnetic mixed spin-(1/ 2,S)\nHeisenberg chains at nonzero temperatures, we have implemented a directed\nloopalgorithminthestochasticseriesexpansionrepresentationof thequantum\nMonte Carlo (QMC) method [21] from Algorithms and Libraries for Phy sics\nSimulations (ALPS) project [22]. The QMC method allows a straightfor ward\ncalculation of the magnetization data at finite temperatures, which will be\nalso confronted with recent zero-temperature magnetization da ta calculated\nwithin the density-matrix renormalization group (DMRG) method [18] serv-\ning as a useful benchmark at low enough temperatures. The susce ptibility of\nthe ferrimagnetic mixed spin-(1/2, S) Heisenberg chains can also be directly\ncalculated from a directed loop algorithm of QMC method, while the isot her-\nmal entropy change can be obtained from the relevant magnetizat ion data\nusing the Maxwell’s relation\n∆ST=/integraldisplayh\n0/parenleftbigg∂m\n∂T/parenrightbigg\nhdh. (2)\nTo avoid an extrapolation due to finite-size effects, we have perfor med QMC\nsimulations for a sufficiently large system size with up to L= 128 units (256\nspins), whereas adequate numerical accuracy was achieved thro ugh 750 000\nMonte Carlo steps used in addition to 150 000 steps for thermalizatio n.\n3 Results and discussion\nLet us proceed to a discussion of the most interesting results for t he magneti-\nzation and susceptibility data of the ferrimagnetic mixed spin-(1/2, S) Heisen-\nberg chains. Fig. 1(a) shows a three-dimensional (3D) plot of the m agneti-\nzation of the ferrimagnetic mixed spin-(1/2,1) Heisenberg chain aga inst the\nmagnetic field and temperature. As one can see, the one-third plat eau due to\nthe gapped Lieb-Mattis ferrimagnetic ground state diminishes upon increas-\ning of temperature until it becomes completely indiscernible above ce rtain\ntemperature kBT/J≈0.5. To explore the temperature effect in more detail,\nFig. 1(b) compares a zero-temperature magnetization curve obt ained within\nDMRG method [18] with low-temperature magnetization data stemmin g from\nQMC simulations. It is quite evident that the square-root singularity of the\nmagnetization, which emerges at both quantum critical points dete rmining an\nupper edge of the one-third plateau and saturation field, is gradua lly rounded\nupon raising temperature.\nA stronger evidence of two quantum phase transitions of the ferr imagnetic\nmixed spin-(1/2,1)Heisenberg chain thus provides the susceptibility , which is\nplotted in Fig. 2. As a matter of fact, the susceptibility still displays a t low4 Jozef Streˇ cka, Taras Verkholyak\n/s48/s49/s50/s51/s52\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s107\n/s66/s32\n/s84\n/s32/s47/s32\n/s74/s109/s32/s47/s32/s109/s115\n/s104/s32/s47/s32/s74/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s109 /s32/s47/s32 /s109\n/s115\n/s40/s97/s41\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s98/s41/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s48/s32/s40/s68/s77/s82/s71/s41\n/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s49/s32/s32/s40/s81/s77/s67/s41\n/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s50/s32/s32/s40/s81/s77/s67/s41\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s51/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s49/s46/s55/s54/s49/s47/s51 /s112/s108/s97/s116/s101/s97/s117/s49/s47/s50 /s49\n/s32/s32/s109/s32/s47/s32/s109\n/s115\n/s104 /s32/s47/s32 /s74\nFig. 1 The total magnetization of the ferrimagnetic mixed spin-(1 /2,1) Heisenberg chain\nnormalized with respect to its saturation value: (a) 3D surf ace plot as a function of temper-\nature and magnetic field constructed from QMC data; (b) Zero- temperature DMRG data\nversus QMC magnetization curves at low enough temperatures .\n/s48\n/s49\n/s50\n/s51\n/s52/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s40/s97/s41\n/s32/s74\n/s107/s66/s32/s84/s32/s47/s32/s74\n/s104\n/s32/s47/s32\n/s74/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s74\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s40/s98/s41/s107\n/s66/s32/s84 /s32/s47/s32 /s74/s32\n/s32/s48/s46/s48/s32/s40/s68/s77/s82/s71/s41\n/s32/s48/s46/s48/s53/s32/s40/s81/s77/s67/s41\n/s32/s48/s46/s49/s48/s32/s40/s81/s77/s67/s41\n/s32/s48/s46/s50/s48/s32/s40/s81/s77/s67/s41\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s51/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s49/s46/s55/s54/s115/s112/s105/s110/s32/s108/s105/s113/s117/s105/s100/s49/s47/s50 /s49\n/s32/s32/s74\n/s104 /s32/s47/s32 /s74\nFig. 2The susceptibility of the ferrimagnetic mixed spin-(1/2,1 ) Heisenberg chain per unit\ncell: (a) 3D surface plot as a function of temperature and mag netic field constructed from\nQMCdata; (b) Zero-temperature DMRGdata versus QMCsimulat ions at lowtemperatures.\nenough temperatures two sharp peaks in a vicinity of the magnetic fi elds driv-\ning the investigated quantum spin system towards the quantum crit ical points\nin addition to a pronounced divergence observable at zero magnetic field. Of\ncourse, these local maxima are gradually suppressed and merge to gether upon\nincreasing of temperature.\nTo illustrate a general validity of the aforedescribed conclusions, F igs. 3\nand 4 show similar plots for the magnetization and susceptibility of ano ther\nferrimagnetic mixed spin-(1/2,3/2) Heisenberg chain. It is quite app arent that\nthe same general trends can be reached as far as the temperatu re effect is con-\ncerned. There are only two quantitative differences. The first diffe rence refers\nto a height of the intermediate Lieb-Mattis plateau, which is in the pre sent\ncase at one-half of the saturation magnetization. The second diffe rence lies\nin absolute values of two field-driven quantum critical points, which d eter-\nmine the end of the intermediate one-half plateau and the beginning o f the\nsaturation magnetization, respectively.Magnetic signatures of quantum critical points of the Heise nberg chains 5\n/s48/s49/s50/s51/s52/s53\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s107\n/s66/s32\n/s84\n/s32/s47/s32\n/s74/s109/s32/s47/s32/s109/s115\n/s104/s32/s47/s32/s74/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s109 /s32/s47/s32 /s109\n/s115\n/s40/s97/s41\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s98/s41/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s48/s32/s40/s68/s77/s82/s71/s41\n/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s49/s32/s32/s40/s81/s77/s67/s41\n/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s50/s32/s32/s40/s81/s77/s67/s41\n/s49/s47/s50 /s112/s108/s97/s116/s101/s97/s117\n/s49/s47/s50 /s51/s47/s50\n/s32/s32/s109/s32/s47/s32/s109\n/s115\n/s104 /s32/s47/s32 /s74/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s52/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s50/s46/s56/s52\nFig. 3The total magnetization of the ferrimagnetic mixed spin-(1 /2,3/2) Heisenberg chain\nnormalized with respect to its saturation value: (a) 3D surf ace plot as a function of temper-\nature and magnetic field constructed from QMC data; (b) Zero- temperature DMRG data\nversus QMC magnetization curves at low enough temperatures .\n/s48\n/s49\n/s50\n/s51\n/s52\n/s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s40/s97/s41\n/s32/s74\n/s107/s66/s32/s84/s32/s47/s32/s74\n/s104\n/s32/s47/s32\n/s74/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s52/s48/s46/s53/s74\n/s48/s46/s48 /s48/s46/s53 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s40/s98/s41/s107\n/s66/s32/s84 /s32/s47/s32 /s74/s32\n/s32/s48/s46/s48/s32/s40/s68/s77/s82/s71/s41\n/s32/s48/s46/s48/s53/s32/s40/s81/s77/s67/s41\n/s32/s48/s46/s49/s48/s32/s40/s81/s77/s67/s41\n/s32/s48/s46/s50/s48/s32/s40/s81/s77/s67/s41\n/s115/s112/s105/s110/s32/s108/s105/s113/s117/s105/s100/s49/s47/s50 /s51/s47/s50\n/s32/s32/s74\n/s104 /s32/s47/s32 /s74/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s52/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s50/s46/s56/s52\nFig. 4 The susceptibility of the ferrimagnetic mixed spin-(1/2,3 /2) Heisenberg chain per\nunit cell: (a) 3D surface plot as a function of temperature an d magnetic field constructed\nfrom QMC data; (b) Zero-temperature DMRG data versus QMC sim ulations at low enough\ntemperatures.\nLast but not least, let us examine the magnetic-field variations of th e\nisothermal entropy change, which represents one of two basic ma gnetocaloric\npotentials. For this purpose, Fig. 5 illustrates typical dependence s of the\nisothermal entropy change of two considered ferrimagnetic mixed spin-(1/2, S)\nHeisenberg chains as a function of the magnetic field at a few differen t temper-\natures. It is quite clear that the isothermal entropy change −∆STexhibits a\nsteep increase when the magnetic field is lifted from zero, which can b e related\nto an abrupt field-induced increase of the magnetization from zero towardsthe\nferrimagnetic Lieb-Mattis plateau m/ms= (2S−1)/(2S+1). The isothermal\nentropy change −∆STthen reaches a constant value within the field range,\nwhich is nearly equal to a width of the intermediate magnetization plat eau\nat a given temperature. This interval of the magnetic fields determ ines range\nof applicability of the ferrimagnetic mixed spin-(1/2, S) Heisenberg chains for\ncooling purposes. Namely, the isothermal entropy change −∆STconsecutively6 Jozef Streˇ cka, Taras Verkholyak\n/s48 /s49 /s50 /s51 /s52/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/s48/s46/s51/s48\n/s48/s46/s50\n/s48/s46/s49\n/s107\n/s66/s32/s84\n/s32/s47/s32 /s74/s32 /s61/s32/s48/s46/s48/s53/s49/s47/s50 /s49\n/s32/s32/s83\n/s84/s32/s47/s32/s107\n/s66\n/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s104 /s32/s47/s32 /s74/s48 /s49 /s50 /s51 /s52 /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/s48/s46/s51/s48\n/s48/s46/s50\n/s48/s46/s49\n/s107\n/s66/s32/s84\n/s32/s47/s32 /s74/s32 /s61/s32/s48/s46/s48/s53/s49/s47/s50 /s51/s47/s50\n/s32/s32/s83\n/s84/s32/s47/s32/s107\n/s66\n/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s104 /s32/s47/s32 /s74\nFig. 5 The isothermal change of the entropy versus the magnetic fiel d at three different\ntemperatures for the ferrimagnetic mixed spin-(1/2, S) Heisenberg chain: (a) S= 1; (b)\nS= 3/2.\nacquiresrelativelydeep minima near both field-driven quantum critica lpoints,\nwhich are gradually lifted and smoothed upon increasing of temperat ure. The\nminima in the isothermal entropy change −∆STcan be thus regarded as an-\nother faithful indicators of the quantum critical points of the fer rimagnetic\nmixed spin-(1/2, S) Heisenberg chains.\n4 Conclusion\nIn the present work we have examined magnetic and thermodynamic proper-\ntiesoftheferrimagneticmixedspin-(1/2, S)Heisenbergchainsatfinitetemper-\natures using QMC simulations. In particular, we have focused our at tention to\na detailed examination of the temperature effect upon the magnetiz ation pro-\ncess in a close vicinity of the field-driven quantum critical points. It h as been\nevidenced that a singularsquare-rootbehaviorof the magnetizat ion, which ac-\ncompanies both quantum phase transitions connected with a break downof the\ngapped Lieb-Mattis ferrimagnetic phase and the gapless Tomonaga -Luttinger\nspin-liquid phase, undergoes a gradual rounding upon increasing of tempera-\nture.Themagnetizationcurveatnon-zerotemperaturesisthus almostwithout\nany clear signature of the quantum critical points unlike other ther modynamic\nresponse functions.\nThe susceptibility and isothermal entropy changes contrarily displa y close\nto quantum phase transitions relatively sharp maxima and minima, res pec-\ntively, which are gradually suppressed and smoothed upon increasin g of tem-\nperature. The respective maximum in the susceptibility and minimum in t he\nisothermal entropy change can be accordingly regarded as a faith ful indicator\nof the field-driven quantum critical point of the ferrimagnetic mixed spin-\n(1/2,S) Heisenberg chains. Although the present study was restricted j ust to\nthe mixed-spin Heisenberg chains with the quantum spin numbers S= 1 and\n3/2, the foregoing studies [9,10,11,12,13,14,15,16,17,18] h ave already proven\nlargely universal behavior of the ferrimagnetic mixed spin-(1/2, S) HeisenbergMagnetic signatures of quantum critical points of the Heise nberg chains 7\nchains also for other spin values for which qualitatively the same beha vior\nshould be expected.\nIt has been also demonstrated that the most efficient cooling throu gh the\nadiabatic demagnetization of the ferrimagnetic mixed spin-(1/2, S) Heisenberg\nchains can be achieved in a range of the magnetic fields from zero up t o nearly\na half of the intermediate magnetization plateau, while the isotherma l entropy\nchange implies for larger magnetic fields a less efficient cooling as it rapid ly\ndrops down near both quantum critical points. It is our hope that t heoretical\nresults of the present work are ofobvious relevance for a series o f bimetallic co-\nordinationcompoundsMM’(pba)(H 2O)3·2H2O [23] andMM’(EDTA) ·6H2O\n[24] (M,M’ = Cu, Ni, Co, Mn), which represent experimental realizatio ns of\nthe ferrimagnetic mixed-spin Heisenberg chains. High-field magnetiz ation and\nmagnetocaloric measurements on these or related series of bimeta llic com-\nplexes are however needed for experimental testing of the prese nt theoretical\npredictions.\nReferences\n1. S. Sachdev, Quantum Phase Transitions . Cambridge University Press, Cambridge\n(1999).\n2. D.C. Mattis, The Many-Body Problem: An Encyclopedia of Exactly Solved Mo dels in\nOne Dimension . World Scientific, Singapore (1993).\n3. J.S. Miller, M. Drillon, Magnetism: Molecules to Materials I . Wiley-VCH Verlag, Wein-\nheim (2001).\n4. J. Simon, W.S. Bakr, R. Ma, M.E.Tai, P.M.Preiss,M. Greine r,Nature472,307 (2011).\n5. A. Honecker, J. Schulenburg, J. Richter, J. Phys.: Condens. Matter 16, S749 (2004).\n6. K. Okunishi, Prog. Theor. Phys. Suppl. 145, 119 (2002).\n7. G. Misguich, Quantum Spin Liquids , inExact Methods in Low-Dimensional Statistical\nPhysics and Quantum Computing , eds. J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban,\nL. F. Cugliandolo. Oxford University Press, Oxford (2008).\n8. Y. Zhou, K. Kanoda, T.-K. Ng, preprint arxiv: 1607.03228\n9. T. Kuramoto, J. Phys. Soc. Jpn. ,67, 1762 (1998).\n10. S. Yamamoto, T. Sakai, J. Phys.: Condens. Matter ,11, 5175 (1999).\n11. T. Sakai, S. Yamamoto, Phys. Rev. B ,60, 4053 (1999).\n12. N.B. Ivanov, Phys. Rev. B ,62, 3271 (2000).\n13. A. Honecker, F. Mila, M. Troyer, Eur. Phys. J. B ,15, 227 (2000).\n14. S. Yamamoto, T. Sakai, Phys. Rev. B ,62, 3795 (2000).\n15. T. Sakai, S. Yamamoto, Phys. Rev. B ,65, 214403 (2002).\n16. A.S.F. Ten´ orio, R.R. Montenegro-Filho, M.D. Coutinho -Filho,J. Phys.: Condens. Mat-\nter,23, 506003 (2011).\n17. N.B. Ivanov, S.I. Petrova, J. Schnack, Eur. Phys. J. B ,89, 121 (2016).\n18. J. Streˇ cka, preprint arxiv: 1607.03617.\n19. E. Lieb, D. Mattis, J. Math. Phys. ,3, 749 (1962).\n20. M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. ,78, 1984 (1997).\n21. A.W. Sandvik, Phys. Rev. B 59, 14157 (1999).\n22. B. Bauer, L.D. Carr, H.G. Evertz, A. Feiguin, J. Freire, S . Fuchs, L. Gamper, J. Gukel-\nberger, E. Gull, S. Guertler, A. Hehn, R. Igarashi, S.V. Isak ov, D. Koop, P.N. Ma, P.\nMates, H. Matsuo, O. Parcollet, G. Pawlowski, J.D. Picon, L. Pollet, E. Santos, V.W.\nScarola, U. Schollw¨ ock, C. Silva, B. Surer, S. Todo, S. Treb st, M. Troyer, M.L. Wall, P.\nWerner, S. Wessel, J. Stat. Mech.: Theor. Exp. ,2011, P05001 (2011).\n23. O. Kahn, Struct. Bonding (Berlin) ,68, 89 (1987).\n24. M. Drillon, E. Coronado, D. Beltran, R. Georges, J. Appl. Phys. ,57, 3353 (1985)."    },    {        "title": "1301.7313v1.Thermally_driven_spin_and_charge_currents_in_thin_NiFe2O4_Pt_films.pdf",        "content": "Thermally driven spin and charge currents in thin NiFe 2O4/Pt \flms\nD. Meier,1,\u0003T. Kuschel,1L. Shen,2A. Gupta,2T. Kikkawa,3K.\nUchida,3, 4E. Saitoh,3, 5, 6, 7J.-M. Schmalhorst,1and G. Reiss1\n1Thin Films and Physics of Nanostructures, Department of Physics, Bielefeld University, D-33501 Germany\n2Center for Materials for Information Technology,\nUniversity of Alabama, Tuscaloosa Alabama 35487, USA\n3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n4PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan\n5WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n6CREST, Japan Science and Technology Agency, Tokyo 102-0076, Japan\n7Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n(Dated: September 18, 2018)\nWe present results on the longitudinal spin Seebeck e\u000bect (LSSE) shown by semiconducting fer-\nrimagnetic NiFe 2O4/Pt \flms from room temperature down to 50 K base temperature. To the best\nof our knowledge, this is the \frst observation of spin caloric e\u000bect in NiFe 2O4thin \flms. The tem-\nperature dependence of the conductivity has been studied in parallel to obtain information about\nthe origin of the electric potentials detected at the Pt coverage of the ferrimagnet in order to dis-\ntinguish the LSSE from the anomalous Nernst e\u000bect. Furthermore, the dependence of the LSSE on\ntemperature gradients as well as the in\ruence of an external magnetic \feld direction is investigated.\nPACS numbers: 85.75.-d, 72.25.-b, 72.20.Pa\nI. INTRODUCTION\nThe spin Seebeck e\u000bect (SSE) generates a spin cur-\nrent induced by a temperature gradient and belongs to\nthe emerging \feld of spin caloritronics1. The e\u000bect was\n\frstly observed in 2008 by Uchida et al. in Ni 81Fe19(Py)\n\flms with Pt stripes2using an in-plane temperature gra-\ndientrTwhich generates a spin current perpendicular\nto the \flm plane. The spin current was detected in the\nPt stripes as an electrommotive force ~EISHE by means of\nthe inverse spin Hall e\u000bect (ISHE)3given by the formula\n~EISHE =DSHE~JS\u0002~ \u001b: (1)\nThe material-dependent constant DSHE describes the\nmagnitude of the spin Hall e\u000bect (SHE)4,5,~JSis the spin\ncurrent and ~ \u001bis the spin polarization vector.\nThe same e\u000bect was also observed in the ferromagnetic\nsemiconductor GaMnAs6and in the Heusler compound\nCo2MnSi7. In 2010 Uchida et al. also demonstrated\nthis e\u000bect in the ferrimagnetic insulator LaY 2Fe5O12\n(La:YIG) on Gd 3Ga5O128with the same geometric con-\n\fguration as for the Py/Pt system, which is now referred\nto as the transversal SSE (TSSE), because the detected\nspin current is transverse to the applied temperature gra-\ndient (~JS?rT). Later Uchida et al. presented an alter-\nnative con\fguration for SSE measurements in Y 3Fe5O12\n(YIG)9and Mn-Zn ferrite (Mn,Zn)Fe 2O410, the so called\nlongitudinal SSE (LSSE).\nFor the LSSE, a temperature gradient rTis applied\nperpendicular to the \flm plane in the z-direction (Fig.\n1 (a)) and a spin current ~JS\rows from the magnetic\nmaterial into a Pt \flm. The external magnetic \feld ~H\nis aligned in the x-direction, hence the spin polarization\nvector~ \u001balso lies in this direction for a su\u000eciently highmagnetic \feld. A voltage is measured across the ends of\nthe Pt \flm along the y-direction (Fig. 1 (a)). Here the\ndetected spin current ~JSis aligned parallel to the applied\ntemperature gradient ( ~JSkrT).\nThe anomalous Nernst e\u000bect (ANE)11can be observed\nin the same con\fguration for ferro- (or ferri-) magnetic\nmetals, which is analog to the anomalous Hall e\u000bect\n(AHE)12, but induced by a temperature gradient. The\nANE is given by the formula\n~EANE=\u000brT\u0002~ m: (2)\nHere,~EANEis the electromotive force, \u000ba coe\u000ecient de-\nscribing the magnitude of the ANE, rTthe temperature\ngradient and ~ mthe unit vector of magnetization13. In\nthe LSSE con\fguration, Eq. (1) and Eq. (2) are very\nsimilar and in general there has been no e\u000bort to sepa-\nrate the longitudinal SSE and the ANE of the measured\nvoltage signal so far14.\nFor this reason magnetic insulators like YIG are re-\nquired to distinguish the voltage signal between longitu-\ndinal SSE and ANE, because no ANE is present in in-\nsulators. Another promising candidate to study is nickel\nferrite (NiFe 2O4). This material grows in the inverse\nspinel structure consisting of two magnetic sublattices15.\nRecent optical spectroscopy measurements show an indi-\nrect bandgap of about 1.6 eV in the minority channel and\na direct bandgap of about 2.4 - 2.8 eV for NiFe 2O4\flms\nof thickness between 150 and 270 nm16. Other experi-\nmental data for the optical band gap range from 0.33 eV\nto 3.7 eV for thin \flms as well as for the bulk17{21.\nStudies relating proximity e\u000bects in YIG/Pt struc-\ntures indicate concern about contributions of the ANE\ndue to spin-polarization of Pt at the interface of the\nPt/ferromagnet \flm22,23. Recently, a study of the ANEarXiv:1301.7313v1  [cond-mat.mtrl-sci]  30 Jan 20132\nV!\n∇T!Cu!Cu!\nPt!NiFe2O4!on MgAl2O4!sapphire!plate!\ny!z!x!\nV!\nA!\nAu !stripes!(a)!(b)!\nFIG. 1. (a) LSSE setup. The sample is sandwiched between\ntwo copper plates and a temperature gradient rTis applied.\n(b) The geometry for 4-point-measurement. Two sputtered\nAu stripes establish electric contact to the ends of the sample\nto hold two lines on the same electrical potential.\nin YIG/Pt compares the magnitudes of the ANE and\nLSSE voltage signals by exchanging the temperature gra-\ndientrTand the external magnetic \feld ~H, but in dif-\nferent con\fgurations24. This study shows that the con-\ntribution of the ANE due to proximity e\u000bect is less than\n5% of the voltage signal in the LSSE con\fguration.\nA contribution due to conductivity at the surface of\nthe ferromagnetic \flm is another possibility to generate\nan ANE, which can be much greater. This has to be\nexcluded in the \frst place. In this work the contribution\nto the measured signal due to a conductive surface of\nNiFe 2O4\flms was investigated.\nII. EXPERIMENTAL\nWe prepared nickel ferrite (NiFe 2O4) \flms on\n8 x 5 mm2MgAl 2O4(100) substrates by direct liquid in-\njection chemical vapor deposition (DLI-CVD) with a\nthickness of 1.1 - 1.2 \u0016m15. In the DLI technique a liquid\nsolution of two precursors (Ni(acac) 2and Fe(acac) 3) in\nthe molar ratio of 1:2 and the solvent N, N-dimethyl for-\nmamide (DMF) are vaporized at a temperature of 175\u000eC\nto obtain a \flm growth rate in a range of 10 - 18 nm/min\nat a deposition temperature of 600\u000eC. In another step a\n10nm Pt \flm was deposited by DC magnetron sputtering\nin a vacuum system with base pressure of 5 \u000110\u00006mbar.\nIn an Ar atmosphere with a pressure of 5 \u000110\u00002mbar a\ndeposition rate of 4.2 nm/min was obtained.\nFor the temperature gradient dependent measurements\nthe NiFe 2O4/Pt sample was clamped between two cop-\nper plates25. The lower copper plate was connected to\na Peltier thermoelectric module to heat or cool the sam-\nple from below, which is electrically insulated from the\ncopper by the substrate. The upper copper plate is con-\nnected to a heat bath leading to a heat \row through\nthe sample to generate a temperature gradient rTin\nthe z-direction (Fig. 1 (a)). The sample is electrically\ninsulated from the upper copper plate by a thin sap-\nphire sheet. The temperature di\u000berence was measured\nbetween the upper and lower copper plates with two con-\nnected T-type thermocouples and were stabilized at a\ncertain \u0001T=j(0;0;\u0001T)j/rT. Two tungsten needleswere contacted at the ends of the Pt \flm with a micro-\nprobing system along the y-direction. The voltage signal\nbetween the two needles was measured with a Keithley\n2182A Nanovoltmeter. An external magnetic \feld ~Hwas\napplied in the x-direction (here de\fned as \u000b= 90\u000e) and\nvaried in the range of \u00061000 Oe. Afterwards the angle \u000b\nbetween~Han the y-direction was varied from \u000b= 0\u000eto\n\u000b= 360\u000efor angle dependent measurements.\nIn a similar setup the sample was clamped between\ntwo copper blocks in a cryostat for low temperature mea-\nsurements. It was possible to heat both sides separately\nto generate a temperature gradient perpendicular to the\nsample plane. Two copper wires were bonded with in-\ndium pads on top of the Pt \flm and connected with a\nNanovoltmeter. In the same way a pure NiFe 2O4\flm\nwithout Pt was connected for reference measurements.\nThe lower temperature was measured by a 27 \n rhodium\niron resistance thermometer for temperatures down to\n1.5 K. The temperature on the upper side was measured\nby a thermocouple. The voltage signal of the thermocou-\nple was calibrated with the rhodium iron sensor. In this\nsetup the copper blocks are also electrically isolated by\nthe substrate and by a sapphire sheet.\nFurthermore, the conductivity of the NiFe 2O4\flm\nwithout Pt was characterized in a 4-point-geometry mea-\nsurement for di\u000berent temperatures. Fig. 1 (b) shows a\nsample with two Au-stripes at the ends of the \flm to hold\ntwo point contacts at the same electrical potential. The\nresistance was measured with a Keithley 2000 in the 4-\nwire resistance measurement mode. For higher resistance\nvalues a voltage of 20 V was applied and the current was\ndetected with a Keithley 6487 Picoammeter in a 2-point-\ngeometry.\nIII. RESULTS\nFig. 2 (a) and Fig. 2 (b) show the voltage signal Vat\nthe ends of the Pt \flm as a function of the external mag-\nnetic \feld~H. An asymmetric behavior is observed which\ncan be compared to measurements from other groups on\nYIG/Pt \flms9,14which had the same angle \u000b= 90\u000ebe-\ntween~Hand the y-direction. The magnitude Vsatof\nthese curves (saturation value) de\fned by ( Vmax\u0000Vmin)=2\nis proportional to the temperature di\u000berence \u0001 T(Fig. 2\n(c)).\nThe voltage signal Vsatalso depends on the angle \u000b.\nFig. 2 (f) shows a sinusoidal shape where Vsatvanishes\nat\u000b= 0\u000eand 180\u000eand has its maximum value at \u000b=\n90\u000eand\u000b= 270\u000e. Here, we emphasize, that the line\ndrawn in Fig. 2 (f) is a pure sine-function. Thus, the\nangle dependence of Vperfectly corresponds to the cross-\nproduct of Eq. (1) and Eq. (2). In Fig. 2 (d) and 2 (e)\nmeasurements of Vas a function of ~Hwith a \fxed angle\n\u000bare presented. The magnitude of Vsatgets smaller with\ndecreasing angle \u000band the asymmetric behavior nearly\nvanishes for \u000b= 0\u000e. An explanation for the symmetric3\n-6-4-20246V [µV]-1000-50005001000H [Oe]-10K∆T = 0K-5K-6-4-20246V [µV]-1000-50005001000H [Oe]∆T = 10K5K0K\n-6-4-20246Vsat [µV]-10-50510∆T [K](a)\n(b)\n(c)(d)\n-6-4-20246Vsat [µV]360270180900α [°]∆T = 10K-6-4-20246V [µV]-1000-50005001000H [Oe]α = 0.0°-7.5°-15.0°-45.0°∆T = 10K-6-4-20246V [µV]-1000-50005001000H [Oe]α = 45.0°15.0°7.5°0.0°∆T = 10K\n(e)\n(f)\nFIG. 2. (a), (b) voltage Vas a function of the external\nmagnetic \feld Hfor various temperature di\u000berences \u0001 Tand\n\u000b= 90\u000eat room temperature. (c) the magnitude Vsatis pro-\nportional to the temperature di\u000berence \u0001 T. (d), (e) voltage\nVas a function of Hfor various angles \u000bbetween~Hand\nthe y-direction for \u0001 T= 10 K at room temperature. (f) the\nmagnitude Vsatdepending on \u000bfor \u0001T= 10 K.\n 1.0 0.5 0.0-0.5-1.0µ/µsat-150001500H [Oe]-6-4-20246V [µV]190180170160coercive field [Oe]\n36031527022518013590450α [°]0.700.680.660.640.620.60squareness(a)(b)\nFIG. 3. (a) voltage Vas a function of the external magnetic\n\feldHfor a temperature di\u000berence \u0001 T= 10K (left axis)\nand the magnetic moment \u0016measured via VSM normalized\nto the saturation value \u0016sat(right axis) at room temperature.\n(b) squareness and coercive \feld as a function of the angle \u000b\nbetween the external magnetic \feld ~Hand the y-direction at\nroom temperature. The solid and the dashed curves are lines\nto guide the eye.\ncontribution for \u000b= 0\u000ecould be the magnetic anisotropy\nof the NiFe 2O4\flms as was also demonstrated for YIG\n\flms14.\nTherefore, measurements using vibrating sample mag-\nnetometry (VSM) are performed and presented in Fig. 3.\nThe observed magnetic moment \u0016normalized to the sat-\nuration value \u0016satis compared in Fig. 3 (a) to the ob-\ntained voltage signal Vfor \u0001T= 10K at room tem-\nperature. One can see that the coercive \feld of about\n(165\u000625) Oe is identical with the coercive \feld of the\nvoltage signal. In Fig. 3 (b) the squareness ( \u0016rem=\u0016sat)\nand the coercive \feld of a pure NiFe 2O4\flm is shown as\na function of the angle \u000b. Two magnetic easy axes are\nobtained induced by the fourfold magnetic anisotropy of\n-10-50510V [µV]-5000500H [Oe]T0 = 50KT0 = 100KT0 = 150KT0 = 200K∆T = 17K∆T = 11K∆T = 9K∆T = 8K12840Vsat [µV]403020100∆T [K]T0 = 50KT0 = 150KT0 = 100KT0 = 200K-4-2024V [µV]-5000500H [Oe]∆T = 9K∆T = 13K∆T = 25K∆T = 0KT0 = 100K-2-1012V [µV]-5000500H [Oe]∆T = 40K∆T = 8K∆T = 27K∆T = 0KT0 = 50K(a)(b)\n(c)(d)FIG. 4. (a),(b) voltage Vas a function of the external mag-\nnetic \feld Hfor the mean temperature T0= 50 K and\nT0= 100 K for various temperature di\u000berences \u0001 T. (c) volt-\nageVin saturation ( Vsat) as a function of \u0001 Tfor di\u000berent\nmean temperatures T0. The slope of the linear \fts estimate\nthe ratioVsat=\u0001Tfor a certain mean temperature T0. (d)\nvoltageVas a function of Hfor di\u000berent mean temperatures\nT0and \u0001T= (12\u00065) K.\n10-610-810-1010-12Vsat /∆T [V/K]350300250200150100500T0 [K]10-5 10-3 10-1 101 conductivity σ [1/Ωm]10110-110-310-5conductivity σ [1/Ωm]25201510501/T0 [K-1] x 10-3(a)(b)\nFIG. 5. (a) conductivity \u001bof a NiFe 2O4\flm as a function of\nthe inverted temperature 1 =T0which shows a semi-conductive\nbehavior and the linear \ft in the intrinsic region with a slope\nof about (-1031\u00068) K\u0001\n\u00001m\u00001. (b) the conductivity \u001band\nthe ratio between the saturated voltage signal Vsatand the\ntemperature di\u000berence \u0001 Tas a function of the mean temper-\natureT0.\n-6-4-20246V [µV]-1000-50005001000H [Oe]T0 = 300KNFONFO/Pt∆T = 10K-0.50.00.5 V [µV]-1000-50005001000H [Oe]NFOT0 = 300K∆T = 10K(a)(b)\nFIG. 6. (a) voltage Vas a function of the external magnetic\n\feldHforT0= 300K and \u0001 T= 10K for a pure NiFe 2O4\flm\n(NFO) and a NiFe 2O4/Pt system (NFO/Pt). (b) the curve\nfor the NFO \flm in a larger scale.\nthe cubic inverse spinel structure of the NiFe 2O4\flms.\nThe magnetic easy axes are aligned in \u000b= 45\u000e=225\u000e\nand\u000b= 135\u000e=315\u000edirections due to maximum square-\nness and coercive \feld. These directions correspond to\nthe<110>directions of the crystallographic NiFe 2O4\nstructure. Hence, for \u000b= 0\u000ethe voltage signal for small\nexternal magnetic \felds ~Hgets a symmetric contribution4\nas already discussed for Fig. 2 (d) and 2 (e) when the\nmagnetic moment of the NiFe 2O4changes its in-plane ori-\nentation and \rips into the direction of the next magnetic\neasy axis. For a complete interpretation of the symmetric\ncontribution further investigations are required.\nFigs. 4 (a) and (b) show the voltage Vas a function of\nthe external magnetic \feld ~Hfor the mean temperatures\nT0= 50 K and T0= 100 K and for various temperature\ndi\u000berences \u0001 T. The voltage signal in saturation Vsatin-\ncreases for higher temperature di\u000berences \u0001 T. This is\ncomparable to Fig. 2 (a) and Fig. 2 (b) at room tem-\nperature, but the proportionality between Vsatand \u0001T\nis less obvious because of the large experimental error\nin the determination of \u0001 Tat low mean temperatures in\nthis setup. In Fig. 4 (c) Vsatis shown as a function of \u0001 T\nfor various mean temperatures T0. For eachT0the slope\nof the \ft curve describes the ratio Vsat=\u0001T. For the \ft\ncurve a linear function ( y=a\u0001x) was assumed consider-\ning that for \u0001 T= 0 K the voltage signal should vanish.\nThe obtained slopes decrease for lower mean tempera-\ntures. In Fig. 4 (d) the voltage Vis shown as a function\nof the external magnetic \feld ~Hfor various mean tem-\nperaturesT0. The temperature di\u000berence is in the range\nof \u0001T= 8 K and \u0001 T= 17 K and an experimental error\nof\u0006\u0001T=2 has to be assumed.\nThe conductivity \u001bof a pure NiFe 2O4\flm (Fig. 5\n(a)) shows a linear slope with 1 =T026{28which corre-\nsponds in a semiconductor model with a band gap of\n(0.18\u00060.01) eV and is smaller than the expected value\nof 1.6 -2.4 eV16. Perhaps, the smaller band gap originates\nfrom impurities in the sample. The estimation of the\nband gap from conductivity measurements is sensitive to\nfree charge carriers, which can cause the ANE. There-\nfore, this technique is suitable to estimate the strength\nof the ANE. The conductivity decreases from \u001b=\n4:54 \n\u00001m\u00001forT0= 300 K to \u001b= 19:2\u000210\u00006\n\u00001m\u00001\nforT0= 50 K. This suggests that for high temperatures\nan electrical conduction in the NiFe 2O4\flms and there-\nfore possible occurrence of ANE is present.\nFig. 5 (b) compares the conductivity \u001bas a function of\nthe temperature T0with the ratio of the saturation volt-\nage signalVsatand the temperature di\u000berence \u0001 Talso as\na function of T0. The curves strongly di\u000ber for low tem-\nperatureT0with the conductivity decreasing much faster\nthanVsat=\u0001T. Therefore, the ANE due to the conduc-\ntivity of the NiFe 2O4\flm can be neglected at low tem-\nperatures. Even by assuming an error in the temperature\ndi\u000berence of\u0006\u0001T=2 and\u000610 K forT0, the measured val-\nues ofVsat=\u0001Tcannot be explained by the decrease of\nthe conductivity \u001bwithT0.\nIn Fig. 6 (a) and 6 (b) the voltage signal for a pure\nNiFe 2O4\flm at room temperature is presented. A small\nhysteresis is obtained which corresponds to the ANE in\nthe absence of a Pt \flm on top (Fig. 6 (b)), but the\nmagnitude of the voltage signal in saturation is about 10\ntimes smaller than the signal for the NiFe 2O4/Pt system.\nThe small signal to noise ratio is due to the strong tem-\nperature dependence of the conductivity of the NiFe 2O4\flm. Small variations in the temperature di\u000berence cause\nlarge variations in the o\u000bset voltage compared to the volt-\nage signal due to the investigated e\u000bects. Detailed mea-\nsurements at low temperature are required. However, for\nlow temperatures the conductivity of the pure NiFe 2O4\ndecreases very rapidly.\nFor a quantitative comparison of the ratio of the satu-\nration voltage signal Vsatand the temperature di\u000berence\n\u0001T, Kikkawa et al. introduce the ratio ~Vsat=\u0001Tinclud-\ning geometric factors of the sample24. This coe\u000ecient is\ngiven by the formula ~Vsat=VsatLz=Ly, whereLzis the\nsample thickness in the direction of the temperature gra-\ndient andLyis the length of the sample in the y-direction.\nThe sample thickness is dominated by the substrate and\nis aboutLz= 500\u0016m. The distance between the voltage\ncontacts is about Ly= 8 mm. For T0= 300K a value of\n~Vsat=\u0001T= (30\u000620) nV/K is found which is an order of\nmagnitude smaller than the values Kikkawa et al. found\nfor YIG/Pt with ~Vsat=\u0001T= 220 nV/K.24This coe\u000ecient\ndecreases to ~Vsat=\u0001T= (2:8\u00060:7) nV/K for T0= 50K\nin the NiFe 2O4/Pt system. Uchida et al. have shown\nthat for single-crystalline YIG/Pt an enhanced magni-\ntude ofVsat=\u0001TaroundT0= 50 K appears but not\nfor polycrystalline YIG/Pt.25An enhanced magnitude\nofVsat=\u0001Tcould not be observed for NiFe 2O4/Pt using\ntemperature steps of 50 K for T0.\nFig. 6 (a) shows that the contribution of the ANE in\nthe NiFe 2O4\flm is very small even when the NiFe 2O4\n\flm exhibits a \fnite electrical conduction. Therefore,\nthe ANE in the NiFe 2O4is not the origin of the strong\ntemperature dependence which is shown in Fig. 5 (b).\nOne possible explanation are proximity e\u000bects at the\nNiFe 2O4/Pt interface as described by Kikkawa et al. for\nYIG/Pt24. Another possible origin is a contribution from\nthe spin-dependent Seebeck e\u000bect1. Spin-polarized con-\nduction electrons can also generate a spin current which\nis detected due to the ISHE in the Pt \flm. This e\u000bect\ndisappears in the insulating region of the NiFe 2O4\flm\nat low temperatures.\nIV. CONCLUSION\nIn conclusion, we investigated NiFe 2O4/Pt \flms in the\nLSSE/ANE con\fguration and found similar results as\npreviously observed for YIG/Pt \flms9. A systematic\nstudy of the origin of the e\u000bect was carried out by vary-\ning the temperature gradient and the angle \u000bbetween\nthe external magnetic \feld and the voltage contacts. We\nobserved that the measured voltage signal varies pro-\nportional to the applied temperature gradient and is si-\nnusoidal with the angle \u000b. Conductivity measurement\nof a pure NiFe 2O4\flm shows semiconducting behavior.\nTherefore, the observed voltage in the LSSE con\fgura-\ntion is explained as a superposition of two e\u000bects. On the\none hand, the LSSE caused by a generated spin current\nand detected by the ISHE in the Pt. On the other hand,\nthe ANE caused by the conductive behavior of the mag-5\nnetic NiFe 2O4\flm, which should increase with tempera-\nture similar to the conductivity. Therefore, conductivity\nmeasurements at low temperatures were carried out. A\nvoltage signal and a hysteretical behavior was observed\nat low temperatures as well as at room temperature, but\nwith a smaller magnitude. A comparison between the ra-\ntio of the saturation voltage signal and the temperature\ndi\u000berence on the one hand and the temperature depen-\ndence of the conductivity of the NiFe 2O4on the other\nhand shows a divergence between the two curves. There-\nfore, contribution of the ANE due to the conductivity of\nthe NiFe 2O4at low temperature can be neglected.\nFurthermore, measurements of the ANE on NiFe 2O4\n\flms without Pt showed a voltage signal 10 times smaller\nthan the signal obtained with Pt. Thus, we conclude that\nthe voltage induced in the Pt is mainly due to a spin\ncurrent from the NiFe 2O4into the Pt \flm.\nV. ACKNOWLEDGEMENTS\nThis work was supported by PRESTO-JST \\Phase\nInterfaces for Highly E\u000ecient Energy Utilization\",CREST-JST \\Creation of Nanosystems with Novel\nFunctions through Process Integration\", a Grant-in-\nAid for Research Activity Start-up (24860003) from\nMEXT, Japan, a Grant-in-Aid for Scienti\fc Research (A)\n(24244051) from MEXT, Japan, LC-IMR of Tohoku Uni-\nversity, the Murata Science Foundation, and the Mazda\nFoundation. The work at the University of Alabama\nwas supported by NSF-ECCS Grant No. 1102263. Fur-\nthermore, the authors gratefully acknowledge \fnancial\nsupport by the Deutsche Forschungsgemeinschaft (DFG)\nwithin the priority programme SpinCat (RE 1052/24-1)\nand the Bundesministerium f ur Bildung und Forschung\n(BMBF).\nVI. REFERENCES\n\u0003dmeier@physik.uni-bielefeld.de; www.spinelectronics.de\n1G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature\nMaterials, 11, 391 (2012).\n2K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. 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The total\nmagnetization approaches Lieb's prediction as U=tbecomes large. As we move away from half-\n\flling, this ferrimagnetic phase becomes a ( q1;q1) spiral phase with q1\u0019\u0019and then undergoes a\nseries of \frst-order phase transitions, ( q1;q1)!(q1;q2)!(q1;0), withq2\u0019\u0019=2, before becoming\nferromagnetic at large U=tor paramagnetic at low U=t.\nI. INTRODUCTION\nThe Hubbard model is one of the most studied models\nin the area of strongly correlated electron systems [3, 4].\nHowever, it remains unsolved for dimensionality larger\nthan one. For the one-dimensional (1D) case, the exact\nsolution is given by the Bethe Ansatz [5], while in the\ncase of two dimensions (2D), the solution is known only\nin some limiting cases or by means of approximations,\nsuch as mean-\feld. The fermionic Hubbard model in a\nsquare lattice has long been known to display antiferro-\nmagnetism (AF) at half-\flling [6]. However, away from\nhalf-\flling, the ground state magnetic ordering is still an\nopen problem [7].\nExtensions of the Hubbard model to 2D decorated lat-\ntices also show interesting features, such as \rat band fer-\nromagnetism (F) [2] and Dirac cones [8]. These decorated\n2D lattices fall into three classes: Lieb's [2], Mielke's\n[9] and Tasaki's [10]. The pursuit for metallic ferro-\nmagnetism has motivated the search of crystal structures\nmatching these decorated lattices. However, there are ex-\nperimental obstacles, such as the lifting of the \rat-band\ndegeneracy by the Jahn-Teller e\u000bect or the di\u000eculty in\ncontrolling the \flling of the lattice. An alternative exper-\nimental approach is to study quantum dot arrays with\nthese geometries [11]. Decorated lattices can also be re-\nalized by manipulating cold atoms in optical lattices [12].\nHere, we study one example of a 2D decorated lattice,\nthe Lieb lattice, i.e., a line-centered square lattice [13].\nThis kind of lattice can be obtained from the usual 2D\nsquare lattice by removing a quarter of its atoms (see\nFig. 1). Each unit cell contains one atom of each kind:\nA, B and C. As a matter of fact, real materials can have\ntheir atoms arranged in a fashion resembling the Lieb\nlattice. Examples include the well-known high- Tcsuper-\nconductors with weakly coupled CuO 2planes [14, 15],\nsuch as La 2\u0000xSrxCuO 4and YBa 2Cu3O7, which can be\nstudied using the perovskite lattice, a three-dimensional\n(3D) generalization of the Lieb lattice [16].\nExact results for magnetism in the Lieb lattice are\nknown. For example, an important theorem proven by\nLieb [2] states that bipartite lattices (lattices with two\nsublattices, A and B, such that each site on sublattice A\nB D \nC A FIG. 1. A square lattice can be divided into four sublattices\nA, B, C and D. The circles represent atomic nuclei and the\narrows represent spins. The Lieb lattice can be obtained by\nremoving one of the sublattices.\nhas its nearest neighbors on sublattice B, and vice versa)\nwhose unit cell contains a di\u000berent number of each kind\nof atom, have ferromagnetic ground states at half-\flling.\nThis is the case of the Lieb lattice [17], as each unit cell\ncontains one A atom and two B-like atoms. One com-\nmon argument is that these states are in fact ferrimag-\nnetic [18], in the sense that although each sublattice is\nferromagnetic, the full lattice is antiferromagnetic, but\nthe magnetization is \fnite due to the di\u000berent number of\natoms in each sublattice. This contrasts with the antifer-\nromagnetic ordering of the square lattice Hubbard model\nin this limit. Note that Lieb's theorem only mentions the\ntotal magnetization per unit cell, not on-site magnetiza-\ntion amplitudes, which can be calculated using numerical\nmethods, such as mean-\feld. This has been done for the\nmulti-layer Lieb optical lattice at half-\flling [19].\nIn this work, we use a mean-\feld approach to com-\npute the magnetic phase diagram of the Lieb lattice as a\nfunction of the average electron density nand Hubbard\ninteraction U, thus going away from both half-\flling and\nthe tight-binding limit. The allowed magnetic phases are\nparamagnetism and spin spiral phases [20]. Ferro- and\nferrimagnetism can be obtained as particular cases of spi-\nral phases. Note that we do not consider spatial phase\nseparation. In order to \fnd such regions in the phase\ndiagram, one needs to use the chemical potential as anarXiv:1505.00578v1  [cond-mat.str-el]  4 May 20152\n(a)\n01/32/314/35/3200.511.52\nnnA, nB=nC\n  \nnA\nnB = nC (b)\nFIG. 2. (a) Plot of the tight-binding dispersion relation of\nthe Hubbard model in the Lieb lattice and (b) the respective\nparticle density of each sublattice, A, B or C, as a function of\nthe total particle density.\nindependent variable [21{23], rather than using the par-\nticle density.\nThe tight-binding Hamiltonian of the Lieb lattice, Ht,\nis given by [24]\ntLxP\nx=1LyP\ny=1\u0002\n(Ay\nx;yBx;y+Ay\nx;yCx;y+H:c:)\n+(Ay\nx;yBx;y\u00001+Ay\nx;yCx\u00001;y+H:c:)\u0003\n:(1)\nLx(Ly) is the number of unit cells along the x(y) direc-\ntion. The hopping terms in the \frst line are intra-unit\ncell and the remaining ones are inter-unit cell. The eigen-\nvalues ofHtoriginate three energy bands, one of which is\n\rat. The dispersion relation for periodic boundary con-\nditions is\n\"\u0006=\u00062tr\ncos2kx\n2+ cos2ky\n2; (2)\nfor the non-\rat energy bands, where k\u000b= 2\u0019n\u000b=L\u000bwith\nn\u000b= 0;1;\u0001\u0001\u0001;L\u000band\u000b2fx;yg. The \rat band is LxLy-\nfold degenerate with zero energy. The one-particle local-\nized states associated with the \rat bands can be written\nas\njloc;x;yi=1\n2\u0010\nBy\nx;y\u0000Cy\nx;y+By\nx;y\u00001\u0000Cy\nx\u00001;y\u0011\njvaci:\n(3)\nThese states form a non-orthogonal basis of the \rat band\nsubspace.\nThe three tight-binding energy bands of the Lieb lat-\ntice energy bands are shown in Fig. 2a. At the point\n(kx;ky) = (\u0019;\u0019), the three branches touch each other.\nExpanding the dispersion relation in Eq. 2 around this\npoint, we \fnd the Dirac cones \"2=t2(k2\nx+k2\ny). The \rat\nband is built up from B- and C-type orbitals in equalshares, while the lower and upper bands involve all three\nsublattices A, B and C.\nThe particle density of a sublattice equals the number\nof electrons at that sublattice divided by the number of\natoms the sublattice comprises, or the number of unit\ncells,\nnA=NA\nLxLy\nnB=NB\nLxLy\nnC=NC\nLxLy:(4)\nIn the non-interacting limit and at zero temperature, the\nparticle density on sublattices A, B and C as a function\nof the global particle density (number of electrons in the\nwhole lattice divided by the total number of sites) is as\nplotted in Fig. 2b. The plot can be interpreted as follows.\nHalf of the probability density of the states in the lower\ndispersive band correspond to the sublattice A, while the\nother half is evenly distributed among sublattices B and\nC. Therefore, starting at n= 0, as we insert electrons in\nthe system, half \\choose\" sublattice A, while the other\nhalf go to sublattices B or C. At n= 2=3, we reach the\n\rat band at \"= 0. At this point, all sites A are singly\noccupied, while sites B and C are quarter-\flled. Any\nnewly-added electrons will only go to sublattices B or C,\nbecause the \rat band only comprises these two kinds of\natoms and going to sublattice A would imply going to\nthe upper dispersive band, which would lead to higher\ntotal energy. At n= 4=3, the \rat band is completely\n\flled, so that for n > 4=3 electrons occupy the upper\ndispersive band going to sites A or B/C at a ratio of\n2:1, as in the lower dispersive band, up to the maximum\n\fllingnA=nB=nC= 2.\nIn this work, we address magnetism in the Lieb lat-\ntice by considering a \fnite on-site Coulomb repulsion U\nusing a mean-\feld approach, and build a n\u0000Uphase di-\nagram. In the case of a square lattice, one assumes that\nthe occupation number is the same in the whole lattice.\nHere, in the case of the Lieb lattice, we assume that the\noccupation number on each sublattice is the same as in\nthe tight-binding limit, for any U(see Fig. 2b). This is\nthe correct assumption for small U=t. Moreover, for large\nU=t, the results of Fig. 5 remain qualitatively the same\nfornA=nB=nC=n.\nII. CALCULATIONS\nThe interaction term of the Hubbard Hamiltonian is\nHU=UX\nsitesn\"n#; (5)\nthat is, the on-site Coulomb repulsion Utimes the num-\nber of double occupancies in the lattice. Applying the\nmean-\feld approximation to the Hubbard Hamiltonian\ngives single-particle energies given by the eigenvalues of\nthe 6\u00026 single-particle Hamiltonian HMF[1, 25, 26],3\n0\nBBBBBB@UnA\n2\u0000t(1 +eiky)\u0000t(1 +eikx)\u0000mU\n20 0\n\u0000t(1 +e\u0000iky)UnB\n20 0 \u0000mU\n2e\u0000iqy 0\n\u0000t(1 +e\u0000ikx) 0UnC\n20 0 \u0000mU\n2e\u0000iqx\n\u0000mU\n20 0UnA\n2\u0000t(1 +ei(ky+2qy))\u0000t(1 +ei(kx+2qx))\n0\u0000mU\n2eiqy 0\u0000t(1 +e\u0000i(ky+2qy))UnB\n20\n0 0 \u0000mU\n2eiqx\u0000t(1 +e\u0000i(kx+2qx)) 0UnC\n21\nCCCCCCA;(6)\nplus the diagonal term\nULuc\n4(3m2\u0000n2\nA\u0000n2\nB\u0000n2\nC): (7)\nThis is a generalization of the Hamiltonian obtained in\nprevious studies of the 2D square lattice Hubbard model\n[1, 25, 26], which did not allow for di\u000berent occupations\nin the sublattices. The magnetic phase of the system is\nde\fned by two order parameters: the vector ~ qand the\nnumberm, as in the works by Dzierzawa [1] and Singh\n[25]. The vector ~ q= (qx;qy) de\fnes the orientation of\nthe spins. For example, qx= 0 is a ferromagnetic phase\nalong thexdirection,qy=\u0019represents antiferromag-\nnetism along the ydirection, and other values of qxor\nqygive spin spiral phases. The paramagnetic phase is\n~ q-degenerate and is characterized by zero magnetization\namplitude. The magnetization amplitude mcan be iden-\nti\fed as the amplitude of the spin spiral wave,\nh~S~ ri=m\n2(cos(~ q\u0001~ r);sin(~ q\u0001~ r);0); (8)\nand appears during the mean-\feld calculations, when\ncomputing averages such as\nhAy\n\"A#i=hS+\nAi=hSA;x+iSA;yi=m\n2ei~ q\u0001~ rA;(9)\nfor sublattice A. Fig. 3 shows what the con\fguration of\nthe Lieb lattice looks like when ~ q= (\u0019;\u0019) andmis \fnite.\nB \nC A \nFIG. 3. According to our de\fnition of ~ q(the change in spin\norientation between two consecutive lattice sites), a Lieb lat-\ntice with~ q= (\u0019;\u0019) is ferromagnetic within each sublattice.\nMoreover, because the magnetization amplitude mis the same\nin every site and each unit cell has two spins in the same di-\nrection and one spin in the opposite direction, the total spin\nper unit cell is nonzero, as predicted by Lieb [2].\nFrom this point forward, we consider t= 1, so that Uis\ngiven in units of t. It is important to remark that, exper-\nimentally, although we cannot directly control the valueofU(the on-site interaction), we can control the ratio\nU=t, for example by applying pressure on the sample.\nIII. RESULTS AND DISCUSSION\nThen\u0000Uphase diagram is computed in the following\nway. For each point ( n;U), the number of electrons N\nis well de\fned, so that we can add the lowest Nmean-\n\feld energies and \fnd the total energy of the system.\nBy numerically minimizing this total energy (using, for\ninstance, the algorithm in Ref. [27]) with respect to qx,\nqyandm, we \fnd the values of these three magnetic\norder parameters which lead to the ground state for this\npair (n;U). Repeating this process for all desired pairs,\none obtains the phase diagram.\nA. Magnetization for high U=t\nThe plot in Fig. 4 shows the mean-\feld ground state\nmagnetization amplitude mas a function of nandU.\nThis result is similar to that of the square lattice in most\nregions of the diagram. Indeed, for high U, the magneti-\nzation is proportional to nbetweenn= 0 andn= 1, and\nproportional to 2\u0000nbetweenn= 1 andn= 2, re\recting\nparticle-hole symmetry.\nThis proportionality can be justi\fed analytically in the\nfollowing way. For very high U, the tight-binding terms\nof the mean-\feld Hamiltonian given by Eq. 6 are but\na small perturbation, which can be neglected as a \frst\napproximation. In this case, all sublattices become equiv-\nalent, implying nA=nB=nC=n, witht= 0. Diag-\nonalizing the new Hamiltonian gives two \rat bands. A\nthree-fold degenerate band atU\n2(n\u0000m) and a three-fold\ndegenerate band atU\n2(n+m). Distributing the electrons\nin the bands and adding up their energies, one obtains\nthe total energy of the system by adding the diagonal\nterm3ULuc\n4(m2\u0000n2) (see Eq. 7), so that having positive\nmyields the same bands as negative m. Let us assume\nm > 0. Forn2[0;1], electrons occupy only the lowest\n(degenerate) energy band, with energyU\n2(n\u0000m). The\ntotal energy is then given by\n3ULuc\n4(m2\u0000n2) +U\n2X\nN(n\u0000m): (10)\nThe result of the summation is simply N(n\u0000m) =\n3nLuc(n\u0000m). Minimizing this with respect to mgives4\nm\nnU\nFIG. 4. Mean-\feld ground state magnetization ( m) of the\nLieb lattice as a function of nandU. The plot is very similar\nto that of the 2D square lattice. The most noticeable di\u000ber-\nence between the two is that, while the square lattice has zero\nmin the vicinity of the point ( n;U) = (1;0), the Lieb lattice\nhas \fnitemin this region of the diagram, more speci\fcally\nbetweenn= 2=3 andn= 4=3.\nthe expected result m=n. Performing an analogous cal-\nculation assuming n > 1 yields the relation m= 2\u0000n,\ni.e., the other half of the plot in Fig. 4 for high U.\nB. Magnetization for low U=t\nThe results for the magnetization amplitude min the\nlimitU!0 can be explained using \frst-order pertur-\nbation theory. Let us denote by H0(the unperturbed\nHamiltonian) the tight-binding terms of Eq. 6, that is,\nHamiltonian HMFwithU= 0. Its eigenvalues are\n\u00062tq\ncos2kx\n2+ cos2ky\n2;\n\u00062tr\ncos2\u0000kx\n2+qx\u0001\n+ cos2\u0010\nky\n2+qy\u0011\n;(11)\nand two coincident \rat bands at \"= 0. Using the inter-\naction terms of Hamiltonian 6 as a perturbation yields,\nto \frst order, two key results. Firstly, the \rat bands\nare split into two non-degenerate nearly \rat bands. One\nof them is shifted to positive energy by an amount pro-\nportional to mU, while the other is shifted to negative\nenergy, by the same amount, at each point of the Bril-\nlouin zone. Secondly, the four non-\rat bands are shifted\nbyU\n4(nA+nB). These two conclusions allow us to pre-\ndict the behaviour of mnearU= 0. For the following\ncalculations, one must keep in mind that the diagonal\nterm in Eq. 7 is also to be accounted for.\nWe begin by \flling up the lower bands (which cor-\nrespond to the bands with minus signs in Eq. 11), dis-\ntributing the particles among the sublattices according to\nFig. 2b. For nlower than 2 =3, it is best to keep m= 0\nbecause, up to \frst order, the energy of the two lower-\nenergy bands (associated with the Hamiltonian H0) doesnot depend on m, and having \fnite mwould only increase\nthe total energy due to the diagonal term in Eq. 7. As\nthe total particle density reaches n\u00192=3 (getting closer\nto 2=3 asUapproaches 0), we start to \fll the nearly \rat\nbands at\"= 0. This is the point at which a \fnite mcan\nbe used to lower the energy of one of the \rat bands, thus\nlowering the total energy of the system. After the lower\n\rat band has been \flled (note that \fnite Uinduces some\nmodulation of the \rat bands, but the argument is valid\nfor small perturbations), we are at n= 1 and start \flling\nthe upper \rat band. Now, it becomes advantageous to\nlower the value of m, so as to reduce the energy of this\nband. Finally, at n\u00194=3, only the two higher-energy\ndispersive bands remain empty and between n= 4=3\nandn= 2, the value of mgoes back to zero, for the same\nreason as when \flling the two lowest-energy bands.\nLet us now compare these assertions with our numeri-\ncal results in Fig. 4. At small Uand far from half-\flling\n(outside the ninterval [2=3; 4=3]), the ground state of\nthe system is paramagnetic ( m= 0), coinciding with the\nsquare lattice result (see Fig. 5a). On the other hand,\ninside the interval n2[2=3; 4=3], the square lattice be-\ncomes paramagnetic (except at exactly n= 1, where it\nis antiferromagnetic, and in a very small region around\nn= 1, where a spiral phase arises; the width of this re-\ngion goes to zero as U=t!1 ) while our result suggests\nthat the Lieb lattice has a magnetic ordering other than\nparamagnetism. To know which ordering it is, one needs\nto look at the results for qxandqy.\nC. Magnetic ordering\nFig. 5 shows both the mean-\feld magnetic phase di-\nagram of the Lieb lattice (bottom plot) and that of the\nsquare lattice (top plot), for comparison. The phase di-\nagrams were computed using the independent variables\nnandU, in the range ( n;U)2[0;2]\u0002[0;20], and were\nobtained by joining the results for the three order param-\neters:qx,qyandm. Nearn= 0 andn= 2, the system\nis ferromagnetic for large Uand paramagnetic for low U,\nlike the square lattice, albeit with a wider ferromagnetic\nregion. At intermediate Uandnnear 0.5 or 1.5, the\nsystem displays a (0 ;q2) spiral phase, characterized by\nq2\u0019\u0019=2.\nThe spiral phase characterized by ~ q= (\u0019;\u0019) only oc-\ncurs at exactly n= 1, for any U, as in the square lattice.\nIn the latter, this would be called antiferromagnetism.\nNonetheless, in the Lieb lattice, a ( \u0019;\u0019) phase should\nbe identi\fed with ferrimagnetic ordering [2, 18] (see Fig.\n3). Indeed, the spin-spin correlation in a ( \u0019;\u0019) phase is\nferromagnetic in each sublattice, but antiferromagnetic\nbetween di\u000berent sublattices. The total spin per unit\ncell is \fnite, because mis \fnite at half-\flling (see Fig. 4)\nand the number of sites per unit cell is odd.\nWhen slightly doped away from half-\flling (0 :95.\nn.1:05), bothqxandqycontinuously deviate from \u0019\nand become a ( q1;q1) phase with q1\u0019\u0019. This area be-5\n0.0 0.2 0.4 0.6 0.8 1.005101520\nU\nnFerro\nParam(q,q)\n(0,q)\n(0,π)\n(q,π)\n2.0 1.8 1.6 1.4 1.2\nFerro\nParam(q,q)\n(0,q)\n(0,π)\n(q,π)AF\n(a)\n0.0 0.2 0.4 0.6 0.8 1.005101520\nU\nnFerro\nParam\n2.0 1.8 1.6 1.4 1.2Ferro\nParam(q1,q1)(π,π)\n(q1,q1)(q1,q2) (q1,q2)(q1,0) (q1,0)\n(0,q2) (0,q2)\n(b)\nFIG. 5. Mean-\feld magnetic phase diagrams of (a) the square lattice and (b) the Lieb lattice. In the case of the square lattice,\nthe value of qvaries continuously in the range [0 ;\u0019], in each region labelled as such. In the case of the Lieb lattice, q1\u0019\u0019and\nq2\u0019\u0019=2, and the transitions between regions labelled using q1orq2are discontinuous.\ncomes narrower in the ndirection as Ugrows larger. This\nphase can be interpreted as a ( \u0019\u0000\u000e;\u0019\u0000\u000e) phase with\nsmall\u000e, that is, a local (looking at only a few unit cells)\nferrimagnet with a slow modulation in the direction of\nspins along the lattice. At large U, when further doped,\nthe system undergoes a \frst-order phase transition from\n~ q\u0019(\u0019;\u0019) to~ q\u0019(q1;q2) withq2\u0019\u0019=2, re\recting lo-\ncal antiferromagnetic correlations in the xdirection, and\nsublattice-wise antiferromagnetism in the ydirection. In\nother words, each sublattice is ferromagnetic in the xdi-\nrection and antiferromagnetic in the ydirection. If doped\neven further away from half-\flling, two more \frst-order\nphase transitions occur: \frst to ( q1;0) and \fnally to (0 ;0)\n(ferromagnetism). At regular intervals in n(namely 0.11,\n0.22, 0.33 and their symmetric counterparts), we \fnd fer-\nromagnetic dips into the paramagnetic region. These can\nmost likely be explained using the symmetry of the lat-tice and higher-order corrections.\nIV. CONCLUSION\nIn summary, we have computed and analysed the n-\nUmean-\feld magnetic phase diagram of the Lieb lat-\ntice, and compared it to that of the square lattice. Far\nfrom half-\flling, the two phase diagrams display ferro-\nmagnetism [ ~ q= (0;0)] for high Uand paramagnetism\n(m= 0) for low U, while at exactly half-\flling (one elec-\ntron per lattice site) the ground state is a ( \u0019;\u0019) spiral\nphase for both lattices.\nAlthough the diagrams coincide at n= 1, it is close to\nthat line that their most remarkable di\u000berences arise. In\nfact, at large U, as we move away from half-\flling [(the\n(\u0019;\u0019) phase)], the Lieb lattice undergoes three \frst-order6\nphase transitions ( \u0019;\u0019)!(\u0019;\u0019= 2)!(\u0019;0)!(0;0),\nunlike in the case of the square lattice, where the transi-\ntion from antiferromagnetism to ferromagnetism is con-\ntinuous in~ q. On the other hand, near the tight-binding\nlimit and within the interval n2[2=3;4=3], the magneti-\nzation of the Lieb lattice is \fnite and the ground state is\ncharacterised by spin spirals, contrasting with the para-\nmagnetic ordering of the square lattice in this region of\nthe diagram.\nOur numerical results are in agreement with a theo-\nrem by Lieb [2], which applies to Hubbard models which\ncomprise sublattices with di\u000berent number of sites (in\nour case, we have twice as many B/C sites as we have A\nsites). The theorem states that the ground state of such\na system at half-\flling is ferrimagnetic [18]. According\nto our results, the ground state at half-\flling is charac-\nterized by~ q= (\u0019;\u0019) and \fnite m, which translates into\nferromagnetic sublattices and \fnite total spin on each\nunit cell (see Fig. 3), which is qualitatively consistent\nwith the theorem. According to this theorem, however,\nthe total magnetization per unit cell in the Lieb lattice\nat half-\flling should be 1 for any U. Our mean-\feld\napproach yields that value as Ugrows large but deviates\nfrom 1 at low U(see Fig. 4). On the other hand, this the-\norem is also applicable to a square lattice Hubbard model\nif one divides the lattice into two sublattices. This has\nbeen done before [26] with a square lattice divided into\ntwo sublattices, A and B, with the same number of sites\neach. In consonance with the aforementioned theorem by\nLieb, this square lattice has zero total spin per unit cell\nat half-\flling, for any U, regardless of mbeing zero or\nnot. Therefore, it stands to reason to conjecture that themean-\feld calculations performed for the square lattice\nalso return wrong values for mat lowU, even though the\ncorrect values cannot be deduced from Lieb's theorem, as\nit only predicts the total spin per unit cell.\nThe disparity between our mean-\feld results at half-\n\flling and the prediction of Lieb's theorem may be due\nto two important restrictions that we imposed in order\nto simplify our calculations. Firstly, we assumed that the\noccupation numbers for any Uremain the same as in the\ntight-binding limit ( U= 0), and secondly, we assumed\nthat the magnetization is the same on every sublattice.\nIf it turns out that these two assumptions are indeed the\nreason for the discrepancy, that is, if the Lieb's theorem\ncan be satis\fed in a mean-\feld approach applied to this\npaper's model, albeit with more relaxed constraints, such\na result should be taken into account in any other mean-\n\feld study of interacting fermions in bipartite lattices or\neven more complex lattices whose unit cells contain more\nthan two types of atoms. This is an open question that\nwe intend to address in the future.\nACKNOWLEDGEMENTS\nR. G. Dias acknowledges the \fnancial support from the\nPortuguese Science and Technology Foundation (FCT)\nthrough the program PEst-C/CTM/LA0025/2013.\nJ. D. Gouveia acknowledges the \fnancial support\nfrom the Portuguese Science and Technology Foundation\n(FCT) through the grant SFRH/BD/73057/2010.\n[1] M. Dzierzawa, Z. Phys. B 86, 49 (1992).\n[2] E. H. Lieb, Physical Review Letters 62, 1201 (1989).\n[3] J. G. Bednorz and K. A. M uller, Zeitschrift fur Physik B\nCondensed Matter 64, 189 (1986).\n[4] M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y. En-\ndoh, Rev. Mod. Phys. 70, 897 (1998).\n[5] H. Bethe, Zeitschrift fr Physik 71, 205 (1931).\n[6] R. G. Dias and J. M. B. Lopes dos Santos, Journal de\nPhysique I 2, 1889 (1992).\n[7] M. P. Marder, Condensed Matter Physics (John Wiley\nand Sons, 2000).\n[8] P. R. Wallace, Physical Review 71, 622 (1947).\n[9] A. Mielke, Journal of Physics A 25, 4335 (1992).\n[10] H. Tasaki, Physical Review Letters 69, 1608 (1992).\n[11] H. Tamura, K. Shiraishi, and H. Takayanagi, Jpn. J.\nAppl. Phys. 39, L241 (2000).\n[12] N. Goldman, D. F. Urban, and D. Bercioux, Physical\nReview A 83, 063601 (2011).\n[13] H. Wang, S.-L. Yu, and J.-X. Li, Physics Letters A 378,\n3360 (2014).\n[14] V. J. Emery, Physical Review Letters 58, 2794 (1987).\n[15] R. T. Scalettar, D. J. Scalapino, R. L. Sugar, and S. R.\nWhite, Physical Review B 44, 770 (1991).\n[16] C. Weeks and M. Franz, Physical Review B 82, 085310(2010).\n[17] H. Tamura, K. Shiraishi, and H. Takayanagi, physica\nstatus solidi (b) 224, 723 (2001).\n[18] A. Mielke and H. Tasaki, Communications in Mathemat-\nical Physics 158, 341 (1993).\n[19] K. Noda, K. Inaba, and M. Yamashita, Phys. Rev. A\n90, 043624 (2014).\n[20] S. Sarker, C. Jayaprakash, H. R. Krishnamurthy, and\nW. Wenzel, Physical Review B 43, 8775 (1991).\n[21] E. Langmann and M. Wallin, Journal of Statistical\nPhysics 127, 825 (2007).\n[22] P. A. Igoshev, M. A. Timirgazin, A. A. Katanin, A. K.\nArzhnikov, and V. Y. Irkhin, Phys. Rev. B 81, 094407\n(2010).\n[23] W. Schumacher, Physica Status Solidi (b) 119, 235\n(1983).\n[24] M. Nita, B. Ostahie, and A. Aldea, Physical Review B\n87, 125428 (2013).\n[25] A. Singh, Z. Tesanovic, and H. H. Kim, Pramana - J.\nPhys. 38, 211 (1992).\n[26] J. D. Gouveia and R. G. Dias, Solid State Communica-\ntions 185, 21 (2014).\n[27] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E.\nWright, SIAM Journal of Optimization 9, 112 (1998)."    },    {        "title": "0808.2015v2.Co_resonant_enhancement_of_spin_torque_critical_currents_in_spin_valves_with_synthetic_ferrimagnet_free_layer.pdf",        "content": "Co-resonant enhancement of spin-torque  critical currents in spin-valves \nwith synthetic-ferrimagnet free-layer \n \nNeil Smith, Stefan Maat, Matthew J. Carey, Jeffrey R. Childress . \nSan Jose Research Center, Hitachi Globa l Storage Technologies, San Jose, CA 95135 \n \nIt is experimentally shown that the critical current for onset of spin-torque instability in \ncurrent-perpendicular-to-plane spin-valves can be strongly enhanced using \"synthetic \nferrimagnet\" free-layers of form FM 1/Ru/FM 2 (FM=ferrromagnet). However, this \nenhancement occurs for only one polarity of bi as current. A two-macrospin model is shown \nto reproduce the observations. The model sugge sts that this phenomenon is related to a \npolarity-dependent, spin-torque induced co-resonance betw een the two natural dynamic \nmodes of the FM 1/FM 2 couple. The resonance c ondition facilitates energy  transfer out of the \nspin-torque destabilized mode into the othe r stable mode whose effective damping is \nactually enhanced by spin-torques, thereby delayi ng the onset of instability of this coupled \nsystem to larger critical currents. \n Spin-torque phenom ena, as m anifested in  giant-m agnetores istive spin-v alves film stacks \nlithographically patterned in to ~100 nm  nanopillars and driven with electrical cu rrents perpendicular to \nthe film  plane have in recen t years been the activ e study of num erous theoretical and experim ental \npapers,1 both for their novel physics as well as potential  applications for m agnetic m emory elem ents, \nmicrowave oscillato rs, and m agnetic recording  read heads. In essen tially a ll of  these s tudies, the  \ndynam ically active m agnetic layer, or \"free layer\" of  the spin-valve film stack, is either theoretically \nmodeled or experim entally fabricat ed as a single ferrom agnetic layer.  This paper investigates, through \nboth experim ental m easurem ent and theoretical modeling, the novel spin -torque dynam ics of a  \n\"synthetic-ferrim agnetic\" free-layer of the for m FM1/Ru/FM2, consis ting of two ferrom agnetic (FM) \nfilms of une qual thickness  separated by a thin (0.8 nm ) Ru spacer which prom otes well-\nknown2FM 1MF t t >>\n2, strong antiparallel coupling betw een the two FM layers. C ompared to the sim ple free-layer  \nsystem , the FM1/Ru/FM2 couple has two addition al spin-torque-pro ducing (Ru/FM) interfaces, and  \nperm its (in the sim ple macrospin pi cture) two independent, non-degenerate, natural modes of oscillation. \nAs will be d iscussed  below, these features  can lead to a novel conditio n of spin-to rque-induced \"quasi-\nco-resonance\" of these two m odes which greatly im pacts the spin-torqu e-stab ility o f such devices, and  \nwhich carries poten tially im portan t practical im plicatio ns f or the ir use  in the af orem entioned  \napplic ation s.  \nThe presen t exper iments use m ultilayer f ilms of form AFM/PL/Ru/RL/Cu/FL1/Ru/FL2 (exc luding \nseed and cap layers). The first ferrom agnetic pi nned-layer (PL) is exchange-pinned to the  \nantiferrom agnetic (AF M) layer, and is  also strongly antiparallel-couple d to a second FM reference-layer \n(RL) across a thin Ru spacer. The PL and RL layers  are clo sely m oment-m atched, for ming a \"synthetic-\nantiferromagnetic \" cou ple (as is c ommon practic e for such devices) which consequently does not \nrespond to a m odest external m agnetic fields. More uni que to the present structur es, the first free-layer \n(FL1) is also antiparallel-coupled to a second fr ee-layer (FL2), for ming the \"synthetic-ferrimagnetic-\nfree-lay er\" (SFM-FL) with shee t-film M-H  behavior (at modest external fields) equivalent to a single \nFM f ilm of thickness .  2FL 1FL t t−\nA first set of experim ental m easurem ents, desc ribed in Fig. 1, uses Ni Fe f ree-lay ers with  \n including a contro l with  = 0, and two SFM-FL designs with  = 2nm  and 3nm . \nThe devices tested have been patterned into 75-nm  circular pillars using E-beam  lithography.2FL 1FL nm4 t t + =2FLt2FLt\n3 \nResistan ce (R-H) loops are m easured (at -5m V bias) in fields collin ear  and transverse  to \nthe IrMn pinning direction. All chosen  devices have nonhysteretic, square  (for )(xH )(yH\nxHR- kOe1≤xH ) and \nnear-symm etric (about ) . Accom panying each R-H data set ar e two  loops, which  0=yHyHR-eIN-Fig. 1. (a) cartoon of devi ce geom etry. (b-d) R-H loops (as % δR/R) an d  N-I e  loops (as rm s power spectral \ndensity at 75 M Hz); for  tFL2 = 0 (b) , 2nm  (c), an d 3nm (d) .  Spin-valve stack structure: \nIrMn(7)/Co Fe(3)a/Ru (0.7)/CoFe(3)/Cu(4)/ NiFe(4+  tFL2)/Ru(0 .7)/NiFe (tFL2);    ( ) de notes film thickness  in nm. \n \nmeasure narrow-band noise N vs electron  curren t  with constant applied fields of either \n or -600 Oe to align FL1 m agnetization either an tiparallel (A P) or para llel (P) with that of \nthe RL. Positiv e ele ctron curren t travels f rom RL to  FL (Fig. 1a). Th e curr ent is  driven by a 2-Hz \nsawtooth generator with s ync pulse triggering the 0.5- sec sw eep of a (zero-span ) spectrum  analyzer. The \n loops are averaged over  sweeps. The eI\nOe600+≅xH\neIN- 50≈ 0≅eI  electronics n oise ( Hz nV/8.0~ ) is \nsubtracted out.  \nThe  techniqueeIN-4 measures the 1/ f-like noise as sociated with therm al perturbations of well-\nknown precessional m otion of a unid irectionally stable FL once sp in-torque instability begins.5. This \nonset is readily observed by the sh arp increase in noise above the /mAHz nV/03.0≈  residual \nelectronics noise for th ese  devices . The \"critical currents\",  , for this onset are found by \nsimple inspection. The  were typically insensitive to  few hundred Oe variations in . Ω≈11crit\neI\ncrit\neIxH\nFor all th icknesses of  FL2, there  is an observed  AP-state  negative  critical poin t  \nwhich was previously shown to be sp in-to rque-instab ility o f the RL/PLmA5.22crit\nAP -≈ −eI\n5,6. The SFM-FL devices alone \nshow an addition al positive  critical p oint  in the P-state, which is discussed further below. For the crit\nPeI+(a) \n0123\nvs. Hx\nvs. HytFL1=4 nm\ntFL2=0 nm\n(%)δR\nR\n0123\nvs. Hx\nvs. Hy(%)δR\nRtFL1=6 nm\ntFL2=2 nm\n-2 -1.5 -1 -0.5 0 0.5 1 1.5 20123\nH (kOe )vs. Hx\nvs. Hy(%)δR\nRtFL1=7 nm\ntFL2=3 nm00.2AFMx\nyzHa\nIe\nPL RL FL1 FL275 nm\n0.40.60.81\nnV\nHzPSD\nPAPH  =  ! 600 OextFL1=4 nm\ntFL2=0 nm\n00.20.8(b) \n0.40.61\ntFL1=6 nm\ntFL2=2 nmnV\nHzPSD\nPAP\nH  = ! 600 Oex\n-6-5-4-3-2-1 012345600.20.8(c) \n0.40.61\nnV\nHzPSD\nI   (mA)ePAPH  =  ! 600  OextFL1=7 nm\ntFL2=3 nm\n(d) 02FL=t  control, the polarity asymmetry ratio  is s imilar to earlier \nobservations,35.2 ) /() (1FL 1FLcrit\nAPcrit\nP - ≈ + −e e I I\n4-6 and is a known consequence of the intrinsic angular dependenc e of electrical transport in \nthese a ll-metallic dev ices. It is unre lated to th e unexpected polarity asym metry discussed below.  \nFig. 2 shows a summary of si milar  measurements for a second, modified film stack which \nincludes thin CoFe layers at the Cu/FL1 and FL1,2/Ru interfaces.eIN-\n7 Referenced to Ni 80Fe20 film s of equa l \nmom ent, the m agnetic thicknesses of FL1,2 ar e sim ilar to those shown in Fig. 1.  \nFig. 2.  P-state N-I e  loops (rms po wer sp ectral d ensity); for spin-valve stack ::  \nIrMn(7)/CoFe(3 )/Ru(0 .6)/CoFe(3 )/Cu(5)/CoFe(0.6)NiFe(4+ tFL2) /CoFe(0.2)/Ru(0.6)/CoFe(0.2)/NiFe(tFL2);      \n( ) denotes film th ickness in nm.. In Ni80 Fe 20 equivalent Angstroms thickness, tFL1 = tFL2+45, tFL2 as indicated. \n \nFig. 3 summ arizes the experim ental results for  for both stack structures, which includes 3 or 4 \ndevices for each value of  , for which the data in F igs. 1 and 2 are representative.  crit\neI\n2FLt\nFig. 3. Critical current  vs.  tFL2  for film stacks from  Fig. 1 (a),  and  Fig. 2 (b).  Solid circles (squares) for     \nP-state (AP-sta te).  Solid curves are m odel results as described in text, using η-coefficien ts as ind icated in  \nfigures. Dashe d curves are for η2 = η3 = 0,   but which exclude IePcrit  > 0. \n  -9-8-7-6-5-4-3-2-1012345678900.20.40.60.81\nnV\nHzPSD\nH  =  -600 Oex\nI   (mA)eP-state\n00\n5515152525\n0 0.5 1 1.5 2 2.5 3 3.5-8-6-4-2024\n(nm)Iecrit\n(mA)\ntFL2P-state\nP AP η   =0.25, η    =0.75, η  =0.75, η  =0.61 1 2 3\nH  =  ! 600 OexAP-state\n0 0.5 1 1.5 2 2.5 3 3.5-10-4-2024(a) \n-8-6Iecrit\n(mA)\n(nm) tFL2P-state\nη   =0.26, η    =0.65, η  =0.4, η  =0.451P\n1AP\n2 3\nH  = ! 600 OexAP-state\n(b) In both cases , there is a striking increase in the magnitude of with incre asing , which is \neven more dramatically demons trated for the second, Co Fe-int erfaced FL stack structure.  In stark \ncontrast to these observations regarding , it is  seen that  shows rather  little abso lute \nchange with . These tw o counter-intuitive resu lts agree well with analogous dV/dI measurem entscrit\nPeI−2FLt\ncrit\nPeI−crit\nAPeI+\n2FLt7 \nusing test devices (sam e film  stack as in Fig. 2) of  similar area but asymm etric non-circular geom etry. \nFor a theoretical insight into this phenom enon, we start w ith a sim ple two-m acrospin m odel for  \nFL1/FL2 which tr eats the RL/PL as an iner t, spin-current polarizer. Magne tostatic, Ru-couplin g and \nZeem an term s for the system  free energy E are taken to be \n] ) /() ( [ˆˆ ) /(\n2 12 1 ru2\n1,;21\n1FL 2FL1FL\nx s s x aku\nkjuu\njk ju s\nm tM tM mHH mHm VME\n+ −⋅ + =∑\n=mm\n                                            (1) \nwhere  are the unit m agnetization vectors fo r FL1,2, the set of  tri-ind icied  are \nmagnetostatic energy coefficients,  is th e interfacial cou pling s trength, , \n is the  app lied f ield, and  is th e volum e of FL1. Slonczewski-typ e2,1ˆmzyxu\njkH,,\n2,1=\n=\nruJ1FL) /(ru ru tM J Hs ≡\nx H ˆa aH=1FLV8 spin-torques \n at the Cu/F L1, FL1/Ru, and Ru/FL2 interfaces are included as follows:  mH ˆ ) (ST\n1FL × = VMs τ\n \nST\n1FL1FL ST STSTST STST\nˆ/ ) (/1) /()2/( ;ˆ ˆˆ ˆ )ˆ(ˆ\neff\n2,11 2 3 21 2 2 1 1 1\nj j s js e\nE VMVMIe H HH H\nHm Hmm Hmm x m H\n+ ∂∂ −=≡ × η=× η+ ±× η=\n=h                                  (2) \n \nto form the tota l effective f ield . In (2), eff\n2,1=jH xˆ± refers  to (P and AP-states), and the  RLˆm 0=eI  \nequilibrium  state now defined to be x m ˆ ˆ01=, x m ˆ ˆ02 −= . The η-coef ficients will b e discuss ed below. \nThe additional (two) degrees of freedom  of th e second m acrospin substantially com plicates the \nalgebraic description relative to the well-known 1- macrospin ca se.. As described previously,5,9 \nindividual, local  coordinates where zyx′′′02 01ˆ ˆ ˆ mx m =′=  are u sed to constru ct the f ollowin g matrix \nformulation  of the linearized  Gilbe rt equ ations of  motion f or the two-dim ensional vectors \n: ) , (21 zj yj ,j m m ′ ′ == ′my z\nay z\naz yvu vkkv\nkvju\njuuj\njk j jvu\njkjkj\njk jkj\njk\nH H H H Htt H HH H H H H H HH H H H H HH HH HHH HH HHH HH HHH HH HHmm\nmH\nmm\nHtt\nDtt\nGtdtd\n22 22 7 8 21 3 711 11 5 6 3 512 ru 4 12 ru 3 2 18 33 7\n22\n4 33 3\n214 22 3\n12\n65\n11,eff\neff1 121\n, )/(,, ,,,)ˆ (1001,0110,0)( ) (\nSTST\nSTSTSTST\nSTST\n− + ≡ − ≡− + ≡ + ≡+ = + ≡ η+η±≡η′⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nη−η≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nηη −≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nη−η− −≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nη′η′−≡′∂∂\n∂∂\n∂′∂\n−δ ⋅ ≡δ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nγα\n≡ δ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−\nγ≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n′′≡′ =′⋅+′⋅ +\n∑\n′′ ′′\nt tt tt tt tt\nmHmmm mHmGD\n                                              (3) \n \nwhere gyrom agnetic ratio Oe) Mrad/(sec19 - ≅γ , and 0>α  is the Gilb ert dam ping param eter. In  \n(2),vu\njkH′′⇔Ht\n is a  tensor-m atrix f ormed from  the 2D Cartesian tensors 44×jkHt\n given  explicit in (3 ), \nand sim ilar for Dt\n and G. The expressions for  assum e the symm etry . t\n7,5Hy\njkx\njkH H =\nThe natural modes for this system  are nont rivial solutions of  (3) of the for m . The 4 \nroots  satisfy stet−∝′)(m\n2 2 1 1 , ω±σω±σ= i i s 0|) ( |det = + − GDHttt\ns , but are more generally found f rom the \neigenvalues of the matrix .1) (−+⋅GDHttt9 Only two of  these roo ts, 2,1 2,1 2,1 ω+σ= i s , describe physically \ndistinct m odes. The spin-torque terms in (2) yield a nonreciprocal Ht\n (i.e., ), which can be \nshownuv\nkjvu\njk H H′′ ′′≠\n10 to perm it unstable  modes with 0) Re( <s . This corresponds physically to the spin-torque-\ninstab ility s een in the  of Figs. 1,2. Com puting , the least  value of eIN- ) ;(2FL 2,1 tIse eI± where \n yields the model-predicted  curves shown in Fig. 3. 0), min(2 1 →σσ ≡σ< ) (2FLcritt Ie\nThe com puted  vs.  assum ed a NiFe satu ration  magnetization , AF/Ru-\ncoupling , , and crit\neI2FLt emu/cc800=sM\n2\nru erg/cm1=J Oe 600=aH 02.0=α . The  were com puted (analytically) from  the \ninteraction energy between uniform ly magnetized rectangular solids (u\njkH\n65≅  nm squares with area equal to 75 nm  circles). Valu es of ,  describe the spin-torque am plitude at the Cu/FL1 interf ace (in P or \nAP states ), and were cho sen to  match the m easured  for con trol devices with . The P-s tate \nvalues are close to th e theoretical estim ateP\n1ηAP\n1η\ncrit\neI 02FL→t\n5,8 , with 2/P\n1β≈η ≈β0.65 roughly the m ean s pin-\npolarization between NiFe and CoFe.11  \nThe (non-unique) values of  and 2η3η, which govern the spin-torque am plitude at the FL1/Ru and \nRu/FL2 interfaces, were chosen to obt ain an eyeball fit with all o ther  vs.  data. The good \nagreem ent between data and m odel in Fig. 3 is obtained with constant  coefficients that are \nindependent of  (  also independent of P or AP) , while  usin g a sim ple magnetic m odel with  \nsome imprecisely known param eter values, e.g.,  and crit\neI2,1FLt\n31−η\n2,1FLt3,2η\nruJ α. The level of agreem ent is perh aps \nfortuitous, though the essential phys ics is believed robust to any s hortcom ings of the model.  \nAn  model can perhaps provide a plausible fit to the  and  NiFe-FL \ndata in Fig. 3a, but this is not so f or the CoFe/NiFe/CoF e-FL data o f Fig. 3b. In eithe r case, it is \nincapab le of accountin g for the S FM-FL data with , which originates from  spin-torque-\ninstab ility at the Ru/FL2 inte rface. Fitting the full m odel t o this data required a value 03 2 =η=η 0crit\nP<eI 0crit\nAP>eI\n0crit\nP>eI\n4.03≥η .  In \naddition, the m odel fails for the  data in Fig. 3b at 0crit\nP>eI nm5.02FL=t , which m ay reflect excess  \ndamping as , and/or a breakdown (due to finite transverse  spin-diffusion length 02FL→t12) of the purely \ninterfacial f orm of spin-torque im plicitly assum ed in (2). \nIn the lim it z z y y m m m m J ′ ′ ′ ′ −→ → ⇒∞→1 2 1 2 ru ,  one m ay obtain from  (3) the following \nanalytica l solutions f or :  crit\neI\n∑\n=+− − + −≡η+ η−α\n⎥⎦⎤\n⎢⎣⎡\n−+→\n2\n1,2101 10 crit\nAP)or(P\n) ()1( )/ 1() or ( e2/) (\n1FL 2FLAP P1FL\n2FL 1FL2FL 1FL\nkjy\njkz\njkkj\nas\ne\nH H Ht t HH VM\nt tt tIh\n                                    (4) \nwhich also a pplies when  With .02FL=t2FL 1FL t t−  fixed, (4) indicates (excluding any -dependence of \n) a  scalin g, but equa lly so for  eithe r  or . The la tter sharp ly \ncontradicts experim ent. Further, (4) ex cludes the additional observed cases of  when FLt\n0H1FL 2FL 1FL ) ( t t t ⋅ +crit\nPeI−crit\nAPeI+\n0crit\nP>eI.02FL>t The underlying physics behind the asymmetric , strong super linear (or weak) -dependence \nof  (or ) would thus appear connect ed with the finiteness of . 2FLt\n0crit\nP<eI 0crit\nAP>eIruJ\nThis is f urther elucid ated in Fig.4. Com puted a s a continuous function of  are critical currents  \n, as well as natura l-mode param eters ruJ\n)(rucritJ Ie ) (21\n< >ω−ω ≡∆πf  and ), max(2 1σσ ≡σ>  evaluated at \n. In the case of negative   in Figs. 4a ,b, the spin -torque term s radic ally alter th e \nnatural oscillation frequencies , even to the point of inducing a literal co-resonance , i.e., )(rucritJ I Ie e= 0crit\nP<eI\n2 1,ωω\n0) (21→ω−ω ≡∆< > πf , between th e two m odes at a finite . Closely accom panying the co-resonance  \nis a broad peak in both  and ruJ\n)(rucrit\nPJ Ie− )(ruJ>σ , with a large  maximum at .  max\nru ruJ J≡\nThe  are th e tem poral decay rate s (or line-wid ths) of the stable  mode. Here, spin-torques can \nincrease  the rate of energy loss from  that m ode of osci llation well beyond that of intrinsic dam ping (e.g., \n at .). The th ird (dam ping) m atrix >σ\n1Gsec4−\n>≈σ 0=eI Dt\n in (3), along w ith nonreciprocal spin-torque  \n0 02 04 06 08 1 12 1403691215\n01224364860\n(mA)−IecrittFL1=6.0 nm\ntFL2=1.5 n m\n(GHz)∆f\nor\nσ>P-state\nH  =  -600 Oex\n0 02 04 06 08 1 12 1(a) \n0510152025\nFig. 4. Mo deled  values for critical cu rrent (solid line), ∆f (dashe d line), and σ> (dotted  line)  v s Jru ,  for  \nparameter values ind icted in tex t and/or figure. η-coefficients sam e as use d in Fig. 3b. \n 401020304050\n(mA)−IecrittFL1=7.0 nm\ntFL2=2.5 nm\n(GHz)∆f\nor\nσ>P-state\nH  =  -600 Oex\n0 02 04 06 08 1 12 1(b) \n012345\n40510152025\n(mA)tFL1=7.0 nm\ntFL2=2.5 nm\n(GHz)∆f\nor\nσ>AP-state\n+IecritH  =   +600 O ex\n0 0.2 0.4 0.6 0.8 1 1.2 1.4048121620(c) \n01020304050\n(mA)−Iecrit\nJru(erg/cm  )2tFL1=7.0 nm\ntFL2=2.5 nmP-state\nH    =H    =  600 O ek1 k2\n(GHz)∆f\nor\nσ>\n(d) contributions to Ht\n, imply that the  two natura l modes are  non-orthogonal, and are coupled both by spin-\ntorques and weakly by intrinsic dam ping. This dynam ic coupling allows energy tr ansfer between m odes, \nwhich is further strongly enhanced at and near  the condition of  co-resonance. However, this \nenhancem ent does not strictly require , but can occur under more general conditions of \"quasi -\nco-reson ance\" where , i.e., when the difference in  the m odes' resonant frequencies is sm aller  0→∆f\n1 /<σ∆>f\nthan the ef fective line -width  of the dam ped mode. This enhanced inter-m ode coupling provides  >σ\nanother energy loss path (in addition to intrinsic dam ping) to counter the positive rate of work by spin-\ntorques on the destabilized m ode, delaying onset  of spin-torque-instability and increasing . It is \nthus not surprising that the -depend ence of  closely follows  that of . Further,  the \nbroad peaks of  in Figs.  4a,b approxim ately coincide  with the quasi-co-resonant condition \n. Finally, since , the broad tail of  and the m onotonic increase \nwith  of  (e.g., com pare Figs. 4a and 4b) yields an explanation for the supe rlinear inc rease of \n with .  crit\nPeI−\nruJcrit\nPeI− )(ruJ>σ\n)(rucrit\nPJ Ie−\n1 /<σ∆>fdevice\nrumax\nru J J < ) (max\nru ruJ J> σ>\n2FLtmax\nruJ\ncrit\nPeI−2FLt\nBy contras t, for positiv e  modeled in Fig. 4c, the co-res onance condition does not occur, and \nthe spin -torque term s actua lly r educe  below  that of  intrinsic dam ping ( ). \nAccordingly ,  shows little  enhanc ement with the SFM -FL design,  and the  model add itionally \nindicates a moderate reduction  in  relative to th e fictitious c ase crit\nAPeI\n) (crit\neI>σ1Gsec4−≈\ncrit\nAPeI\ncrit\nAPeI 03,2= η  (Fig. 3b). Finally, Fig. 4d \nshows results for a bi-stable  SFM-FL with uniaxial anisotropy Oe600=kH  in FL 1,2 replacing the \nextern al field (i.e.,  for t he  term , k a H H→5Hk a H H −→  for  in (3)). Although  \nagain resembles  in shape, there is no co-reson ance nor superlinear enhancem ent with of \n, which at  is about 10% less than predicted by the  model of Fig. 3b. \nThis m odeling resu lt is consis tent with 7H )(rucrit\nPJ Ie−\n)(ruJ>σ2FLt\ncrit\nPeI−2 device\nru erg/cm1≅ J 03,2= η\n0=aH  dV/dI measurem ents7 on non-circular devices with \nmagnetostatic shape anisotropy. \nThe las t result in Fig. 4d clea rly indicates a connection between observable qu asi-co -reson ant \nenhancem ent of , and the presence o f an external field antipa rallel to  (although all states in \nFig. 4 are m agnetostatically  stable at crit\nPeI−2FLˆm\n0=eI  with ). This situa tion would n aturally 2\nru erg/cm1.0>Joccur in practice for a current-per pendicular-to-plane giant-m agnetoresistive m agnetic read sensor, \nwhere the FL is conventionally stabilized by uni directional fields from  abutted perm anent m agnet \nlayers.13 The increase in bias cu rrent (while m aintaini ng device stability) afforded by use of the SFM-FL  \nthus has ready application for im proving sensor output signal for future read heads in hard disk drives.7 \n \nREFERENCES \n \n1 D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mat. 320, 1190 (2008) and m any references therein. \n2 S.S.P. Parkin, N. More and K.P. Roche, Phys. Rev. Lett., 64, 2304 (1990). \n3 D. Lacour, J.A. Katine, N. Sm ith, M.J. Carey and J.R. Childress, Appl. Phys. Lett. 85, 4681 (2004) \n4 N. Sm ith, J.A. Katine, J.R. Childress, M.J. Carey, IEEE Trans. Magn. 41, 2935,(2005). \n5 N. Sm ith, Phys. Rev. B 74, 026401 (2006) . \n6 J.R. Childr ess,  M.J. Carey, S.I Kis elev, J.A. Katine, S.Ma at, and N. Sm ith, J. Appl. Phys. 99, 08S305 \n(2006). \n7 M.J. Carey, N. Sm ith, S. Maat, and J.R. Childress, arXiv:cond-m at/0808.2001. \n8 J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);  247, 324 (2002). \n9 N. Sm ith, J. Magn. Magn. Mater. (2008), doi: 10.1016/j.jm mm.2008.05.009  \n10 N. Sm ith,  arXiv:cond-m at/0406486. \n11 J. Bass, and W . P. Pra tt Jr., J. Phys.: Conden. Matter 19,  183201, (2007) \n12 W. Chen, M. J. Rooks, N. Ruiz, J. Z.  Sun, and A. D. Kent, Phys. Rev. B, 74, 144408 (2006). \n13 C. H. Tsang, R. E. Font ana, T. Lin, D. E. Heim . B. A. Gurney , M. L. W illiam s, IBM J. Res. Develpt. , \n42 103 (1998). \n "    },    {        "title": "1107.5891v1.Ferrimagnetism_and_spontaneous_ordering_of_transition_metals_in_La2CrFeO6_double_perovskite_films.pdf",        "content": "1 Ferrimagnetism and spontaneous ordering of transition -metals in La 2CrFeO 6 \ndouble -perovskite films  \n \nS. Chakraverty,1 A. Ohtomo,2,* D. Okuyama,3 M. Saito,4 M. Okude,1 R. Kumai,5 T. Arima,6 Y. \nTokura,3,7 S. Tsukimoto,4 Y. Ikuhara,4,8 and M. Kawasaki1,3,4, 9 \n \n1Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan  \n2Department of Applied Chemistry, Tokyo Institute of Technology, Tokyo 152-8552, Japan  \n3Cross -Correlated Materials Research Group, RIKEN Advanced Science Institute, Wako 351 -0198, \nJapan  \n4WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan  \n5National Institute of Advanced Industrial Science and Technology, Tsukuba 305 -8562, Japan  \n6Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980 -8577 , \nJapan  \n7Department of Applied Physics, University of Tokyo, Tokyo 113 -8656, Japan  \n8Institute of Engineering Innovati on, University of Tokyo, Tokyo 113 -8656, Japan and \nNanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 456 -8587, Japan  \n9CREST, Japan Science and Technology Agency, Tokyo 102 -0075, Japan  \n \n(Dated: 7/29/2011 ) \n \nAbstract  \nWe report on atomic ordering  of B -site transition -metals and magnetic \nproperties of  epitaxial La 2CrFeO 6 double -perovskite films  grown by \npulsed -laser deposition under various conditions .  The highest ordered sample \nexhibited a fraction of antisite -disorder of only 0.05 and a saturation \nmagnetization of ~2B per formula unit at 5 K.  The result is consistent with \nthe antiferromagnetic ordering  of local spin moment (3 d3\n↓3d5\n↑; S = -3/2+5/2 = \n1).  Therefore,  the magnetic ground state of La 2CrFeO 6 double -perovskite that \nhas been long debate is unambiguously revealed to be ferrimagnetic .  Our \nresults present a wide opportunity to explore novel magnetic properties of \nbinary transition -metal perovskites upon epitaxial stabilization of the ordered \nphase.  \n \n \n                                                           \n* Electronic mail: aohtomo @apc.titech. ac.jp  2 I. INTRODUCTION  \nComplex oxides exhibiting high -Curie temperature ( TC) ferromagnetism have been attracting \nrenewed attention  for various spin -coupled device applications.1  When 3 d and 4 d or 5 d \ntransition -metals are combined in a perovskite oxide, they tend to occupy the octahedral site \nalternately along the [111] direction to form a NaCl -type lattice.  This so -called double -perovskite \nstructure is expressed as A2M’M”O6 where A is an alkaline - or rare -earth element and M’ and M” are \ndifferent transition -metal elements.2  Sr2Fe3+Mo5+O6 and Sr 2Cr3+Re5+O6 are well -known examples \nowing to their half -metallic nature as well as exceptionally high  TC.3,4  For such ideal \ndouble -perovsk ites, a large difference in the formal valence (FV)  permits spontaneous ordering of \ntransition -metal elements, thus facile to synthesize in a bulk form.  Among the 3 d-3d combinations, \nhowever, only a few are known to form double -perovskites suc h as La 2Mn4+M”O6 (M” = Co2+, Ni2+, \nFe2+)5,6 where difference in the ionic radius  as well as FV is exceptionally large.  \nAlthough  the spontaneous ordering is not expected for the combination of isovalent 3 d-3d ions,2 \nLa2CrFeO 6 (LCFO) with Cr3+/Fe3+ (3d3/3d5) has been intensively studied to examine possible \nferromagnetism through the 3 d3-3d5 superexchange according to the Kanamori -Goodenough rule.7,8  \nIn recent years, a number of theoretical and experimental studies on its ordered form have been \nreported.9-13  Based on the local -spin-density calculation, Pickett et al.  showed that ferrimagnetic \nground state with a net spin moment of 2 B/f.u. (a formul a unit) is more stable than ferromagnetic \none with ~7 B/f.u.9  On the other hand, Ueda et al.  postulated that their (111) oriented films, grown \nby pulsed -laser deposition (PLD) technique in a fashion of the LaCrO 3/LaFeO 3 superlattice, exhibited \nferromagnet ism though the measured saturation magnetization is much less than the expected \nvalue.10,11  Until now, there is no critical  evidence to support either  scenario .12,13  \nIn this communication, we report epitaxial synthesis of LCFO double -perovskite films with out \nusing the artificial superlattice technique and their structural and magnetic properties as a function of \ndegree of the Cr/Fe order.  Contrary to the common expectation, well -ordered phase is obtained \nfrom a single target by the PLD growth under a wide  range of growth temperature ( Tg) and oxygen \npartial pressure ( PO2).  The highest ordered sample exhibits the degree of order ~90% and a \nsaturation magnetization ( Ms) of 2.0 ± 0.15 B/f.u. at 5 K.  Therefore, the ground -state magnetic \norder of this compound has been unambiguously verified to be ferrimagnetic .  The Ms is found to \ndepend on oxidation state as well as the Cr/Fe order, while TC is nearly constant around 45 ~  50 K.  \n \nII. EXPERIMENT  \nApproximately 60 nm -thick LCFO films were grown on atomically flat (111) S rTiO 3 substrates \nusing a  PLD system with KrF excimer laser pulses (5 Hz) focused on a target (a LaCr 0.5Fe0.5O3 \ndisordered ceramic tablet, 99.99% purity) at a fluence of 1 .1 J cm-2 (laser spot 0.35 x 0.10 cm2).14  \nThe growth was performed with in-situ monitoring of reflection high -energy electron diffraction \n(RHEED)  pattern under conditions shown in Fig . 1 (a).  After the growth, samples were quenched to 3 room temperature, keeping PO2 constant.  Some of the samples were furnace -annealed in air at \n400°C or 800°C for 3 h to refill residual oxygen vacancies in films as well as substrates.  \nThe film composition was  analyzed for those grown on (001) MgO substrates under the same \nconditions by an electron probe microanalyzer  (JED -2300F/JSM -6701F, JEOL) .  The composition \nof cations was confirmed to be identical to that of the target regardless of growth condition.   Surface \nand crystalline structure of the as -grown films was characterized at room temperature  in air by using \natomic force microscopy (SPI -400, SII NanoTechnology) and four -circle x -ray diffraction (X ’pert \nMRD, PANalytical) with CuK  radiation (  = 1.541838 Å), respectively.  \nX-ray diffraction measurements were also performed using a four -circle diffractometer at the \nsynchrotron beamline BL-3A on the Photon Factory, KEK, Japan .  The photon energy of the \nincident x -ray was tuned at 12 keV.  All diff raction measurements were carried out at room \ntemperature.  The x-ray diffraction intensity of the LCFO was corrected at \n21\n21\n21 , (1 1 1) , \n23\n23\n23 , \n(2 2 2), \n25\n25\n25 , and (3 3 3) reciprocal lattice points.   The integrated intensity w as estimated by \nmeasuring rocking curves along the -axis (perpendic ular to the scattering plane).  \nThe microstructures and interface structures of the samples were observed using the atomically \nresolved HAADF -STEM (JEM -2100F, JEOL) e quipped with an aberration corrector (CEOS Gmbh) \nand electron energy loss spectroscopy (GIF Tridiem, Gatan), operating at an acceleration voltage of \n200 kV.  Moreover, the crystallographic property of the LCFO film was characterized by analyzing \nselected a rea diffraction patterns obtained using a conventional 200 kV -TEM (JEM -2010F, JEOL).  \nThe electron -transparent thin foil specimens for the TEM and STEM observations were prepared by \nthe common procedures of cutting, mechanically grinding, and dimpling foll owed by Ar ion milling \nprocess (accelerating voltage of 2 -4 kV and incidence angle of 6 °) to reduce thickne ss down to \nseveral nanometers.  \nHysteresis and field -cooled magnetization measurements were performed using a SQUID \nmagnetometer  (MPMS -XL, Quantum Design).  The hysteresis measurements were done in the field \nrange ± 1 T at a temperature 5 K and the FC measurements were done over the temperature ranging \nfrom 5 K to 300 K under a magnetic field of 0.1 T.  For clarity, diamagnetic sig nal of the STO \nsubstrates was subtracted.  \n \nIII. RESULTS AND DISCUSSION  \nA. Thin film growth  \nStrong growth -condition dependence of the degree of order has been found  [Fig. 1 (a)].  Highly \nordered films were reproducibly obtained at high -Tg-PO2 region and the Cr/Fe order reached up to \n90% in the films grown at Tg ~ 1000°C  and PO2 = 1 x 10-4 Torr (condition A).  Moreover , film \ncrystallinity gradually degraded in going away from the condition A [Fig. 1 (b)] .  This tendency was \nalso observed in the  initial growth.  Temporal variations of  RHEED  intensity is shown in Figs. 1 (c) \nand (d) for two representative samples grown under conditions A and D, respectively.  In the former 4 case, clear oscillation was observed reflecting a layer -by-layer growth mode.  As a result, atomically \nflat surface with monolayer -high (~0.23 nm) steps was seen (inset).  In the latter case, intensity \ndamped quickly indicating a three -dimensional growth mode.  Correspondingly, film surface \nbecame rough exhibiting lateral face ts and domain boundaries.  \n \nB. Structural characterizations  \nIn order to evaluate the degree of order , we calculate the x-ray diffraction  intensity by using simple \ncrystal model with transition -metal -site disorder and lattice distortion along the [111] direction as \nshown in Fig. 2.  Hereafter we wish to employ the notion of antisite -disorder ( AS) instead of the \nCr/Fe order, which is defined as the percentage of misplaced Cr at Fe site and vice versa.16  From \nthe model with the AS fraction and atomic displacement of La and O ions towards Fe -rich \ntransition -metal -site plane ( La and O in the unit of Å), we calculated the s tructure factor at each \nreciprocal lattice point as follows : \n \n) 3sin(6) 3sin(2) )(21(O O La La Fe Cr\n21\n21\n21   f f f f AS F   \n\n,     (1) \n) 6sin(6) 6sin(2O O La La Fe Cr )111(   f f f f F  \n,      (2) \n) 9sin(6) 9sin(2) )(21(O O La La Fe Cr\n23\n23\n23   f f f f AS F   \n\n,     (3) \n) 12sin(6) 12sin(2O O La La Fe Cr )222(   f f f f F  \n,      (4) \n) 15sin(6) 15sin(2) )(21(O O La La Fe Cr\n25\n25\n25   f f f f AS F   \n\n,    (5) \n))21(9sin( ) 18sin(La La La La Fe Cr )333(      f f f f F\n \n))21(9sin(3) 18sin(3O O O O      f f\n.   (6) \n \nHere, fCr, fFe, fLa, fO, and F(hhh) represent atomic form factor s of Cr, Fe, La, O, and structure factor of \n(h h h ) reflection, respectively.  The calculated intensity I was obtaine d from \nNpL FFI * .  \nL, p, and N are Lorentz factor, polarization, factor, and scale factor, respectively.  The \nexperimentally obtained integrated intensity was fitted by this calculated intensity.  The results are \ngiven in Figs 3 (a), (b) and (c)  for samples grown under conditions A, B and D, respectively.  The \nbest results were obtained when ( AS, La, O) = (0.051, 0.029, -0.020), (0.145, 0.030, -0.035), and \n(0.34, -0.038, 0.108) for the samples shown in Figs. 3 (a), (b) and (c),  respectively.   Apparent \nmagnitude of half -integer reflections with respect to integer reflections systematically decreases with \nincreasing AS fraction.  It shou ld be mentioned that we constructed the model structures so that all \nof the half -integer reflections yielded mutual agreements.  This is because intensity of the integer \nreflections has some ambiguity due to insufficient separation from the substrate refle ctions.  As 5 shown in Fig. 3 (d), sharp x -ray rocking curves confirm good crystallinity of the sample grown under \ncondition A.  From the (1 1 1) and \n21\n21\n21  reflections, lateral -coherence length of fundamental \nperovskite lattice and the ordered phase is evaluated to be 250 nm and 220 nm, respectively.  As for \nsamples grown under condition B and D, each value is 210 nm/65 nm and 180 nm/40 nm, \nrespectively.  \nTheoretical aspects of LCFO double -perovskite have been studied in terms of subtle competition \nbetween ferromagnetic and antiferromagnetic coupling of local spin moments, to which distance \nbetween the nearest -neighbor Fe and Cr ions ( dFe-Cr) is considered in connection with experimental \nresults.12  Thus, it will be important to determine lattice parameters for our samples.  Judging from \nthe position of asymmetric reflections, we have concluded that all the films investigated in this study \nwere pseudomorphicall y grown on (111) STO [Fig. 3 (e) for example ].  From this fact, average \ndistance between the transition -metal ions (i.e., B-O-B along the {100} direction) is deduced through \ndFe-Cr = \n2\n2112\n111 ) 2(d d  where  d111 is the out-of-plane spacing  for LCFO and d11-2 is the in-plane \nspacing for STO  (1.594 Å).  As shown in Fig . 3 (f), dFe-Cr distributes over AS fraction in the \nintermediate range between those of a powder sample (3.9073 Å; AS = 0.5)16 and superlattice films \n(3.922  ~ 3.926 Å, assuming the coherent growth on STO;  AS is unknown).10,13  Miura et al. have \nfound that ground energy minima exists at this range of dFe-Cr in both cases of ferromagnetic and \nantiferromagnetic states.12  Since we carried out magnetic measurements on annealed samples as \nwell, it is worth mentioning that the lattice parameters remained intact after the annealing at 400°C.  \nOur results described so far present striking demonstration of spontaneous ordering o f Cr and Fe in \nLCFO films, which is unexpected taking similar ionic radius of Cr and Fe into account ( rCr3+ = 0.615 \nÅ and rFe3+(HS) = 0.645 Å in an octahedral coordination).  In our growth conditions, FV of Cr is \nexpected to be fixed 3+, while that of Fe c an be either 2+ or 3+.  In Fig . 1 (a), variation in degree of \norder does not coincide with the difference in FV of Fe and Cr, which is indicated by shaded regions \nwith phase equilibrium lines of simple ferrates.  Although actual phase equilibrium  of Fe ion s must \nbe drawn  lower considering the strong er crystal field in perovskite lattice , this contrast implies that \ntuning FV of Fe is probably not critical for enhancement of the ordering.17  Although the mechanism \nof spontaneous ordering is unclear at the moment, we anticipate that lattice disorder has to be reduced \nto obtain higher Cr/Fe ordering.  In fact , variations in degree of order and in rocking curve width \nshow close correlation [Figs. 1 (a) and (b )].  We note that Tg applied for the growth of superlattices \n(600 ~ 700°C)10,13 was much lower than Tg applied here.  The lattice mismatch to substrate is a \ndeleterious factor because the Cr/Fe ordering as well as film crystallinity  became worse when we \nused (111) LaAlO 3 substrates.  \nNow we would like to discuss the microstructures of our samples .  As shown in F ig. 4, atomically \nabrupt and coherent interface was clearly observed for a sample grown under condition B ( AS ~ \n0.12), indicating formati on of elastically distorted region near the interface.  Moreover, a SAD  \npattern into the LCFO layer showed visible (h h h) and (h h k) diffraction spots with half -integer h \nand k, indicating the NaCl -type ordering of Cr and Fe.  In addition, the valence state of Cr and Fe 6 ions was confirmed to be both 3+  by the electron energy loss spectroscopy  analysis .  These results \nare consistent with the x-ray diffraction  analyses.  However, we have seen the intensity of the \nhalf-integer reflections much weaker near the interface than in bulk region.  The disordered region \nhas been found to extend a few nanometers apart from the interface, of which fraction corresponds to \nat most 5% of the whole sample volume.  The presence of interfacial disordered layer is also \nreflected to the fact that one period of RHEED intensity oscillation corresponded to a charge neutral \nunit cell (~0.23 nm), but not the double -perovskite unit cell.   If the Cr/Fe order had occurred from \nthe beginning, the period would have been twice longer because all deposits must be consumed in \norder to complete the double -perovskite unit cell.  Incomplete surface termination of the STO \nsubstrate likely causes for mation of the disordered layer, while it is interesting to reveal what triggers \nthe following spontaneous ordering.  \n \nC. Magnetic properties  \nUsing a SQUID magnetometer, magnetization  versus applied magnetic field and temperature were \nmeasured  for samples grown  under conditions A to D.  For clarity, diamagnetic signal of the STO \nsubstrates was subtracted.   As shown in Fig . 5 (a), the in -plane magnetization loops taken at 5 K for \nthe samples annealed at 400°C traced well -defined hysteresis, of which magnitude systematically \nevolved with decreasing AS fraction.  As for a sample grown under condition A with AS = 0.06, Ms \nat 1 T was 2.0 B/f.u. with an error arising from inaccuracy of the film volume (± 0.15 B/f.u.).  The \nmagnitude of Ms is consistent with antiferromagnetic ordering of local spin moment (3 d3\n↓3d5\n↑; S = \n-3/2+5/2 = 1) and seems remain up to AS ~ 0.37.  The temperature de pendence of field -cooled \nmagnetization showed a clear magnetic transition at TC = 45 ~ 50 K for all the samples [Fig. 5 (b)].  \nHere we shall emphasis that oxygen annealing led to a drastic enhancement of low -temperature \nmagnetization  [Fig. 5 (c)].  In contrast to the annealed samples, as -grown samples indicated broad \nM-T curves without the magnetic transition at TC (not shown).  We believe that suppression of \nferrimagnetic ground state is attributed to the presence of residual oxygen vacancies.  Ot her origin \nlike magnetic contamination during the furnace -annealing could be excluded.  In fact, no change in \nMs was seen for the sample annealed at 800°C (inset).  \nFinally we briefly discuss the observed systematic dependence of Ms on AS fraction.  Conside ring \nthat LaCrO 3 and LaFeO 3 show G-type antiferromagnetic ordering, the spin moment at the antisite \nwill be aligned antiparallel with respect to the magnetization of the host.  In this case, each \nmisplaced Cr ion reduces Ms by 2B (3d5\n↑→3d3\n↑; S = -5/2+2/3 = -1), while each misplaced Fe ion \nalso reduces Ms by 2B (3d3\n↓→3d5\n↓; S = 3/2 -5/2 = -1), giving a loss of the spin magnetization per \nantisite of 4B.  Then, one expects that Ms varies as (2 -4AS)B/f.u. and vanishes at AS = 0.5.  In \nreality, magnetic frustration arising from relative coordination between misplaced ions would \ncontribute to the deviation of Ms.18 \n \nIV. CONCLUSION  7 We have prepared a n umber  of LCFO films on (111) STO substrates at  a wide range of growth \nparameters to  study how the ordering as well as physical properties depend on  the growth parameters .  \nIt has been found that Cr and Fe ions  are spontaneously ordered in the NaCl -type double -perovskite \nstructure.  Highly ordered phase  (up to 90%) could reproducibly be stabilized.  Through systematic \ncharacterization of the samples having a wide range of the degree of order and lattice parameter, we \nhave unambiguously  revealed that m agnetic ground state of La 2CrFeO 6 double -perovskite is  \nferrimagnetic . \n \nACKNOW LEDFMENTS  \nS.C. is supported by the Global COE program (Materials Integration), Tohoku University and a \nGrant -in-Aid for Scientific Research (JSPS).  A.O. is supported by the GCOE program (Chemistry), \nTokyo Institute of Technology and JSPS.  The synchrotron x -ray study was performed with the \napproval of the Photon Factory Program Advisory Committee (No.2009S2 -003).  The work was \npartly supported by JSPS through its \"Funding Program for World -Leading Innovativ e R&D on \nScience and Technology (FIRST Program) ”. 8 References  \n[1] E. Dagotto and Y. Tokura, MRS Bull. 33, 1037 ( 2008 ). \n[2] M. T. Anderson, K.  B. Greenwood, G.  A. Taylor, and K. R. Poeppelmeier, Prog. Solid State \nChem. 22, 197 ( 1993 ). \n[3] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature 395, 677 ( 1998 ). \n[4] H. Kato, T. Okuda, Y. Okimoto, Y. Tomioka, Y. Takenoya, A. Ohkubo, M. Kawasaki, and Y. \nTokura, Appl. Phys. Lett. 81, 328 ( 2002 ). \n[5] G. Blasse, J. Phys. Chem. Solids 26, 1969 ( 1965 ). \n[6] K. Ueda, Y. Muraoka, H. Tabata, and T. Kawai, Appl. Phys. Lett. 78, 512 ( 2001 ). \n[7] J. Kanamori, J. Phys. Chem. Solids 10, 87 ( 1959 ). \n[8] J. B. Goodenough, Phys. Rev. 100, 564 ( 1955 ). \n[9] W. E. Pickett, Phys. Rev. B 57, 10613 ( 1998 ). \n[10] K. Ueda, H. Tabata, and T. Kawai, Science 280, 1064 ( 1998 ). \n[11] W. E. Pickett, G.  I. Meijer, K. Ueda, H. Tabata, and T. Kawai, Science 281, 1571a ( 1998 ). \n[12] K. Miura and K. Terakura, Phys. Rev. B 63, 104402 ( 2001 ). \n[13] B. Gray, H.  N. Lee,  J. Liu, J. Chakhalian, and J.  W. Freeland, Appl. Phys. Lett. 97, 013105 \n(2010 ) and references therein.  \n[14] S. Chakraverty, A. Ohtomo, M. Okude, K. Ueno, and M. Kawasaki, Cryst. Growth Des. 10, \n1725 (2010).  \n[15] The range of the Cr/Fe order or AS fraction shown in Figs. 1  (a), 3 (f), and 5 (c) is determined \nfrom intensity ratio of the LCFO \n21\n21\n21  reflection to the (1 1 1) reflection, which was taken by using \nan in -house XRD and calibrated with the data taken by a synchrotron  XRD [ shown in Figs. 3 (a) to 3 \n(c)]. \n[16] A. K. Azad, A.  Mellergård, S. -G. Eriksson, S.  A. Ivanov, S.  M. Yunus, F. Lindberg, G. \nSvensson, and R. Mathieu, Mater. Res. Bull. 40, 1633 ( 2005 ). \n[17] T. Manako, M. Izumi, Y. Konishi, K. -I. Kobayashi, M. Kawasaki, and Y. Tokura, Appl. Phys. \nLett. 74, 2215 ( 1999 ). \n[18] D. Serrate, J. M. De Teresa, and M.  R. Ibarra, J. Phys.: Condens. Matter 19, 023201 ( 2007 ). 9 Figures:  \n \nFIG. 1: (Color online) (a) Epitaxial growth conditions of LCFO films mapped in a Tg-PO2 diagram.  \nCircles with different size and color represent the growth conditions, where degree of order \ndetermined by XRD are > 80% (large and red), 60 ~ 80% (middle and orange), 40 ~ 60% (small and \ngreen) and < 20% (the smallest and blue).  Solid lines an d shaded regions indicate phase equilibrium \nof Fe -O binary system and stable conditions for coexistence of Cr 2O3 and Fe, FeO, Fe 3O4, or Fe 2O3 \n(ordered from lower left to upper right), respectively.  (b) Contour mapping of the FWHM of \nrocking curves for the  LCFO (1 1 1) reflection.  (c) and (d) RHEED intensity oscillations for the \nspecular beam during the initial growth of LCFO films on STO (111) substrates under conditions A \nand D, respectively.  The insets depict AFM images of each film.  The range of colo r code \ncorresponds to ~ 3 nm in height.  \n \n \nFIG. 2: (Color online) Schematic crystal -structure -model of La 2CrFeO 6.  In this model, the atomic \ndisplacement of La and O are indicated by the La and O.  AS is the degree of antisite -disorder.  10 \n \nFIG. 3: (Color online) (a-c) X-ray diffraction profiles of the samples grown under conditions A, B \nand D, respectively.  The experimentally observed intensity and calculated one are indicated by \ncircles and bars, respectively.  (d) X -ray rocking curves of LCFO (1 1 1) and \n21\n21\n21  reflections \ntaken for sample grown under condition A.  (e) X -ray reciprocal space mapping around STO (3 3 0) \nreflection taken for a sample grown under condition A.  Double reflections are due to CuK1 and \nCuK2 radiation s.  (f) Distance between the nearest -neighbor Fe and Cr ions as a function of \nantisite - disorder ( AS) fraction.  SLs and bulk are referred as to values reported for artificial \nsuperlattices10,13 and a powder sample with AS = 0.5.16 \n \n 11 FIG. 4: (Color online) (a) HAADF -STEM image observed along a substrate [110] zone axis for a \nsample grown under condition B .  Circles highlight positions of La (bright spots), Fe/Cr (weak \nspots), and O (background), respectively.  Horizontal arrow indicates position of the interface \nbetween LCFO film and STO substrate.  Selected area diffraction patterns of STO ( b) and LCFO ( c).  \nThe presence of (h h h) and (h h k) diffraction spots with half -integer h and k in the la tter indicates the \nNaCl -type ordering of B -site cations.  \n \n \nFIG. 5: (Color online) (a) Magnetization hysteresis curves taken at 5 K for samples grown under \nconditions A to D and annealed at 400°C.  (b) The temperature dependence of field -cooled \nmagnetization for the same samples taken during warming under 0.1 T.  Inset shows the temperatu re \ndependence of inverse magnetization.  Solid lines are linear fits to the plots above Tc.  (c) \nSaturation magnetization ( Ms = M at 5 K under 1 T) for as -grown and annealed (400°C) samples as a \nfunction of antisite -disorder fraction.  Inset depicts the an nealing temperature ( TA) dependence of Ms \nfor sample A.  The error bars reflect inaccuracy of the sample volume.  \n "    },    {        "title": "1103.2939v1.Oxygen_hyperstoichiometric_hexagonal_ferrite_CaBaFe4O7_δ_δ__approx_0_14____coexistence_of_ferrimagnetism_and_spin_glass_behavior.pdf",        "content": "1/23  \n \n \nOxygen hyperstoichiometric  hexagonal ferrite CaBaFe 4O7+ (  0.14)  : \ncoexistence of ferrimagnetism and spin glass behaviour  \n \n \nTapati Sarkar *, V. Duffort, V. Pralong, V. Caignaert and B. Raveau  \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,  \n6 bd Maréchal Juin, 14050 CAEN, France  \nAbstract  \n \n An oxygen hyperstoichiometric ferrite CaBaFe 4O7+ (  0.14) has been \nsynthesized using “soft” reduction of CaBaFe 4O8. Like the oxygen stoichiometric \nferrimagnet CaBaFe 4O7, this oxide also keeps the hexagonal symmetry (space group: \nP63mc), and exhibits the same high Curie temperature of 270 K. However, the \nintroduction of extra oxygen into the system weakens the ferrimagnetic interaction \nsignificantly at the cost of increased magnetic frustration at low tempera ture. Moreover, \nthis canonical spin glass (T g ~ 166 K) exhibits an intriguing cross -over from de Almeida -\nThouless type to Gabay -Toulouse type critical line in the field temperature plane above a \ncertain field strength, which can be identified as the anisot ropy field. Domain wall \npinning is also observed below 110 K. These results are interpreted on the basis of \ncationic disordering on the iron sites.  \n \n \n \n \n \n \n \nPACS number: 75.47.Lx  \n \nKeywords  : “114” ferrites, ferrimagnetism and magnetic frustration, spin glass  Ising -\nHeisenberg competition, magnetic anisotropy, domain wall pinning.  \n \n * Corresponding author: Tapati Sarkar  \ne-mail: tapati.sarkar @ensicaen.fr  2/23  Introduction  \n \nThe recent discovery of the new series of “114” oxides, (the cobaltites – \n(Ln,Ca) 1BaCo 4O7 [1 – 8] and the ferrites – (Ln,Ca) 1BaFe 4O7 [9 – 11]) have opened up a \nnew field for the investigation of strongly correlated electron systems. These oxides \nconsist of CoO 4 (or FeO 4) tetrahedra sitting in alternating layers of kagomé and triangular \narrays [10].  The structure can also be described as the stacking of close -packed [BaO 3] \nand [O 4] layers whose tetrahedral cavities are occupied by Co2+/Co3+ (or Fe2+/Fe3+) \nspecies, forming triangular and kagomé layers of CoO 4 (or FeO 4) tetrahedra. This \nstructure has  been primarily responsible for the wide variety of magnetic states that has \nbeen observed in this group of oxides, ranging from a spin glass for cubic LnBaFe 4O7 [9, \n10] to a ferrimagnet for orthorhombic CaBaCo 4O7 [5] and hexagonal CaBaFe 4O7 [9] \noxides.  \nRecent studies of the “114” cobaltites [12 – 17] have revealed the existence of \nclosely related structures with various crystallographic symmetries, and possibility of \noxygen non -stoichiometry in the range “O 7” to “O 8.5” in those systems. This change of \noxygen stoichiometry, which induces the variation of Co2+:Co3+ ratio in the system, is \nexpected to influence the physical properties of these compounds considerably. This is \nthe case of the oxygen rich “114” cobaltites YBaCo 4O8.1 [15] and YbBaCo 4O7.2 [17], \nwhich were shown to be magnetically frustrated rather than magnetically ordered at low \ntemperatures.  \nIn contrast to the cobalt oxides, no report of oxygen hyperstoichiometric “114” \nferrites exists till date, probably due to the fact that Fe2+ gets too easily  oxidized into Fe3+, \nthereby destabilizing the “114” structure at the benefit of pure “Fe3+” oxides. We have, \nthus, investigated the possibility to stabilize the mixed valence Fe2+/Fe3+ in the “114” \noxygen hyperstoichiometric CaBaFe 4O7+ ferrite by reducin g the fully oxidized \ncompound CaBaFe 4O8 [18] at low temperature in an argon -hydrogen atmosphere.  We \nreport herein on the magnetic properties of the “114” oxygen hyperstoichiometric \nCaBaFe 4O7.14 hexagonal ferrite. We show that, like the stoichiometric phas e CaBaFe 4O7, \nthis oxide also exhibits ferrimagnetism with a T C of 270 K, but that the competition \nbetween ferrimagnetism and magnetic frustration is much more pronounced than for the \nstoichiometric phase, as seen from the decrease of the magnetization . Mor e importantly, \nwe observe that CaBaFe 4O7.14 is characterized by a canonical spin glass behaviour with 3/23 Tg  166 K, and an intriguing cross -over from an Ising to a Heisenberg spin glass type \nbehaviour in the external magnetic field at low temperature. Beside s this competition \nbetween ferrimagnetism and spin glass  behaviour , one also observes domain wall pinning \nbelow 110 K. This very different magnetic behaviour of CaBaFe 4O7.14 is explained in \nterms of cationic deficiency and disordering on the iron sites, th e “barium -oxygen” \nhexagonal close packing remaining  untouched.  \n \nExperimental  \n \nThe precursor CaBaFe 4O8 [18] was prepared by the sol gel method. Stoichiometric \namounts of calcium carbonate (Prolabo, 99%) and barium carbonate (Alfa Aesar, 99%) \nwere dissolved in a large excess of melted citric acid monohydrate at ~ 200°C. Iron \ncitrate  (Alfa Aesar, 20% of Fe)  was separately dissolved in hot water leading to a dark \nbrown solution which was poured on the citrate mixture. The water was then evaporated \nfollowed by d ecomposition of the gel. The gel was calcined at 450 °C under air to obtain \nan amorphous precursor, which was then pressed into pellets before firing at 1200 °C to \nobtain CaBaFe 4O8.  \nThe oxygen hyperstoichiometric “114” ferrite, CaBaFe 4O7+ was then obtain ed by \nreducing CaBaFe 4O8 under an Ar/H 2 10% mix at 610 °C for 24  hrs. \nThe oxygen content of the sample was determined by redox titration. The sample \nwas dissolved in hot HCl (3M) flushed with argon to remove the dissolved oxygen. After \ncooling down the sol ution, Fe2+ cations were titrated using 2  10-2 M cerium(IV) sulfate \n(Riedel -de Haën) and 1.10 -phenantroline iron(II) sulfate (Alfa Aesar) as an indica tor \nunder constant argon flow . We obtained  = 0.14.  \nThe X -ray diffraction patterns were registered wit h a Panalytical X’Pert Pro \ndiffractometer under a continuous scanning mode in the 2  range 10° - 120° and step size \n2=0.017°. The d.c. magnetization measurements were performed using a \nsuperconducting quantum interference device (SQUID) magnetometer with  variable \ntemperature cryostat (Quantum Design, San Diego, USA). The a.c. susceptibility, ac(T) \nwas measured with a Physical Property Measurement System ( PPMS ) from Quantum \nDesign with the frequency ranging from 10 Hz to 10 kHz (H dc = 0 Oe and H ac = 10 Oe ). \nAll the magnetic properties were registered on dense ceramic bars of dimensions ~ 4  2  \n2 mm3. \n 4/23  \nResults and discussion  \n \nStructural Characterization  \n \nThe X -ray diffraction pattern (Fig. 1) revealed that CaBaFe 4O7.14 stabilized in the \nsame hexagonal sy mmetry (space group: P63mc) as the “O 7” phase [9]. The Rietveld \nanalysis from the XRD data was done using the FULLPROF refinement program [19]. \nThe fit is also shown in Fig. 1 (red curve). The bottom blue curve corresponds to the \ndifference between the obs erved and calculated diffraction patterns. Satisfactory \nmatching of the experimental  data with the calculated profile of the XRD pattern and the \ncorresponding reliability factors RF = 3.88 % and RBragg = 5.01 % confirm that the fit \nobtained is reasonably a ccurate. The extracted lattice parameters ( a = 6.355 Å, c = 10.372 \nÅ) show a very marginal increase over the “O 7” phase – a increases by ~ 0.11 % while c \nremains virtually unchanged. The refinements of the atomic coordinates, thus, lead to \nresults similar to those previously obtained for CaBaFe 4O7 [9]. The low  value and the \nlow scattering factor of oxygen do not allow any oxygen excess or cationic deficiency to \nbe detected from X -ray powder diffraction data. A very careful neutron diffraction study \nmight perhaps allow the issue to be sorted out, but will really be at the limit of accuracy, \nand consequently, is not within the scope of this paper.  \n \nD. C. magnetization study  \n \nIn Fig. 2, we show the Zero Field Cooled (ZFC) and Field Cooled (FC) \nmagnetization o f CaBaFe 4O7+ recorded under a magnetizing field of 0.3 T. The sample \nshows the same increase in magnetization below ~ 270 K as the oxygen stoichiometric \noxide indicating a similar transition to an ordered magnetic state below 270 K. However, \na careful loo k at the magnetization values reached at the lowermost measured \ntemperature (5 K) immediately reveals a striking difference in the magnetic behaviour of \nCaBaFe 4O7.14 vis – à – vis that of CaBaFe 4O7. While the F.C. magnetization of the \noxygen stoichiometric  compound reaches a value of more than 2.5 µ B/f.u. at T = 5 K, the \nmaximum magnetization value of our oxygen rich sample is only 0.93 µ B/f.u., which is \nless by more than a factor of ½.  \nThis large difference in the magnetization value at low temperature be tween the \ntwo samples (  = 0 and  > 0) prompted us to record the hysteresis curve of our oxygen 5/23 rich sample at low temperature (T = 5 K) and compare it with that of the oxygen \nstoichiometric sample. This is shown in Fig. 3. The magnetization value obtaine d at the \nhighest measuring field of 5 T (2.5 µ B/f.u.) is again different from the oxygen \nstoichiometric sample (3.1 µ B/f.u.), as expected. More importantly, a rather striking \ndifference is seen in the shape of the hysteresis loop. The coercive field (H C) and \nremanent magnetization (M r) of our sample at T = 5 K are 0.77 T and 0.63 µ B/f.u. \nrespectively. While the value of the coercive field compares well with that of the oxygen \nstoichiometric sample, the value of the remanent magnetization is much lower than that \nobtained for CaBaFe 4O7 (M r ~ 1.8 µ B/f.u.). This results in the overall shape of the \nhysteresis loop of our sample (Fig. 3) to be very different from that of the oxygen \nstoichiometric sample (inset of Fig. 3). The degree of magnetic saturation in a sam ple can \nbe roughly quantified from the M -H loop by calculating \n5\nrM H T\nM . While the \noxygen stoichiometric sample had  = 1.7, our oxygen rich sample yields  = 4.0. This \nhigher value of  for the oxygen rich sample indicates an increased lack  of magnetic \nsaturation in the sample, or in other words, a weakening of long range order.  \nAnother important difference with the oxygen stoichiometric sample is in the \nvirgin curve of the M(H) loop. While the virgin curve of the  = 0 sample lies entirely \nwithin the main loop, our oxygen rich sample shows an unusual magnetic behaviour \nwhere a major portion of the virgin curve lies outside the hysteresis loop and meets the \nmain loop only at very high fields.  \n \nA. C. magnetic susceptibility study  \n \nThe tempera ture dependence of the a.c. susceptibility of CaBaFe 4O7.14 in the \ntemperature range 20 K – 280 K and at 4 measuring frequencies ranging from 10 Hz to \n10 kHz is shown in Fig. 4. The sample shows several interesting features which we will \nnow proceed to disc uss separately:  \n(a) At T = 272 K, one can see a sharp peak which is frequency independent i.e. the \nposition of the peak maximum does not shift with a change in the measuring \nfrequency. This peak corresponds to the paramagnetic to ferrimagnetic (PM -FM) \ntransiti on occurring in the sample as it is cooled below 272 K. We note here that \nthe oxygen stoichiometric sample also showed a similar peak at around the same 6/23 temperature. However, in the latter sample, the peak corresponding to the PM -FM \ntransition was the stro ngest (maximum amplitude) compared to the other peaks. In \nour sample, this peak is much smaller in magnitude which corresponds to a \nsignificant weakening of the magnetic ordering (or a smaller volume fraction of \nferrimagnetic domains) which we had mentione d earlier in connection with the \nM(H) loop.  \n(b) The oxide CaBaFe 4O7.14 shows a broader peak at lower temperature (~ 166 K) \nwhich shows pronounced frequency dependence. The peak temperature shifts from \n166 K (for a measuring frequency of 10 Hz) to 176 K (for a  measuring frequency \nof 10 kHz). This corresponds to a peak shift of 0.02 per decade of frequency shift \n(\nlogf\nfT\nTpf\n = 0.02). This value of the parameter p lies within the range for \ncanonical spin glasses, which indicates that this peak is a sig nature of the sample \nundergoing a spin glass transition. We confirm this by analyzing the frequency \ndependence of this peak using the power law form \n0z\nf SG\nSGTT\nT\n\n , where, 0 is \nthe shortest relaxation time available to the system, TSG is the unde rlying spin -\nglass transition temperature determined by the interactions in the system, z is the \ndynamic critical exponent and  is the critical exponent of the correlation length. \nThe actual fittings were done using the equivalent form of the power law: \n\n\n\n\nSGSG f\nTT Tzln ln ln0\n. The fit parameters ( 0 = 7.3  10-10 sec, z = 5.01 and \nTSG = 162.2 K) give a good linear fit (as can be seen in the inset of Fig. 4), and \nconfirms that this peak does correspond to a spin glass transition in the sample. \nWhether the magnetic order disappears in the spin glass phase is not clear at the \nmoment. Our data shows that the spin glass transition occurs within the \nferrimagnetically ordered phase. Whether this transition is accompanied by the \ndestruction of ferrimagnetic long r ange order is an open issue as of now. The \npossibility of coexistence of ferrimagnetic and spin glass orders cannot, however, \nbe ruled out.  7/23 (c) At a lower temperature of ~ 110 K a very broad peak is seen (which broadens out \nto almost kind of a shoulder at lowe r measuring frequency). We will come back to \nthe nature of this feature in the following section.  \n \nMagnetic field dependence of the spin glass freezing temperature  \n \nIn Fig. 5, a.c. susceptibility at 10 kHz driving frequency is plotted as a function of \ntemp erature for different external magnetic fields H dc ranging from 0 to 0.3 T. As can be \nseen from the figure, χ' is suppressed by the magnetic field. First, we focus our discussion \non the evolution of the spin glass freezing temperature (marked by a black ar row in the \nfigure), which occurs at ~ 176 K at the lowest applied field. This peak temperature shows \na continual shift towards lower temperature as the external magnetic field is increased, \nand reaches a value of 155 K at an external magnetic field of 0.3 T. Concomitantly, the \npeak amplitude keeps decreasing as the external magnetic field is increased from 0 to 0.3 \nT. A further increase of the external magnetic field should eventually suppress the spin \nglass transition completely.  \n The purpose of exploring  how the freezing temperature responds to external \nmagnetic field is to check the stability of the spin glass system. This is done by \nexamining the field versus temperature phase diagram obtained from the a.c. \nsusceptibility measurement as shown in Fig. 6.  As was mentioned above, the spin glass \nfreezing temperature is suppressed by increasing the external magnetic field.  \n From a theoretical perspective, de Almeida and Thouless [20] studied the Ising \nspin glass system, and predicted that the spin freezing te mperature ( Tg) depends on H. In \nthe low H range, Tg follows the so -called de Almeida -Thouless (AT) line, expressed as  \n\n23\n001\n\n\n\n\n\ngg\nTHTH H\n. In addition, Gabay and Toulouse [21] investigated the H \ndependence of the spin freezing temperature for the Heise nberg spin glass system. This \nled to the so -called Gabay -Toulouse (GT) line, expressed as  \n\n21\n001\n\n\n\n\n\ngg\nTHTH H . The \nAT line and the GT line are the two critical lines predicted in the presence of field on the \nH-T plane, which mark the phase transition. The first one occurs for an anisotropic Ising \nspin glass while the second is valid for an isotropic Heisenberg spin glass.  8/23 Our sample shows a very interesting behaviour. At low field values (H dc < 0.15 T), \nTg follows the AT line. This can be seen in Fig. 6, where the red line denotes the AT line. \nSince the AT line predicts that \n32H Tg , so we have plotted H2/3 in the H -T phase \ndiagram. However, with an increase in the field (H > 0.15 T), we find deviation from the \nAT line. Remarkably, it is  found that at high field, the variation of Tg(H) agrees with the \nGT line. This can be seen in the inset of Fig. 6, where the blue line denotes the GT line. \nSince the GT line predicts that \n2H Tg , so the H -T phase diagram in the inset is p lotted \nwith H2 in the y -axis.  \nThese experimental results can be explained using the theoretical calculation by \nKotliar and Sompolinsky [22], who have predicted that in the presence of random \nanisotropy, the critical behaviour for a spin glass in fields lo wer than the anisotropy field \nis close to Ising type following the AT line, and crosses over to Heisenberg behaviour in \nhigh fields. The fact that we see a crossover in critical lines on the H -T plane for our \nsample indicates the existence of magnetic anis otropy in the system. At higher applied \nfields, the system behaves like a Heisenberg spin glass, where the spins can freeze along \nany direction with respect to the applied magnetic field. However, when the applied field \nis lower than the anisotropy field, the spins are forced to be aligned along the local \nanisotropy axis. The preference of the spin alignment adds an Ising character to the \nassociated spin cluster.  \n \nDomain wall pinning at lower temperature  \n \nIn this section, we discuss the third feature seen i n the χ'(T) curve – the broad peak \nat ~ 110 K (Fig. 4). This peak at 110 K does not shift (i.e. the peak maximum occurs at \nthe same temperature) with a change in the external magnetic field H dc (see the red line in \nFig. 5). Based on this behaviour, we attr ibute the origin of the feature seen at ~ 110 K to \nenhanced domain wall pinning. The signature of this domain wall pinning can also be \nseen in Fig. 7, where we plot the variation of the coercivity (H C) with temperature. As is \nclear from the figure, the coe rcivity is majorly enhanced below 110 K, which occurs due \nto the domain wall pinning. A close look at the high temperature region, which is \nenlarged and shown in the inset of Fig. 7, reveals that the coercivity also shows an \nenhancement below the paramagne tic to ferrimagnetic phase transition temperature 9/23 (shown by a black arrow), and another enhancement below the spin glass freezing \ntemperature (shown by a blue arrow), as expected.  \n At this stage, we need to go back to our earlier observation of an unusual initial \nmagnetization curve in the M(H) loop measured at low temperature (Fig. 3). Such \nunusual magnetic hysteresis behaviour, with the virgin curve lying outside the main \nhysteresis loop, was earlier associated with irreversible domain wall motion in spin el \noxides [23]. Thus, this unusual magnetization curve is an additional confirmation of the \ndomain wall pinning at ~ 110 K that we had mentioned earlier. In fact, we find that the \nvirgin curve lies outside the main M(H) loop for temperatures below 110 K, b ut above \n110 K, it lies completely inside the main hysteresis loop. This is shown in Fig. 8, where \nwe plot the M(H) loops at temperatures slightly below (Fig. 8 (a)), and slightly above \n(Fig. 8 (b)) 110 K. In the figures, the virgin curves are shown in red  for the sake of \nclarity.  \n \nOrigin of the competition between ferrimagnetism and spin glass behaviour  \n \n In order to understand the different magnetic behaviour of CaBaFe 4O7.14 with \nrespect to CaBaFe 4O7, we must keep in mind that the oxygen excess in the for mer \ninduces an increase of the Fe3+ content in the structure i.e., the Fe3+:Fe2+ ratio increases \nfrom 1 in the stoichiometric phase to 1.32 in the oxygen hyperstoichiometric phase. As a \nconsequence, the Fe3+-Fe3+ antiferromagnetic interactions increase in the oxygen rich \nphase, and may decrease the ferrimagnetism in the structure.  Bearing in mind the model \npreviously proposed by Chapon et. al. [4] to explain the competition between 1D \nferromagnetism and 2D magnetic frustration in the cobaltite YBaCo 4O7 which has the \nsame hexagonal structure, we must consider the iron framework of our compound. The \nlatter consists of corner -sharing [Fe 5] bipyramids running along “ c” interconnected \nthrough “Fe 3” triangles (Fig. 9). In other words, in both oxides, CaBaFe 4O7 and \nCaBaFe 4O7.14, we can expect, similarly to the hexagonal cobalt oxides LnBaCo 4O7, that \nthe system exhibits an unidimensional magnetic order in the bipyramidal rows along “ c”, \nwhereas the triangular geometry of the iron lattice in the (001) plane induces magnetic \nfrustration as soon as the iron species are coupled antiferromagnetically. Such a model \ncan account for the competition between 1D ferrimagnetism and 2D magnetic frustration \nin both oxides, CaBaFe 4O7 and CaBaFe 4O7.14, and explain that the magnetic  frustration 10/23 may be larger in the latter owing to the appearance of larger short range \nantiferromagnetic interactions in the (001) plane . \n Nevertheless, the valency effect alone is not sufficient to explain the appearance of \nthe spin glass behaviour.   Two hypotheses  can be considered to explain this particular \nbehaviour. The first scenario deals with the fact that CaBaFe 4O7.14 contains interstitial \noxygen in spite of the apparent close packed character of the structure, leading to a local \npuckering of the “ O4” and “BaO 3” layers. As a result, the distribution of iron in the \ncationic sites would be locally disordered, leading to a spin glass behaviour. This local \ndistortion would also change the crystal field and would be responsible for the domain \nwall pinnin g. The second scenario deals with the fact that the “barium oxygen” \nframework remains close packed, but that the compound exhibits a  cationic deficiency \naccording to the formula Ca 0.98(Ba 0.98O0.02)Fe 3.93O7. Such an effect would be similar to \nthat observed for “oxidized” spinels -Fe2O3 and Co 3-xO4, which do not contain \ninterstitial oxygen, but were found to be iron or cobalt deficient [24, 25]. This second \nscenario would explain the magnetic behaviour of this phase, which is close to that \nobserved for CaBaF e4-xLixO7 [26]. In both the systems, the doping of the Fe sites with \nlithium or vacancies respectively introduces disordering on the Fe sites, which is in turn,  \nat the origin of the appearance of spin glass behaviour at lower temperature. Thus, the \ncompeti tion between 1D ferrimagnetism and spin glass behaviour appears normal. \nSubsequently, the competition between anisotropic (Ising) and isotropic (Heisenberg) \nspin glass can be understood from the peculiar geometry of the [Fe 4] lattice. Finally, the \niron va cancies would change the nature of the crystal field in the structure, playing the \nrole of pinning centres. This explains both, the broad peak at 110 K and the enhanced \ncoercivity below this temperature, which are the signatures of domain wall pinning.   \n The small deviation from the stoichiometry does not allow to distinguish the \npossibility of interstitial oxygen vis à vis that of cationic deficiency from a structural \nstudy. Attempts are being  made to synthesize similar hexagonal ferrites with larger \noxyge n excess in order to answer this question.  \n \n \n \n \n \n 11/23 Conclusion s \n \n This study illustrates the extraordinarily rich physics of the “114” CaBaFe 4O7+ \nferrite, in connection with its ability to accommodate oxygen excess, similar to what is \nobserved for the spine l family, Fe 3O4 – -Fe2O3. The remarkable feature of this “114” \noxide deals with the competition between ferrimagnetism and spin glass behaviour that \ncan be induced by varying the oxygen content, without changing the hexagonal symmetry \nof the structure. Su ch a behaviour can be explained, like for the “114” cobaltites, as due \nto the competition between 1D magnetic ordering along “ c” and 2D magnetic frustration \nin the triangular (001) lattice. Nevertheless, CaBaFe 4O7 differs significantly from \nCaBaCo 4O7, the latter’s ferrimagnetism originating mainly from a lifting of its 2D \ngeometrical frustration through a strong orthorhombic distortion of its initial hexagonal \nlattice. We believe that the scenario of cation disordering  on iron sites  is the key for \nunderstan ding the magnetism of these materials. Further investigations, especially using \nneutron diffraction and X -ray synchrotron have to be performed in order to further \nunderstand this phenomenon.  \n \nAcknowledgement s \n \nThe authors acknowledge the CNRS and the Conse il Regional of Basse Normandie \nfor financial support in the frame of Emergence Program. V. P. acknowledges support by \nthe ANR -09-JCJC -0017 -01 (Ref: JC09_442369).  \n \n \n \n \n \n 12/23 References  \n \n      [1]     Martin Valldor and Magnus Andersson, Solid State Sciences , 2002, 4, 923  \n      [2]     Martin Valldor, J. Phys.: Condens. Matter ., 2004 , 16, 9209  \n      [3]     L. C. Chapon, P. G. Radaelli, H. Zheng and J. F. Mitchell, Phys. Rev. B , 2006 , 74,  \n               172401  \n      [4]     P. Manuel, L. C. Chapon, P. G. Radaell i, H. Zheng a,d J. F. Mitchell, Phys. Rev.  \n               Lett., 2009 , 103, 037202  \n      [5]    V. Caignaert, V. Pralong, V. Hardy, C. Ritter and B. Raveau, Phys. Rev. B , 2010 , 81,  \n               094417  \n      [6]     E. A. Juarez -Arellano, A. Friedrich, D. J. Wilson, L. Wiehl, W. Morgenroth, B.  \n                Winkler, M. Avdeev, R. B. Macquart and C. D. Ling, Phys. Rev. B , 2009 , 79, 064109   \n      [7]     N. Hollmann, Z. Hu, M. Valldor, A. Maignan, A. Tanaka, H. H. Hsieh, H. -J. 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Lett ., 1981 , 47, 201 (1981)  \n      [22]   G. Kotliar and H. Sompolinsky, Phys. Rev. Lett ., 1984 , 53, 1751 (1984)  \n      [23]   P. A. Joy and S. K. Date, J. Magn. Magn. Mater ., 2000 , 210, 31 (2000)  \n      [24]   J.-E. Jørgensen, L. Mosegaard, L. E. Thomsen, T. R. Jensen and J. C. Hanson, J.  \n               Solid State Chem ., 2007 , 180, 180  \n      [25]   F. Kh. Chibirova, Physics of the Solid State , 2001 , 43, 1291  \n      [26]   K. Vijayanandhini, Ch. Simon,  V. Pralong, V. Caignaert and B. Raveau , Phys. Rev.  \n                B, 2009, 79, 224407  \n \n 14/23 Figure Captions  \n \nFig. 1 X-ray diffraction pattern along with the fit for CaBaFe 4O7+. \nFig. 2 MZFC (T) and M FC (T) curves of CaBaFe 4O7+ measured at H = 0.3 T.  \nFig. 3 M(H) curve of CaBaFe 4O7+ measured at T = 5 K. The virgin curve is shown in red    \n          circles, while the rest of the hysteresis loop is shown in black triangles. The inset  \n          shows the M(H) curve of the oxygen stoichiometric sample (CaBaFe 4O7) measured  \n          at T = 5 K.  \nFig. 4 The real (in -phase) component of a.c. susceptibility for CaBaFe 4O7+ as a function  \n           of temperature in the frequency range f = 10 Hz – 10 kHz, at zero static magnetic  \n           field (H dc) and at a dr iving ac field (H ac) of 10 Oe. The inset shows the plot of ln   \n           vs \n\n\n\n\nSGSG f\nTTTln  for the peak at 166 K.  \nFig. 5 The real (in -phase) component of a.c. susceptibility for CaBaFe 4O7+ as a function  \n          of temperature. The driving f requency was fixed at f = 10 kHz and h ac = 10 Oe.  \n         Each curve was obtained under different applied static magnetic field (H dc) ranging  \n         from 0 T to 0.3 T.  \nFig. 6 Field vs temperature phase diagram of CaBaFe 4O7+. In order to show the AT l ine,  \n          we have plotted H2/3 vs Tg. The inset shows H2 vs Tg and the GT line.  \nFig. 7  Temperature dependence of coercive field for CaBaFe 4O7+. The inset is an  \n           enlarged version of the high temperature region.  \nFig.  8 M(H) loops of CaBa Fe4O7+ at (a) T = 75 K and (b) T = 135 K.  \nFig.  9 Schematic representation of the [Fe 4] tetrahedral framework of hexagonal  \n           CaBaFe 4O7+ showing the Fe 5 bipyramids sharing corners with Fe 3 triangular  \n           groups (adapted from Ref. 11).  \n \n \n \n \n \n \n 15/23 \n20 40 60 80 1000.02.0x1034.0x1036.0x1038.0x1031.0x104P63mc\na=6.3546(2) Å\nc=10.3721(4) Å\n2 = 1.94\nRBragg = 5.01 %\nRF = 3.88 %\n  Intensity (arb. units)\n2 (degree) \nFig. 1 . X-ray diffraction pattern along with the fit for CaBaFe 4O7+. \n \n 16/23  \n0 100 200 300 4000.00.20.40.60.81.0\nH = 0.3 T ZFC\n FCMagnetization ( B/f.u.)\nTemperature (K)\n  \n \nFig. 2 . MZFC (T) and M FC (T) curves of CaBaFe 4O7+ measured at H = 0.3 T.  \n 17/23  \n \n-4 -2 0 2 4-2-1012\n-4 -2 0 2 4-3-2-10123CaBaFe4O7Magnetization ( B/f.u.)\nMagnetic field (T)\n \nT = 5 KMagnetization ( B/f.u.)\nMagnetic field (T)\n \n \nFig. 3.  M(H) curve of CaBaFe 4O7+ measured at T = 5 K. The virgin curve is shown in  \nred ci rcles, while the rest of the hysteresis loop is shown in black triangles. The  inset \nshows the M(H) curve of the oxygen stoichiometric sample (CaBaFe 4O7) measured at T \n= 5 K.  \n 18/23 \n50 100 150 200 2504.0x10-36.0x10-38.0x10-31.0x10-2\n110 K166 K\n272 K\n-3.6 -3.3 -3.0 -2.7 -2.4-9-8-7-6-5-4-3-20 = (7.3 ± 0.1) X 10-10 sec\nTSG = 162.2 K\nz = 5.01 ± 0.01ln \nln{(Tf-TSG)/TSG}\n  Hac = 10 Oe\nHdc = 0 10 Hz\n 80 Hz\n 1 kHz\n 10 kHz' (emu/gm)\nTemperature (K)\n  \n \nFig. 4 . The real (in -phase) component of a.c. susceptibility for CaBaFe 4O7+ as a function             \nof temperature in the frequency range f = 10 Hz – 10 kHz, at zero static magnetic            \nfield (H dc) and at a driving ac fie ld (H ac) of 10 Oe. The inset shows the plot of ln             \nvs \n\n\n\n\nSGSG f\nTTTln  for the peak at 166 K.  \n \n \n \n \n \n \n \n \n \n 19/23  \n \n50 100 150 200 250 3002.0x10-34.0x10-36.0x10-38.0x10-31.0x10-2\n 0.2 T\n 0.225 T\n 0.25 T\n 0.275 T\n 0.3 T 0 T\n 0.025 T\n 0.05 T\n 0.075 T\n 0.1 T\n 0.125 T\n 0.15 T\n 0.175 T\nTemperature (K)' (emu/gm)\n  \n \n \nFig. 5 . The real (in -phase) component of a.c. susceptibility for CaBaFe 4O7+ as a function            \nof temperature. The driving frequency was fixed at f = 10 kHz and h ac = 10 Oe.          \nEach curve was obtained under different applied static magnetic field (H dc) ranging          \nfrom 0 T to 0.3 T.  \n \n \n \n \n \n \n \n \n 20/23  \n155 160 165 170 175 1800.0750.1500.2250.3000.3750.450\n155 160 165 170 1750.000.020.040.060.080.10\nGT line H2 (T2)\nT (K)\nAT lineFM\nSpin glass\n  H2/3 (T2/3)\nT (K)\n \n \nFig. 6 . Field vs temperature phase diagram of CaBaFe 4O7+. In order to show the AT line,           \nwe have plotted H2/3 vs Tg. The inset shows H2 vs Tg and the GT line.  \n \n \n \n \n \n \n \n \n 21/23 \n0 50 100 150 200 250 3000.00.10.20.30.40.50.60.70.8\n120 150 180 210 240 270 3000.020.040.060.080.10\n T (K)HC (T)\n113 K\nTemperature (K)\n  HC (T) \n \nFig. 7 . Temperature dependence of coercive field for CaB aFe 4O7+. The inset is an            \nenlarged version of the high temperature region.  \n \n \n \n \n \n \n \n \n \n \n \n \n 22/23 \n-5 -4 -3 -2 -1 0 1 2 3 4 5-2-1012-2-1012\n(b)Magnetization ( B/f.u.)\nMagnetic field (T)T = 135 KT = 75 K(a)\n   \n  \n \nFig. 8 . M(H) loops of CaBaFe 4O7+ at (a) T = 75 K and (b) T = 135 K.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n 23/23  \n \n \n \nFig. 9 . Schematic representation of the  [Fe 4] tetrahedral framework of hexagonal            \nCaBaFe 4O7+ showing the Fe 5 bipyramids sharing corners with Fe 3 triangular            \ngroups (adapted from Ref. 11).  \n \n \n \n \n \n \n \n "    },    {        "title": "2208.00179v1.On_field_driven_domain_wall_motion_in_compensated_ferrimagnetic_nanowires.pdf",        "content": "On field-driven domain wall motion in compensated ferrimagnetic nanowires\nK. Y . Jing,1X. Gong,1and X. R. Wang1, 2,\u0003\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2HKUST Shenzhen Research Institute, Shenzhen 518057, China\n(Dated: August 2, 2022)\nThe fascinating high-speed field-driven domain wall (DW) motion along ferrimagnetic nanowires near the\nangular momentum compensation point (AMCP) is solved based on the generic ferrimagnetic dynamics. The\nphysics of the absences of precessional torque and infinite high Walker breakdown field at the AMCP is proved\nunder general conditions. Based on the energy conservation principle, an almost exact DW velocity formula,\nvalid beyond the Walker breakdown field, is obtained. Our results agree with all existing experiments and\nsimulations. This theory provides useful guidances to DW manipulation.\nIntroduction .— Magnetic domain wall (DW) dynamics in\nnanowires have attracted much attention for its rich physics\n[1, 2] and promising device applications such as racetrack\nmemories [3]. One critical issue in applications is the real-\nization of high stable DW speed under external forces such\nas magnetic fields and electrical currents. This requires a de-\nlay or removal of so-called Walker breakdown [4]. The en-\ndeavour of increasing DW speed leads to studying DW mo-\ntion in antiferromagnetic nanowires [5–7], and, very recently,\nto that in ferrimagnetic nanowires [8–19]. A ferrimagnet has\nat least two spin sublattices antiferromagnetically interacting\nwith each other. It has two special states called the angu-\nlar momentum compensation point (AMCP) at which the an-\ngular momenta of the two sublattices cancel each other and\nthe magnetization compensation point at which the magneti-\nzations cancel each other. One class of ferrimagnets is rare-\nearth-transition-metal alloys whose AMCP and magnetization\ncompensation point are di \u000berent in general and can be tuned\nby compositions, other than the temperature. Unlike an an-\ntiferromagnet, ferrimagnetic states can be manipulated by a\nmagnetic field, a spin transfer torque, and a spin-orbit torque.\nAlso, unlike a ferromagnet, the net magnetization of a ferri-\nmagnet can be very small but not zero, especially around an\nAMCP such that it is susceptible to the magnetic field with\nsmall Zeeman energy. One fascinating discovery is the very\nhigh DW speed of thousands meters per second in compen-\nsated ferrimagnetic (FiM) nanowires near the AMCP [9–11].\nHere we show that high DW speed near the AMCP is related\nto the absence of precessional torque and Walker breakdown\nphenomenon at the AMCP.\nAlthough FiM dynamics should be described by coupled\npartial di \u000berential equations for magnetizations on at least two\nantiferromagnetically coupled sublattices, existing theoreti-\ncal studies treat a ferrimagnet either as a ferromagnet whose\ndynamics follows Landau-Lifshitz-Gilbert (LLG) equation\n[10, 11] or an antiferromagnet with the N ´eel order governed\nby a second-order partial di \u000berential equation [9, 12, 16, 20–\n22]. DW dynamics is then obtained from converting the par-\ntial di \u000berential equations into ordinary di \u000berential equations\nfor the collective coordinates of DW center and DW-plane\ncanting angle [9, 12, 16, 20–22]. Indeed, existing theories\nhave enriched our understanding of DW dynamics in ferri-magnets in many aspects. However, there are some drawbacks\nin these approaches. These approaches fail to provide a quan-\ntitative explanation to both experiments and simulations since\nthey rely on the existence of a DW plane and a rigid body\nassumption for the Thiele equation [23]. It often needs to as-\nsume also certain DW structure such that the approaches are\ndi\u000ecult, if not impossible, to generalize to situations where\nthe assumptions are not valid such as for vortex DWs and\nDWs in chiral magnets. Furthermore, the physical picture be-\nhind the FiM DW motion is unclear in these approaches and\nan accurate description of the DW speed beyond the Walker\nbreakdown field is still challenging.\nIn this work, the origin of the high DW speed and absence\nof Walker breakdown field at the AMCP of a FiM nanowire\nare explained based on generic dynamics for coupled sublat-\ntice magnetizations of a ferrimagnet with a general Rayleigh\ndissipation. We show that a static DW between two domains\nwith di \u000berent energy densities does not exist. Spins in the\nDW must move in a field that creates such an energy density\ndi\u000berence. Moving spins must dissipate energy due to the in-\nevitable coupling between spins and its environment described\nby Gilbert damping in magnetization dynamics. The dissi-\npated energy must be compensated by the Zeeman energy re-\nleased from the DW propagation toward domain of the higher\nenergy density. At the AMCP, precessional torque vanishes\ndue to the zero angular momentum and the Walker breakdown\nfield become infinity, leading to the high DW speed. Further-\nmore, a universal relationship between DW speed and DW\nstructure is obtained, and an almost exact formula for high-\nfield DW velocity is derived.\nModel .— We consider a head-to-head (HH) DW in a FiM\nnanowire, whose easy axis is along the wire defined as the z-\naxis as shown in Fig. 1. M1andM2are the magnetizations\non two sublattices with M1andM2being their saturation mag-\nnetization. The total magnetic energy of the wire in the pres-\nence of a uniform magnetic field HisE=R\n\"d3xwith the\nenergy density of\n\"=JM1\u0001M2+X\ni=1;2h\nAi(rMi)2+fi(Mi)\u0000\u00160Mi\u0001Hi\n;(1)\nwhere J>0 is the antiferromagnetic interlattice-spin coupling\nconstant. Aiand fiare the ferromagnetic exchange sti \u000bnessarXiv:2208.00179v1  [cond-mat.mes-hall]  30 Jul 20222\nzxy\nM1\nM2DW\nRegion I Region II Region III\nFIG. 1. Schematic of a HH FiM DW in a nanowire. Region I and III\nare two uniform FiM domains, separated by a DW (region II) whose\nwidth is \u0001. DW structure can be very complicated. His the external\nfield. Colours denote the spin orientations: The red for spins along ˆ z\nand the light-blue for spins along \u0000ˆz.\nand anisotropic magnetic energy density for sublattice i(i=\n1;2).fiis assumed to have two equal minima at Mi=\u0006Miˆz.\nThe FiM magnetization dynamics is generically governed\nby the following equations [24, 25]\n1\n\r1@M1\n@t=\u0000M1\u0002 \nH1\u0000\u000b11\n\r1M1@M1\n@t\u0000\u000b12\n\r1M1@M2\n@t!\n1\n\r2@M2\n@t=\u0000M2\u0002 \nH2\u0000\u000b22\n\r2M2@M2\n@t\u0000\u000b21\n\r2M2@M1\n@t!\n;(2)\nwhereHi=\u0000\u0016\u00001\n0\u000eE=\u000eMiand\ri=gi\u0016B=~(i=1;2) are\nthe e\u000bective field and the gyromagnetic ratio for Mi, respec-\ntively. gi,\u0016B, and ~are the Land ´eg-factor of sublattice i\n(i=1;2), the Bohr magneton, and the Planck constant, re-\nspectively.\u000b11;\u000b22and\u000b12;\u000b21are intra-sublattice and inter-\nsublattice damping coe \u000ecients. We have\u000b12\n\r1M1=\u000b21\n\r2M2due to\nthe action-reaction law. si=Mi=\riis the spin density of sub-\nlattice i(i=1;2).\r1,\r2in a general ferrimagnet because of\nthe di \u000berence in Land ´eg-factors of sublattices. For example,\nin GdFeCo alloys, gGd'2,gFeCo'2:2 [9].\nResults .— We prove first that no static DW is allowed in\nthe presence of a magnetic field along the z\u0000direction, except\nat magnetization compensation point. If a static DW solution\nexists, the DW structure should satisfy equations Mi\u0002Hi=0\n(i=1;2). As illustrated in the Supplemental Materials [26], it\nimpliesMi(x;t) (i=1;2) satisfying following equation\n\t\n@\n26666664\" 1\u0000j=x;y;zX\ni=1;22Ai(rMi;j)\n(rMi;j)37777775\u0001d\u001b=const:(3)\nwhere@\nis any closed surface of the system, 1is the 3\u00023\nunit matrix, and\ndenotes the dyadic product. Eq. (3) cannot\nbe true for a DW with M1=M1ˆz;M2=\u0000M2ˆzon its left and\nM1=\u0000M1ˆz;M2=M2ˆzon its right as shown in Fig. 1, or\nvice versa, because it requires ( M1\u0000M2)H=0. Thus, a static\nDW can only exist either with H=0 or M1=M2. In other\nwords, a static DW cannot exist between two domains with\ndi\u000berent energy density. This result can also be understood\nfrom following argument: Assume Mi(x) is a static DW that\nseparate a left domain with a lower energy density \"1from\nthe right domain with a higher energy density \"2(> \" 1). The\nenergy change by shifting DW to the right by a distance L, i.e.Mi(x)!Mi(x+Lˆz), isLS(\"1\u0000\"2)<0, here Sis the cross\nsection area of the wire. The DW is not stable against a rigid\nshift to the right because this small change in spin structure\nalways lower the system energy. Thus a DW must vary with\ntime under a magnetic field.\nWhen Jis much larger than the Zeeman energy, M1and\nM2are always anti-parallel to each other. We define Me\u000b=\n(M1\u0000M2)m, wheremis the unit vector of M1. Thenm\nsatisfies the following equation\n(s1\u0000s2)@m\n@t=\u0000(M1\u0000M2)m\u0002He\u000b+\u000bm\u0002@m\n@t;(4)\nwhereHe\u000b=M1H1\u0000M2H2\nM1\u0000M2. In terms of m, the total energy\nisE[m]=Rh\nA(rm)2+f(m)\u0000\u00160(M1\u0000M2)m\u0001Hi\nd3x\nwith A=A1M2\n1+A2M2\n2, where ais the lattice constant. De-\nnote\u000b=\u000b11s1+\u000b22s2\u0000\u000b12s2\r2\n\r1\u0000\u000b21s1\r1\n\r2, the thermodynamic\nsecond law requires \u000b > 0 to ensure the Rayleigh dissipation\nfunctionalR=\u00160\u000b\n2R\u0010@m\n@t\u00112d3x[24, 25, 27] to be positive-\ndefinite. Equation (4) says that the change of spin angular\nmomentum (left-hand side) equals the net torque (right-hand\nside) that is the sum of a torque from an e \u000bective field on the\nnet magnetization ( M1\u0000M2,0) and a dissipative torque from\nthe motion ofm. At the AMCP, the dissipative torque cancels\nthe field torque.\nEquation (4) can be recast as an e \u000bective LLG equation\n[11, 20, 28–30]\n@m\n@t=\u0000\re\u000bm\u0002He\u000b+\u000be\u000bm\u0002@m\n@t; (5)\nwith an e \u000bective gyromagnetic ratio \re\u000b=jM1\u0000M2j=(s1\u0000s2)\nand an e \u000bective Gilbert damping \u000be\u000b=\u000b=(s1\u0000s2).\re\u000b\u000be\u000bis\nalways positive because a moving magnetization must dissi-\npate its energy to its environment (See Eq. (6) below). s1>s2\nands1<s2correspond to lattice-1 and lattice-2 dominate\ncases. Following a similar derivation in the literature [31, 32],\nthe energy dissipation rate is [33, 34],\ndE\ndt=\u0000\u000be\u000b\re\u000b\u00160\u0010\n1+\u000b2\ne\u000b\u0011\n(M1\u0000M2)Z\n(Me\u000b\u0002He\u000b)2d3x:(6)\nWe divide the wire into three regions as shown in Fig. 1:\nI for the domain with Me\u000bparallel toH, II for the DW, and\nIII for the domain with Me\u000banti-parallel to H. Energy dis-\nsipation occurs only in the DW region (region II) where Me\u000b\nandHe\u000bare not collinear [33, 34]. The change of energies, EI\nandEIII, of region I and III comes from the DW propagation\nalong the wire, and should bed(EI+EIII)\ndt=\u00002\u00160(M1\u0000M2)HvS,\nwhere vis the DW velocity. DW energy EIImust be around\na certain value. Thus the time averaged energy change rate\nmust be zero. In another word,dEII\ndtis either zero or oscillates\nwith zero average. The energy conservation requires\nv=\u000be\u000b\re\u000b\n2HS\u0010\n1+\u000b2\ne\u000b\u0011Z\n(m\u0002He\u000b)2d3x\n+1\n2\u00160(M1\u0000M2)HSdEII\ndt:(7)3\nThis is a universal relationship between DW velocity and the\nDW structure and can serve as a proper definition of instan-\ntaneous DW velocity. The second term on the right should\nbe identical zero in the case of a rigid DW motion such that\nthe DW speed is constant. In the case that a DW deforms\nitself during its propagation, the energy dissipation rate and\nDW energy EIIoscillate with time anddEII\ndt=0, where the bar\ndenotes the time average. This results in an oscillating DW\nspeed whose time-averaged value is\n¯v=\u000be\u000b\re\u000b\n2HS\u0010\n1+\u000b2\ne\u000b\u0011Z\n(m\u0002He\u000b)2d3x: (8)\nWe note (m\u0002He\u000b)2=H2\ne\u000b;\u0012+H2\ne\u000b;\u001e, where He\u000b;\u0012andHe\u000b;\u001e\nare two field components perpendicular to min the local coor-\ndinate framework\u0010\nem;e\u0012;e\u001e\u0011\n.\u0012(x;t) and\u001e(x;t) are the polar\nand the azimuthal angles of m. Below we consider anisotropy\nenergy of f(m)=\u0000Kzm2\nz+Kym2\ny.\nHe\u000b;\u0012=Hsin\u0012\u0000G;\nHe\u000b;\u001e=\u00001\n\u00160(M1\u0000M2) sin\u0012@f\n@\u001e\n+2A\n\u00160(M1\u0000M2) sin\u0012@\n@z \nsin2\u0012@\u001e\n@z!\n;(9)\nwhere G=1\n\u00160(M1\u0000M2)\u0014\n2A@2\u0012\n@z2\u0000@f\n@\u0012\u00002Asin\u0012cos\u0012\u0010@\u001e\n@z\u00112\u0015\n.\nEquation (5) along e\u0012;e\u001ebecomes\n@\u0012\n@t=\re\u000bHe\u000b;\u001e\u0000\u000be\u000bsin\u0012@\u001e\n@t\nsin\u0012@\u001e\n@t=\u0000\re\u000bHe\u000b;\u0012+\u000be\u000b@\u0012\n@t:(10)\nEliminate time-derivative of \u0012from equation (10), we have\n\u0010\n1+\u000b2\ne\u000b\u0011\nsin\u0012@\u001e\n@t=\re\u000b\u0010\n\u000be\u000bHe\u000b;\u001e\u0000He\u000b;\u0012\u0011\n: (11)\nIf the DW propagates as a rigid-body along the z-direction,\nthe case of a field below the Walker breakdown [4], i.e.\n@\u001e\n@z=0,@2\u001e\n@z2=0, and@\u001e\n@t=0, using Eq. (9), we have\n2A@2\u0012\n@z2\u0000@f\n@\u0012=0, so that He\u000b;\u0012=\u0000Hsin\u0012, whose max-\nimal allowed external field is the Walker breakdown field.\nFor ( M1\u0000M2);(s1\u0000s2),0,@\u001e\n@t=0 obviously requires\n\u000be\u000bHe\u000b;\u001e=He\u000b;\u0012. This means that the DW-plane cants an an-\ngle to generate a non-zero He\u000b;\u001eto coherently vary \u0012such that\nthe DW propagates along the wire. Recall our biaxial model\nf(\u0012;\u001e)=\u0000Kzcos2\u0012+Kysin2\u0012sin2\u001e, the Walker breakdown\nfield is HW=max\u0010\u000be\u000bKysin 2\u001e\n\u00160(M1\u0000M2)\u0011\n=\u000bKy\n\u00160j(M1\u0000M2)(s1\u0000s2)j. Both\re\u000b\nand\u000be\u000bdiverge as ( s1\u0000s2)\u00001near the AMCP. The limit of\nEq. (11) under s1\u0000s2!0 (\u000b2\ne\u000b;\re\u000b\u000be\u000b\u0018(s1\u0000s2)\u00002) gives\nHe\u000b;\u001e=0 and\u001e=0 when@\u001e\n@t=0. Thus the DW-plane re-\nmains in the xz-plane and never rotates, leading to an infinite\nHWat the AMCP. One can also see this point by consider-\ning an equivalent form of Eq. (5),@m\n@t=\u0000\re\u000b\n1+\u000b2\ne\u000bm\u0002He\u000b\u0000\re\u000b\u000be\u000b\n1+\u000b2\ne\u000bm\u0002(m\u0002He\u000b). At the AMCP, the precessional torque\nvanishes since\re\u000b\n1+\u000b2\ne\u000b!0 as ( s1\u0000s2)!0 while the damp-\ning torque is finite because of lim (s1\u0000s2)!0\re\u000b\u000be\u000b\n1+\u000b2\ne\u000b=M1\u0000M2\n\u000b,0.\nThis means that the precessional motion is completely pro-\nhibited, andmat any point inside the DW rotates coherently\ntoward external field, leading to a rigid DW propagation along\nthe wire (see the Supplemental Materials [26]).\nEquation (5) with our biaxial magnetic anisotropy has\nthe well-known Walker DW solution [4] of \u0012(z;t)=\n2 arctan\u0010\nexp\u0010\u0010\nz\u0000Rt\n0v(\u001c)d\u001c\u0011\n=\u0001(t)\u0011\u0011\n, where \u0001is the DW\nwidth. It givesdEII\ndt=\u00004AS\n\u00012d\u0001\ndt=0 for a rigid-body DW\npropagation. He\u000b;\u0012=\u000be\u000bHe\u000b;\u001e=\u0000Hsin\u0012,(m\u0002He\u000b)2=\nH2\ne\u000b;\u0012+H2\ne\u000b;\u001e=\u0010\n1+\u000b2\ne\u000b\u0011\nH2sin2\u0012=\u000b2\ne\u000b. Substituting DW\nwidth definition ofR\nsin2\u0012d3x=2S\u0001into Eq. (7), one has\nv=(M1\u0000M2)\u0001\n\u000bHand DW speed at Walker breakdown field\nvW=Ky\u0001\n\u00160(s1\u0000s2), independent of the damping coe \u000ecient and\ndivergent at the AMCP.\nAway from the AMCP, HWis finite. A DW shall precess\naround wire axis while it propagates along the wire when H>\nHW. From Eqs. (8) and (9), we have\n¯v=\u000be\u000b\re\u000b\n2HS\u0010\n1+\u000b2\ne\u000b\u0011Zh\n(Hsin\u0012\u0000G)2+H2\ne\u000b;\u001ei\nd3x:(12)\nAverage DW velocity is (see the Supplemental Materials [26]\nfor detailed derivation),\n¯v=c1H+c1\n\u000b2\ne\u000b\u0012\nH\u0000q\nH2\u0000H2\nW\u0013\n(13)\nwhere c1=\u000be\u000b\re\u000b\n2S(1+\u000b2\ne\u000b)R\nsin2\u0012d3x=(M1\u0000M2)\u000b¯\u0001\n(s1\u0000s2)2+\u000b2is peaked at the\nAMCP. Equation (13) is exact under very sensible assump-\ntions, and all coe \u000ecients in Eq. (13) are fully determined by\nthe model parameters.\nEquation (13) predicts a negative di \u000berential DW mobility\nin the range of HW<H<\u000b2\ne\u000b+1p\n\u000b4\ne\u000b+2\u000b2\ne\u000bHW. This prediction is\nalso true for ferromagnetic case. In order to find out how ac-\ncurate of Eq. (13) is for H>HW, we use MuMax3 [35] to nu-\nmerically solve Eq. (2) for a synthetic ferrimagnetic strip wire\nas shown in Fig. 1 that consist of two antiferromagnetically-\ncoupled ferromagnetic-layers of 1nm thick each. The strip\nsize is 16 nm\u00022 nm\u00021024 nm. The cell size in simulations\nis chosen to be 1 nm \u00021 nm\u00021 nm. To mimic a GdFeCo al-\nloy [9], the model parameters are J=1:2\u000210\u00004J\u0001A\u00002m\u00001,\nA1=9:8\u000210\u000024J\u0001m\u0001A\u00002,A2=1:23\u000210\u000023J\u0001m\u0001A\u00002,\nbiaxial anisotropy are considered for each sublattice, fi=\n\u0000Kz;i\nM2\niM2\ni;z+Ky;i\nM2\niM2\ni;y,i=1;2,Kz;1=Kz;2=0:65 MJ=m3,\n\u000b12=\u000b21=0.Ky;iand\u000bii(i=1;2) are used for simulat-\ning di \u000berent systems as labelled by Set 1-6 in Table I. The\ngyromagnetic ratios \r1=\r2=1:76\u00021011s\u00001T\u00001, the satura-\ntion magnetizations are M1=1010 kA=m,M2=900 kA=m.\nThe coupling field between two sublattices is of hundreds of\nTesla to guarantee collinearity of two spin sublattices. Dif-\nferent from a natural ferimagnet, inter-sublattice coupling is4\nData set Set 1 Set 2 Set 3 Set 4 Set 5 Set 6\nKy;1( MJ=m3) 0.05 0.035 0.02 0.1 0.1 0.1\nKy;2( MJ=m3) 0.05 0.035 0.02 0.1 0.1 0.1\n\u000b11 0.02 0.02 0.02 0.005 0.01 0.015\n\u000b22 0.02 0.02 0.02 0.005 0.01 0.015\n\u000be\u000b 0.3473 0.3473 0.3473 0.0868 0.1736 0.2605\nKy( MJ=m3) 0.1 0.07 0.04 0.2 0.2 0.2\n\u00160HW(T) 0.3157 0.2210 0.1263 0.1579 0.3157 0.4736\n¯\u0001(nm) 3.85 3.87 3.89 3.79 3.75 3.79\nc1(\u00160\u0001m\u0001s\u00001T\u00001) 210.00 211.13 212.34 57.48 111.37 162.69\nTABLE I. Ky;1,Ky;2,\u000b11, and\u000b22are model parameters. \u000be\u000b,Ky,\u00160HW,¯\u0001, and c1are computed quantities.\nalong the y-direction in our synthetic ferrimagnet. In the sim-\nulation, a DW is first created at the center of nanowire, then a\nuniform magnetic field is applied in the +ˆzdirection. The ve-\nlocity is obtained from the linear fit of time-evolution curve of\nthe DW center (where mz=0). For high fields above Walker\nbreakdown, the average velocities are obtained from data ac-\ncumulated for more than 4 velocity oscillating periods.\nWe consider six di \u000berent systems with various Ky;iand\u000bii\n(i=1;2). The detail values of the model parameters are given\nin Table I. Because of large speed di \u000berence, Fig. 2(a) plot ¯ v\nvs.Hfor three systems with the same \u000bii=0:02 and di \u000berent\nKy;i, label as Set 1, 2, 3. Figure 2(b) is the similar plots for\nthree systems with the same Ky;i=0:1 MJ=m3, but di \u000berent\n\u000bii, label as Set 4, 5, 6. The corresponding values of c1,\u000be\u000b,\nandHWcomputed from this theory are also given in Table\nI. The perfect agreements between the simulation results (the\nsymbols) and theoretical prediction (the solid curves) demon-\nstrate that Eq. (13) is almost exact.\nDiscussion and Conclusion .— Before conclusion, we\nwould like to make a few remarks. 1) The relationship be-\ntween the instantaneous DW speed and the DW structure is ex-\nact that explains why our high-field DW speed formula with-\nout any fitting parameters agree perfectly with simulation re-\nsults. 2) Since no collective-mode approximation is used, the\ntheory is applicable to all types of DWs. 3) High DW speed\nis a result of the absence of Walker breakdown field at the\nAMCP. This explains the observed high DW speed of more\nthan 1.5km /s at the AMCP although the mobility \u0016=(M1\u0000M2)\u0001\n\u000b\nforH<HWitself is comparable to or even smaller than that\nfor a ferromagnetic wire [36, 37].\nIn summary, a generic theory of field-driven DW motion\nin FiM wires is presented. A static DW cannot exist in a\nhomogeneous ferrimagnetic nanowire when a uniform static\nmagnetic field or any other external force creates an energy\ndensity di \u000berence between two domains separated by the DW.\nSpins in the DW must vary with time under the external mag-\nnetic field such that the system energy is dissipated due to the\nGilbert damping. The dissipated energy must be compensated\nby the Zeeman energy released from moving the DW toward\nthe domain with the higher energy density. High DW speed\nnear the AMCP is the consequence of the absences of preces-\nsional torque and infinite high Walker breakdown field at the\n(a)\n(b)FIG. 2. Average DW speed of a head-to-head DW as a function of\nan applied field along the +ˆz-direction. Symbols are MuMax3 sim-\nulation results and solid curves are theoretical formula without any\nfitting parameter. a) Three systems (denoted as Set 1, 2, 3) with the\nsame\u000b11=\u000b22=0:02 but di \u000berent Ky;i. b) Three systems with (de-\nnoted as Set 4, 5, 6) with the same Ky;i=0:1 MJ=m3but di \u000berent\u000bii\n(i=1;2). Their values are listed in Table I.\nAMCP. A lower Zeeman energy density and a high energy dis-\nsipation rate contribute also to the high DW speed at a reason-\nable lower field near the AMCP. Away from the AMCP, our\napproach can not only obtain the exact DW velocity below the\nWalker breakdown field, but also an almost exact velocity for-\nmula beyond the Walker breakdown field. 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Yamada\nDepartment of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan\nTakuya Satoh∗\nDepartment of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan and\nQuantum Research Center for Chirality, Institute for Molecular Science, Okazaki, Aichi 444-8585, Japan\n(Dated: December 29, 2023)\nWe performed time-resolved pump–probe measurements using rare-earth iron garnet\nGd3/2Yb1/2BiFe 5O12as a two-sublattice ferrimagnet. We measured the initial phases of the mag-\nnetic resonance modes below and above the magnetization compensation temperature to clarify the\nsublattice selectivity of the inverse Faraday effect in ferrimagnets. A comparison of the time evo-\nlution of magnetization estimated using the equations of motion revealed that the inverse Faraday\neffect occurring in ferrimagnetic materials has sublattice selectivity. This is in striking contrast to\nantiferromagnets, in which the inverse Faraday effect acts on each sublattice identically. The initial\nphase analysis can be applied to other ferrimagnets with compensation temperatures.\nThe ultrafast control of magnetic materials using light\npulses has attracted considerable interest over the years.\nIn magnetic insulators, the inverse Faraday effect (IFE)\nand Faraday effect (FE) have been applied to excite and\ndetect magnetization dynamics, respectively.[1] The IFE,\nwhich is the reverse of the magneto-optical FE, was the-\noretically proposed by Pitaevskii[2] and Pershan [3] and\ndemonstrated by van der Ziel et al.[4] Ultrafast magneti-\nzation control has been demonstrated in weak ferromag-\nnets, such as DyFeO3[1] and FeBO3,[5] and pure antifer-\nromagnets, such as NiO,[6] via the IFE using femtosecond\nlight pulses. Here, an effective magnetic field pulse HIFE\nwas induced by a circularly polarized light pulse. The\ndirection of HIFEis determined by the helicity of the cir-\ncularly polarized light. The impulsive HIFEacts on the\nsublattice magnetization of transition metal ions (e.g.,\nFe3+ions in DyFeO3and FeBO3and Ni2+ions in NiO),\nfollowed by precession as free induction decay. In this\ntheory, the magnitude and direction of HIFEacting on\neach sublattice are identical when HIFEis perpendicular\nto the sublattice magnetizations.[7]\nFerrimagnets consist of magnetic sublattices aligned\nin opposite directions with different magnetic ions of\ndifferent magnitudes, such that the vector sum is not\nzero. Thus, ferrimagnets exhibit a net magnetiza-\ntion in the ground state. Strong antiferromagnetic ex-\nchange interactions occur between the two magnetic\nsublattices. Accordingly, the magnetic resonance in\nferrimagnets has two modes: the ferromagnetic reso-\nnance (FMR) mode (GHz range), which is similar to\nthat in ferromagnets, and the exchange resonance mode\n(sub-THz range), which is unique to ferrimagnets.[8–\n10] Furthermore, when the temperature dependence of\nthe sublattice magnetization differs, the net magneti-\nzation can vanish at the magnetization compensation\ntemperature TM. Magnetization dynamics in ferrimag-\n∗satoh@phys.titech.ac.jp\n0 5 10 15 20 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 \n0 100 200 300 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0 5 10 15 20 -0.4 -0.2 0.0 0.2 0.4 0.6 \n0 50 100 150 200 250 300 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 \nFaraday rotation (mrad) \nTime (ps) +sin ωHF t\n−sin ωHF t\n(d) (c) (e) +H0Faraday rotation (mrad) \nTime (ps) +sin ωLF t\n−sin ωLF t(b) \n(a) \n+H0\nFaraday rotation (mrad) \nTime (ps) +sin ωHF t\n−sin ωHF t−H0\nσ+\nσ− σ−σ+Faraday rotation (mrad) \nTime (ps) +sin ωLF t\n−sin ωLF t−H0\nσ+\nσ−σ−σ+FIG. 1. (Color online) Experimental geometry. The GdYb-\nBIG single crystal was placed in an electromagnet, which pro-\nduces an in-plane magnetic field ±H0.\nnets have been actively studied in the field of ferrimag-\nnetic spintronics, which combines controllability, sim-\nilar to ferromagnetism, with ultrafastness, similar to\nantiferromagnetism.[11–13]\nUltrafast IFE using circularly polarized light pulses\nhas been reported in ferrimagnets.[14–20] However, the\nmagnetic field and temperature dependence of the initial\nphase were not explicitly discussed; moreover, whether\nthe IFE generated HIFEin the opposite or same direction\nfor each sublattice has not been studied. We attempt to\nclarify this issue by measuring the initial phase of the co-\nherently excited resonance modes using a two-sublattice\nrare-earth (RE) iron garnet.\nThe bismuth-doped RE iron garnet\nGd3/2Yb1/2BiFe5O12(GdYb-BIG) is a ferrimagnet\nwith a Curie temperature of 573 K and TM= 96 K,\nwhich was obtained from the temperature dependence\nof magnetization and the sign change of Faraday\nrotation.[21] The exchange interaction between Fe ions\nin the tetragonal and octahedral sites is much stronger\nthan that between Fe and RE ions. Therefore, in\nthe sub-THz frequency range, one can regard the two\nmagnetic sublattices to be composed of Fe magnetizationarXiv:2312.16553v1  [cond-mat.mtrl-sci]  27 Dec 20232\n0 5 10 15 20-0.4-0.20.00.20.40.60.8\n0 100 200 300-0.6-0.4-0.20.00.20.40.60.80 5 10 15 20-0.4-0.20.00.20.40.6\n050100 150 200 250 300-0.6-0.4-0.20.00.20.40.60.8Faraday rotation (mrad)\nTime (ps)+sinωHFt\n−sinωHFt\n(c)(b)\n(d)+H0Faraday rotation (mrad)\nTime (ps)+sinωLFt\n−sinωLFt(a)\n+H0\nFaraday rotation (mrad)\nTime (ps)+sinωHFt\n−sinωHFt−H0\nσ+\nσ− σ−σ+Faraday rotation (mrad)\nTime (ps)+sinωLFt\n−sinωLFt−H0\nσ+\nσ−σ−σ+\nFIG. 2. (Color online) Magnetization dynamics at 300 K. HF\nmode at external field: + H0(a) and −H0(b). LF mode at\nexternal field: + H0(c) and −H0(d). Pump helicity: σ+(red\ndots) and σ−(blue dots). The solid lines are the damped\nsinusoidal functions with αHF∼0.03 and αLF∼0.05 for the\nHF and LF modes, respectively.\nMFeand RE magnetization MRE(Ref. [20]).\nWe used a GdYb-BIG single crystal with a (111) plane\norientation and a thickness of 140 µm, which was grown\nby the liquid-phase epitaxy method. As shown in Fig. 1,\nwe performed time-resolved pump–probe measurements\nin transmission geometry. The polarization of pump\npulse (wavelength 1300 nm, pulse energy 4 µJ, repeti-\ntion rate 500 Hz, pulse width 60 fs, spot diameter 280\nµm) was circular ( σ±) to excite the magnetic resonance\nmodes via the IFE. The Faraday rotation angle of the\nlinearly polarized probe pulse (wavelength 800 nm, pulse\nenergy less than 10 nJ, repetition rate 1 kHz, pulse width\n60 fs, spot diameter 80 µm) is mainly sensitive to the\nout-of-plane component MFe\nzover the entire temperature\nrange.[21, 22] We applied an external in-plane magnetic\nfieldH0= 2 kOe in the positive and negative directions,\nwhich was sufficient to align MFeandMREin the plane,\nresulting in a single-domain structure below 75 K and\nabove 130 K. Thus, we measured the dependences of the\nmagnetization dynamics on the helicity ( σ±) of the circu-\nlar pump polarization, the direction of the external mag-\nnetic field ( ±H0), and the temperature ( T≷TM). The\ndirection of HIFEcan be controlled by σ±; the directions\nofMFeandMREcan be controlled by ±H0; and the rel-\native magnitudes of MFeandMREcan be controlled by\nT.\nFigures 2(a)–2(d) and 3(a)–3(d) show the Faraday\nrotation of the probe as a function of delay tat\nT= 300 K ( > T M) and 60 K ( < T M), respec-\ntively, in the external magnetic fields + H0[(a): t≤\n20 ps, (c): t≤300 ps] and −H0[(b): t≤20\nps, (d): t≤300 ps], together with fitting with\ndamped sinusoidal functions sin( ωHFt) exp(−αHFωHFt)\nand sin( ωLFt) exp(−αLFωLFt). At 300 K, a high-\n0 10 20 30-1.0-0.50.00.51.01.5\n0 50 100 150 200-1.0-0.50.00.51.00 10 20 30-1.5-1.0-0.50.00.51.01.52.0\n0 50 100 150 200-1.5-1.0-0.50.00.51.01.5+H0Faraday rotation (mrad)\nTime (ps)+sinωHFt\n−sinωHFt\n(c)(b)\n(d)σ+\nσ−Faraday rotation (mrad)\nTime (ps)+sinωLFt\n−sinωLFt(a)\nσ+\nσ−\nσ+\nσ−+H0\nFaraday rotation (mrad)\nTime (ps)+sinωHFt\n−sinωHFt−H0Faraday rotation (mrad)\nTime (ps)+sinωLFt\n−sinωLFtσ+\nσ−−H0FIG. 3. (Color online) Magnetization dynamics at 60 K. HF\nmode at external field: + H0(a) and −H0(b). LF mode at\nexternal field: + H0(c) and −H0(d). Pump helicity: σ+(red\ndots) and σ−(blue dots). The solid lines are the damped\nsinusoidal functions with αHF∼0.01 and αLF∼0.35 for the\nHF and LF modes, respectively.\nfrequency (HF) mode at 403 GHz and a low-frequency\n(LF) mode at 5.8 GHz were observed. At 60 K, an HF\nmode at 222 GHz and an LF mode at 10.7 GHz were ob-\nserved. The HF and LF modes were attributed to the ex-\nchange resonance and spatially propagating FMR (mag-\nnetostatic) modes, respectively.[17] From Figs. 2(a)–2(d)\nand 3(a)–3(d), we observe that the initial phases of the\nsinusoidal functions [sin( ωt) or−sin(ωt)] of the LF and\nHF modes do not depend on the temperature or the direc-\ntion of the external field. In contrast, the pump helicity\nchanged the phases of both modes by 180◦. The results\nare summarized in Table I. The peak at approximately\n6–7 ps in Figs. 2(a) and 2(b) is due to the reflection of\nthe pump pulse from the second face of the sample.[23]\nIt should be noted that for T < T M, measurements\nwere made at 40 and 50 K in addition to 60 K, with qual-\nitatively the same results. For T < 40 K, the analysis is\ncomplicated by the contribution of Yb ions. For TM< T,\nmeasurements were performed at 140–300 K and similar\nresults were obtained. For 75 K < T < 130 K, the ap-\nplied field was not sufficient to align the magnetization\nin-plane, and the two modes could not be excited.\nTo understand the initial phases, two cases can be con-\nsidered for the sublattice selectivity of the IFE. The di-\nrections of HFe\nIFEacting on MFeandHRE\nIFEacting on MRE\nare opposite (Case 1 in Fig. 4) and the same (Case 2 in\nFig. 5), respectively. In each case, MFeandMRErotate\ninstantaneously according to the following equations of\nmotion under the impulsive actions of HFe\nIFEandHRE\nIFE,\nrespectively.\ndMFe\ndt=−γMFe×HFe\nIFE, (1)\ndMRE\ndt=−γMRE×HRE\nIFE. (2)3\nTABLE I. Time evolutions of the HF and LF modes (mea-\nsurement and Case 1)\nTemperature T > T M\nPump helicity σ+σ−\nExternal field + H0 −H0 +H0 −H0\nLF mode + sin ωLFt+ sin ωLFt−sinωLFt−sinωLFt\nHF mode + sin ωHFt+ sin ωHFt−sinωHFt−sinωHFt\nT < T M\nσ+σ−\n+H0 −H0 +H0 −H0\n+ sin ωLFt+ sin ωLFt−sinωLFt−sinωLFt\n+ sin ωHFt+ sin ωHFt−sinωHFt−sinωHFt\nH0 a\nb\nccM1 M2 M2\nM2M1\nM1LF mode\nHF mode\nFIG. 4. Case 1: snapshot of sublattice magnetization devi-\nations by the action of HIFEpulses with the opposite direc-\ntions on each magnetic sublattice. The cross and dot circles\nindicate that the directions of HIFEare from the front to the\nback and from the back to the front of the plane, respectively.\nThese deviations can be decomposed into those of the LF and\nHF modes. M1andM2rotate counterclockwise and clock-\nwise around the H0axis in LF and HF modes, respectively.\nHere, γis the gyromagnetic ratio, which is equal for Gd\nand Fe ions. Even if they were different, the qualita-\ntive argument in this paper would still hold. Equations\n(1) and (2) are valid only during laser-pulse excitation,\nwhere the interactions between the sublattices, magnetic\nanisotropy field, external magnetic field, and damping\ncan be neglected.[24] After the pulses of HFe\nIFEandHRE\nIFE\ndisappear, MFeandMREcontinue to rotate as a super-\nposition of the counterclockwise LF and clockwise HF\nmodes around the H0axis, which determine the initial\nphases of the two modes in Figs. 2(a)–2(d) and 3(a)–\n3(d). Let the two sublattice magnetizations be M1and\nM2, where |M1|>|M2|.M1points toward the external\nmagnetic field. At 300 K, M1=MFeandM2=MRE.\nAt 60 K, M1=MREandM2=MFe.\nIn Case 1, let aandbbe the in-plane displacements of\nM1andM2in the LF mode, respectively. The ratio a/b\ncan be approximated as |M1|/|M2|. In the HF mode,\nthe in-plane displacements of M1andM2are regarded\nas identical and are denoted by c.[10] The in-plane dis-\nplacements of M1andM2can be represented by the\nsuperposition of the LF and HF modes if a≥c≥bholds\ntrue. In special cases where HIFEis generated only in\nH0\nM1\nM2M2\nM2M1\nM1LF mode\nHF mode\nH0\nM1\nM2LF mode\nHF mode(a)\n(b)M2M1\nM2M1\nM2M1M2\nM1FIG. 5. Case 2: snapshot of sublattice magnetization devia-\ntions by the action of HIFEpulses with the same directions\non each magnetic sublattice. The dot circles indicate that the\ndirections of HIFEare from the back to the front of the plane.\nThese deviations can be decomposed into those of the LF and\nHF modes. The dominant motion is in the LF mode (a) or HF\nmode (b). M1andM2rotate counterclockwise and clockwise\naround the H0axis in LF and HF modes, respectively.\nM1orM2,a > b =cora=c > b hold true, respec-\ntively. The dependences of the time evolution of MFe\nz\non the pump helicity, the direction of the external mag-\nnetic field, and the temperature were consistent with the\nexperimental results, as presented in Table I.\nIn Case 2, the time evolution of MFe\nzdiffers depending\non whether the dominant motion is in the LF mode (Case\n2a) or HF mode (Case 2b), because the sense of rotation\nis opposite between the two modes. In Case 2a [Fig.\n5(a)], the MFe\nzfor the LF mode can be determined, but\nnot that for the HF mode. In Case 2b [Fig. 5(b)], the\nMFe\nzfor the HF mode can be determined, but not that for\nthe LF mode. The time evolutions of MFe\nzare presented\nin Tables II and III for Cases 2a and 2b, respectively;\nhowever, they do not match the experimental results in\nTable I.\nWe conclude that the IFE in GdYb-BIG operates in\nopposite directions for each sublattice (Case 1). In spe-\ncial cases, HIFEis generated only in MFeorMRE. This\nis consistent with the sublattice selectivity of the FE, in\nwhich the signs of the contributions of Fe and RE ions to\nthe FE are opposite and the Fe ion’s contribution is dom-\ninant in the near-infrared range.[22] This is in contrast to\nantiferromagnets, in which the magnitude and direction\nofHIFEacting on each sublattice are identical.[7] The ini-\ntial phase analysis can be applied to other ferrimagnets\nby measuring below and above the compensation temper-\natures. A clarification of the sublattice selectivity of the\nIFE is expected to promote the development of devices\nthat utilize ferrimagnetic materials and magnetooptical4\nTABLE II. Time evolutions of the HF and LF modes (Case\n2a). The ±signs of the HF modes are in the same order.\nTemperature T > T M\nPump helicity σ+σ−\nExternal field + H0 −H0 +H0 −H0\nLF mode + sin ωLFt+ sin ωLFt−sinωLFt−sinωLFt\nHF mode ±sinωHFt±sinωHFt∓sinωHFt∓sinωHFt\nT < T M\nσ+σ−\n+H0 −H0 +H0 −H0\n−sinωLFt−sinωLFt+ sin ωLFt+ sin ωLFt\n∓sinωHFt∓sinωHFt±sinωHFt±sinωHFt\neffects, such as the study of magnetization reversal using\nthe IFE.[25]\nACKNOWLEDGMENTS\nWe are grateful to Kouki Mikuni for his technical as-\nsistance. This study was financially supported by the\nJapan Society for the Promotion of Science KAKENHI(grant Nos. JP19H01828, JP19H05618, JP21H01032,\nJP22H01154, and JP22K14588), the Frontier Pho-\ntonic Sciences Project (NINS grant Nos. 01212002\nand 01213004), and OML Project (NINS grant No.\nOML012301) of the National Institutes of Natural Sci-\nences (NINS), and MEXT Initiative to Establish NeXt-\ngeneration Novel Integrated Circuits CenterS (X-NICS)\n(grant No. JPJ011438).\nTABLE III. Time evolutions of the HF and LF modes (Case\n2b). The ±signs of the LF modes are in the same order.\nTemperature T > T M\nPump helicity σ+σ−\nExternal field + H0 −H0 +H0 −H0\nLF mode ±sinωLFt±sinωLFt∓sinωLFt∓sinωLFt\nHF mode + sin ωHFt+ sin ωHFt−sinωHFt−sinωHFt\nT < T M\nσ+σ−\n+H0 −H0 +H0 −H0\n∓sinωLFt∓sinωLFt±sinωLFt±sinωLFt\n−sinωHFt−sinωHFt+ sin ωHFt+ sin ωHFt\n[1] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pis-\narev, A. M. Balbashov, and Th. Rasing, Nature 435,\n655 (2005).\n[2] L. P. Pitaevskii, Sov. Phys. JETP 12, 1008 (1961).\n[3] P. S. Pershan, Phys. Rev. 130, 919 (1963).\n[4] J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom,\nPhys. Rev. Lett. 15, 190 (1965).\n[5] A. M. Kalashnikova, A. V. Kimel, R. V. Pisarev, V. N.\nGridnev, A. Kirilyuk, and Th. Rasing, Phys. Rev. Lett.\n99, 167205 (2007).\n[6] T. Satoh, S.-J. Cho, R. Iida, T. Shimura, K. Kuroda,\nH. Ueda, Y. Ueda, B. A. Ivanov, F. Nori, and M. Fiebig,\nPhys. Rev. Lett. 105, 077402 (2010).\n[7] C. Tzschaschel, K. Otani, R. Iida, T. Shimura, H. Ueda,\nS. G¨ unther, M. Fiebig, and T. Satoh, Phys. Rev. B 95,\n174407 (2017).\n[8] J. Kaplan and C. Kittel, J. Chem. Phys. 21, 760 (1953).\n[9] R. K. Wangsness, Phys. Rev. 91, 1085 (1953).\n[10] B. Lax and K. J. Button,\nMicrowave Ferrites and Ferrimagnetics (McGraw-Hill,\n1962).\n[11] B. A. Ivanov, Low Temp. Phys. 45, 935 (2019).\n[12] J. Finley and L. Liu, Appl. Phys. Lett. 116, 110501\n(2020).\n[13] S. K. Kim, G. S. Beach, K.-J. Lee, T. Ono, Th. Rasing,\nand H. Yang, Nat. Mater. 21, 24 (2022).\n[14] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Tsukamoto,\nA. Itoh, A. Kirilyuk, and Th. Rasing, Phys. Rev. Lett.\n98, 207401 (2007).[15] A. H. M. Reid, A. V. Kimel, A. Kirilyuk, J. F. Gregg,\nand Th. Rasing, Phys. Rev. Lett. 105, 107402 (2010).\n[16] T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando,\nE. Saitoh, T. Shimura, and K. Kuroda, Nat. Photon. 6,\n662 (2012).\n[17] S. Parchenko, A. Stupakiewicz, I. Yoshimine, T. Satoh,\nand A. Maziewski, Appl. Phys. Lett. 103, 172402 (2013).\n[18] S. Parchenko, T. Satoh, I. Yoshimine, F. Stobiecki,\nA. Maziewski, and A. Stupakiewicz, Appl. Phys. Lett.\n108, 032404 (2016).\n[19] M. Deb, P. Molho, B. Barbara, and J.-Y. Bigot, Phys.\nRev. B 94, 054422 (2016).\n[20] A. Stupakiewicz and T. Satoh, J. Phys. Soc. Jpn 90,\n081008 (2021).\n[21] S. Parchenko, M. Tekielak, I. Yoshimine, T. Satoh, A.\nMaziewski, and A. Stupakiewicz, IEEE Trans. Magn.\n50, 6000904 (2014).\n[22] A. R. Khorsand, M. Savoini, A. Kirilyuk, A. V. Kimel,\nA. Tsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett.\n110, 107205 (2013).\n[23] P. Khan, M. Kanamaru, W.-H. Hsu, M. Kichise, Y. Fujii,\nA. Koreeda, and T. Satoh, J. Phys.: Condens. Matter\n31, 275402 (2019).\n[24] I. Yoshimine, T. Satoh, R. Iida, A. Stupakiewicz,\nA. Maziewski, and T. Shimura, J. Appl. Phys. 116,\n043907 (2014).\n[25] T. Dannegger, M. Berritta, K. Carva, S. Selzer, U. Ritz-\nmann, P. M. Oppeneer, and U. Nowak, Phys. Rev. B\n104, L060413 (2021)."    },    {        "title": "1908.06450v1.Anomalous_magnetic_behavior_and_complex_magnetic_structure_of_proximate_LaCrO3_LaFeO3_system.pdf",        "content": " 1  \nAnoma lous magnetic behavior and complex magnetic structure of proximate LaCrO 3 – \nLaFeO 3 system  \nBrajesh Tiwari1*,  Ambesh  Dixit2, M. S. Ramachandra Rao3 \n1 Department of Physics, Institute of Infrastructure Technology Research and Management, \nAhmedabad -380026 , India . \n2Department of Physics & Center for Solar Energy, Indian Institute of Technology Jodhpur , \nKarwad 342037, India . \n3Department of Physics, Indian Institute of Technology Madras, Chennai, 600036 , India . \n*brajeshtiwari@iitram.ac.in   \n \n \nAbstract:  \nWe investigated complex magnetic properties of multifunctional LaCr O3-LaFeO 3 system . \nThe magnetic measurements substantiate the presence of competing  complex magnetic \nordering against temperature , showing  paramagnetic to ferrimagnetic transition  at ~ 300 K, \nfollowed by antiferromagnetic  (AFM) transition near ~250 K superimposed on ferri magnetic \nphase. The onset of weak ferrimagnetic ordering is attributed to the competing complex \ninteraction between two AFM LaCrO 3-LaFeO 3 sublattices. The low-temperature  AFM \nordering is also substantiated by  temperature -dependent  Raman measurements, where  the \nintensity ratio of 700 cm-1 Raman active mode  show ed the clear enhancement with lowering \nthe temperature. The non -saturating nature of magnetic moments in LaCr O3-LaFeO 6 suggests \nthe predominating AFM ordering in conjunction with ferrimagnetic  ordering between 250 K \n– 300 K up to 5 T magnetic field.  A complex magnetic structure of LaCrO 3-LaFeO 3 is \nconstructed, emphasizing the metastable magnetic phase near room temperature and low \ntemperature antiferromagnetic state.    \n \n \n \nKeywords:  Magnetism; proximity effect; ternary magnetic oxides, Raman Spectra,  \ntransition temperature; ferrmimagntism; antiferromagentism.   2 Introduction:  \nThe multifunctional materials, especially complex oxide materials, are not only attracting \nattention due to their potential but also providing rich understanding of the fundamentals, \nwhich allows designing novel materials with desired functional properties . LaTMO 3 (TM = \nTransition Metal) is one such family of complex oxide materials, having perovskite  structure \nwith TM magnetic ions.  Direct or indire ct cross –coupling among spin, orbital , lattice and \ncharge degrees of freedom provide avenue  for potential ap plication and fundamental study in \nthese materials.  The canted spin structure of TM ions also exhibits  competing  magnetic \ninteraction  and thus,  giving rise to the intricate magnetic structure s. LaCrO 3 and LaFeO 3 \noxide systems exhibit antiferromagnetic transitions at ~ 290 K and 740 K respectively  [1-3, \n4, 5] and more interestingly, also exhibit weak ferromagnetism s near room temperature  [6-9]. \nThe search of more than one ferroic ordering in oxide systems is always attracting attent ion \nfor their potential in new class of electronic devices such as four state memories, voltage \ncontrolled magnetic switches and sensors, electric  field controlled spintronic devices. The \nrecent studies provide evidences about the room temperature magnetoelectric coupling in \nLaCrO 3 [6] and probably the ferroelectric ordering in LaFeO 3 perovskite systems  [5]. In spite \nof magnetic and magnetodielectric properties, these oxide systems exhibit enhanced oxygen \nionic conductivity and their electronic cond uctivity can be tailored by manipulating the \nsuitable dopant at different cation sites. For example, the calcium doping at La site in LaCrO 3 \nmakes it highly conducting and is a potential candidate for high temperature solid oxide fuel \ncell el ectrode materi al. The distorted TMO 6 octah edra in LaTMO 3 systems is the driving \ncause for the complex physical properties such as variation of transition temperature, strong \nelectron -phonon coupling, weak ferromagnetism, electrical conductivity by manipulating the \nexcha nge and hopping strengths  [10]–[12]. The order and amplitude of such changes in \nphysical properties are associated with the degree of distortion. The divalent doped  3 lanthanum manganite system shows charge –ordering , which is closely related to \nantiferromagnetic phase , while charge delo calization i.e. metallic state  coincides with \nferromagnetism.  The screened potential energy becomes large in certain TM oxide materials , \ndue to the various external factors (doping, temperature, pressure etc),  causing electron  \nlocaliz ation  and thus, inhibiting the electrical conduction. This is known as Mott – transition \nin such transition metal -based  perovskite systems  [13], [14] . In addition , the hole-doped  \nLa2CuO 4 antiferromagnetic  become s high temperature superconductor because of the  strong \nelectron -electron correlation. These systems provide a wide avenue to understand the \nunderlying physics and related mechanism governing such functional properties and their \npossible tunability. The double perovskite materials with and without transition metals are \ngaining attention for designing multifunctional material systems  [15] [16]–[18]. Way back in \n1960’s, Goodenough -Kanamuri  predicted th at double perovskite  La2CrFeO 6 should have a \nferromagnetic  ground state with Tc close to room temperature   [19]–[21]. Since then a lot \nmore effort s are made exper imentally and theoretically with varying conclusions. Some \nconcluded ferromagnetic ground state [20] while other s ferromagnetic [ 21].  This prompted \nus to explore La 2CrFeO 6 double perovskite  while synthesizing  LaCrO 3-LaFeO 3  system. This \nsystem is usually showing intermediate properties  of both perovskites, however, some  unique \nmagnetic and optical phonon properties  are also noticed  due to close proximity of these two \npredominantly antiferromagnetic compounds. It is a fundamental challenge in materials ’ \ndesign to control and understand the change in materials behavior in close proximity to other \nmaterials under varying external conditions. In the present work , we investigate d the effect of \nspin ordering on the magnetic and ele ctronic properties of closely proximated LaCrO 3-\nLaFeO 3 system.    \n \n  4 Experimental Details:  \nAll oxide p recursors were heated at ~ 800 oC to remove any residual oxide and ground in a \nstoichiometric  ratio to homogenize the pre -synthesized material s. This homogeneous material \nwas heated at 950 oC for 48 hours with intermediate grinding to ensure the homogeneity of a \nsolid  solution. For structural and phase identification, powder X-ray diffraction (XRD) data \nof the samples were recorded using  a PANalytical X’Pert Pro X-ray diffractometer with Cu \nKα radiation. DC magnetic measurements were performed using  a vibrating sample \nmagnetometer as an attachment in Physical Property Measurement System (Model 6000, \nQuantum Design, USA) in the temperature ra nge of 100 –350 K. Magnetization \nmeasurements were performed  as a function  of temperature in zero-field cooled (ZFC) and \nfield cooled  (FC)  modes. Various m agnetization  isotherms were recorded at different \ntemperatures up to an  applied magnetic field of 50  kOe in the vicinity of magnetic transitions. \nRaman spectra were recorded for LaCrO 3, LaFeO 3 and proximate  LaCrO 3-LaFeO 3 at room \ntemperature with the hel p of 532 nm green laser source. Temperature -dependent  Raman \nspectra were also recorded for LaCrO 3-LaFeO 3 system down to 100 K in order to understand \nthe spin-phonon coupling if any , following the magnetic behavior .  \n \nResults and discussion:  \nThe phase identification of synthesized materials was confirmed using X -ray diffraction and \nthe respective diffractograms are shown  in Fig. 1 in conjunction with LaCrO 3 and LaFeO 3 \nperovskite structure to understand the phase evolution of LaCrO 3-LaFeO 3  system.  The XRD \npatterns , Fig. 1 (lower and middle panel) , confirm the phase purity of pristine LaCrO 3 and \nLaFeO3 bulk materials and results are consisten t with the reported literature [6], [7], [22] . All \nthe peaks are in agreement and representative ( h k l) planes are marked for LaCrO 3 system. \nLaCrO 3 and LaFeO 3 systems crystallize in distorted orthorhombic perovskite system with  5 almost similar lattice parameters. The structure consists of corner shared tilted TMO 6 (TM = \nCr, Fe) oct ahedra . The structural and  magnetic details as reported earlier suggest that the \ntilted octahedral  may induce non -collinearity in the spin structure, giving rise to the weak \nferromagnetism in these systems [2], [6] . The  XRD diffractogram, Fig. 1 (upper panel) , can \nbe visualized as the superimposed XRD spectrum of these  pristine materials. These closely \nspaced doublets confirm the formation of mixed phase  LaCrO 3-LaFeO 3 system  without any \nadditional impurities.   \nThe room temperature Raman  spectrographs of these systems  are shown in Fig. 2 and \nanalyzed  to understand the mi croscopic phase evaluation  for LaCrO 3-LaFeO 3 system . The \nfactor group analysis of orthorhombic (Pnma space group) suggests that there are 24  ( = 7Ag \n+ 5B1g + 7B2g + 5B3g) Raman active modes in this distorted perovskite  LaCrO 3 and d etail of \nmode assignments are given in references  [6], [23], [24] . The identical crystallographic \nstructure of LaCrO 3 and LaFeO 3, space group Pnma, gives rise to the similar vibrational \nmodes for both systems with a small deviation for different atom ic masses of Cr and Fe \natoms . Thus, it was difficult to separate out the vibrational  contribution of one  from other, as \nevident in XRD graph s, Fig. 1 , as well as from Raman spectra , Fig 2 . The temperature \ndependent Raman spectra are shown in Fig 2, where s ome of the Raman active modes in \nproximate LaCrO 3-LaFeO 3 system show  peculiar  temperature dependence  as compared to \npristine systems . After  careful analysis of first and second order  optical phonon B2g(1) \nmodes, it is observed that intensity ratio (I 1/I2) of these modes show a sharp increase near  \nsecond magnetic transition , as shown in Fig 2(b) . This increa se in the intensity of first order \nB2g(1) mode as compared to second order mode near magnetic anomaly indicates a spin -\ndependence of optical phonon mode in Raman scattering that can be a manifestation of \nelectron transfer with lattice vibrations and/or ani sotropic exchange interactions.  \n  6  LaCrO 3 and LaFeO 3 materials are known antiferromagnetic with Neel temperature 290 K \nand 710 K, respectively [6], [9], [25] . The tilted Cr/Fe -O6 octahedra  lead to the canted TM \nelectron spins and thus , causing weak ferromagnetism in these systems. In conjunction with \nthe observed weak ferromagnetism in these systems, the near room temperature \nmagnetodielectric coupling has also been reported in both systems. Considering the complex \nmagnetic intera ctions in pristine systems, DC magnetization as function of temperature fro m \n350 K to 100 K has been  recorded under zero field cooled (ZFC) and field cooled (FC) \ncondition at 1000 Oe  for LaCrO 3-LaFeO 3 system. The measured temperature dependent \nmagnetic moment is shown in F ig. 3. The observed sudden rise in magnetization near 290 K \nin this sample is due to the antiferromagnetic ordering of LaCrO 3, superimposed with weak \nferromagnetism  because of spin canting . The proximate presence of LaFeO 3 tries to reorient \nthe magnetic spin stru cture of LaCrO 3 sublattice along the weak ferromagnetic structure of \nLaFeO 3, causing relatively larger ferromagnetic component below 290 K. This ferromagnetic \nstate of LaCrO 3-LaFeO 3 system preserves down to 250 K, after that the antiferromagnetic \nordering of LaCrO 3 starts dominating. This LaCrO 3 antiferromagnetic dominance, leads to \nanother  antiferromagnetic transition for LaCrO 3-LaFeO 3 system. Thus, two magnetic \ntransition s are observed clearly at 290 K (weak ferromagnetic) and 250 K (antiferromagn etic) \nin LaCrO 3-LaFeO 3 system. Further, t o understand the dynamic nature of these magnetic \ntransitions , we carried out temperature dependent AC magnetic measurements at 100, 300, \n1000, 3000 and 10000 Hz frequencies in the temperature same range and plots are shown  in \nFig. 3 . The first magnetic transition around  290 K is coinciding  to that of LaCrO 3 long-range  \nantiferromagnetic Neel temperature , superimposed with weak ferromagnetic ordering of both  \npristine  LaCrO 3 and LaFeO 3 systems  [4]. The onset of additional magnetic transition at ~ 250 \nK may be the consequence of competing spin interaction between close proximity of \nmagnetic Cr and Fe  sublattices . To our surprise even transition at 250 K  is also frequency  7 independent , suggest ing a long-range  spin ordering in LaCrO 3-LaFeO 3 system . The observed  \nfrequency independence magnetic transitions also rule out th e possibility of any \nclustering/spin glassy impurities in the synthesized LaCrO 3-LaFeO 3 system.  The \nsimultaneous pre sence of two AFM transitions in  this system  may be the consequence of \nproximity of Fe and Cr magnetic ions and complex magnetic interaction between them. The \nXRD results confirm the synthesis of LaCrO 3-LaFeO 3 mixed phase system and observed \ncomplex magnetic properties suggest the presence of compet ing magnetic interaction \nbetween different Cr and Fe ion sites in LaCrO 3-LaFeO 3 system . The magnetization \nisotherms are measured near these transition temperatures to probe the nature of magnetic \nordering in conjunction with the measured temperature depend ent magnetization \nmeasurements. The measured magnetic isotherms are shown in Fig. 4 for temperatures 200, \n250, 280 and 315 K. The weak ferromagnetic component at 315 K is lower than that of 280 \nK, suggesting that at higher temperature, only LaFeO 3 weak fer romagnetic component is \ncontributing in this  system. However, at lower temperatures, the contribution of LaCrO 3 \nweak ferromagnetic component is also added, as can be observed in Fig. 4. In contrast to the \nobserved weak ferromagnetism, the magnetization curves are not saturating up to 50 kOe  \nmagnetic field suggesting the dominance of antiferromagnetic state  in LaCrO 3 – LaFeO 3 \nsystem. The magnified magnetization curves are shown in Fig. 4  (lower panel) , showing \nnearly temperature insen sitive weak remnance  ~ 3x10-3 emu/g. However, the respective \ncoercive field is much larger ~ 1 kOe for LaCrO 3 – LaFeO 3 system, suggesti ng the robust \nmetastable ferromagnetic state.  \nThe magnetic phase diagram against temperature is constructed for LaCrO 3-LaFeO 3 system, \nand is shown in Fig. 5. The high-temperature  phase (>740 K) is paramagnetic, changing into \nantiferromagnetic phase dominated by LaFeO 3 in the proximate  LaCrO 3-LaFeO 3 system. \nThis antiferromagnetic phase is superimposed with the weak ferromagnetic phase because of  8 the canted iron spin s in FeO 6 octahe dron in LaFeO 3 sublattice, as marked in Fig. 5. Further \nreducing the temperature below 290 K, this  changes into a metastable ferromagnetic phase at \nthe onset of LaCrO 3 antiferromagnetic transition . This phase persists until 250 K, where the \nsystem shows  another  antiferromagnetic phase with weak ferromagnetism simultaneously.  \n \nConclusion:  \nWe studied the complex magnetic behavior of proximate LaCrO 3- LaFeO 3 system with \ndifferent magnetic phases and intertwining of optical phonons with magnetic ordering.  These \nstudies may lead to the materials engineering to design complex magnetic structured \nmaterials with competing magnetic phases at or above room temperatures in mixed phase \nsystems. The observed spin -lattice coupling from temperature dependent Raman sp ectra \nshows a possibility of inducing magnetodielectric coupling in such mixed phase systems.  \nFurther investigations are required to understand the microscopic origin of observed complex \nmagnetic structure and spin -lattice coupling proximate LaCrO 3-LaFeO 3 system for possible \ntunability of spin -lattice functional properties.  \n \nAcknowledgement :  \nAuthor Brajesh Tiwari acknowledges Prof essor Shiva Prasad for technical discussions for the \nmanuscript . Ambesh Dixit acknowledges UGC -DAE  Consortium For  Scientific Research, \nGov. of India through project number CRS -M-221 for this work.  \n \nReference:  \n[1] I. V. Solovyev, N. Hamada, and K. Terakura, “Non -collinear magnetism in distorted \nperovskite compounds,” Phys. B Condens. Matter , vol. 237 –238, pp. 44 –45, Jul. 1997.  \n[2] B. Tiwari, A. Dixit, R. Naik, G. Lawes, and M. S. R. 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Rabe, “Large spin -phonon coupling and magnetically induced \nphonon anisotropy in SrMO3 perovskites (M=V,Cr,Mn,Fe,Co) ,” Phys. Rev. B - \nCondens. Matter Mater. Phys. , vol. 84, no. 10, pp. 1 –6, 2011.  \n[12] C. Weingart, N. Spaldin, and E. Bousquet, “Noncollinear magnetism and single -ion \nanisotropy in multiferroic perovskites,” Phys. Rev. B , vol. 86, no. 9, p. 094413, Sep. \n2012. \n[13] N. F. Mott, “Metal -insulator transition,” Reviews of Modern Physics , vol. 40, no. 4. pp. \n677–683, 1968.  \n[14] H. Park, A. Millis, and C. Marianetti, “Site -Selective Mott Transition in Rare -Earth -\nElement Nickelates,” Phys. Rev. Lett. , vol. 109, no. 1 5, pp. 1 –5, Oct. 2012.  \n[15] J. B. Goodenough, “An interpretation of the magnetic properties of the perovskite -type \nmixed crystals La1−xSrxCoO3−λ,” J. Phys. Chem. Solids , vol. 6, no. 2 –3, pp. 287 –\n297, Aug. 1958.  \n[16] M. P. Ghimire, L. Wu, and X. Hu, “Possib le Half Metallic Antiferromagnetism in a \nDouble Perovskite Material with Strong Spin -Orbit Couplings,” pp. 1 –8, 2014.  \n[17] B. Gray, H. N. Lee, J. Liu, J. Chakhalian, and J. W. Freeland, “Local electronic and \nmagnetic studies of an artificial La2FeCrO6 doub le perovskite,” Appl. Phys. Lett. , vol. \n97, no. 1, p. 013105, 2010.  \n[18] S. Chakraverty et al. , “Ferrimagnetism and spontaneous ordering of transition -metals \nin La 2 CrFeO 6 double -perovskite films,” pp. 1 –11. \n[19] K. Miura and K. Terakura, “Electronic and  magnetic properties of La2FeCrO6: \nSuperexchange interaction for a d5 -d3 system,” Phys. Rev. B , vol. 63, no. 10, p. \n104402, Feb. 2001.  \n[20] W. E. Pickett, “Ferromagnetic Superlattices,” Science (80 -. )., vol. 281, no. 5383, p.  11 1571a, 1998.  \n[21] B. Gray, H.  N. Lee, J. Liu, J. Chakhalian, and J. W. Freeland, “Local Electronic and \nMagnetic Studies of an Artificial La2FeCrO6 Double Perovskite,” vol. 013105, pp. \n14–17, 2010.  \n[22] A. A. Cristóbal, P. M. Botta, P. G. Bercoff, and J. M. Porto López, “Mechanosynthes is \nand magnetic properties of nanocrystalline LaFeO3 using different iron oxides,” \nMater. Res. Bull. , vol. 44, no. 5, pp. 1036 –1040, 2009.  \n[23] M. Iliev et al. , “Raman spectroscopy of low -temperature (Pnma) and high -temperature \n(R3¯c) phases of LaCrO3,” Phys. Rev. B , vol. 74, no. 21, pp. 1 –7, Dec. 2006.  \n[24] M. Iliev et al. , “Distortion -dependent Raman spectra and mode mixing in RMnO3 \nperovskites (R=La,Pr,Nd,Sm,Eu,Gd,Tb,Dy,Ho,Y),” Phys. Rev. B , vol. 73, no. 6, pp. 3 –\n8, Feb. 2006.  \n[25] C. Chen, K. B. Xu, Y . M. Cui, and C. C. Wang, “Polaronic relaxation in LaFeO3,” \nMater. Lett. , vol. 89, pp. 153 –155, Dec. 2012.  \n \nFigure Captions:  \n1. X-ray diffraction of bulk LaCrO 3 and LaFeO 3  material with LaCrO 3 (Bottom panel) \nand LaFeO 3 (Middle panel) . \n2. (a) Raman Spectra o f LaCrO 3, LaFeO 3 and  La 2FeCrO 6 at room temperature recorded \nusing 532 nm laser source. ( b) Temperature -dependent  Raman spectra of La 2FeCrO 6 \nusing 514 nm laser source. ( c) Intensity ratio of first order and second order Raman \npeak as a function  of temperature suggesting suppression of second order peak upon \nmagnetic ordering.  \n3. Temperature -dependent  DC (Zero -Field Cooled and Field Cooled) and AC (at \ndifferent frequencies)  magnetic susceptibility plots for La 2FeCrO 3 .   12 4. Representative isothermal magnetization loops close to magnetic transitions.  \n \nFigures:  \n \n \nFigure 1.  \n \n \n20 30 40 50 60 70 80 90(242)\n(323)(412)(341)(400)(321)(113)(141)(202)(220)(211)(200)(121)(111)\n2q (degree)LaCrO3(101) Intensity (a.u.)\nLaFeO3 La2CrFeO6 13  \nFigure 2(a)  \n \n100 200 300 400 500 600 700 800 900024004800720096000160032004800640001000200030004000100 200 300 400 500 600 700 800 900\n Intensity (counts)\nRaman Shift (cm-1) LaCrO3\n Intensity (counts)  LaFeO3 \n Intensity (counts)  La2FeCrO6 14  \n \nFigure2 b  \n \nFigure 2 (c)  \n100 150 200 250 300 3501.21.31.41.5\nT (K)I1/I2 15  \n  \nFigure 3  \n100 150 200 250 300 3509.0µ10.0µ11.0µ12.0µ13.0µ14.0µ' (emu/g-Oe)\nT (K) 100Hz\n 300Hz\n 1000Hz\n 3000Hz\n 10000Hz\n100 150 200 250 300 35018.0µ20.0µ22.0µ24.0µ26.0µ (emu/g-Oe )\nT (K) ZFC\n FC 16  \nFigure 4  \n \n \n \nFigure 5  \n-4 -2 0 2 4-0.010-0.0050.0000.0050.010M (emu/g)\nH (kOe) 200 K\n 250 K\n 280 K\n 325 K\n-50 -40 -30 -20 -10 010 20 30 40 50-0.06-0.04-0.020.000.020.040.06M (emu/g)\nH (kOe) 200 K\n 250 K\n 280 K\n 325 K"    },    {        "title": "1502.05738v3.Ferrimagnetism_in_2D_networks_of_porphyrin_X_and__XO__X_Sc_____Zn__with_acetylene_bridges.pdf",        "content": "arXiv:1502.05738v3  [cond-mat.mtrl-sci]  23 Oct 2015Ferrimagnetism in 2D networks of porphyrin-X and -XO (X=Sc, ...,Zn) with acetylene\nbridges.\nMa/suppress lgorzata Wierzbowska and Andrzej L. Sobolewski\nInstitut of Physics, Polish Academy of Science (PAS), Al. Lo tnik´ ow 32/46, 02-668 Warszawa, Poland\n(Dated: August 7, 2021)\nMagnetism in 2D networks of the acetylene-bridged transiti on metal porphyrins M(P)-2(C-C)-2\n(denoted P-TM), and oxo-TM-porphyrins OM(P)-2(C-C)-2 (de noted P-TMO), is studied with the\ndensity functional theory (DFT) and the self-interaction c orrected pseudopotential scheme (pSIC).\nAddition of oxygen lowers magnetism of P-TMO with respect to the corresponding P-TM for most\nof the first-half 3 d-row TMs. In contrast, binding O with the second-half 3 d-row TMs or Sc increases\nthe magnetic moments. Ferrimagnetism is found for the porph yrin networks with the TMs from\nV to Co and also for these cases with oxygen. This is a long-ran ge effect of the delocalized spin-\npolarization, extended even to the acetylene bridges.\nPACS numbers: 68.35.bm, 68.35.Dv, 68.65.Cd, 75.10.Lp, 75. 25.Dk\nI. INTRODUCTION\nPorphyrins are popular molecules in living organisms.\nTheir ability to bind the transition metals is utilized in\nbiological processes significant for the human-blood func-\ntions - employing Fe in heme. It is also responsible for\nthe photosynthesis in plants - building a complex with\nMg, namely chlorofile.1The oxo-metal-porphyrins, such\nas oxotitanium porphyrin (P-TiO), can be used for solar-\nactivated water splitting - what has been theoretically\nproposed to occur due to low-lying ligand-to-metal in-\ntramolecular charge-transfer states2and recently experi-\nmentally confirmed.3\nPorphyrins or phthalocyanines can be assembled at\nmetals4,5and other surfaces6forming 2D networks.7In\nturn, the surfaces or additional ligands change the charge\nstate of the transition metal bound to porphyrins with\nrespect to the bare 2D network. This is because different\nvalence states of a metal, metal-oxygen moiety, or metal-\nLi complex lay closely in the energy.8–11Not only the\nsingle layers of the metal-organic networks exist, also the\ndouble- and triple-decker layers of metal-phthalocyanines\nhave been grown, on both the ferro- and antiferromag-\nnetic substrates.12\nFor catalytic reactions with the intramolecular charge\ntransfer, such as the water splitting, the intermolecu-\nlar connections must be very weakly conducting (non-\ncovalent); for example the hydrogen bridges (O-H...O).\nFor the spintronic applications, on the other hand, these\nconnections should be conducting. For this reason, we\nhave chosen the acethylene bridges. Various choices of\nthe intermolecular-bridging type were examined experi-\nmentally for the energy transfer rates,13,14and a role of\nthe frontier molecular-orbitals was theoretically studied\nfor the donor-acceptor dimers.15\nA new branch of material science grows on top of po-\ntential utilization of the 2D organic layers16in the elec-\ntronic devices; as well as the magnetic nanoparticles de-\nposited on various substrates.17Selective incorporation\nof metal atoms into organic templates enables wide func-\ntionalization of the layers,18hence many applications areplausible. The 4-fold symmetry of porphyrins and ph-\nthalocyanines with the transition metals makes their net-\nworks similar to the Heusler compounds, which are very\nstrong magnets.19There are review articles on the spin-\ntronic topics in the 2D networks.20,21The photovoltaic\nthin films based on the metal-phthalocyanine 2D net-\nworks - organized in the AA-stacking order - conduct the\ncurrent very well in the columnar direction.22Similar ef-\nfect should be observed in metal-porphyrins. For tech-\nnological reasons, the conductive properties of the single\nmetal-porphyrin molecules between the electrodes,23and\nthe 1D metal-porphyrin chains24were also investigated.\nRecently, high-mobility field-effect transistors have been\nconstructed by building the ABAB-type stacking of the\nphthalocyanine-VO and -TiO layers.25\nIn this work, we focus on the covalent 2D networks of\nporphyrins with all 3 d-row transition metals and TMO,\nbridged with the acetylene moiety. The model struc-\ntures of our systems are presented in Fig. 1. We want to\nobtain a material with strong and long-range magnetic-\nexchange interactions. Thus the covalent intermolecular\nconnection was chosen, since it has been found that the\ncovalent inorganic materials - as hosts for the transition\nmetal impurities - are strongly magnetically coupled for\nlong distances.26,27\nFIG. 1. 2D network of porphyrin-MnO with acetylene\nbridges; top view and side view.\nInterestingly, another molecular system - similar to2\nthat studied in this work, but assembled at Cu and\ncontaining Fe - was demonstrated to switch the easy\nmagnetization-direction after adding the oxygen.28We\nreport the ferrimagnetism, similar to that theoreti-\ncally predicted by other authors in porphyrins with Mn\nand connected by 4-bromophenyl to form 1D magnetic\nchains.29This effect can be mapped on the atomic scale\nby the measurement of magnetic resonance spectra at gi-\ngahertz frequencies using X-ray magnetic circular dichro-\nism (XMCD).30\nII. THEORETICAL DETAILS\nWe performed the density-functional theory (DFT)\ncalculations using the Quantum ESPRESSO (QE)\nsuite of codes,31which employs the plane-wave basis\nset and the pseudopotentials to describe the core elec-\ntrons. The exchange-correlation functional was cho-\nsen for the gradient corrected Perdew-Burke-Ernzerhof\n(PBE) parametrization.32The ultrasoft pseudopotentials\nwere used with the energy cutoffs 35 Ry and 400 Ry\nfor the plane-waves and the density, respectively. The\nMonkhorst-Pack uniform k-mesh in the Brillouin zone\n(BZ) has been set to 10 ×10×1, and the Fermi-surface\nenergy broadening parameter 0.02 Ry was chosen for\na better convergence. The vacuum separation between\nthe periodic slabs was 40 ˚A. The pseudopotentials for\nthe atoms from Sc to Co were modeled with the semi-\ncore states (3s and 3p) included in the valence band, in\ncontrast, the Cu and Zn valence shells were constructed\nwithout the deeper states.\nWe started with the geometry optimization of a sin-\ngle molecule, namely porphyrin-Ti, using the B3LYP\nmethod which is equivalent to the DFT scheme of the\nBLYP-type with 20% of the exact exchange (EXX).33–35\nThis step was done with the quantum chemistry package\nTURBOMOLE,36which represents the wavefuntions in\nthe gaussian basis set; we used the correlation-consistent\nvalence double-zeta atomic basis set with polarization\nfunctions for all atoms (cc-pVDZ).37Obtained geome-\ntry and the lattice vectors, derived from the molecular\nsize, were inserted as an input for the geometry optimiza-\ntion of the periodic structures, calculated by means of the\ngeneralized gradient approximation (GGA) scheme (with\nthe PBE parametrization) using the QE tool. The above\nprocedure is accurate enough, since the central ’squares’\nof porphyrin and phthalocyanine derivatives do not relax\nmuch after binding a metal.38\nIt is known, that the DFT approach underestimates\nthe energy gaps and the energetic positions, with respect\nto the Fermi level, of the localized d- or f-shells. This\nfact has consequences in the description of magnetiza-\ntion. In order to improve the treatment with the lack of\nthe exact exchange, we applied the pseudopotential self-\ninteraction correction (pSIC) method proposed by Filip-\npetti and Spaldin,39and implemented by us in the QE\npackage.40The pSIC method is superior to the DFT+Uapproach41for two reasons: (i) the correction is param-\neter free, unlike the DFT+U parameters: the Coulomb\nU and the exchange J, (ii) the correction is applied to\nall atomic shells, not only d- or f-shell of the transition\nmetals or the rare earths. The pSIC-kernel includes the\nHartree and the exchange-correlation potentials, Vσ\nHXC ,\ncalculated on the orbital-density nσ\ni(r), dependent on\nspinσ, and built using the atomic pseudopotential or-\nbital,ϕi(r). The main equations are:\nˆVσ\nSIC=/summationdisplay\ni|ϕi(r)Vσ\nHXC [nσ\ni(r)]/angbracketright /angbracketleftVσ\nHXC [nσ\ni(r)]ϕi(r)|\n/angbracketleftϕi(r)|Vσ\nHXC [nσ\ni(r)]|ϕi(r)/angbracketright\nnσ\ni(r) =pσ\ni|ϕi(r)|2\npσ\ni=/summationdisplay\nnkfσ\nnk/angbracketleftψσ\nnk|ϕi/angbracketright/angbracketleftϕi|ψσ\nnk/angbracketright\nThe occupation numbers pσ\niare obtained from the pro-\njection of the Kohn-Sham states ψσ\nnkonto the pseudopo-\ntential atomic-orbitals ϕi(r) andfσ\nnkare the Fermi-Dirac\noccupations. Usefulness of this method has been demon-\nstrated in a number of various applications.42–44For this\nwork, the most important are the relations between the\nenergetic positions of the shells: 3 d-TM, 4s-TM and 4p-\nTM, 2p-O and 2p-N. Although, we included the SIC for\nall atomic shells, these corrections are ’on-site’ - the same\nlike in the DFT+U approach and contrary to the Na-\ngaoka model, which includes the ’inter-site’ strong corre-\nlations on the parametric grounds.45,46Using the pSIC\napproach instead of the DFT+U scheme is especially jus-\ntified for the molecular crystals and 2D metal-organic\nframeworks, due to the fact they possess the flat band\nstructures.47In constrast, the ordinary strongly corre-\nlated systems such as the metal oxides and diluted mag-\nnetic semiconductors possess only d-type orf-type flat\nbands and the rest of projected atomic states have usual\nsemiconducting bandwidths.\nIII. RESULTS AND DISCUSSION\nIn this study, we focus our attention on magnetism,\nwhich is a phenomenon sensitive to the system geometry.\nAs mentioned in the theoretical details section, the ge-\nometry was optimized for all studied cases. Positions of\ntransition metal atoms above the plane of the nitrogens\nsquare and the oxygen-TM bond lengths are collected\nin the Table S1 in the supplemental information.48\nInspecting the atomic gradients varied during the geom-\netry optimization and the corresponding magnetizations\nof the systems, we noticed that a tiny change of the\nTM-positions or the TM-O bond-lengths can influence\nmagnetizations. This effect is similar to the phenomenon\nobserved in the perovskites49and also for metal atoms\nat graphene and graphite.50In our cases, however, the\ngeometric effect does not alter main trends in the results\nreported below.3\nFIG. 2. Total and absolute magnetizations (in µB) for metal-porphyrins and oxo-metal-porphyrins with the a cethylene bridges,\nobtained with the GGA and pSIC approaches using the GGA-opti mized geometry.\nA. Magnetizations\nIn Fig. 2, the total and absolute magnetizations in the\nelementary cells, obtained within the GGA and the pSIC\nframeworks, are presented for porphyrins with all TM\nand TMO additions. The total magnetization, defined\nvia the up and down spin-densities, n↑(r) andn↓(r), as\nMtotal =/integraldisplay\nn↑(r)dr−/integraldisplay\nn↓(r)dr,\nreflects the summed magnetization of the sample seen\nfrom a distance. In contrast, the absolute magnetization,\ndefined as\nMabsolute =/integraldisplay\n|n↑(r)−n↓(r)|dr,\nmeans a sum of the local magnetizations regardless their\nsignum. Large difference between the total and the ab-\nsolute magnetizations gives an information on the space\nseparation of the alterred magnetic moments, i.e. the\nferrimagnetic character of the sample.\nFerrimagnetism is well pronounced for the porphyrins\nwith metals from V to Co, and it is the strongest in\nthe Mn, Fe and FeO cases for both calculation meth-\nods, GGA and pSIC, and additionally in the V and VO\ncases obtained with the pSIC scheme. The spin-density\nmap for the porphyrin-Mn network is presented in Fig. 3\nfor the pSIC method. The vanadium spin-density map\nis similar and included in Fig. S2 in the supplemental\ninformation.48It is clear that the nitrogen atoms are\nspin-polarized in the TM vicinity.\nThe effect of oxygen addition on magnetization is very\nstrong. In the case of the TM atoms from the first halfof the 3d-row, the oxygen addition strongly reduces the\nmagnetic moments; except the porphyrin-ScO network.\nIn contrast, for the TM atoms from the second half of\nthe 3d-row, addition of oxygen causes an increase of the\nmagnetic moments. This is an interesting result - similar\nto the results for co-doping of metal-phthalocyanines\nwith Li11- and might be utilized in spintronic devices.\nFIG. 3. Spin-density map of porphyrin-Mn, obtained with\nthe pSIC approach.\nTo get further insight into the electronic structure and\nmagnetism, it is useful to examine the L¨ owdin popula-\ntion numbers. The differences between the L¨ owdin pop-\nulation numbers for the spin up and spin down are the4\nTABLE I. L¨ owdin spin-polarizations (in µB) for the 2D porphyrin networks with TMO (upper part) and TM (l ower part);\ncalculated with the pSIC scheme. Different carbon atoms are i ndexed as follows: C1 denotes carbons adjacent to N, C2\ncarbons terminated with H, C3 carbons adjacent to acetylene , and C4 carbons of acetylene - they are presented in Fig. 4. Th e\ncorresponding GGA numbers are given in parenthesis.\nshell Sc Ti V Cr Mn Fe Co Ni Cu Zn\nOsp 0.720 0.0 -1.570 -0.215 0.043 1.670 2.265 1.546 1.636 1.491\n(0.577) (0.0) (-0.149) (-0.062) (0.577) (0.728) (1.392) (1 .403) (1.435) (1.389)\nTMd0.003 0.0 0.940 2.036 2.991 2.612 3.026 1.562 0.506 0.001\n(-0.016) (0.0) (1.154) (1.684) (2.424) (1.128) (1.579) (1. 749) (0.677) (0.025)\nTMsp0.007 0.0 0.00510.038 0.087 0.015 0.008 -0.005 -0.044 -0.006\n(-0.001) (0.0) (0.020) (0.021) (0.024) (0.006) (0.013) (-0 .010) (-0.043) (-0.009)\nNsp 0.000 0.0 -0.270 -0.047 -0.036 0.051 0.057 0.101 0.158 0.008\n(0.004) (0.0) (-0.019) (-0.032) (-0.030) (-0.007) (0.003) (0.135) (0.164) (0.011)\nC1sp 0.000 0.0 -0.120 0.024 0.006 0.109 0.117 -0.005 -0.007 -0.00 1\nC2sp 0.000 0.0 -0.057 0.007 0.001 0.056 0.047 0.007 0.007 0.001\nC3sp 0.003 0.0 0.052 0.003 -0.005 -0.051 -0.057 -0.004 0.007 0.00 1\nC4sp 0.001 0.0 0.006 0.003 -0.001 -0.006 -0.003 -0.003 0.001 0.00 1\nTMd 0.0 1.570 1.630 3.820 4.158 3.940 3.046 0.0 0.420 0.0\n(0.0) (0.989) (2.510) (3.656) (3.543) (2.092) (-1.044) (0. 0) (0.490) (0.0)\nTMsp 0.0 0.014 -0.97120.140 -0.23530.074 0.046 0.0 -0.027 0.0\n(0.0) (0.199) (0.197) (0.163) (0.146) (0.013) (-0.092) (0. 0) (-0.018) (0.0)\nNsp 0.0 -0.063 -0.290 -0.074 -0.470 -0.109 0.062 0.0 0.136 0.0\n(0.0) (-0.019) (-0.057) (-0.068) (-0.044) (-0.036) (0.011 ) (0.0) (0.129) (0.0)\nC1sp 0.0 0.057 0.210 0.034 0.093 0.172 0.109 0.0 -0.004 0.0\nC2sp 0.0 0.027 -0.019 0.010 0.009 0.050 0.079 0.0 0.005 0.0\nC3sp 0.0 0.002 -0.002 -0.009 -0.039 -0.071 -0.056 0.0 0.002 0.0\nC4sp 0.0 0.003 -0.095 -0.004 -0.004 -0.004 -0.007 0.0 0.000 0.0\n10.032(s), -0.27(p)20.006(s), -0.977(p)30.099(s), -0.334(p)\nFIG. 4. The indices for carbon atoms used in the Table 1.\nspin polarizations of chosen atomic orbitals. They are\nlisted in Table 1, for all porphyrins with TM and TMO.\nWe present the numbers obtained mainly with the pSIC\nscheme. The corresponding GGA results are also listed,\nfor a comparison, for the 2 p-states of the oxygen and\nnitrogen atoms and the TM atomic orbitals.\nThe pSIC method usually tends to localize the\nelectrons42and delocalize the holes.44With the DFT+U\nmethod, the 3 d-TM orbital occupations often increase\nand thesp-orbital occupations of the TM-neighbours de-\ncrease, with respect to the DFT result. For the pSIC\nmethod this is also usually true, due to the fact that\nthe self-interaction correction is stronger for the localizedshells. Netherveless, this is not a ’golden rule’ and for\nsome cases it might be broken. The cases of porphyrin-V\nand -VO are special. Without oxygen, the sp-orbitals of\nV are substantially occupied, because they overlap much\nwith the neighbouring nitrogens. It is interesting that\nwith the pSIC aproach, the polarization of sp-V is anti-\nferromagnetic (AF) to 3 d-V. Large spin-polarizarions of\nnitrogens - around -0.3 µB, which gives -1.2 µBin total -\nare not surprising. Nitrogen is an element which usually\ncouples antiferromagnetically to TM - as for instance in\n(GaN,Mn) dilute magnetic semiconductor.51Moreover,\nthesp-V orbitals overlap and couple ferromagnetically\n(FM) with the sp-N shells. The fact that the 4 p-shell\nof V in porphyrins is substantially occupied can be com-\npared to the similar effect observed in (GaAs,Mn)44and\nSi:Mn.52On the other hand, in the GGA approach, the\nsp-V shell couples ferromagnetically to 3 d-V, since the\nstrong SI error is not corrected for the sp-N orbitals.\nFor the porphyrin-VO, the sp-N is also AF-coupled\nwith respect to 3 d-V. With the difference that the strong\ncoupling of sp-V with N- spis replaced by the V inter-\naction with oxygen, which is more strongly polarizable\nthan nitrogen. The spin polarization of oxygen is around\n-1.6µBand antiferromagnetic with respect to 3 d-V. The\nsp-N localized spin is -0.27 µB, which is similar to that5\nfor porphyrin-V. However, the largest spin polarization of\nthesp-N orbitals is in the case of porphyrin-Mn, around\n-0.47µB. Interestingly, the coupling of sp-N with re-\nspect to 3d-TM is antiferromagnetic not for all studied\n2D networks - the exceptions are: porphyrin-Co, -Cu,\n-FeO, -CoO, -NiO and -CuO.\nThis is a consequence of the 4-fold crystal symmetry\nand the electronic filling of the 3 d-TM orbitals. The TM\natoms possess valence electrons of the s−andd−type\nand need to make the bonds with 4 N atoms and oxygen.\nFormation of the local magnetic moment at the TM atom\nmeans less electrons involved in the chemical bonds, and\na hole might be delocalized over these bonds. The lack of\nTM electrons with the spin up at the TM-N bonds - for\nthe cases with the d-shell less occupied than it is for Mn\n- causes the domination of the spin down polarization at\nthese bonds, and the magnetic coupling of N atoms to the\nTM is antiferromagnetic. The coupling changes signum\nwith the growing number of electrons at the d−-shell of\nTM, and the cases with Ni and Cu are ferromagnetically\noriented to the N atoms. Large spin polarization of the\nsp-O orbitals is very promissing for tailoring spintronic\ndevices. It is worth to notice that the orientation of the\nspin coupling at oxygen with respect to 3 d-TM is usually\nthe same as that of the coupling between sp-N and 3d-\nTM.\nLet us have a look at carbons of the porphyrin\nmolecules and bridges; these adjacent to nitrogens, de-\nnoted C1, and farther, denoted: C2, C3 and C4 - see the\ncaption of Table 1 and Fig. 4. The carbon atom denoted\nC1 bears a spin which couples AF to 3 d-V (with local C1\nmoment of -0.12 µB) for the VO case, and FM (0.21 µB)\nfor the porphyrin-V. Other spin polarizations at C1 also\ndeserve our attention: for porphyrin-Fe (0.17 µB), P-Mn\n(0.09µB), P-FeO and P-Co (0.11 µB), and P-CoO (0.12\nµB). The carbon atoms denoted C2 polarize mostly in\nthe case of porphyrin-Co (0.08 µB). Even far from the\nTM atom, C3 atoms, are spin-polarized around 0.04-0.07\nµBfor porphyrin-Mn, -Fe, -FeO, -Co, -CoO. Remarkably,\nin porphyrin-V, the carbon atom of the acetylenic-bridge,\nnamely C4, couples antiferromagnetically to C1, and it\nis spin-polarized of about -0.1 µB.\nThe results presented in this subsection align with the\nexperimental and theoretical conclusions of the work on\nporphyrin-Cr adsorbed at the Co surface.9Namely, the\nlocal spin-moments induced at nitrogens are AF-coupled\nto Cr, and the spin of less than halfly-occupied 3 d-shell\nof Cr is AF-coupled to the more than halfly-occupied 3 d-\nshell of Co substrate - as in our case, it holds for the Cr\nand O localized magnetic moments.\nAs for the difference between the pSIC and the GGA\nresults, the pSIC magnetizations are always larger than\nthe GGA ones. This is due to the fact that the Hund’s\nrule is strongly obeyed when the electron localization is\nhigher like in the pSIC and DFT+U case. Moreover,\nfor the separate molecules, the magnetizations are local-\nized only at the transition metals and oxygen. In con-\ntrast, for the periodic systems, the magnetizations arealso partially localized on N, O, and even carbons of the\nacethylene bridges.\nEven if the magnetization is not localized at the acety-\nlene bridges, the covalent bonds between the molecules\nare essential for the spin delocalization within the\nnetwork-building molecules. Without these bonds, the\nmolecules would be spin-polarized with the well localized\ninteger spin multiplicity, which would not be present at\nthe nitrogen atoms. In contrast, for the periodic sys-\ntems containing the N atoms and magnetic impurities -\nsuch as already mentioned diluted magnetic semiconduc-\ntor (Ga,Mn)N - the spin polarization is delocalized to the\nnitrogen atoms, with quite substantial contribution.51\nThe more extensive view on the magnetization can be\nderived from the projected densities of states (PDOS)\nonto the 2p-N, 3d-TM and 2 p-O states, and the total\nDOS, which are presented in Fig. 5; for the results ob-\ntained with the pSIC approach. Similar PDOS plots for\nthe GGA method are included in Fig. S3 in the supple-\nmental information.48The numbers presented in Table 1\nare obtained from the quadrature of the atomic spin-\ndensities presented in Fig. 5. It is in the energy range\nfrom the bottom of the valence band - not presented in\nthe figure - up to the Fermi level. The spin-asymmetry\nof the PDOS is visible for the cases: V, Cr, Mn, Fe, and\nCo without and with the oxygen. The spin-asymmetry of\nthe 2p-N PDOS is pronounced even for the cases where\nthe L¨ owdin spin-polarizations are rather small. Interest-\ningly, for some cases - like Cr - the spin polarization at\nthe Fermi level does not show up, but the total contri-\nbution to magnetism originates from the deeper states.\nIn this case, mostly from Cr and less from the nitrogen\natoms.\nB. Metallicity\nIn Fig. 5, we see that most of P-TM and P-TMO net-\nworks are metallic. The Fermi levels of some cases -\nnamely Cr, MnO, TiO, Ni, Zn - are placed within a little\nenergy gap. The half-metallicity is plausible in the cases:\nTi, Cu, VO, CrO, FeO, CoO, where the more accurate\nGW-calculations would be necessary.53It is very usual\nfor the TM-doped metallic systems, that the Fermi level\ncuts the 3d-states. The effect of such methods like the\nDFT+U or the pSIC with respect to the GGA results\nmoves the 3 d-states away from the Fermi level, if these\nstates are not pinned to it. In our systems, however, some\ncases calculated with the GGA approach show clearly the\nTM-based metallicity, while in the pSIC approach, the\n3dstates disappear from the Fermi level - e.g. for the\nporphyrin-Mn, -Fe, -VO, -MnO, -CoO and -Ni networks.\nPurely carbon-based metallicities in such magnetic sys-\ntems like P-VO, P-Mn and P-FeO networks seem to be\npromissing for spintronic devices, because the magnetic\ncouplings in these cases might be long range. Interest-\ningly, there are also the metallic systems where the oxy-\ngen states dominate at the Fermi level. These systems6\n-303\n-303\n-303Density of States [1/eV]\n-303\n-4 -2 0 2 4\nEnergy [eV]-303\n-4 -2 0 2 4VO\nCo CoOFe FeOMn MnOV\nCr CrO-303\n-303\n-303Density of States [1/eV]\n-303\n-4 -2 0 2 4-303\n-4 -2 0 2 4\nEnergy [eV]ScO\nZn ZnOCu CuONi NiOSc\nTi TiO\nFIG. 5. Projected densities of states (PDOS) of the metal- an d oxo-metal-porphyrin 2D networks obtained with the pSIC\napproach. Grey color depicts the total DOS, the brown color i s the PDOS of 3 d-TM states, the blue line is the PDOS of 2 p-N\n(summed for 4 atoms), the red line is the PDOS of 2 p-O states. Fermi level is at zero energy.\nare almost half metallic - for instance in the porphyrin-\nNiO, -CuO and -ZnO networks.\nIV. CONCLUSIONS\n2D organic networks are promissing for spintronic\napplications. In this work, we searched for magnets\namong oxo-metal-porphyrins and metal-porphyrins con-\nnected with the acetylene bridges. For magnetism, the\nrelative energetic positions of the 3 d−, 4s−and 4p−\ntransition-metal states with respect to the 2 p-states of\nthe neighbouring atoms are very important. Therefore,\nwe used the self-interaction corrected pseudopotential\nscheme (pSIC), within the DFT framework for the cal-\nculations of the electronic structure.\nMost of the porphyrin networks with TM and TMO are\nmagnetic; except the Sc, TiO, Ni and Zn cases. Addition\nof oxygen increases the magnetizations for the second-\nhalf 3d-row TMs and the ScO case. The rest of cases: Ti,\nV, Cr and Mn decrease magnetizations after the oxygenaddition.\nDifference between the total and absolute magnetiza-\ntions indicate ferrimagnetism, which is the strongest for\nthe porphyrin-V, -VO and -Mn networks. Further exam-\nination of the systems with the L¨ owdin populations tool\nshows the AF-coupling of the oxygen magnetic moment\nwith the TM local moment for the P-VO and P-CrO\ncases, and FM-coupling for the P-FeO, P-CoO, P-NiO,\nP-CuO and P-ZnO systems. Interestingly, in the P-ScO\ncase, the magnetic moment is localized at oxygen and\nnot at TM, and in the MnO case the situation is oppo-\nsite. The strongest spin-polarization localized at nitro-\ngens, and AF-coupled to the 3 d-TM local moment, was\nfound for the P-Mn, P-V, P-VO and P-Fe cases. The\nsame couplings are ferromagnetic for the P-NiO, P-Cu\nand P-CuO 2D networks. The spin-polarization effects\nare very long range for the porphyrin networks with the\nTMs from V to Co. In these cases, the spin-asymmetry\nof the atomic shells was found very far from the TM\natoms, and it was well pronounced even on the acetylenic\nbridges.\nIn summary, the ferrimagnetic cases found in our7\nstudies are characterized by the induced long-range\nspin-polarizations. These systems certainly would be\ngood candidates for the high-T cmagnets and will show\nup new properties when are deposited at the surfaces of\nother materials.\nAcknowledgementsThis work has been supported by the the Na-\ntional Science Center in Poland (the Project No.\n2013/11/B/ST3/04041). Calculations have been per-\nformed in the Interdisciplinary Centre of Mathematical\nand Computer Modeling (ICM) of the University of\nWarsaw within the grant G59-16.\n1C. A. Villee, Biology, 5th ed., 1967, London: Harvard Uni-\nversity.\n2A. L. Sobolewski and W. Domcke, Phys. Chem. 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Rinke, arXiv:1412.7507"    },    {        "title": "1111.3430v1.Frustration_Induced_Ferrimagnetism_in_Heisenberg_Spin_Chains.pdf",        "content": "arXiv:1111.3430v1  [cond-mat.str-el]  15 Nov 2011Typeset with jpsj3.cls <ver.1.1> Full Paper\nFrustration-Induced Ferrimagnetism in Heisenberg Spin Ch ains\nTokuro Shimokawa∗and Hiroki Nakano†\nGraduate School of Material Science, University of Hyogo, K amigori, Hyogo 678-1297, Japan\n(Received October 15, 2019)\nWe study ground-stateproperties of the Heisenberg frustrate d spin chain with interactions\nup to fourth nearest neighbors by the exact-diagonalization meth od and the density matrix\nrenormalizationgroupmethod. We find that ferrimagnetismis realize dnot onlyin the caseof\nS=1/2 but also S=1 despite that there is only a single spin site in each unit cell determine d\nfrom the shape of the Hamiltonian. Our numerical results suggest t hat a “multi-sublattice\nstructure” is not required for the occurrence offerrimagnetism in quantum spin systems with\nisotropic interactions.\nKEYWORDS: quantum spin chain, frustration, ferrimagnetis m, DMRG, exact diagonalization\nFerrimagnetism is a fundamental phenomenon in the field of ma gnetism. One of the most\ntypical examples of ferrimagnetism is the ( S,s) = (1,1/2) mixed spin chain with a nearest-\nneighbor antiferromagnetic (AF) interaction.1)In this system, the so-called Lieb-Mattis-type\nferrimagnetism2,3)is realized in the ground state because two different spins are arranged\nalternately in a line owing to the AF interaction. This syste m includes two spins in a unit\ncell of the system. In other known ferrimagnetic cases of qua ntum spin systems except the\nS= 1/2 Heisenberg frustrated spin chain studied in ref. 4, the sit uation that the system has\nmorespinsthanonein each unit cell hasbeenthesame. Until o urrecent study4)demonstrated\nthe occurrence of ferrimagnetism in the ground state of the S= 1/2 Heisenberg frustrated\nspin chain despite the fact that a unit cell of the chain inclu des only a single spin, namely,\nit has no sublattice structure, it had been unclear whether t he “multi-sublattice structure”\nis required for the occurrence of the ferrimagnetism in a qua ntum spin system composed of\nisotropic interactions. The Hamiltonian examined in ref. 4 is given by\nH=J/summationdisplay\ni[Si·Si+1+1\n2Si·Si+2]\n−J′/summationdisplay\ni[Si·Si+3+1\n2(Si·Si+2+ASi·Si+4)], (1)\nwhere the real constant Ais fixed to be unity. Here, Siis theS= 1/2 spin operator at the\n∗E-mail address: rk09s002@stkt.u-hyogo.ac.jp\n†E-mail address: hnakano@sci.u-hyogo.ac.jp\n1/6J. Phys. Soc. Jpn. Full Paper\nsitei. The numerical study of this system clarified the existence o f the ferrimagnetic ground\nstate when the controllable parameter J′/Jis changed. In addition, research confirmed that\nthere are two types of ferrimagnetic phases: the phase of the Lieb-Mattis (LM) type and the\nphase of the non-Lieb-Mattis (NLM) type, which has been foun d in several frustrated spin\nsystems.5–8)\nThe purpose of this study is to confirm that the above example i s not a special or rare\ncase by investigating other models. In this study, we discus s the ground state of Hamiltonian\n(1) not only in the case of S= 1/2, but also in the case of Sibeing an S= 1 spin operator.\nMoreover, we focus on the case of A= 0.4, which is different from A= 1. Note that energies\nare measured in units of J; we setJ= 1 hereafter.\nWe employ two reliable numerical methods, i.e., the density matrix renormalization group\n(DMRG) method9,10)and the exact-diagonalization (ED) method. Both methods ca n give\nprecise physical quantities for finite-size clusters. The D MRG method is very powerful for\na one-dimensional system under the open-boundary conditio n. On the other hand, the ED\nmethod does not suffer from the limitation posed by the shape of the clusters; there is no\nlimitation of boundary conditions, although the ED method c an treat only systems smaller\nthan those that the DMRG method can treat. Note that, in the pr esent research, we use the\n“finite-system” DMRG method.\nIn the present study, two quantities are calculated. One is t he lowest energy in each\nsubspace divided by Sz\ntotto determine the spontaneous magnetization M, whereSz\ntotis the\nzcomponent of the total spin. We obtain the lowest energy E(N,Sz\ntot,J′) for a system size\nNand a given J′. For example, the Sz\ntotdependence of E(N,Sz\ntot,J′) in a specific case of J′\nis presented in the inset of Fig. 1(a). This inset shows the re sults obtained by our DMRG\ncalculations of the system of N= 72 with the maximum number of retained states ( MS) of\n600, and a number of sweeps ( SW) of 10. One can find the spontaneous magnetization Mfor\nagivenJ′as the highest Sz\ntotamong those at the lowest common energy. (See thearrowhead i n\nthe inset.) The other quantity is the local magnetization in the ground state for investigating\nthe spin structure of the highest- Sz\ntotstate. The local magnetization is obtained by calculating\n/angbracketleftSz\ni/angbracketright, whereSz\niis thez-component of the spin at the site iand/angbracketleftO/angbracketrightdenotes the expectation\nvalue of the physical quantity Owith respect to the state of interest.\nFirst, let us show the results of the J′dependence of M/Msin Fig. 1, where Msis the\nsaturated magnetization. Irrespective of S= 1/2 orS= 1, we find the nonmagnetic phase\n(M/Ms= 0) and ferromagnetic phase( M/Ms= 1). Between thetwo phases, wealso findthree\nregions: the regions of 0 < M/M s<1/3,M/Ms= 1/3, and 1/3< M/M s<1. ForS= 1/2,\n2/6J. Phys. Soc. Jpn. Full Paper\n\tB\n\tC\n 0 5 10 15 Stot z–43–42.5–42E nergy S=1/2 N=72 \nDMR G \nMS=600, SW=10 \nJ’ =2.2 \n0 1 2 \nJ’ 00.51M/M sN=24 DMR G \nN=48 DMR G \nN=72 DMR G S=1 A=0.40 1 2 \nJ’ 00.51M/M s\nN=24 E D periodic\nN=24 E D open\nN=72 DMR G S=1/2 A=0.4\nFig. 1. (Color) (a) J′dependence of the normalized magnetization M/Msin the ground state in the\ncase ofS= 1/2 withA= 0.4. In the inset of (a), the lowest energy in each subspace divided by\nSz\ntotis shown. Results of the DMRG calculations are presented when the s ystem size is N= 72\nforJ′= 2.2. The arrowhead indicates the spontaneous magnetization Mfor a given J′;Mis\ndetermined to be the highest Sz\ntotamong the values with the lowest common energy. (b) J′\ndependence of M/Msin the ground state in the case of S= 1 with A= 0.4.\none can see that the region of 0 < M/M s<1/3 is much narrower than the distinctly existing\nregion of NLM ferrimagnetism4)in the case of S= 1/2 withA= 1. The width of the present\nregion for A= 0.4 seems to vanish in the limit of N→ ∞. One finds that the occurrence\nof the NLM ferrimagnetism in Hamiltonian (1) requires a four th-neighbor interaction with A\nthat is larger than the specific value between A= 0.4 andA= 1. The width of the region of\nM/Ms= 1/3 in both cases of S= 1/2 withA= 0.4 andS= 1 with A= 0.4 seems to survive\nin the limit of N→ ∞. The region of 1 /3< M/M s<1 is presumably considered to merge\nwith the ferromagnetic (FM) phase in the thermodynamic limi t. The reason for this is that\nthis region appears only near M/Ms= 1 and that M/Msin this region becomes progressively\nlarger with increasing N. In addition, we cannot confirm this region in the calculatio ns within\nN≤30 of the S= 1/2 system under the periodic-boundary condition irrespecti ve of the\nvalues of A. The issue of whether or not the region of 1 /3< M/M s<1 survives should be\nclarified in future studies; hereafter, we do not pay further attention to this issue.\n3/6J. Phys. Soc. Jpn. Full Paper\n\tB\n \tC\n0 0.02 0.04\n1/N12J’ J’ 4\nJ’ 3\nJ’ 2\nJ’ 1\n0 0.02 0.04\n1/N00.511.5Width of phase 0<M/M s<1/3 \nM/M s=1/3 \nFig. 2. (a) Size dependences of the boundaries of the regions in the case ofS= 1 with A= 0.4. The\nresults presented are those of N= 24,36,48,60, and 72 from the DMRG calculations. (b) Size\ndependence of the width of each region in the case of S= 1 with A= 0.4. The width of the region\nof 0< M/M s<1/3 and that of M/Ms= 1/3 are defined as |J′\n2−J′\n1|and|J′\n3−J′\n2|, respectively.\n0 20 40 60\ni–1 –0.5 00.51<S iz>J’ =2.1, M=24 \nFig. 3. Local magnetization /angbracketleftSz\ni/angbracketrightunder the open-boundary condition: for J′= 2.1 in the case of\nS= 1 with A= 0.4 from the DMRG calculation for N= 72. The site number is denoted by i,\nwhich is classified into i= 3n−2, 3n−1, and 3n, wherenis an integer. Squares, circles, and\ntriangles mean i= 3n−2, 3n−1, and 3n, respectively.\nNext, we study the size dependences of the phase boundaries i n the case of S= 1 with\nA= 0.4 depicted in Fig. 2(a). We present results of four boundarie s:J′=J′\n1between the\nnonmagnetic phase and the region of 0 < M/M s<1/3,J′=J′\n2between the regions of\n0< M/M s<1/3 andM/Ms= 1/3,J′=J′\n3between the regions of M/Ms= 1/3 and\n1/3< M/M s<1, andJ′=J′\n4between the region of 1 /3< M/M s<1 and the FM phase. To\nconfirm the behavior up to the thermodynamic limit, we also ex amine the N−1dependences\nof the two widths of the regions of M/Ms= 1/3 and 0< M/M s<1/3 in Fig. 2(b). Although\nthe width of the region of M/Ms= 1/3 decreases with increasing N, this dependence shows\na behavior that is convex-downwards for large sizes; the wid th seems to converge to 0.3.\nTherefore, the phase of M/Ms= 1/3 definitely survives in the limit of N→ ∞. On the\nother hand, the width of 0 < M/M s<1/3 obviously disappears in the limit of N→ ∞. An\nappropriate tuning of the parameters in Hamiltonian (1) of t heS= 1 system might cause the\n4/6J. Phys. Soc. Jpn. Full Paper\nNLM ferrimagnetism; such parameter sets should be searched for in future studies.\nFinally, we examine the local magnetization /angbracketleftSz\ni/angbracketrightin the phase of M/Ms= 1/3 in the\ncase ofS= 1 with A= 0.4. In Fig. 3, we present our DMRG result of /angbracketleftSz\ni/angbracketrightof the system\nofN= 72. We confirm the up-down-up spin behavior, and this spin st ructure is consistent\nwithM/Ms=1/3 in the parameter region near approximately J′= 2.1 in Fig. 1(b). Thus, this\nphase is considered to be the LM-type ferrimagnetic phase.\nIn summary, we study the ground-state properties of a frustr ated Heisenberg spin chain by\nthe ED and DMRG methods. Despite the fact that this system con sists of only a single spin\nsiteineach unitcell determinedfromtheshapeof theHamilt onian, theLM-typeferrimagnetic\nground state is realized in a finite region not only in the case ofS= 1/2 but also of S= 1.\nThe present models showing ferrimagnetism indicate that a “ multi-sublattice structure” is\nnot required for the occurrence of ferrimagnetism in quantu m spin systems with isotropic\ninteractions as a general circumstance.\nWe are grateful to Professor Y. Hasegawa for his critical rea ding of the manuscript. This\nwork was partly supported by Grants-in-Aid (Nos. 20340096, 23340109, and 23540388) from\nthe Ministry of Education, Culture, Sports, Science and Tec hnology (MEXT) of Japan. This\nwork was partly supported by a Grant-in-Aid (No. 22014012) f or Scientific Research and\nPriority Areas “Novel States of Matter Induced by Frustrati on” from the MEXT of Japan.\nSome of the calculations were carried out at the Supercomput er Center, Institute for Solid\nState Physics, University of Tokyo. Exact-diagonalizatio n calculations in the present work\nwere carried out based on TITPACK Version 2 coded by H. Nishim ori. DMRG calculations\nwere carried out using the ALPS DMRG application.11)\n5/6J. Phys. Soc. Jpn. Full Paper\nReferences\n1) T. Sakai and K. Okamoto: Phys. Rev. B. 65(2002) 214403.\n2) E. Lieb and D. Mattis: J. Math. Phys. 3(1962) 749.\n3) W. Marshall: Proc. Roy. Soc. A 232(1955) 48.\n4) T. Shimokawa and H. Nakano: J. Phys. Soc. Jpn. 80(2011) 043703.\n5) S. Yoshikawa and S. Miyashita: J. Phys. Soc. Jpn. 74(2005) Suppl. 71.\n6) K. Hida: J. Phys.: Condens. Matter 19(2007) 145225.\n7) H. Nakano, T. Shimokawa, and T. Sakai: J. Phys. Soc. Jpn. 80(2011) 033709.\n8) T. Shimokawa and H. Nakano: J. Phys.: Conf. Ser. 320(2011) 012007.\n9) S. R. White: Phys. Rev. Lett. 69(1992) 2863.\n10) S. R. White: Phys. Rev. B. 48(1993) 10345.\n11) A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A. Feiguin, L. G amper, E. Gull, S. Gurtler, A.\nHonecker, R. Igarashi, M. Korner, A. Kozhevnikov, A. Lauchli, S. R. Manmana, M. Matsumoto,\nI. P. McCulloch, F. Michel, R. M. Noack, G. Pawlowski, L. Pollet, T. Pru schke, U. Schollwock, S.\nTodo, S. Trebst, M. Troyer, P. Werner, and S. Wessel: J. Magn. M agn. Mater. 310(2007) 1187\n(see also http://alps.comp-phys.org).\n6/6"    },    {        "title": "1103.1967v2.Phase_diagram_of_the_XXZ_ferrimagnetic_spin__1_2__1__chain_in_the_presence_of_transverse_magnetic_field.pdf",        "content": "arXiv:1103.1967v2  [cond-mat.str-el]  7 Jun 2011Phase diagram of the XXZ ferrimagnetic spin-(1/2, 1) chain i n the presence of\ntransverse magnetic field\nA. Langari1, J. Abouie2,3, M. Z. Asadzadeh1and M. Rezai1\n1Department of Physics, Sharif University of Technology, Te hran 11155-9161, Iran\n2Department of Physics, Shahrood University of Technology, Shahrood 36199-95161, Iran and\n3School of Physics, Institute for Research in Fundamental Sc iences (IPM), Tehran 19395-5531, Iran\n(Dated: October 22, 2018)\nWe investigate the phase diagram of an anisotropic ferrimag netic spin-(1 /2,1) in the presence of a\nnon-commuting (transverse) magnetic field. We find a magneti zation plateau for the isotropic case\nwhile there is noplateau for the anisotropic ferrimagnet. T he magnetization plateau can appear only\nwhen the Hamiltonian has the U(1) symmetry in the presence of the magnetic field. The anisotropic\nmodel is driven by the magnetic field from the N´ eel phase for l ow fields to the spin-flop phase for\nintermediate fields and then to the paramagnetic phase for hi gh fields. We find the quantum critical\npoints and their dependence on the anisotropy of the aforeme ntioned field-induced quantum phase\ntransitions. The spin-flop phase corresponds to the spontan eous breaking of Z 2symmetry. We use\nthe numerical density matrix renormalization group and ana lytic spin wave theory to find the phase\ndiagram of the model. The energy gap, sublattice magnetizat ion, and total magnetization parallel\nand perpendicular to the magnetic field are also calculated. The elementary excitation spectrums\nare obtained via the spin wave theory in the three different re gimes depending on the strength of\nthe magnetic field.\nPACS numbers: 75.10.Jm, 75.50.Gg, 75.30.Ds, 64.70.Tg\nI. INTRODUCTION\nQuantum ferrimagnets are a general class of strongly correlated magnetism, which have attracted much interest\nin experimental as well as theoretical investigations. Examples of s uch realizations are the bimetallic molecular\nmagnets like CuMn(S 2C2O2)2(H2O)3·4.5H2O and numerous bimetallic chain compounds which have been synthesiz ed\nsystematically1,2. In these materials, the unit cell of the magnetic system is compose d of two spins, the smaller one\nisσ= 1/2 and the larger one ( ρ) is changed from 1 /2 to 5/2. The magnetic and thermodynamic properties of these\nmodels are different from the homogeneous spin counterparts. Fo r instance, the one dimensional mixed-spin model\nrepresents a ferromagneticbehavior for the low temperature re gime while a crossoverappears to the antiferromagnetic\nbehavior as temperature increases3–7. The crossover can be explained in terms of the two elementary exc itations\nwhere the lower one has the ferromagnetic nature and a gapped sp ectrum above it with antiferromagnetic property8.\nMoreover, the mixed spin models have shown interesting behavior fo r the quasi one dimensional lattices (ferrimagnetic\nladders). Despite that the two-leg spin-1/2 ladder is gapful, repre senting a Haldane phase, the two-leg (mixed spin)\nferrimagnet is always gapless with the ferromagnetic nature in the lo w energy spectrum. However, a special kind of\ndimerization can drive the ferrimagnetic ladder to a gapped phase9,10.\nThe presence of a longitudinal magnetic field preserves the U(1) sy mmetry of the XXZ interactions and creates a\nnonzero magnetization plateau in a one-dimensional ferrimagnet fo r small magnetic fields in addition to the saturation\nplateau for large magnetic fields11–13. The former plateau corresponds to the opening of the Zeeman en ergy gap\nwhich removes the high degeneracy of the ground state subspace . The ferrimagnets on ladder geometry present a rich\nstructure of plateaus depending on the ratio and dimerization of ex change couplings14. In both one-dimensional and\ntwo-leg ferrimagnets the magnetization plateaus can be understo od in terms of the Oshikawa, Yamanaka and Affleck\n(OYA) argument15because the longitudinal magnetic field commutes with the rest of th e Hamiltonian and the models\nhave U(1) symmetry. However, the situation is different when a tra nsverse magnetic field is applied on the system,\nbecause the transverse field does not commute with the XXZ intera ction and breaks the U(1) symmetry of the model.\nThe onset of a transverse field develops an energy gap in a spin-1/2 chain which initiates an antiferromagnetic order\nperpendicular to the field direction16–19. The ordered phase is a spin-flop phase because of nonzero magne tization in\nthe field direction; however, there is no magnetization plateau even in the gapped phase20. The lack of U(1) symmetry\nprohibits the use of the OYA argument, thus prompts the question of a magnetization plateau and the presence of an\nenergy gap21in the spectrum.\nThe structure of the paper is as follows. First we study the anisotr opic ferrimagnetic chain in the presence of a\ntransverse magnetic field by using the density matrix renormalizatio n group (DMRG)22and exact diagonalization\nLanczos methods. The energy gap, sublattice magnetization, and total magnetization in both parallel and perpendic-\nular to the field direction are presented in Sec. II. We further addr ess the energy gap behavior versus the magnetic\nfield and the magnetization plateau. The phase diagram of the model is also presented in the same section. We then2\n0 0.5 1 1.5 2 2.5 3 3.5 4\nh (transverse magnetic field)00.511.522.5Energy Gap∆=0.0\n∆=0.5\n∆=1.0\n0 0.02 0.04 0.06 0.08\n(2N)-100.10.20.30.40.5\nhc1\nhc2\nh=2.0∆=0.0\nFIG. 1: The energy gap versusthe transverse magnetic field. D ifferentplots belong tovarious values of theanisotropy par ameter\n∆ = 0.0,0.5.1.0. Inset: The scaling of gap versus (2 N)−1for ∆ = 0 .0 at two critical points hc1andhc2confirms the vanishing\nof gap at these points while its scaling at h= 2.0 verifies a finite gap in the thermodynamic limit ( N→ ∞).\nuse an analytical tool, the spin wave theory (SWT), to obtain the low energy excitation spectrum of the model in Sec.\nIII. The SWT is applied in three different regions depending on the str ength of the magnetic field. The qualitative\nbehavior of the model is explained in terms of SWT and the magnetizat ion is compared with DMRG results. The\nresults of SWT help to explain the energy gap behavior of DMRG data. We finally summarize our results in Sec. IV,\nwhere we put together both quantitative DMRG and qualitative SWT r esults to analyze the different phases of the\nmodel in the presence of a transverse magnetic field.\nII. DENSITY MATRIX RENORMALIZATION GROUP RESULTS\nWe have implemented the numerical DMRG technique to study the mag netic properties of the anisotropic ferri-\nmagnetic spin-(1 /2,1) chain in the presence of a transverse magnetic field given by the H amiltonian (1):\nH=JN/summationdisplay\ni=1[σx\niρx\ni+σy\niρy\ni+σx\niρx\ni+1+σy\niρy\ni+1+∆(σz\niρz\ni+σz\niρz\ni+1)−h(σx\ni+ρx\ni)], (1)\nwhereσα\ni(ρα\ni) represents the α-component of spin operators at site ifor spin amplitude σ= 1/2 (ρ= 1). The\nantiferromagnetic exchange coupling is J >0, the anisotropy is defined by ∆, and his proportional to the strength\nof the transverse magnetic field.\nThe DMRG computations have been done on an open chain of length 10 8 spins (N= 54 unit cells) and the number\nof states kept in each step of DMRG is 300 ≤m≤500. We have also studied the chains with larger lengths (up to\nN= 100) and observed no significant changes on the data of magnetiz ation and staggered magnetization within 5\ndigits of accuracy.\nThe energy gap is defined as the difference between the first excite d state energy and the ground state energy. It\nshows whether the model is gapless or gapful depending on its zero or nonzero value, respectively. Using the DMRG\ncomputations, we have plotted in Fig. 1 the energy gap of the model versus the transverse magnetic field for different\nvalues of anisotropy parameter, ∆ = 0 ,0.5,1.0. All plots show a gapped phase for small values of the magnetic field ,\nh<hc1(∆), andaparamagneticgappedphasefor h>hc2(∆). Thegapvanishesattwocriticalpoints, h=hc1(∆)and\nh=hc2(∆). The isotropic case (∆ = 1) remains gapless in the intermediate re gionhc1(∆)<h<h c2(∆), while the\nanisotropic case (∆ ∝ne}ationslash= 1) is gapful. The gap behaves differently for various ∆ in the small-fie ld and intermediate-field\ngapped phase.\nIn the isotropic case ∆ = 1, the behavior of gap versus hcan be explained in terms of the elementary excitations of\nthe model. For ∆ = 1, the U(1) symmetry of the model is restored an d the magnetic field operator commutes with the\nrest of the Hamiltonian. Thus, the energy spectrum for ∆ = 1 is expr essed in terms of the spectrum at h= 0 plus a3\n0 0.5 1 1.5 2 2.5 3 3.5 4\nh (transverse magnetic field)-0.500.511.5x component spin σ=1/2\nρ=1\nσ+ρ∆=0(a)\n0 0.5 1 1.5 2 2.5 3 3.5 4\nh (transverse magnetic field)00.511.5Mx(x-component magnetization)∆=0.0\n∆=0.5\n∆=1.0(b)\nFIG. 2: (a) The x-component sublattice magnetization versus a transverse m agnetic field for both σ= 1/2 andρ= 1 spins\nand their sum as the unit cell magnetization in x-direction for anisotropy parameter ∆ = 0. (b) The x-component unit\ncell magnetization versus a transverse magnetic field for di fferent anisotropies ∆ = 0 .0,0.5,1.0. The dashed line shows the\nsaturation value at Mx= 3/2.\nshift of energy which depends on h. Ath= 0 the model has SU(2) symmetry and the ground state is a ferrom agnetic\nstate with total spin SG=N(|ρ−σ|), which is highly degenerate and the lowest ferromagnetic spectru m is a gapless\none, namely ν−(k) [see Eq. (17)]. An antiferromagnetic spectrum ( ν+(k)) exists above the ferromagnetic one, and\nthe lowest state of the antiferromagnetic spectrum has total sp inSAF=SG+1 with a finite gap 2 J|ρ−σ|, measured\nfrom the ground state. Upon adding a commuting magnetic field to th e ferrimagnetic chain the symmetry is lowered\nto U(1) and the energy levels are affected by a Zeemann term, i.e., −hSx. For the magnetic fields smaller than h\n(which will be defined later), the Zeeman energy gain of the ground s tate is larger than all of the other states in\nthe ferromagnetic spectrum; thus the ground state remains rob ust, and the first excited state is the first state in the\nferromagnetic spectrum (with energy J(ν−(k) +h)), which leads to the energy gap equal to Jh. This explanation\nremains valid until the gain of the Zeeman term of the lowest state of the antiferromagnetic spectrum ( J(ν+(k)−h))\ndominates the gain of the first excited state in the ferromagnetic s pectrum. It defines hby the following equation:\nν−(k)+h=ν+(k)−h, (2)\nwhich gives h=|ρ−σ|within linear approximation of SWT (from which both ν±(k) will be derived in the next\nsections). At this point, the first excited state is the lowest state of the antiferromagnetic spectrum. Thus, the energy\ngap behaves as (2 J|ρ−σ| −Jh) before it vanishes at h=hc1(∆ = 1) = 2 |ρ−σ|. The linear increasing behavior\nfor small fields and then linear decreasing of the energy gap are clea r in the DMRG data for ∆ = 1, shown in Fig.\n1. Although the DMRG values for handhc1have some discrepancies with those obtained by linear SWT, the SWT\ngives the qualitative behavior correctly.\nThe energy gap of the anisotropic Hamiltonian (∆ ∝ne}ationslash= 1) is defined as E1−E0for 0<h<h c1andh>hc2, where\nE1is the first excited state energy and E0is the ground state energy. However, the ground state becomes degenerate\n(E1 =E0) forhc1≤h≤hc2, where the energy gap is the difference between the second excite d state energy and\nthe ground state one, E2−E0. For small magnetic fields the scaling behavior of the energy gap can be explained\nusing the quasi-particle excitations of the model as h→0. The leading term of quasi-particle excitations for very\nsmall magnetic fields ( h→0) gives the scaling of energy gap as√\nh, for ∆∝ne}ationslash= 1 [in the weak field SWT, Eq.(18)]. In\na similar manner, the leading term of the strong field SWT [Eq.(21)] lead s to linear dependence of the gap on the\nmagnetic field in the paramagneic phase which explains very well the be havior in Fig. 1. The linear dependence of\ngap versus the magnetic field for h>hc2is confirmed by the DMRG numerical data for any isotropies.\nWe have plotted the energy gap versus (2 N)−1in the inset of Fig. 1 to observe its finite size scaling (where 2 Nis\nthe total number of spins). We have implemented both the Lanczos and DMRG algorithms to calculate the energy\ngap for ∆ = 0. We have plotted the minimum value of gap which occurs at hc1andhc2versus (2N)−1which clearly\nshows that the gap vanishes in the thermodynamic limit ( N→ ∞). It suggests that both hc1andhc2correspond to\nquantum critical points. The different magnetization characterist ic confirms that a quantum phase transition occurs\nat bothhc1andhc2(see Fig. 2). We have also plotted the energy gap for h= 2.0 to justify that the gap of the\nintermediate region is finite in the thermodynamic limit.\nWe have also plotted the x-component magnetization of each sublattice in Fig. 2-(a) for ferr imagnetic spin-(1 /2,1)4\n0 0.2 0.4 0.6 0.8 1\n∆0.470.480.490.50.510.52Mxh=0.2\nh=0.5\nh=0.8\nh=1.0\nh=1.3\nFIG. 3: Unit cell magnetization ( Mx) versus the anisotropy parameter (∆) for some low magnetic fi eld values ( h). Our plots\njustify the plateau only for ∆ = 1. The dashed line represents Mx= 0.5 (the plateau value).\nchain with ∆ = 0 versus hemploying the DMRG technique. The total magnetization has been plo tted in Fig. 2-(b)\nfor different values of anisotropy, ∆ = 0 ,0.5,1.0. To calculate the magnetization we have considered those spins wh ich\nare far from the open ends of the chain to avoid the finite size bound ary conditions. In this respect, ten spins have\nbeen neglected from each side of the open chain and the magnetizat ion has been averaged over the rest of spins.\nFigure 2-(b) shows the possibility of two plateaus in the magnetizatio n along the field direction. For the isotropic\ncase (∆ = 1), it can be explained in terms of the OYA argument15. According to this argument, n(S−m) = integer,\nwherenis the periodicity of the ground state, Sthe total spin of unit cell, and ma possible magnetization plateau\nof the unit cell, the one-dimensional spin-(1 /2,1) chain can show two plateaus at m= 1/2 and 3/2. However, for\n∆∝ne}ationslash= 1 the axial symmetry of the model is broken by the transverse ma gnetic field, and the OYA argument is not\napplicable. Thus, more investigations is required to figure out the diff erence between the anisotropic (∆ ∝ne}ationslash= 1) and\nisotropic (∆ = 1) cases.\nTo get more knowledge on the behavior of magnetization for the anis otropic case, we have plotted the total mag-\nnetization in the magnetic field direction ( Mx) versus the anisotropy parameter (∆) for small magnetic field valu es\nin Fig. 3. The plots have been shown for those values of the magnetic field which seems to exhibit the magnetization\nplateaus. Figure 3 clearly verifies that the magnetization plateau on ly exists for the isotropic case, while there is no\nplateau for ∆ ∝ne}ationslash= 1. The magnetization per unit cell ( Mx) in the direction of magnetic field ( h) is given by\nMx=−1\nN∂E0\n∂h, (3)\nwhereE0is the ground state energy. The above relation for a gapped phase simply states that if the ground\nstate energy is linear in the magnetic field ( E0∝h), the magnetization will be constant, (the presence of plateau);\notherwise the magnetization will depend on the magnetic field, (the a bsence of plateau). Let write the Hamiltonian\nasH=H0−hH1whereH0is the XXZ interacting part and hH1is the magnetic field part. In the presence of\nU(1) symmetry (∆ = 1) the interacting and the magnetic field parts c ommute [H0,H1] = 0. Thus, E0is a linear\nfunction of hwhich leads to the emergence of a magnetization plateau when the en ergy gap is nonzero. This agrees\nwith the OYA statement. However, the transverse magnetic field b reaks the U(1) symmetry in the anisotropic case\n(∆∝ne}ationslash= 1) and [H0,H1]∝ne}ationslash= 0. Therefore, the ground state energy depends on hnon-linearly which gives a change of\nmagnetization when hvaries, i.e. the lack of magnetization plateau even if a finite energy ga p exists.\nAlthough the above general explanation is applied to the strong mag netic field regime the saturated plateau ( Mx=\n1.5) can also be explained from another point of view. An eigenstate wit h full saturation is classified as a factorized\nstate23in whichallspinsalignin thedirectionofthe magneticfield. Asageneral argument, ithasbeenshowninRef.23\nthat the full saturation for an anisotropic Heisenberg type intera ction in the presence of a magnetic field takes place\nat a finite value of the magnetic field if the model is rotationally invarian t around the field direction. Accordingly,\nthe saturation at Mx= 1.5 takes place only for the isotropic case (∆ = 1) and h≥hc2. In the anisotropic case\n(∆∝ne}ationslash= 1), the fully polarized plateau can take place for infinite strong mag netic field while the nearly saturated state,\n(Mx≃1.5), can be observed for large magnetic fields. To justify this argum ent we have plotted the x-component5\nFIG. 4: Schematic of spins’ orientations in different phases of the anisotropic ferrimagnetic spin-(1 /2,1) chain in the presence\nof a transverse magnetic field.\nmagnetization of each unit cell for different values of ∆ in Fig. 2-(b). It is clear that the magnetization in the field\ndirection does not reach the saturation value of Mx= 1.5 for ∆ = 0 and 0 .5, while it obviously touches its saturated\nvalue for ∆ = 1 and h≥3.\nThe antiferromagnetic interactions between the spins in each unit c ell make them to be antiparallel, which leads to\nthe totalx-component magnetization Mx=∝an}bracketle{tσx+ρx∝an}bracketri}ht ≃0.5. This phase has been shown schematically in Fig. 4-(1)\nwhere we have neglected the effects of small quantum fluctuations on the directions of the spins. The non-commuting\ntransverse magnetic field opens a gap which is robust as long as h < h c1. This (gapped) N´ eel phase corresponds\nto the first plateau at Mx= 0.5 for ∆ = 1 and a semi-plateau ( Mx≃0.5) for ∆ ∝ne}ationslash= 1. By further increasing h,\nthe gap is closed at the first critical field hc1(∆) (for ∆ = 0, hc1≃1.6) where the magnetization starts to increase\nobviously. Further increasing of the magnetic field leads to a continu ous change of the ground state property which\ngives a gradual change of the magnetization-Fig. 4-(2-4). For st rong magnetic field ( hc2(∆ = 0) /greaterorsimilar2.4) the spins are\nnearly aligned in the direction of the magnetic field, the semi-plateau a tMx≃1.5 [Fig. 2-(a) and Fig. 4-(5)].\nTo get more insight on the ground state properties of the model, we have plotted the y-component spin expectation\nvalue versus the transversemagnetic field in Fig. 5 for ∆ = 0. The mag netization in the ydirection for both sublattice\nspins is zerofor h/lessorsimilar1.6andh/greaterorsimilar2.4; however, it becomes nonzeroin the intermediate region1 .6/lessorsimilarh/lessorsimilar2.4. The values\nof theycomponent spins in the unit cell are equal, and their directions are op posite to each other, ∝an}bracketle{tσy∝an}bracketri}ht=−∝an}bracketle{tρy∝an}bracketri}ht. It is\nsurprising that for any value of the magnetic field 1 .6/lessorsimilarh/lessorsimilar2.4 we get ∝an}bracketle{tσy∝an}bracketri}ht=−∝an}bracketle{tρy∝an}bracketri}htwhereas the spin magnitude on\nthe sublattices are different ( σ∝ne}ationslash=ρ). At the factorizing field, hf≃2.24 (which will be explained in the next section),\nwhere the condition σsin|θ|=−ρsin|β|should be satisfied, the mentioned relation is obtained ∝an}bracketle{tσy∝an}bracketri}ht=−∝an}bracketle{tρy∝an}bracketri}ht. The\nstaggered magnetization in the ydirection,SMy=∝an}bracketle{tσy−ρy∝an}bracketri}ht, is nonzero for this region. Moreover, our numerical\ndata verifies that the zcomponent magnetization on both sublattices is zero for any value o f the magnetic field.\nGenerally, let us consider the ycomponent staggered magnetization as an order parameter, whic h is nonzero for\nhc1(∆)<h<h c2(∆) and zero elsewhere. Nonzero SMycorresponds to a spontaneous breaking of Z 2symmetry. In\nfact, foranyvalue of∆ the model hasa Z 2symmetry whichcan be expressedby the parity operator P=⊗iσx\niρx\ni. This\nsymmetry, whichcanalsobeconsideredasa πrotationaroundthemagneticfielddirection( x), leadstovanishingvalue\nfor thezandycomponents of the spins. However, the symmetry is spontaneous ly broken for hc1(∆)<h <h c2(∆)\nwhich selects one of the parity eigenkets to give nonzero sublattice magnetization in the ydirection.\nIII. SPIN WAVE ANALYSIS\nWe have applied the spin wave theory to get more knowledge and a qua litative picture of the phase diagram. The\nSWT is a method to describe a spin model in terms of boson operators . The elementary excitations of the spin model\nare given by bosonic quasi-particles which are constructed on a give n background. Based on this fact the SWT can be6\n0 0.5 1 1.5 2 2.5 3 3.5 4\nh (transverse magnetic field)-0.4-0.200.20.40.60.81y-component spinσ=1/2\nρ=1\n|σ−ρ|∆=0\nFIG. 5: The y-component sublattice magnetization versus the transvers e magnetic field for ∆ = 0. This component is nonzero\nonly in the intermediate phase 1 .6/lessorsimilarh/lessorsimilar2.4. Moreover, the y-component spins are exactly equal and antiparallel for bot h\nsublattices. The staggered magnetization of the unit cell i n theydirection is nonzero within the intermediate region.\nconsidered on different backgrounds to build up a bosonic system. T ypically, a state in the Hamiltonian Hilbert space\nis considered as the background which is supposed to be the ground state within an approximation. However, there\nexists some spin models such as the isotropic antiferromagnetic Heis enberg spin-1/2 chain that do not have an ordered\nground state, and thus the SWT fails to explain the properties of th e model correctly. Therefore, the existence of an\nexact ground state is a good starting point to initiate a spin wave ana lysis.\nThe ferrimagnetic chain both in the absence and presence of longitu dinal magnetic fields has been studied by the\nSWT3–5,8,12,13. Although theN´ eelstateis notthe exactgroundstateforafer rimagnetin the presenceofalongitudinal\nfield, the SWT gives a good description of the model which justifies th at the quantum fluctuations are not strong\nenough to ruin the whole picture. It would be more interesting to initia te a spin wave theory based on an exact\nground state for a ferrimagnet in the presence of a transverse m agnetic field. According to Ref.23, the exact ground\nstate of a general class of ferrimagnets can be found at the fact orizing magnetic field, h=hf. This ground state is\na factorized state, which is a perfect background to implement SWT . It gives a reliable analysis around h=hf(see\nnext subsection). We will also study the SWT for small and large magn itudes of the magnetic field. Our analysis\nis limited to the linear spin wave approximation to get the magnetic prop erties of the spin-( σ,ρ) ferrimagnets in the\npresence of a transverse magnetic field.\nA. SWT at h=hf\nLet us briefly introduce the exact factorized ground state of a fe rrimagnet in the presence of a magnetic field23.\nThe factorized ground state for the Hamiltonian of Eq. (1) can be w ritten in the following form:\n|ψ0∝an}bracketri}ht=/circlemultiplydisplay\ni∈Aσ,j∈Bρ|σ′\ni∝an}bracketri}ht|ρ′′\nj∝an}bracketri}ht, (4)\nwhere|σ′\ni∝an}bracketri}htand|ρ′′\nj∝an}bracketri}htare the eigenstates of /vector σi·ˆn′\niand/vector ρj·ˆn′′\njwith the largest eigenvalues, respectively, with ˆ n′\niand ˆn′′\nj\nbeing unit vectors pointing in polar angles ( θ,ϕ= 0) and (β,α= 0).AσandBρrepresent the two sublattices which\ncontain the two different spins. The factorized state is called a bi-angle state , defined by the two angles ( θ,β) and\nrepresents the ground state of the model at h=hf, where\ncosβ=ρ+∆σ/radicalbig\nρ2+σ2+2∆ρσ,\ncosθ=σ+∆ρ/radicalbig\nρ2+σ2+2∆ρσ, (5)7\nand\nhf= 2/radicalbig\nρ2+σ2+2∆σρ,\nǫf=−(σ2+ρ2+∆σρ), (6)\nwithǫfbeing the ground state energy per site at the factorizing field.\nTo perform the spin wave analysis around h=hf, we first implement a rotation on the original Hamiltonian ( H).\nThe rotated Hamiltonian ( ˜H) is the result of rotations on all lattice points of H, and is given by the following relations\n˜H=˜D†H˜D,\n˜D=/circlemultiplydisplay\ni∈Aσ,j∈BρDσ\ni(0,θ,0)Dρ\nj(0,β,0). (7)\nThe rotation operator\nDρ(0,β,0) =D(α= 0,β,γ= 0) =Dz(α)Dy(β)Dz(γ),\nis defined in terms of Euler angles, and a similar expression is considere d forDσ(0,θ,0).\nIn the rotated basis defined by ( x′,y′,z′) and (x′′,y′′,z′′), the bi-angle state becomes the fully polarized ground\nstate of ˜H. In the next step, the rotated Hamiltonian is bosonized via a Holstein -Primakoff (HP) transformation,\nσ+\ni=/radicalBig\n2σ−a†\niaiai, σx′\ni=σ−a†\niai,\nρ+\nj=/radicalBig\n2ρ−b†\njbjbj, ρx′′\nj=ρ−b†\njbj, (8)\nwhereai(a†\ni) andbj(b†\nj) are two types of annihilation (creation) boson operators, satisf ying the commutation relations:\n[ai,a†\nj] =δi,j, [bi,b†\nj] =δi,j, [ai,b†\nj] = 0 and [ai,bj] = 0.\nThe Hamiltonian in the momentum ( k) space and in the linear spin wave approximation is written as\n˜H=E1+˜H1+˜H2,\nE1=NJσρcos(β−θ)+NJh(ρcosβ+σcosθ),\n˜H1=J/summationdisplay\nk/braceleftbigg\n2√ρσ∆cosk\n2(akb†\nk+bka†\nk)\n+/parenleftBigg\nh2\nf\n2ρ+hfσcosθ\nρ+(hf−h)cosβ−2∆σ/parenrightBigg\nb†\nkbk\n+/parenleftBigg\nh2\nf\n2σ+hfρcosβ\nσ+(hf−h)cosθ−2∆ρ/parenrightBigg\na†\nkak/bracerightbigg\n,\n˜H2=J√\nN(hf−h)√\n2/bracketleftBig√ρsinβ(b0+b†\n0)\n+√σsinθ(a0+a†\n0)/bracketrightBig\n,\n(9)\nwhereNis the total number of spins in each sublattice. The unitary transfo rmation that diagonalizes ˜H1is given by\nχk=akcosηk−bksinηk,\nψk=bkcosηk+aksinηk, (10)\nwhereχkandψkare the quasi-particle boson operators that preserve the boson ic commutation relations. In this\nrepresentation, we obtain\n˜H1=/summationdisplay\nk(ω−(k)χ†\nkχk+ω+(k)ψ†\nkψk), (11)8\nin whichω±are the quasi-particle excitation modes. The dispersion relations ar e given by\nω±(k) =JD+±JD−+2∆√σρtan(2ηk)cosk\n2/radicalbig\n1+tan2(2ηk),\ntan(2ηk) =2∆√ρσcosk\n2\nD−, (12)\nin which\nD±≡h2\nf\n4(1\nρ±1\nσ)+hf(σ\n2ρcosθ±ρ\n2σcosβ)\n−∆(σ±ρ)+hf−h\n2(cosβ±cosθ). (13)\nA shift on the zero momentum component of boson operators, defi ned by two constants t±, diagonalizes the full\nHamiltonian; i.e., χ0→χ0+t−, ψ0→ψ0+t+, where\nt+=√\n2N(h−hf)/parenleftbig√σsinη0sinθ+√ρcosη0sinβ/parenrightbig\n2ω+,\nt−=√\n2N(h−hf)/parenleftbig√σcosη0sinθ−√ρsinη0sinβ/parenrightbig\n2ω−.\n(14)\nThe diagonalized Hamiltonian is given by\n˜H=Egs+/summationdisplay\nk/parenleftbigg\nω−(k)χ†\nkχk+ω+(k)ψ†\nkψk/parenrightbigg\n,\nEgs=E1+ω−t−2+ω+t+2\n+√\n2NJ(hf−h)/parenleftbigg\n(√σsinθcosη0−√ρsinη0sinβ)t−\n+(√σsinθsinη0+√ρcosη0sinβ)t+/parenrightbigg\n,\nEf= 2NJǫf (15)\nwhereEgsis the ground state energy which reduces to Efat the factorizing field ( hf) (i.e., the energy of the exact\nbi-angle state ).\nThe magnetic properties of model (1) can be studied through the lin ear spin wave theory-Eq. (15). In Fig. 6-(a),\nwe have plotted the sublattice magnetization of the anisotropic fer rimagnetic spin-(1 /2,1) chain for ∆ = 0 .5. Thex\nandycomponents of sublattice magnetization are nonzero [Fig. 6-(a)]; h owever, the z-component of the sublattice\nmagnetization is zero, denoting that the spins are located in the xyplane. It should be noted that the values of\nsublattice magnetization is exact at the factorizing field while it is appr oximately correct for the magnetic field close\nto the factorizingfield. The α-component of total magnetizationper unit cell is Mα=∝an}bracketle{tρα+σα∝an}bracketri}htand the corresponding\nstaggered magnetization is defined SMα=∝an}bracketle{tρα−σα∝an}bracketri}ht. In Fig. 6-(b), we have plotted the xandycomponents of total\nmagnetization and staggered magnetization. Around the factoriz ing field the model has a considerable magnetization\nin thexdirection and a staggered magnetization in the ydirection, which identifies a spin-flop phase around the\nfactorizing field. The model has a dual character i.e. it behaves like a ferromagnet in the xdirection and like an\nantiferromagnet in the ydirection, it is the result of two branches of excitations, Eq. (15), which are the origin of the\nexistence of two dynamics in the model24,25. We will discuss later the effects of magnetic field on the configuratio n of\nboth spins in more details.\nB. SWT at weak and strong magnetic fields\n(a)Weak Field SWT\nIn the SWT it is assumed that the ground state defines a particular c lassical direction for the spins. In the weak\nmagnetic fields close to h= 0, we expect to have a N´ eel-ordered configuration. Therefore , we use the following9\n2 2.5 3 3.5 4\nh (transverse magnetic field)-0.500.51Sublattice magnetization<σ  >\n<ρ  >\n<σ  >\n<ρ  >∆=0.5\nhfyx\nx\ny(a)\n2 2.5 3 3.5 4\nh (transverse magnetic field)-0.500.511.5Unit cell magnetizationMx\nMy\nSMx\nSMy∆=0.5\nhf(b)\nFIG. 6: (a) The sublattice magnetization. (b) Magnetizatio n and staggered magnetization per unit cell of the anisotrop ic\nferrimagnetic spin-(1 /2,1) chain versus transverse field, for ∆ = 0 .5. The factorized ground state is chosen as the background\nin the linear SWT.\nHolstein-Primakoff (HP) transformations:\nσ+\ni=a†\ni/radicalBig\n2σ−a†\niai, σx\ni=−σ+a†\niai,\nρ+\nj=/radicalBig\n2ρ−b†\njbjbj, ρx\nj=ρ−b†\njbj. (16)\nIn the linear spin wave approximation and within Fourier space repres entation, one can diagonalize the Hamiltonian\nwhich is given by\nH=E0+/summationdisplay\nk{ν−(k)v†\nkvk+ν+(k)w†\nkwk}, (17)\nwhere\nE0=−NJ(2σρ+ρ+σ)−NJh(ρ−σ)+1\n2/summationdisplay\nk(ν−(k)+ν+(k)),\nν±(k) =J/radicalbigg\n2(p2+s2−2∆ρσcos2k\n2±D1),\nD1=/radicalbigg\n(p2−s2)2−4[∆(p2+s2)−ps(1+∆2)]ρσcos2k\n2\np=ρ−h\n2, s =σ+h\n2. (18)10\n0 0.2 0.4 0.6 0.8 1\nh (transverse magnetic field)-0.500.511.5Spin expectation value<σ  >\n<ρ  >\n<ρ  +σ  >\n<ρ  −σ  >∆=0.5x\nx\nxx\nxx\nFIG. 7: The sublattice magnetization, total magnetization and staggered magnetization per unit cell of the anisotropi c ferri-\nmagnetic spin-(1 /2,1) chain versus transverse field, for ∆ = 0 .5, when the N´ eel order is chosen as the background in the line ar\nSWT.\nandv†\nk,w†\nk(vk,wk) are bosonic quasi-particlecreation (annihilation) operators. The procedure of the diagonalization26\ndictates that the bosonic Hamiltonian should be positive definite. This constraint implies that for |∆| ≤1 the\namount of magnetic field obeys the condition h <2(ρ−σ), and for 1 ≤ |∆|<ρ+σ\n2√σρthe magnetic field should be\n|h−ρ+σ|</radicalbig\n(ρ+σ)2−4ρσ∆2.\nLet us consider the special case of ( σ=1\n2,ρ= 1) and ∆ = 0 .5. The Hamiltonian of this system (in the linear SWT\napproximation) is positive definite only for magnetic fields smaller than h0= 1. Accepting this condition we have\nplotted in Fig. 7 the sublattices field-induced magnetization, the tot al magnetization and the staggeredmagnetization\nper cell of the whole chain versus transverse field h. It shows that for 0 <h<1 the model is affected slightly by the\ntransversemagnetic field. In other words, the staggered magne tization in the xdirection is close to its maximum value\n(the N´ eel ordered state). The quantum fluctuations for 0 <h<1 are not strong enough to change the magnetization\nfrom its zero field value. However, upon reaching h0= 1 the quantum fluctuations are suddenly increased so that\nthey destroy the ordered state completely. Thus within this linear S WT, the first critical field is hSWT\nc1=h0= 1\nand for an arbitrary ( σ,ρ)-ferrimagnetic chain it becomes hSWT\nc1= 2|ρ−σ|. Although hSWT\nc1does not depend on\nthe anisotropy parameter ∆ and is slightly different from the DMRG re sults (Fig. 2), the linear SWT describes the\nelementary excitations of the model well.\n(b)Strong Field SWT\nFor the strong magnetic fields the ground state is ordered in the dir ection of the magnetic field. The fully polarized\nground state in which all spins are aligned in the field direction is used as the background for initiating the SWT. In\nthis case the following HP transformation is implemented for the spin o perators\nσ+\ni=/radicalBig\n2σ−a†\niaiai, σx\ni=σ−a†\niai,\nρ+\nj=/radicalBig\n2ρ−b†\njbjbj, ρx\nj=ρ−b†\njbj. (19)\nThe diagonalized Hamiltonian in terms of the Fourier space represent ation and within the linear SWT is\nH=E0+/summationdisplay\nk{Ω−(k)V†\nkVk+Ω+(k)W†\nkWk}, (20)11\n3 3.2 3.4 3.6 3.8 4\nh (transverse magnetic field)00.511.5Spin expectaion value<σ  >\n<ρ  >\n<ρ  +σ  >\n<ρ  −σ  >∆=0.5\nxx\nx\nxx\nx\nFIG.8: Themagnetizationofsublattices, thetotalmagneti zation, andthestaggered magnetization perunitcellofana nisotropic\nferrimagnetic spin-(1 /2,1) chain versus transverse field and for ∆ = 0 .5 and when the background in the linear SWT is the\nfield-induced fully polarized state.\nwhere\nE0=NJ(2σρ+ρ+σ)−NJh(ρ+σ+1)+1\n2/summationdisplay\nk(Ω−(k)+Ω+(k)),\nΩ±(k) =J/radicalbigg\n2(p2+s2+2∆ρσcos2k\n2±D2),\nD2=/radicalbigg\n(p2−s2)2+4[∆(p2+s2)+ps(1+∆2)]ρσcos2k\n2\np=h\n2−ρ, s =h\n2−σ, (21)\nandV†\nk,W†\nk(Vk,Wk) are bosonic quasi-particle creation (annihilation) operators. The condition to have a positive\ndefinite bosonic Hamiltonian implies that for |∆| ≤1 the amount of the magnetic field should be larger than 2( ρ+σ)\nand for|∆| ≥1 the magnetic field should be larger than ρ+σ+/radicalbig\n(ρ−σ)2+4ρσ∆2.\nAgain we consider the special case of ( σ=1\n2,ρ= 1) and ∆ = 0 .5. The Hamiltonian of this system in the linear\nSWT approximation is positive definite only for a magnetic field larger th anhSWT\nc2= 3. The magnetization of each\nsublattice, the total field-induced magnetization, and the stagge red magnetization per unit cell are plotted in Fig.\n8. Forh >3, the model is in the polarized phase. We have already shown in Ref.23,25that the full saturation only\nhappens for the isotropic case ∆ = 1. Thus the model possesses an upper critical field hc2= 3 for ∆ = 1. The\ncomparison with DMRG results shows that hSWT\nc2= 3 is the true value, which is the consequence of weak quantum\nfluctuations for the strong field regimes. For ∆ ∝ne}ationslash= 1, the fully saturated state appears at infinite magnetic field. It\ncan be understood simply by imposing θ= 0 =βin Eq. (5) which can be fulfilled only for ∆ = 1 in the Hamiltonian\ngiven by Eq. (1). In general, the full saturation occurs at a finite m agnetic field if the model has the U(1) symmetry\naround the direction of the magnetic field.\nLet us discuss qualitatively the effects of a non commuting transver se magnetic field on the phase diagram of the\nanisotropic ferrimagnetic spin-(1 /2,1) chain. The SWT gives two branches of quasi-particle excitations f or each of\nthe small, intermediate and large magnetic field regions. At zero magn etic field the lower branch is gapless with\nferromagnetic nature while the upper one is gapped with antiferrom agnetic signature. A nonzero magnetic field opens\na gap in the ferromagnetic branch which remains robust for h≤hc1. Moreover, the staggered magnetization in the\nfield direction is close to its maximum value which implies a N´ eel phase. At h=hc1a quantum phase transition\nfrom the N´ eel phase to the spin-flop phase takes place where the staggered magnetization perpendicular to the field\ndirection becomes nonzero. The quasi-particle excitations for the spin-flop phase are given by ω±(k). In the spin-flop\nphase (hc1< h < h c2) an entanglement phase transition occurs at h=hfwhere the quantum correlations become\nindependent for h < hfandh > hf. The increase of magnetic field causes the second quantum phase t ransition at\nh=hc2to a nearly polarized state in the field direction. The excitations in the field induced polarized phase ( h>hc2)\nare gapful given by Ω±(k), where the gap is proportional to the magnetic field.12\nIV. SUMMARY AND DISCUSSION\nThe ground state phase diagram of the anisotropic ferrimagnetic ( σ,ρ) chain in the presence of a non commuting\ntransversemagneticfieldhasbeenstudied. Thegeneralpictureh asbeenobtainedwithin thespin waveapproximation.\nWe haveappliedthreeschemesoflinearspinwaveapproximationtofin d the magneticphasediagramofthe anisotropic\nferrimagnetic spin-( σ,ρ) chain with anisotropy parameter ∆ and in the presence of the tran sverse magnetic field ( h).\nThe spin wave approximation has been applied close to h= 0 (weak fields), h=hf(intermediate regime), and\nh≫hf(strong fields), where hfis the factorizing magnetic field. The ground state is known exactly a th=hf\nas a product of single spin states. We have studied the magnetizatio n in the field direction. There is a plateau\natMx= 0.5 for isotropic case where the ground state energy is linear in magne tic field while no plateau observed\nfor the anisotropic cases. However, the magnetization along the m agnetic field changes slightly as long as h≤hc1\nand its value is Mx≃0.5, which motivates to recognize it as a N´ eel phase . The model exhib its a quantum phase\ntransition at h=hc1from the N´ eel phase to (i) a spin-flop phase for ∆ ∝ne}ationslash= 1, (ii) a gapless Luttinger liquid for\n∆ = 15,13. The magnetization evolves in the spin-flop phase when the magnetic field is increased. The spin-flop phase\ncontains the factorizing field ( h=hf) where an entanglement phase transition takes place and quantum correlations\nvanish. Further increase of the magnetic field leads to a polarized ph ase which resembles a plateau at the saturated\nmagnetization in the field direction. However, it will be fully saturated only for ∆ = 1 (the presence of a rotational\nsymmetry around the magnetic field) which is represented by a quan tum phase transition at a finite value hc2. The\nvalidity domain of spin wave analysis were introduced and it was shown t hat the corresponding results were in good\nagreement with the DMRG numerical computations.\nTo get more accuratevalues on the magnetizationprocess of spin- (1/2,1)ferrimagnet, we havealso plotted in Fig. 9\nthe DMRG data of the x- andy-component staggered magnetization in addition to the x-component magnetization of\nunit cellversusthetransversemagneticfieldfor∆ = 0. Themagnet izationcurvehasbeendivided tofiveregionswhich\nhas been labeled in fig. 4, fig. 9, and also in Table. I. Region-(1) is defin ed by the N´ eel phase for 0 ≤h<hc1≃1.6\nwhere both MxandSMxare nearly constant while SMyis zero. The spin-flop (gapped) phase, hc1≤h≤hc2,\nwhere a nonzero SMysets up can be distinguished to three parts, namely regions-(2-4) . Forhc1≤h/lessorsimilar1.9 which is\nlabeled region-(2) we observe ∝an}bracketle{tσx∝an}bracketri}ht<0 and∝an}bracketle{tρx∝an}bracketri}ht>0. It is a spin-flop phase which is called spin-flop (I) in Table. I.\nRegion-(3) is defined at h≃1.9 where the projection of smaller spin along the magnetic field become s zero,∝an}bracketle{tσx∝an}bracketri}ht= 0,\ni.e.Mx=SMx. The rest, 1 .9/lessorsimilarh≤hc2≃2.4, labeled by region-(4) where ∝an}bracketle{tσx∝an}bracketri}ht>0 and∝an}bracketle{tρx∝an}bracketri}ht>0 is called spin-flop\n(II). The region-(5) is the polarized phase along the direction of ma gnetic field, i.e. Mx≃1.5 andSMy= 0. It is\nobserved from Fig. 2-(a) that the component of smaller spin in the d irection of the magnetic field is affected strongly\nby the magnetic field while the corresponding component for the larg er one is almost constant.\nThe spin-flop (I) is a characteristic behavior of XXZ ferrimagnets in the presence of transverse magnetic field\nbecause the spin component of the smaller spin along the magnetic fie ld is opposite to the field direction ( ∝an}bracketle{tσx∝an}bracketri}ht<0)\nwhile the spin-flop (II) is similar to the corresponding phase of the homogeneous XXZ spin chain in the presence of\ntransverse magnetic field ( ∝an}bracketle{tσx∝an}bracketri}ht>0)18,25. In the anisotropic ferrimagnetic chain the transverse field first d evelops a\nN´ eel phase and a field-induced quantum phase transition leads to a spin-flop phase. Moreover, the Z 2symmetry is\nspontaneously broken for small-field region in the homogeneous spin chain while it will be broken in the intermediate\nfieldshc1(∆)< h < h c2(∆) for ferrimagnets. A summary of different properties of the ho mogenous XXZ spin 1/2\nchain and the corresponding (1 /2,1) ferrimagnet both for isotropic and anisotropic cases is present ed in Table. II.\nTABLE I: Different configurations of the ground state of the fe rrimagnetic spin-(1 /2,1) chain with ∆ = 0 in the presence of a\ntransverse magnetic field.\nRegion h Phase Order parameters\n(1) 0 ≤h <1.6 N´ eel Mx= 1/2,SMy= 0\n(2) 1 .6≤h <1.9 Spin-Flop(I) /angbracketleftσx/angbracketright<0,SMy>0\n(3) h≃1.9 Spin-Flop /angbracketleftσx/angbracketright= 0,SMy>0\n(4) 1 .9≤h <2.4 Spin-Flop(II) /angbracketleftσx/angbracketright>0,SMy>0\n(5) h >2.4 Nearly Polarized Mx≃3/2,SMy= 0\nIt is also interesting to mention that the low energy effective Hamilton ian of the anisotropic spin-(1 /2,1) chain in\nthe presence of a transverse magnetic field can be represented b y the fully anisotropic (XYZ) spin-1/2 Heisenberg\nchain in an applied field (though we do not report such calculations in th is paper). This helps to get more knowledge\nfrom the results on the effective model27. However, both spin wave approximation and DMRG results show tha t the\nmodel has two nearly constant magnetization in the presence of tr ansverse magnetic field, the small-field plateau at\nMx≃0.5 forh < h c1(∆) and the saturated Mx≃1.5 for large fields ( h > h c2(∆)). The general behavior is the\nsame for any value of the anisotropy parameter (∆); however, th e critical fields hc1(∆) andhc2(∆) depend on ∆. For13\n0 0.5 1 1.5 2 2.5 3 3.5 4\nh (transverse magnetic field)-0.500.511.5spin expectation<ρ  +σ  >\n<ρ  −σ  >\n<σ  −ρ  >∆=0\nxx x\nx\nyy\n(1)(2)\n(3)(4)\n(5)\nFIG. 9: The x-component magnetization, x- andy-components staggered magnetization versus the transvers e field for a\nferrimagnetic spin-(1 /2,1) chain. Effects of the magnetic field on the spins of each subl attice are divided into five different\nregions.\ninstance,hc1(∆ = 0.5)≃1.8 andhc2(∆ = 0.5)≃2.6.\nTABLE II: Different ground state phases are classified for the heterogeneous spin-(1 /2,1) XXZ ferrimagnet along with the\nhomogeneous spin 1 /2 XXZ antiferromagnet. The comparision between isotropic a nd anisotropic cases in the presence of the\ntransverse magnetic field ( h) is presented. The magnetization per unit cell is m. The ferrimagnet has two critical points hc1\nandhc2while the homogeneous antiferromagnet has a critical point athc.\nSpin Region Isotropic case (∆ = 1) Anisotropic case (∆ /negationslash= 1)\n(1/2,1) 0 ≤h < hc1 Gapped N´ eel, plateau at m= 1/2 Gapped N´ eel, no plateau\n(1/2,1) hc1< h < h c2 Gapless Luttinger liquid, no plateau Gapped spin-flop, no pl ateau\n(1/2,1) h > hc2 Gapped paramagnet, plateau at m= 3/2 Gapped paramagnet, no plateau\n1/2 0 ≤h < hc Gapless spin-fluid, no plateau Gapped spin-flop, no plateau\n1/2 h > hc Gapped paramagnet, plateau at m= 1/2 Gapped paramagnet, no plateau\nThe magnetization process can also be viewed as a non-unitary evolution of the system. The entanglement of a\npure state (ground state in our case) is conserved under local un itary operations28. For the ferrimagnetic spin-(1 /2,1)\nchain, the entanglement of the system is decreased by increasing t he magnetic field for h < h f. The entanglement\nvanishesat h=hfwherethe groundstate isgivenby atensorproductstate. Thisis a n entanglementphasetransition.\nIt is thus concluded that the effect of magnetic field is a non-unitary evolution of the ground state.\nV. ACKNOWLEDGMENT\nJ.A thanks H. Movahhedian for his fruitful comments. A. L. would like to thank A. T. Rezakhani for his detailed\ncomments on the final version of the manuscript. A.L and M.R. would lik e to thank the hospitality of physics\ndepartment of the institute for research in fundamental science s (IPM) during part of this collaboration. This work\nwas supported in part by the Center of Excellence in Complex System s and Condensed Matter (www.cscm.ir). The\nDMRG computation has been done by using ALPS package29which is acknowledged.14\nReferences\n1Gleizes A and Verdaguer M, Ordered magnetic bimetallic chains: a novel class of one-di mensional compounds , 1981J.\nAm. Chem. 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A 54, 3824\n29Albuquerque F et. al, The ALPS project release 1.3: Open-source software for stro ngly correlated systems , 2007Journal of\nMagnetism and Magnetic Materials 310, 1187"    },    {        "title": "1607.06200v1.Ferrimagnetism_in_delta_chain_with_anisotropic_ferromagnetic_and_antiferromagnetic_interactions.pdf",        "content": "arXiv:1607.06200v1  [cond-mat.str-el]  21 Jul 2016Ferrimagnetism in delta chain with anisotropic ferromagne tic and\nantiferromagnetic interactions\nD. V. Dmitriev and V. Ya. Krivnov∗\nInstitute of Biochemical Physics of RAS,\nKosygin str. 4, 119334, Moscow, Russia.\n(Dated:)\nWe consider analytically and numerically an anisotropic sp in-1\n2delta-chain (saw-\ntooth chain) in which exchange interactions between apical and basal spins are ferro-\nmagnetic and those between basal spins are antiferromagnet ic. In the limit of strong\nanisotropy of exchange interactions this model can be consi dered as the Ising delta\nchain with macroscopic degenerate ground state perturbed b y transverse quantum\nfluctuations. These perturbations lift the ground state deg eneracy and the model\nreduces to the basal XXZ spin chain in the magnetic field induc ed by static apical\nspins. We show that the ground state of such model is ferrimag netic. The excita-\ntions of the model are formed by ferrimagnetic domains separ ated by domain walls\nwith a finite energy. At low temperatures the system is effectiv ely divided into two\nindependent subsystems, the apical subsystem described by the Ising spin-1\n2chain\nand the basal subsystem described by the XXZ chain with infini tezzinteractions.\nI. INTRODUCTION\nThe low-dimensional quantum magnets on geometrically frustrated lattices are exten-\nsively studied during last years [1, 2]. An important class of such syst ems is lattices consist-\ning of triangles. An interesting and a typical example of these objec ts is the s=1\n2delta or\nthe sawtooth Heisenberg model consisting of a linear chain of triang les as shown in Fig.1.\nThe interaction J1acts between the apical ( σi) and the basal ( Si) spins, while J2is the\ninteraction between the neighboring basal spins. A direct interact ion between the apical\n∗Electronic address: krivnov@deom.chph.ras.ru2\nJ1 \nJ2 i\niS1i\n1iSapical subsystem \nbasal subsystem \nFIG. 1: The △-chain model.\nspins is absent. The Hamiltonian of this model has a form\nˆH=J1N/summationdisplay\ni=1[Sx\ni(σx\ni+σx\ni+1)+Sy\ni(σy\ni+σy\ni+1)+∆1Sz\ni(σz\ni+σz\ni+1)−∆1\n2]\n+J2N/summationdisplay\ni=1[Sx\niSx\ni+1+Sy\niSy\ni+1+∆2(Sz\niSz\ni+1−1\n4)] (1)\nwhere ∆ 1and ∆ 2are parameters representing the anisotropy of the basal-apical and the\nbasal-basal exchange interactions respectively, Nis the number of triangles. The constants\nin this equation are chosen so that the energy of the ferromagnet ic state with the total spin\nLz\ntot=Sz\ntot+σz\ntot=±Nis zero.\nThe isotropic delta chain (∆ 1= ∆2= 1) with both antiferromagnetic interactions J1>0\nandJ2>0 (AF delta chain) has been studied as a function of the parameterJ2\nJ1[3–5]. In\nspite of the simplicity of this model it exhibits a variety of peculiar prop erties. IfJ2\nJ1= 1 the\nmodel has two-fold degenerate ground state where neighboring p airs of spins form singlet\nconfigurations [4]. WhenJ2\nJ1=1\n2the delta chain supports the independent localized magnon\nstates. These states determine both the ground states proper ties and the low-temperature\nthermodynamics in the vicinity of the saturation magnetic field [6–10 ]. In particular, the\nground state is highly degenerate, the zero-temperature magne tization has a plateau and\nthe specific heat has the extra low-temperature peak.\nIn contrast to the AF delta chain the same model with J1<0 andJ2>0 (the F-AF\ndelta chain) is less studied. It is known [11] that the ground state of the F-AF isotropic delta\nchain is ferromagnetic for α=J2\n|J1|<1\n2. It was argued in Ref.[11] on a base of numerical\ncalculations that the ground state for α >1\n2is a special ferrimagnetic state. The critical\npointα=1\n2is the transition point between these two ground state phases. Th e isotropic\nF-AF delta-chain at the transition point α=1\n2has been studied in Ref.[12]. It was shown\n[12] that the ground state at the transition point (at zero magnet ic field) is macroscopically3\ndegenerate and consists of multi-magnon configurations formed b y independent localized\nmagnons and the special localized multi-magnon complexes.\nThe isotropic F-AF delta chain is a minimal model for the description of several mag-\nnetic compounds such as malonato-bridged copper complexes of fo rmula [Cu(bpy)H2O]×\n[Cu(bpy)(mal)H2O](ClO4)2containing magnetic Cu2+ions[11,13–15]. Fromtheanalysis of\nthe experimental data it was concluded [13] that the ratio of excha nge interactions α=J2\n|J1|\nin this compound is α≃1. It means that this compound is on the ferrimagnetic side of\nthe ground state diagram of the isotropic delta chain. Thus, the st udy of the ferrimagnetic\nstate of the F-AF delta chain is important and interesting problem. N umerical calculations\nused in Ref.[11] suppose that the ground state magnetization per s ite in the ferrimagnetic\nphase in the isotropic model is1\n4. Unfortunately, numerical methods do not allow to obtain\nthe detail information about the structure and the properties of the ferrimagnetic phase. At\nthe same time this model is rather complicated and can not be tracta ble analytically.\nIn this paper we show that the analysis of the anisotropic F-AF mode l in the limit of\nhigh anisotropy helps to understand the origin and the properties o f the ferrimagnetic phase.\nFor simplicity we consider the case of equal basal-apical and the bas al-basal anisotropy\n∆1= ∆2= ∆. In this case with ∆ ≫1 the ferrimagnetic phase can exist in a narrow\ninterval of the value α(close to α= 1) between the ferromagnetic (at α <∆\n1+∆) and the\nantiferromagnetic (at α >1) phases [14]. Therefore, in order to investigate the ferrimagnet ic\nphase we put α= 1. Then the Hamiltonian of the F-AF delta chain can be represented in\na form:\n1\n∆ˆH=1\n∆N/summationdisplay\ni=1(Sx\niSx\ni+1+Sy\niSy\ni+1)−1\n∆N/summationdisplay\ni=1[Sx\ni(σx\ni+σx\ni+1)+Sy\ni(σy\ni+σy\ni+1)] (2)\n+N/summationdisplay\ni=1[Sz\niSz\ni+1−Sz\ni(σz\ni+σz\ni+1)+1\n4]\nwhere we put J1=−1 andJ2= 1.\nThe main aim of this paper is to study the model (2) for ∆ ≫1. We expect that some\nprincipal features of the ferrimagnetic phase of model (2) surviv e in the isotropic case.\nAdditional motivation of this study is related to the problem of ‘order by disorder’. The\nfact is that the model (2) in the limit ∆ → ∞turns into the classical Ising model on the\ndelta chain with equal but opposite in sign apical-basal and basal-bas al interactions:\nˆHI=N/summationdisplay\ni=1[Sz\niSz\ni+1−Sz\ni(σz\ni+σz\ni+1)+1\n4] (3)4\nIt is known [16, 17] that the ground state of this model is macrosco pically degenerate and\nit is separated from the excited states by a finite energy gap. This d egenerate ground state\nis disordered (zero magnetization), and the main question of the ‘or der by disorder’ problem\nis what happens when such disordered system is perturbed by the q uantum fluctuations.\nThe quantum fluctuations can lift the degeneracy and drive the sys tem to either ordered\nor disordered ground state. Generally, there are many different w ays of the introduction of\nsuch perturbations. One of them is given by the transverse terms in Eq.(2) and we will show\nthat it leads to the ordered ground state. On the contrary, the p erturbation of the Ising\nmodel (3) by a transverse magnetic field results in the disordered g round state [16].\nAnother example of influence of quantum dynamics onthe Ising mode l (3) was considered\nin Ref.[17], where the anisotropic F-AF model (1) was studied for a sp ecial choice of the\nexchange interactions and the anisotropies: α= 1/(2∆1) and ∆ 2= (2∆2\n1−1). For such\nchoice of the interactions the F-AF model describes the phase bou ndary between different\nground state phases on the ( α,∆1) plane and reduces to the Ising model (3) at ∆ 1→ ∞.\nThequantum fluctuationsliftthegroundstatedegeneracy ofIsin g model (3) butonlypartly,\nso that the degeneracy remains macroscopic on this phase bounda ry, it does not depend on\n∆1and coincides with that for the isotropic F-AF delta-chain at α=1\n2. The spectrum of\nlow-energy excitations has a highly nontrivial multi-scale structure leading to the specific\nlow-temperature thermodynamics [17]. This special model is anothe r example of ‘disorder\nby disorder’ instead of ‘order by disorder’.\nThe paper is organized as follows. In Section II we study the spectr um of model (2) in\ndifferent sectors of total spin Sz\ntotand show that the ground state is ferrimagnetic one. In\nSection III we study the low-temperature thermodynamics of the system both analytically\nand numerically. In Section IV we give a summary of our results.\nII. FERRIMAGNETIC GROUND STATE\nAt ∆→ ∞the model (2) reduces to the Ising model on the delta-chain descr ibed by\nHamiltonian (3). The total 4Neigenstates of this model is divided in two subsets. The\nfirst one consists of degenerate ground states with zero energy . These states include two\ntypes of the spin configurations on triangles: either three spins in t he triangle have the same\norientation or two basal spins of the triangle are opposite oriented . In each triangle there5\nare three configurations which satisfy these conditions. Because the number of admissible\nconfigurations is the same for each triangle, the total number of t he ground states is 3N.\n(4N−3N) states of the second subspace are separated from the ground states by a ‘big’ gap\nwith the energy E∼1.\nAn infinitesimal perturbation of transverse interactions in Eq.(2) lif ts the macroscopic\ndegeneracy of the ground state. However, a role of the first and the second terms in lifting\nis different. The first term has non-zero matrix elements both betw een the states of the\nfirst and the second subsets while the second term in Eq.(2) has non -zero matrix elements\nbetween the states of the first and the second subsets only. Thu s, only the first term in\nEq.(2) gives contributions to an energy to the first order in1\n∆whereas the second term is\nresponsible for the corrections which are proportional to1\n∆2. Therefore, to the leading order\nin1\n∆we can neglect the second term in Eq.(2) and the Hamiltonian (2) redu ces to that given\nby\nˆH=P[∆ˆHI+N/summationdisplay\ni=1(Sx\niSx\ni+1+Sy\niSy\ni+1)]P (4)\nwherePis a projector onto the first subspace containing 3Nstates and ∆ → ∞is assumed.\nThemodel(4)describesthebasal XXZchainwithinfinite zzinteractionsinthemagnetic\nfield produced by the static apical spins and the magnetic field in the i-th basal site is hi=\n∆(σz\ni+σz\ni+1). As a result, the magnetic field acting on the basal spins depends o n the spin\nconfiguration of apical subsystem. At first we consider the most s imple case when all apical\nspins are up (down) producing the uniform magnetic field on basal su bsystem: hi= ∆\n(hi=−∆). It is easy to check that if all apical spins are up (down), the pro jectorPin\nEq.(4)eliminatesthestatesinwhichtwobasalspinsdown(twospinsu p)occupyneighboring\nsites. The total number of allowable states is (1+√\n5\n2)N[18]. The Hamiltonian (4) for the\ncasehi= ∆ takes the form\nˆH=P0{N/summationdisplay\ni=1(Sx\niSx\ni+1+Sy\niSy\ni+1)}P0 (5)\nwhereP0is the projector onto the states with no neighboring spins down.\nThe model (5) can be mapped onto spinless fermions via the Jordan- Wigner transforma-\ntion\nS+\nm=c+\nmexp(iπ/summationdisplay\nl>mc+\nlcl)\nSz\nm=1\n2−c+\nmcm (6)6\nwherec+\nmis the Fermi-operator and we identify a spin down and a spin up as a par ticle and\na hole, correspondingly.\nIn fermion language the Hamiltonian (5) reads\nˆH=P0{1\n2N/summationdisplay\ni=1(c+\nici+1+c+\ni+1ci)}P0 (7)\nand the projector P0forbids two particles to occupy neighboring sites.\nThe model of the spinless fermions with such constraint (infinite nea rest-neighbor in-\nteraction) can be mapped onto the model of non-interacting ferm ions as follows [19] (for\nsimplicity, we consider an open chain with Nsites). Each configuration of Mfermions on\nNsites with constraint is mapped to the configuration of Mfermions on ( N−M+1) sites\nwithoutconstraint byremovingoneemptysitebetween two occupie dsites. TheHamiltonian\nof such model depends on a number of fermions and has a form\nˆH(M) =1\n2N−M+1/summationdisplay\ni=1(c+\nici+1+c+\ni+1ci) (8)\nBesides, the matrix elements between the corresponding configur ations of Eq.(7) and\nEq.(8) are equal to each other. An equivalence of two models means that the dispersion\nrelation in the spin sector Sz=N\n2−Mis\nε(km) =−coskm (9)\nwhere\nkm=πm\nN−M+2(10)\nwithm= 1,2,...N−M+1.\nAccording to Eq.(9) the ground state energy of model (8) in the limit N,M≫1 but for\na fixed fermion density ρ=M\nNis\nE0(ρ) =N1−ρ\nπsin/parenleftBiggπρ\n1−ρ/parenrightBigg\n(11)\nMinimization of E0(ρ) with respect to ρgives\nρ=ρ0≃0.3008 (12)\nand\nE0(ρ0)≃ −0.217N (13)7\nReturning to the spin language, Eq.(12) means that the ground sta te of Eq.(5) is realized\nin the spin sector Sz=N(1\n2−ρ0). Thus, the total spin of the ground state of delta chain\n(2) is\nLz\n0=N(1−ρ0) (14)\nIt follows from Eq.(11) that the energy of the lowest excitations in t his spin sector is\nε=π(1−ρ0)\nNsin/parenleftBiggπρ0\n1−ρ0/parenrightBigg\n, (15)\ni.e. the excitations are sound-like with the sound velocity\nc= sin(πρ0\n1−ρ0) (16)\nThe case with all apical spins down is considered in a similar way. In this c ase the role\nof the Fermi-particles is played by the basal spins up and the total g round state spin is\nL0\nz=−N(1−ρ0).\nWe note that formulae similar to Eqs.(11) and (12) have been obtaine d earlier by the\nBethe-ansatz method [20] in the problem of an asymmetric diffusion o f molecules with dif-\nferent size.\nEq.(11) with ρ=ρ0defines the ground state energy of the Hamiltonian (4) for the\nferromagnetic configuration of the apical subsystem. Now we nee d to consider other distri-\nbutions of up and down apical spins. This problem can not be solved an alytically and we\nuse numerical calculations of finite chains. These calculations show t hat the most important\nconfigurations of the apical spin subsystem are the states with alt ernating domains of the up\nand down spins. The simplest configuration of such type is a two-dom ain structure consist-\ning oflspins up and ( N−l) spins down separated by two domain walls (for cyclic chains).\nFor the two-domain configuration the magnetic field induced by the a pical spins is: h= ∆\nfor (l−1) basal sites; h=−∆ for (N−l−1) sites; and h= 0 on two basal sites located\nin the center of two domain walls. (The ferromagnetic state of the a pical spins considered\nabove corresponds to l= 0 orl=Nand it can be identified as the one-domain structure). It\nis apparent that the minimal energy of the two-domain state with l,N≫1 is reached when\nthe density of the fermions (in fermionic language) in each domain is ρ=ρ0. The total spin\nof this state is Lz= (2l−N)(1−ρ0). It is clear that the energy of this state is higher than\nthe ground state energy of the one-domain state due to the pres ence of defects (the domain\nwalls). The energy of the domain wall Edw(l) is defined as a half of the energy difference8\n00.020.040.060.080.10.12\n0 2 4 6 8 10 12 14Edw \nl \nFIG. 2: Dependence of the domain wall energy on the domain siz eEdw(l) is calculated for the\ncyclicXXZchain of length N= 24 as a half of the energy difference between the two-domain\nconfiguration with lapical spins down and ( N−l) apical spins up and the one-domain ground\nstate energy.\nbetween the two-domain configuration with lapical spins down and ( N−l) apical spins up\nand the one-domain ground state energy. The numerical calculatio ns on finite chain N= 24\nfor the dependence of the domain wall energy on the domain size Edw(l) are shown in Fig.2.\nThe energies of the one-domain and two-domain states are chosen for the optimal value of\nthe total Sz. As can be seen in Fig.2 the domain wall energy Edwslowly depends on lwhen\nthe domain size l≥2 andN≫1 and rapidly converges to the value Edw≃0.07.\nSimilarly, any apical spin configuration can be represented as many d omain structure\nconsisting of rdomains with spins up and rdomains with spins down domains with 2 r\ndomain walls. Numerical calculations show that the ground state ene rgy of the r-domain\nstate is\nE(r) =E0+2rEdw (17)\nwhereE0is the ground state energy of the one-domain configuration ( r= 0) given by\nEq.(11).\nIn order to study the stability of the one-domain ground state with respect to a creation\nof the two-domain states we consider the dependence of the grou nd state of the one-domain\nconfiguration with all apical spins up, E0(ρ), forρclose toρ0. According to Eq.(11) the9\nenergyE0(ρ) has a minimum at ρ0and can be expanded in |ρ−ρ0| ≪1 as\nE0(ρ) =E0(ρ0)+bN(ρ−ρ0)2(18)\nwhere\nb=π\n2(1−ρ0)3sin/parenleftBiggπρ0\n1−ρ0/parenrightBigg\n≈4.46 (19)\nIn an instability point the energies and the total spins of the one- an d two-domain states\nare equal. The total spins of the one-domain state and two-domain one with lup and (N−l)\ndown apical spins are Lz=N(1−ρ) andLz= (N−2l)(1−ρ0), respectively. As a result\nthe instability point is determined by the relations\nbN(ρ−ρ0)2= 2Edw (20)\n(ρ−ρ0) = 2(1−l\nN)(1−ρ0)\nAs follows from Eqs.(20) the instability occurs for ρ > ρ0and for small deviation from\nthe minimum ( ρ−ρ0)∼N−1/2. Thus, in the thermodynamic limit N→ ∞the ground state\nis realized for the one-domain state in the total spin sectors with |Lz| ≥Lz\n0(see Eq.(14)),\nwhile in the sectors |Lz|< Lz\n0the ground state corresponds to the two-domain structure.\nBut the global ground state of the model (4) is twofold degenerat e ferrimagnetic state with\nLz=±Lz\n0. In these states the magnetization on apical and basal sublattice s are|/an}bracketle{tσz\ni/an}bracketri}ht|= 0.5\nand|/an}bracketle{tSz\ni/an}bracketri}ht| ≃0.2, so that the total magnetization per site is/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nLz\n2N/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle= 0.35. The ground state\nenergy as a function of Lz/Nobtained by numerical calculations of finite delta-chains with\nN= 10 and N= 14 is shown in Fig.3. Irregular form of this dependence is due to finite -size\neffects, which are caused mainly by the deviation of the particle dens ityρ=M/Npossible\nfor a given chain length Nfrom the optimal value ρ0. However, as it can be seen in Fig.3\nthe amplitude of oscillations decreases with Nand the expected thermodynamic limit 2 Edw\nis shown in Fig.3 by thick solid line.\nIII. LOW TEMPERATURE THERMODYNAMICS\nThe partition function Zof the model (4) is a sum of contributions to Zcorresponding\nto all possible configurations of the apical spins. Generally, each co nfiguration of the cyclic\ndelta-chain with 2 rdomain walls is specified by a set of rdomains of the apical spins up with10\n00.10.20.30.40.5\n-1 -0.5 0 0.5 1E(Lz) - Egs \nLz/N N=14\nN=10\nFIG. 3: Lowest energies in different sectors of total spin Lz\ntotfor delta-chains with N= 10 and\nN= 14. Predicted thermodynamic limit is shown by thick solid l ine.\nlengthsl1,l2,...lrandrdomains of the apical spins down of length m1,m2,...mrwhich\nsatisfy the conditions\nr/summationdisplay\ni=1li=N−k,r/summationdisplay\ni=1mi=k (21)\nwherekis a total number of down apical spins.\nThen, the partition function Zis\nZ=/summationdisplay\nrZr(l1,m1,l2,m2,...lr,mr) (22)\nwhere summation is carried out over li,misatisfying relations (21) and it includes two one-\ndomain configurations with r= 0.\nThe calculation of Zin Eq.(22) is a complicated problem. However, it can be simplified\nfor low temperatures. As was noted before the ground state ene rgy of the configurations\nwith 2rdomain walls is higher than the one-domain state on the value 2 rEdw. The same\nholds for the free energies. As an example, we represent in Fig.4 the difference between the\nfree energies of the one-domain ( r= 0) and two-domain ( r= 1) configurations of cyclic\nchain with N= 8 as the function of T. This difference varies only slightly with Tand it\nis close to the energy of two domain walls 2 Edw, so that the deviation from the value 2 Edw\nis less than 7% for T < T 1≃0.5. It means that the two-domain partition function Z1at11\n00.050.10.150.20.250.3\n0 0.2 0.4 0.6 0.8 1Fr=1 - Fr=0 \nT \nFIG. 4: Difference of the free energies of the two-domain ( r= 1) and the one-domain ( r= 0)\nconfigurations as a function of Tfor the XXZ chain of length N= 16.\nT < T 1can be written as\nZ1=Z0exp(−2Edw\nT) (23)\nwhereZ0is the partition function of the model (5) describing the one-domain configuration.\nSimilarly, if all domain sizes are large ( li,mj≫1), the free energy per site is the same\nfor each domain and it is equal to that for the one-domain configura tion. Therefore, the\npartition function of the r- domain configuration can be approximately written as\nZr=Z0exp(−2rEdw\nT) (24)\nThen the partition function (22) takes the form\nZ=Z0N/2/summationdisplay\nr=0exp(−2rEdw\nT)W(r,N) (25)\nwhereW(r,N) forr≥1 is the number of the configurations with 2 rdomain walls. The\nweightsW(r,N) are known [21]\nW(r,N) =N−r/summationdisplay\nm=rN\nmCr\nmCr−1\nN−m−1 (26)\nwhereCk\nnare binomial coefficients and W(0,N) = 2.\nThe sum in Eq.(25) looks like the partition function of the 1D Ising mode l of the apical\nspinsσ=1\n2with the effective nearest-neighbor ferromagnetic interaction J=Edw, i.e. the12\n0.2750.280.2850.290.2950.30.305\n0.01 0.1 1 10 \nT \nFIG. 5: Dependence ρ(T).\npartition function ZatT < T 1is a product of the partition functions of the model (5) and\nthat of the effective 1D Ising model ZI, i.eZ=Z0ZI. It means that the free energy and\nother thermodynamic quantities are sums of those for the 1D Ising model and for the model\n(5). As to the thermodynamics of the latter it can be obtained using the known spectrum\nof this model given by Eq.(9). Then, the free energy F0=−TlnZ0has a form\nF0\nN=−T(1−ρ)\nπ/integraldisplayπ\n0ln[1+exp(cosk+µ\nT)]dk (27)\nThe chemical potential µand the density ρas functions of Tare determined from the\nequations ∂F/∂ρ= 0 and ∂F/∂µ= 0 with F=F0+µρ, which result in\nµ=−T\nπ/integraldisplayπ\n0ln[1+exp(cosk+µ\nT)]dk\nρ=(1−ρ)\nπ/integraldisplayπ\n0[1+exp( −cosk+µ\nT)]−1dk (28)\nIn particular, the temperature dependence of the density ρ(T) is shown in Fig.5. As\nfollows from Fig.5 ρ(T) changes from ρ≃0.3 atT= 0 toρ= (√\n5−1)/2√\n5≃0.276\natT≫1. The formula (27) coincides with that obtained by different method in Ref.[22],\nwhere the XXZchain in the vicinity of the triple point has been studied.\nUsing Eq.(27) and well known thermodynamics of the 1D Ising model w e can obtain\nall thermodynamic quantities of the model (4). As an example, the s pecific heat C(T) =\nCI(T)+C0(T) as a function of Tis shown in Fig.6 together with the contributions CI(T)13\n00.050.10.150.20.25\n0.001 0.01 0.1 1 10C \nT CI(T)\nC0(T)\nCI(T) + C0(T)\nFIG. 6: Two contributions to the specific heat and their sum as a function of T.\nandC0(T). The specific heat has a sharp maximum at T≃0.03 and the main contribution\nto it is given by the Ising term, while the shoulder in C(T) atT≃0.3 is related to the\nmaximum in C0(T). AtT→0 the ‘Ising’ contribution CI(T) is exponentially small and the\nspecific heat is uniquely determined by that for C0(T)\nC\nN= 2(1−ρ0)πTsin−1(πρ0\n1−ρ0), T→0 (29)\nAs we noted before, Eqs. (24) and (25) are valid when the domain siz es in the many\ndomain configurations are large. To determine the temperature re gion for which this is\nthe case we use the steepest descent method for the calculation o f the sum in Eq.(25).\nUsing Stirling’s formula for the binomial coefficients in W(r,N) we found that the main\ncontribution to the sum is given by the terms with\nk=N\n2\nr=N\n2(1+exp(Edw\nT))−1\nl↓=l↑= (1+exp(Edw\nT)) (30)\nwherel↑andl↓are average lengths /an}bracketle{tli/an}bracketri}htand/an}bracketle{tmj/an}bracketri}htof up- and down domains.\nAccording to Eq.(30) the representation of the partition function in the form (25) is valid\nif exp(Edw/T)≫1 (orT < E dw). Because Edw< T1we conclude that the partition function\nin the form (25) secures a correct thermodynamics of the model ( 4) forT < E dw, while for\nT > E dwit can give a qualitative description only.14\n00.050.10.150.20.250.3\n0.001 0.01 0.1 1 10C \nT N=10\nN=8\nN=6\nprediction N=infinity\nFIG. 7: Specific heat C(T) calculated for delta chains with N= 6,8,10 and that predicted by\napproximation (32).\nNevertheless, our calculations of finite systems show that all r- dependence of the par-\ntition function Zr(l1,m1,l2,m2,...lr,mr) for the configurations with small domains is ex-\npressed by the factor exp( −2rEdw/T) whereEdw≃0.07 as before. According to Eq.(30)\nforT > E dwthe average size of domains becomes l↓=l↑≃2. Using these facts we take\nas an approximation for the r- domain partition function Zr(l1,m1,l2,m2,...lr,mr) the\nexpression in a form\nZr=˜Zexp(−2rEdw\nT) (31)\nwhere/tildewideZis the partition function for the up-up-down-down ( ↑↑↓↓↑↑↓↓ ...) configuration of\nthe apical spins.\nThen, the partition function ZatT > E dwis\nZ=˜ZZI (32)\nThe thermodynamics of the up-up-down-down configuration is fou nd by an exact diago-\nnalization (ED) calculation of finite chains. Corresponding results fo r the specific heat are\npresented in Fig.7. In Fig.7 we also represent the results of the ED ca lculations of the model\n(2) with ∆ = 100 for N= 6,8,10. We note that the model (2) with large but finite ∆ and\nthe model (4) are formally non-equivalent because the total numb er of states of these two\nmodels are different and include 4Nand 3Nstates, respectively. However, in the tempera-\nture region T <10 the thermodynamics of the model (2) is governed by exactly 3Nstates15\n00.10.20.30.40.50.60.70.8\n0.001 0.01 0.1 1 10 100 1000S \nT ln(2)/20 ln(3)/2 ln(2) \nFIG. 8: Dependence of entropy per site on temperature S(T) for model (2) with ∆ = 100 and\nN= 10.\nas follows from the temperature dependence of the entropy per s pin (see Fig.8). Thus, at\nT <10 the thermodynamics of the models (4) and (2) is identical. As it can be seen in\nFig.7, the data for C(T) for the model (2) with different Ndeviate at T<∼Edwboth from\neach other and from the results for infinite system obtained from E q.(25). It means that\nthe finite-size effects are essential in this temperature region. On the other hand, the data\nfor different Nare indistinguishable at T>∼1, testifying that the finite-size data correctly\ndescribe the thermodynamic limit. We note also that at T>∼1 these data are close to\nthose obtained from Eq.(31) for the up-up-down-down configura tion. At the same time, the\nthermodynamics based on Eqs.(25) show the qualitatively similar beha vior of the specific\nheat in this temperature region.\nLastly, we consider the temperature dependence of the zero-fie ld susceptibility χ(T). In\nthis case it is necessary to include the external magnetic field hext≪1 in the model (2).\nWe confine ourself by the temperature region T<∼Edwwhere the partition function is the\nproduct of the Ising and the one-domain terms. We do not dwell on t he technical details\nof the corresponding computations. They are related to the solut ions of Eqs.(27) and (28)\nas the functions of the temperature and the magnetic field hext. The final result for the\nzero-field susceptibility χ(T) has the form\nχ(T)\nN= 2χI(T)(1−ρ(T))+χ0(T) (33)16\nwhereχI(T) is the zero-field susceptibility per site of the above-mentioned effe ctive Ising\nmodel:\nχI=1\n4Texp(−Edw\n2T) (34)\nρ(T) is the solution of Eq.(28) with hext= 0 andχ0(T) is the susceptibility of the model (5)\ngiven by\nχ0(T) =(1−ρ(T))3\nπT/integraldisplayπ\n0exp(−cosk+µ(T)\nT)[1+exp( −cosk+µ(T)\nT)]−2dk(35)\nwithµ(T) determined by Eq.(28) with hext= 0.\nThe temperature dependence of the quantity χ(T)Tis shown in Fig.9. The susceptibility\nχIis proportional to1\nTexp(Edw/2T) atT→0 whileχ0(0) is finite\nχ0(0) =(1−ρ0)3\nπsin/parenleftBig\nπρ0\n1−ρ0/parenrightBig (36)\nTherefore, the behavior of the susceptibility at low temperatures is determined by the\n‘Ising’ contribution χIand, therefore, exponentially diverges at T→0. In Fig.9 we also\nrepresent the temperature dependence of χ(T)Tfor finite delta-chains obtained by the ED\ncalculationsofmodel(2). Incontrasttotheanalyticspredictingt heexponentiallydivergence\nofχ(T)Tin the thermodynamic limit, the calculations of finite chains show the fin ite limit\nforχ(T)TatT= 0. Such behavior is related to the fact that the value χ(T)TatT= 0\nfor finite Nequals to the square of the ground state spin which is L2\nz=N2(1−ρ0)2, which\nturns into the divergence of χ(T)Tin the thermodynamic limit.\nIV. SUMMARY\nWe have studied the spin-1\n2F-AF delta chain in the limit of large anisotropy of exchange\ninteractions. In this limit the model reduces to the 1 D XXZ chain on basal sites in the\nstatic magnetic field depending on the domain structure of the apica l spins. The ground\nstate is twofold degenerate and magnetically ordered. In the grou nd state the apical spins\nform a fully polarized state with |/an}bracketle{tσz\ni/an}bracketri}ht|= 0.5 and the magnetization of the basal spins is\n|/an}bracketle{tSz\ni/an}bracketri}ht| ≃0.2. Of particular interest are the excited states which involve the do main walls\nseparating the domains of one or another ground state. Based on the domain statistics we\nreduced the low-temperature thermodynamics problem to those f or the effective 1D Ising\nmodel for the apical subsystem and the 1 D XXZ chain with infinite zzinteractions for the17\n012345\n0 0.02 0.04 0.06 0.08 0.1 0.12 0.14T/N \nT N=6\nN=8\nN=10\nEq.(33)\nFIG. 9: Dependence of the susceptibility per site χ(T)T/NonTfor model (4) with N= 6,8,10.\nAnalytical prediction Eq.(33) is shown by thick solid line.\nbasal subsystem. The correlation functions/angbracketleftBig\nσz\niσz\ni+r/angbracketrightBig\nand/angbracketleftBig\nSz\niSz\ni+r/angbracketrightBig\nbehave similarly to 1D\nIsing ones with a correlation length proportional to exp( Edw/2T) at low temperatures.\nThis simple picture provides a starting point for the qualitative under standing of the\nferrimagnetic phase of the isotropic model. Preliminary numerical re sults indicate that the\nground state magnetization on the apical and the basal sites does not change considerably\nwhen the anisotropy parameter ∆ decreases from the large value t o 1. In the isotropic case\nthey are /an}bracketle{tσz\ni/an}bracketri}ht= 0.414 and/an}bracketle{tSz\ni/an}bracketri}ht= 0.086 [23]. However, additional symmetry of the isotropic\nmodel requires certain modifications of the presented approach.\nAcknowledgments\nWe would like to thank J.Richter for valuable comments on the manuscr ipt. The numer-\nical calculations were carried out with use of the ALPS libraries [24].\n[1] H. T. Diep, ed., Frustrated spin systems (World Scientifi c, Singapore, 2013).\n[2] C. Lacroix, P. Mendels and F. Mila, eds., Intoduction to f rustrated magnetism. Materials,\nExperiments, Theory (Springer-Verlag, Berlin, 2011).18\n[3] D. Sen, B.S. Shastry, R.E. Walstedt and R. Cava, Phys. Rev . B53,6401 (1996).\n[4] T. Nakamura and K. Kubo, Phys. Rev. B 53, 6393 (1996).\n[5] S. A. Blundell and M. D. Nuner-Reguerio, Eur. Phys. J. B 31, 453 (2003).\n[6] O. Derzhko, J. Richter, M. Maksymenko, Int. J. Modern Phy s.29, 1530007 (2015).\n[7] M. E. Zhitomirsky and H. Tsunetsugu, Phys. Rev. B 70, 100403 (2004).\n[8] J. Schnack, H.-J. Schmidt, J. Richter and J. Schulenberg , Eur. Phys. J. B 24, 475 (2001).\n[9] J. Richter, J. Schulenburg, A. Honecker, J. Schnack, and H.J. Schmidt, J. Phys.: Condens.\nMatter16, S779 (2004).\n[10] O. Derzhko and J. Richter, Phys. Rev. B 70, 104415 (2004).\n[11] T. Tonegawa and M. Kaburagi, J. Magn. Magn. Materials, 272, 898 (2004).\n[12] V. Ya. Krivnov, D. V. Dmitriev, S. Nishimoto, S.-L. Drec hsler, and J. Richter, Phys. Rev. B\n90, 014441 (2014).\n[13] Y. Inagaki, Y. Narumi, K. Kindo, H. Kikuchi, T. Kamikawa , T. Kunimoto, S. Okubo, H.\nOhta, T. Saito, H. Ohta, T. Saito, M. Azuma, H. Nojiri, M. Kabu ragi and T. Tonegawa, J.\nPhys. Soc. Jpn. 74, 2831 (2005).\n[14] M. Kaburagi, T. Tonegawa and M. Kang, J.Appl.Phys. 97, 10B306 (2005).\n[15] C. Ruiz-Perez, M. Hernandez-Molina, P. Lorenzo-Luis, F. Lloret, J. Cano, and M. Julve,\nInorg. Chem. 393845 (2000).\n[16] R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).\n[17] D. V. Dmitriev, V. Ya. Krivnov, Phys. Rev. B 92, 184422 (2015).\n[18] C. Domb, Adv. Phys. 9,149 (1960).\n[19] S. -A. Cheong and C. L. Henley, Phys. Rev. B 80, 165124 (2009).\n[20] F. C. Alcaraz and R. Z. Bariev, Phys. Rev. E 60, 79 (1999); cond-mat/9904042.\n[21] M. Gaudin, The Bethe wave function, Cambridge Universi ty Press, 2014.\n[22] C. Trippe, F. Gohman, and A. Klumper, cond-mat/0912.17 39.\n[23] S. Nishimoto, S.-L. Drechsler, and J. Richter, private communications.\n[24] B. Bauer et al., J. Stat. Mech. P05001 (2011)."    },    {        "title": "2103.04105v1.Anomalies_in_the_dynamics_of_ferrimagnets_near_the_angular_momentum_compensation_point.pdf",        "content": "1 \n JMMM 509 (2020) 166876  \nCorrected version  \nAnomalies in the dynamics of ferrimagnets near the angular momentum \ncompensation point  \n \nA.K. Zvezdin1,2, Z.V. Gareeva  3*, K.A. Zvezdin1,4  \n  \n1Prokhorov General Physics Institute, Russian Academy of Sciences, 119991, Moscow, Russia  \n2P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninskiy Prospekt  53, \n119991, Moscow, Russia  \n3Institute of Molecule and Crystal Physics, Subdivision of the Ufa Federal Research Centre of the \nRussian Academy of Sciences , 450075 , Ufa, Russia  \n4Moscow Institute of Physics and Technology (State University), 141700, Dolgoprudny, Russia  \n \nCorresponding authors: gzv@anrb.ru , zvezdin@gmail.com  \n \nAbstract  \nIn this paper, we elaborate analytical theory of domain wall dynamics close to  the angular \nmomentum compensation point based  on non-linear dynamic equations derived from the effective  \nLagrangian of a ferrimagnet. In the framework of the proposed m odel, we explore dynamic \nprocesses in the Walker and post Walker regimes . Analysis of the precession angle and domain \nwall’s velocity oscillations in post Walker regime in a ferrimagnet is performed. We show that \nalthough spin oscillations quench the dynam ics of domain walls near the Walker breakdown field, \na further increase of the driving magnetic field increases domain wall speed and mobility. An \nanomalous behavior of domain wall dynamic properties near the angular momentum compensation \npoint in ferrimag nets is discussed . 2 \n Keyword s: magnetization dynamics, ferrimagnets , domain walls  \n \n \n \n1. Introduction  \nUltrafast spin dynamics in antiferromagnets (AFMs) and ferrimagnets (FiMs) near \ncompensation temperature are receiving much attention owing to the potential applications for spin \nelectron devices [1 - 15]. Recent a dvances in antiferromagnetic spintronics and femtosecond lasers \nsources operating with ultrafast speeds and ultrashort times open new horizons for future \ntechnologies based on high -speed nanoscale elements in memory devices.  \nAn important attributes of spin dynamics in magnets are velocity and mobility of domain \nwalls (DWs). Magnetic DWs, active nanoscale elements, are well – suited for the concept of \nracetrack memory, logic units and spintronic memristors [7 - 15]. In this sense, a lot of expectations \nare related with antiferromagnetic materials where DW dynamics is extremely fast. As known [16 \n- 19], in antiferromagnetic orthof errites (YFeO 3) DW velocity attains 20 km/s and DW mobility is \nof the order of 0.1 km/s Oe.  \nDespite the undoubted advantages of antiferromagnetic dynamics, the difficulties \nassociated with the detection of DWs and their manipulation in AFMs force research ers to turn to \nferrimagnetic materials. However, the Walker breakdown velocities in FiMs are much lower (an \norder of magnitude) than those in AFMs.  \nThe exchange enhancement of dynamic parameters and giant DWs velocities are expected \nin FiMs near the subla ttice compensation point Tcomp [20]. However, there are still no reliable \nexperimental data in this direction. In particular, this can be explained by the fact that the saturation \nmagnetization vanishes \n0sM  near the compensation poin t \ncomp TT , and the corresponding spin \ntorque, which one can call the Zeeman torque, also tends to zero.  \nAnomalous spin dynamics in FiMs with strong sublatices coupling can occur at two \ntemperatures:  \ncompT , sublatti ce compensation temperature where M1=M2, and TA , angular 3 \n momentum compensation temperature where \n12\n12MM\n , \ni is the gyromagnetic ratio for the \nspecific sublattice, Mi  is the sublattice magnetization ( i=1,2) [5 - 8].  \nNear the angular momentum compensation point, the situation to study spin dynamics is \nmore favorable. Recent experiments on domain wall dynamics in ferrimagnetic GdFeCo alloy \nwere reported in [1, 2]. Authors of Ref. [1] measured enhancement  of DW velocity close to the \nangular momentum compensation point. The phenomenon was qualitatively explained in terms of \ncollective coordinates and atomistic spin model simulations.   \nIn our work, we elaborate analytical model based on the effective Lagran gian of FiMs [21] \nto describe dynamics of FiMs near the angular momentum compensation point TA. Proposed \napproach allows us to explore the structure of a moving DW and its featured dynamic properties: \nDW speed, mobility and the angle of precession both for  steady state motion and post – Walker \noscillatory regime.   \n2. Model  \nTo describe dynamics of a  ferrimagnet  (FiM) close to the angular momentum compensation \npoint we consider FiM with two magnetic sublattices  M1, M2 whose magnetization have different \ntemperature dependences. Such situation is realized in intermetallics, the most of rare earth \ntransition metals alloys (e.g. GdFeCo, TbFeCo, CoGd)  and weak ferromagnets (e.g. YFeO 3, \nRFeO 3) [1, 16 –23]. Approaching the FiMs compensation temperature  AFM ordering tends to be \nestablished ; and antiferromagn etic \n12L M M  and ferromagnetic vectors \n12M M M  are \nconsidered as  order paramet ers.  \nDynamics of FiM is used to be described in terms of Landau Lifshitz  pair equations  for each \nof Mi sublattices  [24, 25]. However,  to consider  domain wall (DW) motion it is more convenient \nto use Lagrange formalism in the spherical coordinates. The Lagrangian and the dissipation \nRayleigh function of two –sublatticed  FiM read \n2\n11 cosi\ni i i\ni iMLW   \n         (1) 4 \n \n 2\n22\n1sini\ni i i i\ni iRM \n        (2) \nwhere Mi is the magnitude of the i-th sublattice magnetization ;\n,ii  are the polar and azimuthal \nangles characterizing the orientation of the i-th sublattice magnetization ; \ni and \ni  are the i-th \nsubllatice damping parameter  and gyromagnetic ratio ; Wi is the  i-th subllatice thermodynamic \npotential.     \nTo characterize the canting of magnetic sublattices we introduce additional variables \n,  \ndefined as \n12 ,            , \n12 ,              , where \n,  are the polar and \nazimuthal angles of antiferromagnetic vector L in the spherical coordinate frame with polar axis \noriented along the direction of applied magnetic field.  In our problem  magnetic field is oriented \nalong the normal to a film  (see inset in Figure 1 ), uniaxial magnetic anisotropy of a film is \nsupposed to be strong . In the vicinity of compensation temperature ( TA) the canting angles \n,  \nare assumed to be sufficiently small  \n1, 1   that is valid at H<<Hex, Hex ~50 T (TbFeCo, \nGdFeCo).  \nThe effective Lagrangian of two - sublatticed uniaxial ferrimagnets was suggested and \ndescribed in details in Ref. [21]. We include into consideration the non – uniform exchange energy, \nthe in -plane magnetic anisotropy and rewrite the effective Lagrangian [21] as follows  \n22\n2\n22\n2 2 2 2cos sin22\nsin sin sin sineff\neff eff eff\nuL m H H\nddK K Adx dx      \n   \n     \n                    \n            \n    (3) \nwhere \n21\n2MMm , \n12\n2MMM ;\nexM\nH  is the transverse magnetic susceptibility, Hex is the \nexchange magnetic field acting between sublattices; \n,uKK are the constants of  perpendicular and  \nin-plane magnetic anisotropies, here we assume that \nu\neffHK\n \n ; A is the exchange stiffness 5 \n constant, \n and   are the polar and azimuthal angles of the ferromagnetic vector m,  H=(0,0, Hz) \nis magnetic field applied along the “easy magnetization axis ” (geometry of a problem is shown in \ninsert to Fig.1) . The effective Rayleigh dissipation function is taken in a form [21]  \n 2 2 2sin2eff\neff\neffMR    \n           (4) \nwhere  \n0effm\nmm\n, \n0effm\nmm ,\n12\n12 , \n121 1 1 1\n2  \n , \n12\n0\n12mM\n , \n1\n0\n21effmm\nM\n \nwhere m0 is the magnetization in the compensation point of angular momentum  TA.   \nLet’s consider for the beginning DW dynamics θ(t) in the Walker approach, i.e. assume that \n0\n,  \n0d\ndx , then t he Euler – Lagrange equation for the function θ(x, t) reads  \n2\n2\n222 sin 2 sin 02u\neffdMA K H mHdx             \n     (5) \nAt the boundary conditions \n 0,           we find the particular solution  of eq. ( 5)  \n0222arctg exp\n1/x qt\nqc \n,               (6) \nwhere  \n2\n21qmHM q\nс\n\n\n,           (7) \n2\n2uA\nKH\n\n,  \n2\neffAc  \nwhere  q is the coordinate of DW centre  \nEqs. (6), (7) represent  the exact solution of eq.  (5). One can check  it by the direct substitution 6 \n of eq s. (6), (7) into eq.( 5) [16].  Eq. (6) describe s the spin distribution in side the 1800 domain wall \n(DW) moving with the velocity \nq\n . Eq. ( 5), the double sin -Gordon equation with dissipation, and \nits solution ( 6) were obtained and used for description and investigation of DW dynamics in the \ndifferent systems [16, 2 6, 27].  \nIn the general case\n0\n . We solve the system of Euler -Lagrange equations using  the \nperturbation theory for solitons  \n\n01\n1, , ...,     \n, , ...x t x qt x t\nx t t x t  \n     \n  \n       (8) \nwhere Euler Lagrange equation for φ in which we substitute eq. ( 6) reads  \n0 0 0 0 0 0 22sin sin 2 sin sin cos cos 0\neff eff eff effmMKH                \n\n      \n   (9) \n Accounting ( 8) we rewrite eq. ( 5) in operator form \n1ˆLf  where \n2\n0 21ˆ cos 2x L     , function \nf depends on \n0x qt\n  and the system parameters. Solvability condition of such equations is \ndetermined by the requirement of orthogonality between functions f and \n0d\ndx : \n00df dxdx\n  \n(Fredholm alternative). Implementing the described procedure for eq. ( 5) and integrating ( 9) over \n, x  \n, we obtain the system of dynamic equations similar to the Slonczewski equations [2 5] \nat \n/1qc\n   \n2sin 2 0eff\neff effMq m H\nm q MK\n\n    \n\n\n   \n       (10) \nNote that at \n0 K  eqs. ( 10) give the exact solutions of eqs. ( 5), (9). As follows from eq. ( 10) \nthe DW velocity \nq\n  is proportional to the magnetic torques, the precession rate \n\n  depends on a \npressure on DW exerted by driving field, damping and magnetic anisotropy.  7 \n  \n3. Domain wall dynamics in the steady state (Walker) and post Walker regimes  \n To understand the characteristic features of DW  dynamics close to the angular momentum \ncompensation point we make several steps.   \nI) We start with the case when in -plane magnetic anisotropy is absent\n0 K . Since \n1  and \n10\neff\n close to the angular momentum compensation point \n0 mm  one can neglect the first \nterm in eq. ( 5). In this case in accordance with eqs. ( 10) DW velocity is determined as in Ref. [1]  \nby equation  \n2\n2 0qmHM mm\nM\n \n                        (11) \nThe dependences of DW velocity on the parameter \n/mM  at various magnitudes of \nmagnetic field H are shown in Figure 1 . For calculations we used the parameters of  GdFeCo [1, \n21, 22 ]  \n5 6 3 7\n3~ 1 10 , 1 10 , ~ 2 10 , 1100 , ~ 0.02, ~ 2 10 , 2.2, 2u d ferg ergK A M G g gcm cm  \n        \n 8 \n \nm/M0.00 0.05 0.10 0.15 0.20Domain wall velocity\n          V, m/s\n02004006008001000120014001600\nH=1000 Oe\nH=400 OeH=600 Oe\n \nFigure 1 . Domain wall velocity as the function of the \n12\n12MMm\nM M M  (dimensionless FiM ’s \nmagnetization)  in the case of non – stationary motion  at different values of driving magnetic field: \nblack line stands for H=400 Oe, green line stands for H=600 Oe, red line stands for H=1000 Oe . \nGeometry of t he problem is shown in the inse t. \n \nAs seen in Fig.1 the DW velocity enhances approaching the angular momentum compensation \npoint  \n0  (\n0 mm ) and tends to zero at magnetization compensation temperature \n(M1(Tc)=M2(Tc)) \n0 (\n0 m ). Our results ( Figure 1 ) are in consistence with the findings of \nRef. [1], the dependences shown in Figure 1  can be reconstructed in terms of T accounting the \ntemperature dependences of the sublattices  magnetizations \n12( ),  ( )M T M T . \nII) We a nalyse the dynamic impact given by in -plane magnetic anisotropy \n0 K .  \ni) At first we consider the steady state motion occurring in magnetic fields lower than the \nWalker field \n0w HH . In this case \n00\n  the DW moves with the velocity \nqH\n  (see eq. 9 \n (10)), where \n  is the DW mobility.  \nSteady – state motion velocity linearly depends on magnetic field up to the Walker velocity  \nwwmVHM\n\n           (12) \nestablished at the Walker field  \n22eff\nw effH K M Mm\n\n .                        (13) \n if we take \n22 Km   \nm/M0.03 0.06 0.09 0.12 0.15 0.180.00.51.01.52.0\n=0.0476\n \nFigure 2 . The Walker field and the effective mass as functions of \n12\n12MMm\nM M M (dimensionless \nFiM’s magnetization ).  \n. \nAs seen from eq.(13)  the Walker field diverges  close to the angular momentum \ncompensation point \n0 mm  (Figure 2 ).  \nii) We turn to the non – stationary regime \n0\n  in magnetic fields higher than the Walker field 10 \n \nw HH.  In the case \n1  one can neglect the \n2\neff\n  term in eqs. (10) and  obtain the  solution  \n22 2\n22tan 1 tan 1w w w\neff\neffH H HHtH H H                \n    (14) \n22\n22sin 2eff\neff w\neff w effqHHH      \n     (15) \nEquations (1 4), (15) describe the oscillations of precession angle \n  and  DW velocity \nq\n  . That \nmeans that in post -Walker regime at\nw HH  the angle of in -plane precession \n and DW velocity \nq\n become the periodical functions dependent on system parameters . The dependences of \noscillations amplitude and period on parameter \n/mM  are shown in Figure 3 . One can see \nthat DW velocity tends to be constant in a time close to the angular momentum compensation point \nTA. However, due to an unlimited increase in the oscillation frequency approaching \n0 mm , a \nrapid change in the angle of pre cession at \ntT (green line in Figure  3a) manifests itself in the \nsharpening “beak - like” areas occurring at \ntT  on the dependence \n()qt\n  (green line in Figure  \n3b), where T is the oscillations period  11 \n \nt, ns0.0 0.5 1.0 1.5 2.0Precession angle\n, deg.\n0200400600800 \nt, ns0.0 0.5 1.0 1.5 2.0Domain wall velocity\n           V , m/s\n020040060080010001200\n \nFigure 3  a) Oscillations of in – plane precession angle as the function of a time, b) domain wall \nvelocities as the functions of a time  at different values of the \n12\n12MM\nMM : black lines stand for \n0.025\n, red lines stand for \n0.035 , green lines stand for \n0.04 , H=800 Oe.  \n \nAveraging of DW velocity (1 5) on the precession movement of azimuthal angle \n given 12 \n by eq. (1 4) yields  \n22\n22sin 2eff\neff w\neff w effHqHH       \n     (16) \nDriving magnetic field\n               H, Oe0 200 400 600 800 1000Averaged domain wall velocity\n                  <V>, m/s\n02004006008001000120014001600\n=0.047\n=0.035\n=0.025\n=0.04\n \nFigure 4 . Average domain wall velocity as the function of driving magnetic field H in the Walker \nand post – Walker regimes calculated at different values of  the \n12\n12MM\nMM (dimensionless \nFiM’s magnetization) : black line stands for \n0.025 , red line stands for \n0.035 , green line \nstands for \n0.04 , blue line stands for angular momentum compensation point  \n00.0475 . \n \nDependences of the average DW velocity on magnetic field at different ν values are shown in \nFigure 4 where  the Walker and post Walker regimes are distinguished. In the steady state Walker \nregime DW velocity linearly increases with the driving field H and attains the maximum value \nmax qV\n at the Walker field \nw HH . In the post Walker regime close to the critical point \nw HH\n jump – like  decrease of DW velocity  occurs, however with the enhancement of magnetic 13 \n field \nw HH average DW velocity \nq\n  increases. Close to the angular momentum compensation \npoint \n00.0475  the DW velocity permanently increases.  \nIn contrast to Ref. [1] we show the differences in slope angles of the curves < V>(H) (Figure 4 ) \nboth in Walker and post – Walker  regimes and present more detailed consideration of post Walker \nDW dynamics.  \nEq. ( 7) describes the quasi - relativistic compression of the DW width, which occurs with \nan increase of the DW velocity. This effect, which is typical for the AFMs dynamics, in particular, \nleads to saturation of DW velocity with an increase of the driving magne tic field; in AFMs, the \nDW reaches its ultimate velocity which can be about 20 km/s [16 - 19]. As we showed, in FiMs \nthe situation becomes essentially different: in the vicinity of angular momentum compensation \npoint \n0 mm  the Walker f ield \nWH and ultimate velocity diverge \nc  (see eq. ( 7)).  \n \n4 Discussion and c onclusion  \nTo conclude, we present the analytical model with integrable dynamic equations on the \nbase of effective ferrimagnet Lagrangian given in Ref. [21]. We obtain the exact solutions of \ndynamic equations allowing  describe DW dynamics in ferrimagnets close to the angular \nmomentum compensation point TA. The processes in -between stationary Walker and non – \nstationary oscillatory  regimes have been explored.  \nWe compare results of our calculations with experimental data on DW dynamics and their \nexplanation given in Ref. [1].  In the case when in -plane magnetic anisotropy is absent the \ncalculated DW velocity and its temperature depe ndences ( Figure 1 ) coincide with the findings of \nRef. [1]. We show the principal agreement with results of Ref. [1] on the behavior of DW velocity \nin the driving magnetic field. Magnetic field applied along the “easy magnetization axis” increases \nDW veloci ty up to the Walker breakdown, afterwards the DW quenches due to the transition into \noscillatory post – Walker regime typical for ferromagnets. Further enhancement of a field increases 14 \n DW velocity. The highest values of velocity (up to 1.5 km per second in  a field around 1000 Oe) \nare attained at the temperatures close to the angular momentum compensation point.  \nOur analysis also reveals the differences from findings of Ref. [1]. We show that DW \nvelocity is also the function of the net angular momentum  \n2 1 2 1 ( ) ( ) / ( )T M M M M    related \nwith  magnetization of specific sublattices dependent on the temperature. In contrast to Ref. [1] \nwe include in our consideration in – plane magnetic anisotropy typical e.g. for uniaxial crystals \n(GdFeCo) and demonstrate osci llations of the precession azimuthal angle and DW velocity in the \npost Walker regime. Actually, the in – plane magnetic anisotropy favors outcome of magnetic \nvector from the DW plane described by azimuthal angle  φ. The competition between impacts given \nby the driving magnetic field and magnetic anisotropy results in the oscillations of φ angle and as \na consequence DW velocity close to the Walker breakdown field.   \nOur findings give additional information on the oscillatory post Walker regime. We show \nthat t he amplitude and period of oscillations depend on temperature and material parameters.  \nElaborated model can be used as a tool to study plethora of spin dynamics phenomena in \nferrimagnetic materials such as magnetic – dipole radiation of electromagnetic wa ves excited by \noscillations of the moving DW, phenomenon theoretically predicted in Ref. [ 31]; current induced \nmotion of DWs, experimentally observed in AFM coupled bi -layers [2]; accelerating and focusing \nmagnetic vortices and skyrmions [ 32] attractive for spintronic applications.  \nDeclaration of Competing Interest  \nThe authors declare that they have no known competing financial  interests or personal \nrelationships that could have appeared to influence  the work reported in this paper.  \nAcknowledgments  \nThis work is supported by the Russian Science Foundation grant No. 17-12-01333.  \n \n \n \n 15 \n References  \n \n [1] K. -J Kim, S.K. Kim, Y. Hirata, S.H. Oh, T. Tono, D.H. Kim, T.Okuno,…, T.Ono, Nature \nMaterials  2017 , 16 (12), 1187.  \n[2] S.A. Siddiqui, J. Han, J.T. Finley, C.A. Ross, L. Liu, Phys.Rev.Lett . 2018 , 121, 057701  \n[3] B. A. Ivanov, Low Temp. Phys.  2019 , 45 (9), 935.  \n[4] C. D. Stanciu,  AV Kimel , F Hansteen, A Tsukamoto,  A. Itoh, A. Kirilyuk, Th. Rasing, Phys. \nRev. B  2006 , 73 (22), 220402.  \n[5] F. Schlickeiser,  U Atxitia , S Wienholdt, D Hinzke , O. Ghubykalo -Fesenko, U. Nowak, Phys. \nRev. B  2012 , 86 (21), 214416.  \n[6] M. Binder, A Weber,  O Mosendz , G Woltersdorf , M. Izquierdo, I. Neudecker, J. R. Dahn, T. \nD. Hatchard, J. -U. Thiele, C. H. Back, M. R. Scheinfein, Phys. Rev. B  2006 , 74 (13), 134404.  \n[7] R. F.  Soohoo, A. H. J. Morrish, Appl. Phys . 1979 , 50 (B3), 1639.  \n[8] T. Kato ,  K. Nakazawa, R Komiya , N. Nishizawa, S. Tsunashima, S. Iwata, IEEE Trans. \nMagn.  2008 , 44 (11), 3380.  \n[9] S. S. Parkin, M. Hayashi, L. Thomas, Science  2008 , 320, 190. \n[10] D. A. Allwood , G. Xiong, C.C. 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K. Zvezdin, J. Phys.: Condens. Matter  \n2019 , 32 (1), 01LT01.  \n[22] D.-H. Kim, et al. Phys. Rev. Lett . 2019 , 122 (12), 127203  \n[23] S. K. Kim, Nature Electronics  2020 , 3 (1), 18 –19. \n[24] C. Kittel, Phys. Rev. 1951 , 82, 565  \n[25] A. P. Malozemof f, J. C. Slonczewski, Magnetic Domain Walls in Bubble Materials: \nAdvances in Materials and Device Research (vol.1) , Academic Press, 2016 . \n[26] A.K. Zvezdin, A.F. Popkov, Pis’ma Zh. Eksp. Teor. Fiz . 1985 , 41, 90. \n[27] A.K. Zvezdin, A.A. Mukhin, Pis’ma Zh. Eksp. Teor. Fiz . 1985 , 42, 129.  \n[28] W. Doering. Zs. Naturforsch . 3a, 373 (1948)  \n[29] H. E. Khodenkov, Physica status solidi  (a) 1979 , 53 (1), K103.  \n[30] X.R. Wang, P. Yan, J.Lu, C.He, Annals of Physics  324 (2009) 1815   \n[31] A.K. Zvezdin, Pis’ma Zh. Eksp. Teor. Fiz . 1980 , 31, 508 -510,  \n[32] S. K. Kim, K.J. Lee, Y. Tserkovnyak, Phys. Rev. B 2017,  95, 140404(R)  \n \n "    },    {        "title": "2302.03100v1.Observation_of_Coherently_Coupled_Cation_Spin_Dynamics_in_an_Insulating_Ferrimagnetic_Oxide.pdf",        "content": "Observation of Coherently Coupled Cation Spin Dynamics in an Insulating\nFerrimagnetic Oxide\nC. Klewe,1,a)P. Shafer,1J. E. Shoup,2C. Kons,2Y. Pogoryelov,3R. Knut,3B. A. Gray,4H.-M. Jeon,5\nB. M. Howe,4O. Karis,3Y. Suzuki,6, 7E. Arenholz,1D. A. Arena,2,b)and S. Emori6, 8,c)\n1)Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA,\nUSA\n2)Department of Physics, University of South Florida, Tampa, FL, USA\n3)Department of Physics and Astronomy, Molecular and Condensed Matter Physics, Uppsala University, Uppsala,\nSweden\n4)Materials and Manufacturing Directorate, Air Force Research Lab, Wright Patterson Air Force Base, OH,\nUSA\n5)KBR, Beavercreek, OH, USA\n6)Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA,\nUSA\n7)Department of Applied Physics, Stanford University, Stanford, CA, USA\n8)Department of Physics, Virginia Tech, Blacksburg, VA, USA\n(Dated: 9 January 2023)\nMany technologically useful magnetic oxides are ferrimagnetic insulators, which consist of chemically distinct\ncations. Here, we examine the spin dynamics of di\u000berent magnetic cations in ferrimagnetic NiZnAl-ferrite\n(Ni0:65Zn0:35Al0:8Fe1:2O4) under continuous microwave excitation. Speci\fcally, we employ time-resolved x-ray\nferromagnetic resonance to separately probe Fe2+=3+and Ni2+cations on di\u000berent sublattice sites. Our results\nshow that the precessing cation moments retain a rigid, collinear con\fguration to within \u00192\u000e. Moreover, the\ne\u000bective spin relaxation is identical to within <10% for all magnetic cations in the ferrite. We thus validate\nthe oft-assumed \\ferromagnetic-like\" dynamics in resonantly driven ferrimagnetic oxides, where the magnetic\nmoments from di\u000berent cations precess as a coherent, collective magnetization.\nMagnetic insulators are essential materials for\ncomputing and communications devices that rely\non spin transport without net charge transport1,2.\nMost room-temperature magnetic insulators possess\nantiferromagnetically coupled sublattices3{7. Many\nare true antiferromagnets with prospects for ultrafast\nspintronic devices3{5. Yet, challenges remain in\ncontrolling and probing the magnetic states of\nantiferromagnets8{10. For practical applications,\nperhaps more promising insulators are ferrimagnetic\noxides6,7,11{ such as iron garnets and spinel ferrites {\nthat possess unequal sublattices incorporating di\u000berent\ncations. The magnetization state in such ferrimagnets\ncan be straightforwardly controlled and probed by well-\nestablished methods, i.e., via applied magnetic \felds and\nspin currents7. Further, the properties of ferrimagnetic\noxides (e.g., damping, anisotropy) can be engineered\nby deliberately selecting the cations occupying each\nsublattice6,11,12.\nMost studies to date have e\u000bectively treated\nferrimagnetic oxides as ferro magnets: the cation\nmagnetic moments are presumed to remain collinear\nand coherent while they are excited, such that they\nbehave as one \\net\" magnetization (i.e., the vector sum\nof the cation moments). However, it is reasonable\na)Electronic mail: cklewe@lbl.gov\nb)Electronic mail: darena@usf.edu\nc)Electronic mail: semori@vt.eduto question how much these cation moments can\ndeviate from the ferromagnetic-like dynamics. Such\ndeviations may be plausible, considering that the\ncoupling among the cations may not be perfectly rigid\nor that di\u000berent magnetic cations in the sublattices\nmay exhibit di\u000berent rates of spin relaxation (e\u000bective\ndamping)11,12. Indeed, a recent experimental study on\nbiaxial yttrium iron garnet demonstrates peculiar spin-\ntorque switching results13, suggesting that ferrimagnetic\noxides { even with a large net magnetization { could\ndeviate from the expected ferromagnetic-like dynamics.\nGiven the application potential and fundamental interest,\nit is timely to explore the dynamics of speci\fc sublattices\nand cations in ferrimagnetic oxides.\nIn this Letter, we present unprecedented experimental\ninsight into resonant spin dynamics in a multi-\ncation ferrimagnetic oxide. Speci\fcally, we investigate\nsublattice- and cation-speci\fc dynamics in NiZnAl-\nferrite (Ni 0:65Zn0:35Al0:8Fe1:2O4), a spinel ferrimagnetic\noxide with two magnetic sublattices [Fig. 1]: (i) the\ntetrahedrally coordinated sublattice, Td, predominantly\nconsisting of Fe3+\nTdcations and (ii) the octahedrally\ncoordinated sublattice, Oh, predominantly consisting\nof Fe3+\nOh, Fe2+\nOh, and Ni2+\nOhcations. We utilize x-ray\nferromagnetic resonance (XFMR)14{26, which leverages\nx-ray magnetic circular dichroism (XMCD) that is\nsensitive to chemical elements, site coordination, and\nvalence states. With this XFMR technique, we detect\nthe precessional phase and amplitude for each magnetic\ncation species.\nOur cation-speci\fc XFMR measurements are furtherarXiv:2302.03100v1  [cond-mat.mtrl-sci]  6 Feb 20232\naugmented by the following attributes of NiZnAl-ferrite.\nFirst, the NiZnAl-ferrite \flm exhibits about two orders\nof magnitude lower magnetic damping than the ferrite\nin an earlier XFMR study19, yielding a far greater\nsignal-to-noise ratio in XFMR measurements. This\npermits comprehensive measurements at multiple applied\nmagnetic \felds, which allow precise quanti\fcation of the\nprecessional phase lags among the cation species. Second,\nNiZnAl-ferrite is an intriguing test-bed for exploring\nwhether the excited magnetic cations retain collinear\ncoupling. The nonmagnetic Zn2+and Al3+cations dilute\nthe magnetic exchange coupling in NiZnAl-ferrite27, as\nevidenced by a modest Curie temperature of \u0019450 K28,\nsuch that the magnetic Fe2+=3+and Ni2+cations may\nnot remain rigidly aligned. Lastly, with diverse magnetic\ncations in NiZnAl-ferrite, we address whether cations\nwith di\u000berent spin-orbit coupling can exhibit distinct\nspin relaxation12by quantifying the FMR linewidths and\nprecessional cone angles for the di\u000berent cations. Taken\ntogether, we are able to probe { with high precision { the\npossible deviation from the oft-assumed ferromagnetic-\nlike dynamics in the ferrimagnetic oxide.\nOur study focuses on a 23-nm thick epitaxial NiZnAl-\nferrite \flm grown on (001) oriented, isostructural\nMgAl 2O4substrates by pulsed laser deposition28.\nThe NiZnAl-ferrite \flm, magnetized along the [100]\ndirection, was probed at room temperature with a\ncircularly polarized x-ray beam at Beamline 4.0.2 at\nthe Advanced Light Source (ALS), Lawrence Berkeley\nNational Laboratory. The XFMR measurements follow\na pump-probe method: the RF excitation (4-GHz pump)\nis synchronized to a higher harmonic of the x-ray\npulse frequency (500-MHz probe), and the transverse\ncomponent of the precessing magnetization is probed\nstroboscopically. A variable delay between the RF\npump signal and the timing of the x-ray pulses enables\nmapping of the complete magnetization precession cycle.\nA photodiode mounted behind the sample collects\nthe luminescence yield from the subjacent MgAl 2O4\nsubstrate. The luminescence yield detection enables the\nFIG. 1. Schematic of a portion of the spinel structure,\nshowing two cations (e.g., Fe3+\nOh, Ni2+\nOh) occupying the\noctahedrally-coordinated sublattice (green and purple) and\na cation (e.g., Fe3+\nTd) occupying the tetrahedrally-coordinated\nsublattice (blue). The gray spheres represent oxygen anions.\n858 856 854 852\nenergy (eV)-0.15-0.10-0.050.000.050.10XMCD (arb. units)\n712 710 708 706\nenergy (eV)Fe2+\nOhFe3+\nOh\nFe3+\nTdNi2+\nOhFIG. 2. Static XMCD spectra taken at the L3edge of\nFe and Ni. The characteristic peaks are attributed to\nthe corresponding cation valence states and sublattice site\noccupation.\ninvestigation of high-quality epitaxial \flms on single-\ncrystal substrates20,22,23,26. This is in contrast to\ntransmission detection that is limited to polycrystalline\n\flms on thin membrane substrates17,19. A more detailed\ndescription of the XFMR setup is provided in Refs. 25\nand 26.\nBy tuning the photon energy to the element- and\ncoordination-speci\fc features in the static XMCD\nspectra, we are able to probe the magnetism of\ndi\u000berent elements, valence states, and sublattice sites\nindividually . Static XMCD spectra at the L3edge of\nFe and Ni are shown in Fig. 2. The spectra show\npronounced peaks from di\u000berent cations on the Ohand\nTdsublattices. While an XMCD spectrum is generally\na complicated superposition of di\u000berent coordinations\nand valence states, the three distinct peaks in the Fe\nL3spectrum at 708 :0 eV, 709:2 eV, and 710 :0 eV are\nattributed to Fe2+\nOh, Fe3+\nTd, and Fe3+\nOh, respectively, to a\ngood approximation29,30. The opposite polarities of the\nFe3+\nTdand Fe2+=3+\nOhpeaks re\rect the antiferromagnetic\ncoupling between the TdandOhsublattices at static\nequilibrium. Ni2+cations predominantly occupy the Oh\nsublattice28,29, such that the XMCD peak at 853 :5 eV is\nassigned to Ni2+\nOh.\nXFMR measurements were carried out at the photon\nenergies speci\fc to the cations found above. For\neach cation, we performed phase delay scans to map\nout the precession at di\u000berent \feld values across the\nresonance \feld \u00160Hres. Figure 3(a) displays a set of\nphase delay scans taken at a photon energy of 710 :0 eV,\ncorresponding to Fe3+\nOh. Each scan was taken at a \fxed\nbias \feld between 17 :0 mT and 21 :6 mT. The phase delay\nscans exhibit pronounced oscillations with a periodicity\nof 250 ps in accordance with the 4-GHz excitation.\nFigure 3(b) depicts delay scans for Fe3+\nTd(709:2 eV) and\nFe3+\nOh(710:0 eV) taken at \u00160H= 19.3 mT (center of\nthe resonance curve). The opposite sign of the two\noscillations indicates a phase shift of about 180\u000ebetween\nthe two sublattices. The result in Fig. 3(b) thus suggests\nthat the moments of Fe3+\nTd(709:2 eV) and Fe3+\nOhin NiZnAl-3\nXFMR (arb. units)17.0 mT\n17.9 mT\n18.3 mT\n18.5 mT\n18.7 mT\n18.9 mT\n19.1 mT\n19.3 mT\n19.5 mT\n19.7 mT\n19.9 mT\n20.1 mT\n20.3 mT\n20.7 mT\n21.6 mTexp. data Sine fit Fe3+\nOh\n500 400 300 200 100\nphase delay (ps)(a) XFMR (arb. units)\n500 400 300 200 100\nphase delay (ps)(b)Fe3+\nOhSine fit Fe3+\nTdSine fit\n(19.3    0.1) mT +\nFIG. 3. (a) Bias \feld resolved phase delay scans at the\nresonant core level excitation energy of Fe3+\nOh(710:0 eV). The\ndashed curve highlights the characteristic shift across the\nresonance. (b) Comparison between delay scans of Fe3+\nTd\n(709:2 eV) and Fe3+\nOh(710:0 eV) cations taken at 19 :3 mT.\nferrite maintain an antiferromagnetic alignment during\nresonant precession.\nIn the remainder of this Letter, we quantify the\nprecessional phase and relaxation of each cation\nby analyzing our \feld-dependent XFMR results,\nsummarized in Fig. 4. Figure 4(a) shows that all cations\nin NiZnAl-ferrite exhibit a characteristic 180\u000ephase\nreversal across the resonance of a damped harmonic\noscillator. Quick visual inspection reveals that all Oh\ncations are approximately in phase. Further, the Oh\nandTdcations are approximately 180\u000eout of phase,\nas expected for the precession of antiferromagnetically\ncoupled moments.\nTo quantify the phase lag among the cations precisely,\nthe \feld dependence of the precessional phase \u001efor each\ncation is modeled with\n\u001e=\u001e0+ arctan\u0012\u0001Hhwhm\nH\u0000Hres\u0013\n; (1)where\u001e0is the baseline of the precessional phase (set\nto 0 for Fe3+\nOh) and \u0001Hhwhm is the half-width-at-half-\nmaximum FMR linewidth. Equation 1 is equivalent\nto the expressions in Refs. 17 and 21 and valid when\nthe e\u000bective magnetization (including the out-of-plane\nmagnetic anisotropy) \u00160Me\u000b\u00191 T28is much larger than\nthe applied bias \feld \u00160H\u001920 mT. We quantify Hres\nandHhwhm by simultaneously \ftting the \feld dependence\nof the precessional phase \u001e[Eq. 1] and of the precessional\namplitudeA,\nA/s\n\u0001Hhwhm2\n\u0001Hhwhm2+ (H\u0000Hres)2: (2)\nTo account for reduced sensitivity far from the resonance,\nthe \fts in Fig. 4(a,c) are weighted using the error\nbars from the sinusoidal \fts of the phase delay scans\n(e.g., Fig. 3(b)). The results of the \ftting are shown\nin Fig. 4(a,c) and Table I.\nIf the magnetic moments of the four cations were\nperfectly collinear, the phase lag should be \u001e0= 0 for\ntheOhcation species whereas \u001e0= 180\u000efor theTd\ncation species. Taking Fe3+\nOhas the reference, the results\nin Table I show that \u001e0deviates by\u00191.5\u000efrom the\nperfect collinear scenario. However, we caution that the\nuncertainty of \u001e0in Table I is likely underestimated.\nIndeed, by examining the residuals of the \fts displayed\nin Fig. 4(b), we observe a scatter in the measured\nprecessional phase of at least \u00192\u000e. It is sensible\nto conclude that the Ohcations maintain a relative\nprecessional phase lag of (0 \u00062)\u000e, whereas the Ohand\nTdcations maintain a phase lag of (180 \u00062)\u000e. Even with\nthe diluted exchange coupling from nonmagnetic Zn2+\nand Al3+cations, the magnetic Fe2+=3+and Ni2+cations\nretain a coherent, collinear alignment.\nReducing the experimental uncertainty to well below\n2\u000ewould be extremely challenging. For each cation, a\nsmall drift in the beamline photon energy with respect\nto its XMCD peak (Fig. 2) might shift its apparent\nprecession phase, due to an overlap in the cation speci\fc\nXMCD features. For instance, considering that the\ndi\u000berence between the Fe3+\nTdand Fe3+\nOhpeaks is only\u00190.8\neV, an energy drift of \u00190.01 eV could cause a phase shift\nof\u00192\u000e. The nominal resolution of the electromagnet at\n\u00190.1 mT may also contribute to the scatter in the \feld\ndependence of XFMR phase. Moreover, the timing jitter\nof the master oscillator of up to \u00193 ps limits the time\nresolution of the phase delay scans. Taking all the above\nfactors into account, the resolution of \u00192\u000ein our present\nstudy is in fact at the practical limit.\nWe now provide insight into the spin relaxation of each\nmagnetic cation species by quantifying the cation-speci\fc\nFMR linewidth \u0001 Hhwhm . In particular, we examine\nwhether di\u000berent spin relaxation emerges for magnetic\ncations with di\u000berent strengths of spin-orbit coupling {\ne.g., Fe3+with nominally zero orbital angular momentum\nvs Fe2+with likely nonzero orbital angular momentum12.\nHowever, Fig. 4(c) and Table I show that all magnetic4\n17 18 19 20 21 22-270-180-90090\n      \n \n      phase (deg.)\nfield m0H (mT)Fe𝑂ℎ3+\nFe𝑂ℎ2+\nNi𝑂ℎ2+\nFe𝑇𝑑3+\n18.0 18.5 19.0 19.5 20.0-4-2024residual phase (deg.)\nfield m0H (mT)\n18.0 18.5 19.0 19.5 20.0\nfield m0H (mT)\n18.0 18.5 19.0 19.5 20.0\nfield m0H (mT)\nFe𝑇𝑑3++180O\n17 18 19 20 21 220.00.20.40.6      \n \n \n amplitude (mV)\nfield m0H (mT)Fe𝑂ℎ3+\nFe𝑂ℎ2+\nNi𝑂ℎ2+\nFe𝑇𝑑3+\nFe𝑂ℎ3+Fe𝑂ℎ2+Ni𝑂ℎ2+(a) (c)\n(b)\n18.0 18.5 19.0 19.5 20.0\nfield m0H (mT)\nFIG. 4. (a) Field dependence of the precessional phase for each magnetic cation species. The solid curve indicates the \ft\nresult with Eq. 1. (b) Residuals of the precessional phase (i.e., di\u000berence between the experimentally measured data and the\narctan(\u0001H=(H\u0000Hres)) part of the \ft curve) in the vicinity of the resonance \feld. The dashed horizontal line indicates the\nphase lag\u001e0relative to the precessional phase of Fe3+\nOh. The shaded area indicates the standard deviation of the data points\nshown in each panel. (c) Field dependence of the precessional amplitude. The solid curve indicates the \ft result with Eq. 2.\ncations in NiZnAl-ferrite exhibit essentially the same\nlinewidth, \u00160\u0001Hhwhm = (0:43\u00060:03) mT, consistent\nwith the value obtained from conventional FMR for\nNiZnAl-ferrite28. Our \fnding thus indicates that the\nexchange interaction in NiZnAl-ferrite leads to uniform\nspin relaxation across all magnetic cations.\nTo further characterize cation-speci\fc spin relaxation,\nwe quantify the precessional cone angle \u0012cone of each\nmagnetic cation species. Speci\fcally, \u0012coneis obtained\nfrom the amplitudes of the XFMR signal IXFMR and\nXMCD peak IXMCD via\n\u0012cone= 2 arcsin\u0012IXFMR\nIXMCD\u0013\n: (3)\nWe \fnd that all cation moments precess with a cone\nangle of\u0012\u00191:0\u00001:1\u000e. Our results con\frm the\nexchange interaction in NiZnAl-ferrite is strong enough\nto lock all magnetic cations at the same relaxation rate,\nas evidenced by the invariance of the linewidth and\nprecessional cone angle to within .10%.\nOur \fnding is distinct from recent work on a\nferrimagnetic DyCo alloy, showing di\u000berent damping\nparameters for two magnetic sublattices afterfemtosecond-laser-induced demagnetization31. In\nRef. 31, the laser pulse produces a highly nonequilibrium\ndistribution of spins, which in turn quenches the\nexchange interactions, and the two ferrimagnetic\nsublattices are free to relax quasi-independently of\neach other. In this case, the rare-earth Dy sublattice\nwith stronger spin-orbit coupling exhibits a higher\ndamping parameter than the transition-metal Co\nsublattice. By contrast, the magnetic moments in\nour experiment are forced to oscillate by continuous\nmicrowave excitation, yet remain at near-equilibrium\nacross the entire resonance curve. The near-equilibrium\nforced oscillations { in concert with the exchange\ninteractions { favor the rigid, coherent coupling among\nthe magnetic cations and sublattices.\nIn summary, we have investigated time-resolved,\ncation-speci\fc resonant magnetic prcession at room\ntemperature in an epitaxial thin \flm of NiZnAl-ferrite,\na spinel-structure insulating ferrimagnetic oxide. The\nlow damping of this ferrite \flm yields a large XFMR\nsignal-to-noise ratio, allowing us to resolve precessional\ndynamics with high precision. In particular, we have\nobtained two key \fndings. First, the magnetic cations\nretain a coherent, collinear con\fguration, to within an5\nTABLE I. Resonance \feld Hres, linewidth \u0001 Hhwhm , relative\nprecessional phase lag \u001e0, and precessional cone angle \u0012conefor\neach magnetic cation species, as derived from \ftting the \feld\ndependence of the precessional phase [Eq. 1] and amplitude\n[Eq. 2].\nCation \u00160Hres\u00160\u0001Hhwhm \u001e0\u0012cone\n(mT) (mT) (degree) (degree)\nFe3+\nOh19:21\u00060:03 0:43\u00060:03 0 1 :0\u00060:1\nFe2+\nOh19:22\u00060:03 0:44\u00060:03 1:7\u00061:2 1:0\u00060:1\nNi2+\nOh19:23\u00060:02 0:43\u00060:02 1:5\u00061:2 1:1\u00060:1\nFe3+\nTd19:22\u00060:03 0:43\u00060:03\u0000178:5\u00061:4 1:1\u00060:1\nuncertainty in the precessional phase of \u00192\u000e. Second, the\nstrongly coupled magnetic cations experience the same\nmagnitude of spin relaxation, to within an uncertainty\nof.10%. Thus, the oft-assumed \\ferromagnet-like\"\ndynamics remain robust in the ferrimagnetic oxide,\neven with high contents of nonmagnetic Zn2+and\nAl3+cations that reduce the exchange sti\u000bness. We\nemphasize that our conclusion is speci\fc to the resonant\ndynamics under a continuous-wave excitation. Future\ntime-resolved XMCD measurements may resolve cation-\nspeci\fc dynamics in ferrimagnetic oxides driven by\nsub-nanosecond pulses , e.g., of electric-current-induced\ntorques, with potential implications for ultrafast device\ntechnologies.\nACKNOWLEDGEMENTS\nC.K. acknowledges \fnancial support by the Alexander\nvon Humboldt foundation. S.E. and Y.S. were funded by\nthe Vannevar Bush Faculty Fellowship of the Department\nof Defense under Contract No. N00014-15-1-0045. Work\nby S.E. was also supported in part by the Air Force\nO\u000ece of Scienti\fc Research under Grant No. FA9550-\n21-1-0365. Y.S. was also funded by the Air Force O\u000ece\nof Scienti\fc Research under Grant No. FA9550-20-1-\n0293. D.A.A. acknowledges the support of the National\nScience Foundation under Grant No. ECCS-1952957\nand also the USF Nexus Initiative and the Swedish\nFulbright Commission. The Advanced Light Source is\nsupported by the Director, O\u000ece of Science, O\u000ece of\nBasic Energy Sciences, of the U.S. Department of Energy\nunder Contract No. DE-AC02-05CH11231.\nDATA AVAILABILITY\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.1A. Brataas, B. van Wees, O. Klein, G. de Loubens, and M. Viret,\nPhys. Rep. 885, 1 (2020).\n2A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands,\nNat. Phys. 11, 453 (2015).\n3T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat.\nNanotechnol. 11, 231 (2016).\n4O. Gomonay, T. Jungwirth, and J. Sinova, Phys. status solidi\n^ a€\\ Rapid Res. Lett. 11, 1700022 (2017).\n5V. 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Lett. 15, 2100047 (2021)."    },    {        "title": "1005.5297v2.Haldane_Phases_and_Ferrimagnetic_Phases_with_Spontaneous_Translational_Symmetry_Breakdown_in_Distorted_Mixed_Diamond_Chains_with_Spins_1_and_1_2.pdf",        "content": "arXiv:1005.5297v2  [cond-mat.str-el]  10 Nov 2010Typeset with jpsj3.cls <ver.1.0> Full Paper\nHaldane Phases and Ferrimagnetic Phases with Spontaneous T ranslational\nSymmetry Breakdown in Distorted Mixed Diamond Chains with S pins1and1/2\nKazuoHida∗, Ken’ichi Takano1, and Hidenori Suzuki1†\nDivision of Material Science, Graduate School of Science an d Engineering,\nSaitama University, Saitama, Saitama, 338-8570\n1Toyota Technological Institute, Tenpaku-ku, Nagoya 468-8 51\n(Received February 28, 2018)\nThe ground states of two types of distorted mixed diamond cha ins with spins 1 and 1 /2\nare investigated using exact diagonalization, DMRG, and ma pping onto low-energy effective\nmodels. In the undistorted case, the ground state consists o f an array of independent spin-\n1 clusters separated by singlet dimers. The lattice distort ion induces an effective interaction\nbetween cluster spins. When this effective interaction is an tiferromagnetic, several Haldane\nphases appear with or without spontaneous translational sy mmetry breakdown (STSB). The\ntransition between the Haldane phase without STSB and that w ith (n+1)-fold STSB ( n= 1,\n2, and 3) belongs to the same universality class as the ( n+1)-clock model. In contrast, when\nthe effective interaction is ferromagnetic, the quantized a nd partial ferrimagnetic phases appear\nwith or without STSB. An effective low-energy theory for the p artial ferrimagnetic phase is\npresented.\nKEYWORDS: mixed diamondchain, distortion, frustration, H aldane phase, spontaneoustranslational sym-\nmetry breakdown, partial ferrimagnetism\n1. Introduction\nQuantum magnetism in frustrated spin systems is a\nrapidly developing field of condensed matter physics.1,2)\nAt first glance, one would expect that geometrical frus-\ntration enhances quantum fluctuation and drives an or-\ndered state into a disordered state. However, recent\nprogress in this field of physics has shown that this sim-\nple intuition is not always valid and that geometrical\nfrustration induces a variety of exotic quantum phenom-\nena,whicharenoteasilypredicted.Underanappropriate\ncondition, it even stabilizes an unexpected magnetic long\nrangeordersuch asthe frustration-inducedferrimagnetic\nand spin nematic orders.\nTo understand magnetism under the interplay of geo-\nmetrical frustration and quantum fluctuation, it is desir-\nable to begin with typical spin models with exact solu-\ntions. Among them, there exist a class of models whose\nground states are exactly written down as spin cluster\nsolid (SCS) states because of frustration. A SCS state is\natensorproduct stateofexactlocaleigenstatesofcluster\nspins. Well-known examples are the Majumdar-Ghosh\nmodel3)whose ground state is a prototype of sponta-\nneously dimerized phases in one-dimensional frustrated\nmagnets4)and the Shastry-Sutherland model5)which\ncorresponds to the material SrCu 2(BO3)2.6,7)In these\nmodels, the spin clusters are singlet dimers.\nThe diamond chain is another frustrated spin chain\nwith exact SCS ground states. The lattice structure is\nshown in Fig. 1. In a unit cell, there are two kinds of\nnonequivalent lattice sites occupied by spins with mag-\nnitudesSandτ; we denote the set of magnitudes by ( S,\nτ). One of the authors and coworkers8,9)introduced this\n∗E-mail address: hida@phy.saitama-u.ac.jp\n†Present address: Department of Physics, College of Humanit ies\nand Sciences, Nihon University, Setagaya-ku, Tokyo 156-85 50Sτ\nτ\nFig. 1. Structure ofthe diamond chain. Spin magnitudes ina u nit\ncell are indicated by Sandτ; we denote the set of magnitudes\nby (S,τ). The PDC is the case of S=τ, while the MDC is the\ncase ofS= 2τwith an integer or half-odd integer τ.\nlattice structure and generally investigated the case of\n(S,S), i.e., the pure diamond chain (PDC). Any PDC\nis shown to have at least one exact SCS ground-state\nphase where each spin cluster has spin 0. Particularly,\nin the case of (1/2, 1/2), they determined the full phase\ndiagram of the ground state by combining rigorous ar-\nguments with numerical calculations. After that, Nigge-\nmann et al.10,11)arguedabout a seriesofdiamond chains\nwith (S, 1/2). As for the special case of (1/2, 1/2), they\nreproduced the results of ref. 9.\nThe mixed diamond chain (MDC) is defined as a dia-\nmond chain with ( S,S/2) for the integer S.12)The spe-\ncial case of (1, 1/2) was first investigated by Niggemann\net al.10,11)They considered it asone of the seriesofmod-\nels with ( S,1/2).Recently, extensiveinvestigationonthe\nMDC has been carried out by the present authors.12–14)\nThe MDC is of special interest among diamond chains,\nbecause only the MDC has the Haldane phase in the ab-\nsence offrustration,so that wecan observethe transition\nfrom the Haldane phase to a SCS phase induced by frus-\ntration. In contrast, diamond chains of other types have\nferrimagnetic ground states for weak frustration.\nThe features common to all types of diamond chains\nare their infinite number of local conservation laws and\n12 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\nmore than two different types ofexact SCS groundstates\nthat are realized depending on the strength of frustra-\ntion. For example, S= 1/2 PDC has a nonmagnetic\nphaseaccompaniedbyspontaneoustranslationalsymme-\ntry breakdown (STSB) and a paramagnetic phase with-\nout STSB. This model also has a ferrimagnetic ground\nstate in the less frustrated region.9)On the other hand,\nthe MDC with spins 1 and 1 /2 has 3 different param-\nagnetic phases accompanied by STSB and one paramag-\nnetic phase without STSB. This model also has a non-\nmagnetic Haldane ground state in a less frustrated re-\ngion.10,12)The SCS structures of the ground states are\nalso reflected in characteristic thermal properties, as re-\nported in ref. 13.\nModifications of the PDC and MDC have been exam-\nined by many authors. Among them, the spin 1/2 PDC\nwith distortion has been thoroughly investigated by nu-\nmerical methods.15–17)It is found that azurite, a nat-\nural mineral, consists of distorted PDCs with spin 1/2\nand that the magnetic properties of this material have\nbeen experimentally studied in detail.18,19)Other mate-\nrials have also been reported.20,21)The diamond chain\nis one of the simplest models compatible with the 4-spin\ncyclic interaction. The effects of this type of interaction\non PDC have recently been investigated by Ivanov et\nal.22)The present authors also investigated the MDC\nwith bond-alternating distortion and found an infinite\nseries of ground states with STSB.14)In addition, as re-\nviewed in ref. 14, the MDC is related to other impor-\ntant models of frustrated magnetism such as the dimer-\nplaquette model,23–28)frustrated Heisenberg ladders,29)\nhybrid diamond chains consisting of Heisenberg bonds\nand Ising bonds,30,31)and an Ising model on a hierarchi-\ncaldiamondlattice.32)Amongthem,thedimer-plaquette\nchain with ferromagnetic interplaquette interaction re-\nduces to the MDC in the limit of strong interplaquette\ninteraction.28)\nThus far, in spite of the theoretical relevance of the\nMDC, no materials described by the MDC have been\nfound. Nevertheless, synthesizing MDC materials is not\nan unrealistic expectation in view of the success of the\nsynthesis of many low dimensional bimetallic magnetic\ncompounds33)and organic magnetic compounds.34)In\ngeneral, it is natural to expect that the lattice is pos-\nsibly distorted in real MDC compounds as in azurite.\nFrom this viewpoint, it is important to present theoret-\nical predictions on the ground state of distorted MDCs\nto widen the range of candidate materials of MDC and\nto raise the possibility of their synthesis.\nWe begin by classifying the distortion patterns by\nthe normal modes of each diamond unit. Excluding two\ntranslations and one rigid body rotation, we have 5 nor-\nmal modes as depicted in Fig. 2 within the diamond\nplane. A distorted MDC may be realized as a result of\nthe collective softening of these normal modes. In partic-\nular, the distortion patterns in (a) and (b) break the lo-\ncal conservation laws that hold in the undistorted MDC.\nHence, these distortions induce effective interactions be-\ntween the cluster spins in the whole lattice, and may\nform novel exotic phases. We investigate these interest-\ning cases in the present paper. In what follows, we name(a) (b) (c)\n(d) (e)\nFig. 2. Displacement modes of a diamond unit.\nSl1+δA\nλ\n1+δA1−δA\n1−δAτl(1)\nSl+1\nτl(2)(a)\nSl1+δB\nλ\n1−δB1+δB\n1−δBτl(1)\nSl+1\nτl(2)(b)\nFig. 3. Structures of MDC with S= 1 and τ(1)=τ(2)= 1/2\nwith (a) type A and (b) type B distortions.\nthe distortion patterns in (a) and (b) as type A and type\nB, respectively. The MDCs with type A and type B dis-\ntortions are depicted in Figs. 3(a) and 3(b), respectively.\nThe distortion patterns in Figs. 2(d) and 2(e) do not\nchange the geometry of the original undistorted MDC.\nThe distortion pattern in Fig. 2(c) is of another interest,\nsince it induces the bond alternation in the undistorted\nMDC without breaking the local conservation laws. This\ncase has been investigated separately and published in a\nprevious paper.14)\nThis paper is organized as follows. In §2, the Hamil-\ntonians for the MDCs with the type A and type B dis-\ntortions are presented, and the structure of the ground\nstatesoftheMDCwithoutdistortionissummarized.The\nground-statephasesfor the MDC with the type Adistor-\ntion are discussed in §3, and those for the MDC with the\ntype B distortion are discussed in §4. The last section is\ndevoted to summary and discussion.\n2. Hamiltonian\nThe MDCs with the type A and type B distortions are\ndescribed, respectively, by the following Hamiltonians:\nHA=N/summationdisplay\nl=1/bracketleftBig\n(1+δA)Slτ(1)\nl+(1−δA)τ(1)\nlSl+1\n+(1−δA)Slτ(2)\nl+(1+δA)τ(2)\nlSl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,\n(1)J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki 3\nHB=N/summationdisplay\nl=1/bracketleftBig\n(1+δB)Slτ(1)\nl+(1+δB)τ(1)\nlSl+1\n+(1−δB)Slτ(2)\nl+(1−δB)τ(2)\nlSl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,\n(2)\nwhereSlis the spin-1 operator, and τ(1)\nlandτ(2)\nlare\nthe spin-1/2 operators in the lth unit cell. The param-\neterδA(δB) represents the strength of type A (type B)\ndistortion, and is taken to be nonnegative without spoil-\ning generality. The number of unit cells is denoted by\nN, and then the total number of sites is 3 N. We will\nconsider these systems in the large Nlimit.\nForδA= 0 and δB= 0, both eqs. (1) and (2) reduce\nto the undistorted MDC Hamiltonian,\nH0=N/summationdisplay\nl=1/bracketleftbigg\nSlTl+TlSl+1+λ\n2/parenleftbigg\nT2\nl−3\n2/parenrightbigg/bracketrightbigg\n(3)\nwith the composite spin operators Tl≡τ(1)\nl+τ(2)\nl.\nBefore going into the analysis of the distorted MDC,\nwe briefly summarize the ground-state properties of the\nHamiltonian (3) reported in ref. 12 for convenience.\n(i)T2\nlcommutes with the Hamiltonian H0for anyl.\nTherefore, the composite spin magnitude Tldefined\nbyT2\nl=Tl(Tl+1) is a good quantum number that\ntakes the values 0 or 1. Hence, each energy eigen-\nstate has a definite set of {Tl}, i.e. a sequence of\n0’s and 1’s with length N. A pair of τ(1)\nlandτ(2)\nl\nwithTl= 0 is called a dimer. A cluster including\nnsuccessive Tl= 1 pairs bounded by two Tl= 0\npairs is called a cluster- n. The cluster- nis equiva-\nlent to an antiferromagneticspin-1 Heisenberg chain\noflength2 n+1withopen boundarycondition.Since\na cluster- nis decomposed into a sublattice consist-\ning ofn+1 sites with Sl’s and that consisting of n\nsites with Tl’s, the ground states of a cluster- nare\nspin triplet states with total spin unity on the basis\nof the Lieb-Mattis theorem.35,36)This implies that\neach cluster- ncarries a spin-1 in its ground state.\n(ii) There appear 5 distinct ground-state phases called\ndimer-cluster- n(DCn) phases with n= 0,1,2,3,\nand∞. The DC nstate is an alternating array\nof dimers and cluster- n’s. The phase boundary\nλc(n,n′) between DC nand DCn′phases are\nλc(0,1) = 3,\nλc(1,2)≃2.660425045542 ,\nλc(2,3)≃2.58274585704 ,\nλc(3,∞)≃2.5773403291 , (4)\nwhereλc(0,1)is obtainedanalyticallyandotherval-\nues are calculated numerically.\n(iii) In the DC ∞ground state realized for λ < λc(3,∞),\nTl= 1 for all l. This state is not accompanied by\nSTSB and is equivalent to the Haldane state of an\nantiferromagnetic spin-1 Heisenberg chain with infi-\nnite length.\n(iv) Each of the DC nstates with 0 ≤n≤3 realized forλ > λ c(3,∞) is a uniform array of cluster- n’s with\na common value of nand dimers in between. In the\nDCnphase with 1 ≤n≤3, (n+1)-fold STSB takes\nplace. In the DC0 phase, no translational symmetry\nis broken.\nIn what follows, we numerically examine various aspects\nof the type A and type B distortion effects on the MDC.\nBecause the DC3 phase is only realized within a very\nnarrow interval of λ, it is difficult to analyze the effect\nof distortion numerically in this phase. Hence, we do not\nconsider the DC3 phase in the following numerical anal-\nysis.\n3. Ground-State Properties of the MDC with\nType A Distortion\n3.1 Weak distortion regime\nWe now inspect the nature of the effective interaction\nbetween two cluster- n’s separated by a dimer consisting\nofτ(1)\nlandτ(2)\nlin the presence of the weak type A dis-\ntortion. For δA>0,Sl(Sl+1) tends to be antiparallel to\nτ(1)\nl(τ(2)\nl) rather than to τ(2)\nl(τ(1)\nl), as is known from\nFig. 3(a). The spins τ(1)\nlandτ(2)\nlare antiparallel to each\nother because they form a singlet dimer. Therefore, Sl\nandSl+1tend to be antiparallel to each other. In each\ncluster-n, the number of spins Sl’s is larger than the\nnumber of composite spins Tl’s by one. Hence, from the\nLieb-Mattistheorem,35)thetotalspinofthegroundstate\nof the cluster- npoints to the same direction as the Sl’s\nbelonging to that cluster- n. Therefore, the total spins of\nthe cluster- n’s on both sides of the dimer also tend to be\nantiparallel to each other. Thus, the effective coupling\nbetween the spins of neighboring cluster- n’s is antifer-\nromagnetic. This physical argument will be numerically\nensured below.\nIn general, the interaction between two spins with a\nmagnitude of 1 is the sum of bilinear and biquadratic\nterms. Therefore, the effective Hamiltonian for cluster-\nn’s in the phase that continues to the DC nphase in the\nlimit ofδA→0 is written as\nHeff=Nc/summationdisplay\ni=1Heff(i,i+1), (5)\nHeff(i,i+1) =JeffˆSiˆSi+1+Keff/parenleftBig\nˆSiˆSi+1/parenrightBig2\n,(6)\nwhereˆSiis the total spin of the i-th cluster- nwith a\nmagnitude of 1, Ncis the total number of cluster- n’s,\nandJeffandKeffare effective coupling constants. From\nsymmetry consideration, the signs of δAdoes not affect\nthe sign of the effective coupling constants. Hence, these\ncoupling constants are of the order of δ2\nAfor small δA.\nWe numerically calculated the ground-state energy of a\npair of cluster- n’s with total spin Stot, and compared\nit with the corresponding eigenvalues of Heff(i,i+ 1).\nThen we confirmed that Jeff/δ2\nAandKeff/δ2\nAare almost\nindependent of δAtypically for δA<0.002. The constant\nvaluesof Jeff/δ2\nAandKeff/δ2\nAareshowninFig.4forthree\nphases (n=0, 1, and 2), which will be explained below.\nBecause the effective coupling constants satisfy 0 <\nKeff/Jeff<1, the ground state is the Haldane state for4 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\n3 4 50510\n0.51\nλ λc(0,1)Keff/Jeff\nKeff/δA2Jeff/δA2Keff/JeffHDC0 Jeff/δA2\nKeff/δA2\n2.7 2.8 2.9 302040\n00.51\nλλc(0,1) λc(1,2)Keff/Jeff\nKeff/δA2Jeff/δA2Keff/Jeff HDC1Jeff/δA2\nKeff/δA2\n2.6 2.62 2.64 2.66050100\n00.51\nλc(1,2) λc(2,3)Keff/Jeff\nKeff/δA2Jeff/δA2Keff/Jeff HDC2 Jeff/δA2\nKeff/δA2\nλ\nFig. 4. Effective bilinear interaction ( Jeff) and biquadratic inter-\naction (Keff) between spin clusters for small δAin HDC0, HDC1,\nand HDC2 phases from top to bottom. The ratio Keff/Jeffis also\nshown.\nsmallδA.37)In the Haldane state, each spin-1 degree of\nfreedom is carried by a cluster- nrather than by a sin-\ngle spin. We call the state the Haldane DC n(HDCn)\nstate. In the HDC nstate with n≥1, the (n+ 1)-fold\ntranslational symmetry is spontaneously broken unlike\nthe conventional Haldane state without STSB. Both the\nHDC0 state for λ > λ c(0,1) and the HDC ∞state for\nλ < λ c(3,∞) are the Haldane states without STSB. In\nparticular, the HDC ∞state continues from the Haldane\nstate (DC ∞state) of the undistorted MDC mentioned\nin§2.12)Uniform Haldane\nλ>λc(1,0)\nHDC1\nHDC2\nHDC3Uniform Haldane\nλ>λc(1,0)\nUniform Haldane\nλ<λc(3,∞) or δA » 1(a)\n(b)\n(c)\n(d)\n(e)\nFig. 5. Valence bond structures of the ground states of all ph ases\nfor the MDC with type A distortion. A small filled circle repre -\nsents a spin with a magnitude of 1/2. An original spin with a\nmagnitude of 1 is represented by two decomposed 1/2 spins in\nan open circle indicating the symmetrization. A valence bon d is\nrepresented by a dashed oval.\n3.2 Connection to the strong distortion regime\nIn the strong distortion regime of δA→1 and small\nλ, the three spins τ(2)\nl−1,Sl, andτ(1)\nlform a singlet clus-\nter. Hence, the ground state is a state with spin gap and\nwithout STSB. This nature is common to the HDC0 and\nHDC∞phases in §3.1. Furthermore, the HDC ∞state\nis transformed into the HDC0 state only by rearranging\ntwo valence bonds within each diamond unit, as shown\nin Figs. 5(a) and 5(e). Therefore, the strong distortion,\nHDC0, and HDC ∞regimes are considered to be differ-\nent parts of a single phase. The continuity of the three\nregimes will be confirmed by the numerical analysis dis-\ncussed in §3.3. In what follows, we call this phase the\nuniform Haldane (UH) phase as a whole.\n3.3 Numerical phase diagram\nUnder the periodic boundary condition, even in the\nparameter region where STSB takes place, the ground\nstate of a finite chain is a superposition of the symmetry-\nbroken states, and the translational symmetry is re-\ncovered. Under the open boundary condition, however,\none of the symmetry broken states is selected by the\nboundary effect. Therefore, we employ the DMRG cal-\nculation with the open boundary condition to determine\nthe phase diagram for finite δA. The DMRG calculation\nis carried out using the finite-size algorithm up to 288\nsites keeping 200 states in subsystems. We calculate the\nground-state expectation values/angbracketleftbig\nT2\nl/angbracketrightbig\nand define the ef-\nfective spin magnitude Tlon thel-th diagonal bond by\nTl(Tl+1) =/angbracketleftbig\nT2\nl/angbracketrightbig\n. A typical ldependence of Tlis shown\ninFig.6ineachphase.Withtheincreasein δA,thetrans-\nlational symmetry is recoveredas expected. For finite δA,\nthe ground-state phase is identified from the periodicity\nin the oscillation of Tl. In the HDC nphase, the values ofJ. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki 5\n0 5001\nlTl\nδA=0.04\nδA=0.02λ=2.803N=288(a)\n0 5001\nlTl\nδA=0.009\nδA=0.007λ=2.623N=282(b)\nFig. 6. Profiles of Tlfor (a)λ= 2.8 and (b) λ= 2.62.\n0 0.2 0.4 0.600.20.40.6∆T\nN−1/8δA=0.0304\nδA=0.0306\nδA=0.0308(a)\n0 0.2 0.4 0.600.10.2∆T\nN−2/15δA=0.00824δA=0.00822\nδA=0.00826(b)\nFig. 7. System size dependences of ∆ Tat (a)λ= 2.85 and (b)\nλ= 2.62. The data are plotted against N−β/νwhereβandν\nare the critical exponents of the order parameter and correl ation\nlength, respectively, for the 2-dimensional (a) Ising and ( b) 3-\nclock model.\nTlfollow the sequence\n...TSTL···TL/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nnTS,TL···TL/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn..,(TL> TS).(7)\nThus, we define the order parameter of the HDC nphase\nby ∆T=TL−TS. In DMRG, ∆ Tis measured at the\nsites closest to the center of the chain.\nThe valence bond structures for the HDC nphases as2.6 2.8 300.020.04\nδA\nλλc(2,3)λc(3,∞)Z2STSB\n  Z3STSBNo STSB\nλc(1,2) λc(0,1)HDC1\nHDC2Uniform Haldane\nFig. 8. Phase diagram of the MDC with type A distortion. The\ntriangles indicate the position of the phase boundary for δA= 0.\nwell as the UH phase are shown in Fig. 5. We see the\ntranslational invariance of period n+ 1 in the HDC n\nground state in contrast to the period-1 invariance in\nthe UH ground state. Hence, the Zn+1STSB takes place\nat the HDC n-UH phase boundary. We expect that this\ntransition belongs to the 2-dimensional ( n+ 1)-clock\nmodel universality class. The system size dependence\nof ∆Tforλ= 2.85 is shown in Fig. 7(a) around the\nHDC1-UH phase boundary. Here, the data are plotted\nagainstN−β/νwith the order parameter critical expo-\nnentβ= 1/8 and the correlation length critical expo-\nnentν= 1 for the two-dimensional Ising universality\nclass.This showsthatthecriticalvalueof δAliesbetween\n0.0304 and 0.0308. A similar plot is shown in Fig. 7(b)\nforλ= 2.62 around the HDC2-UH phase boundary, as-\nsuming the critical exponents of two-dimensional 3-clock\nmodel(equivalently3-statePottsmodel38))withβ= 1/9\nandν= 5/6. This shows that the critical value of δA\nlies between 0.00822 and 0.00826. The critical points at\nother values of λare determined similarly. The results\nare shown in the phase diagram of Fig. 8. The error bars\nare within the size of the symbols.\nTo confirm the consistency of the universality class,\nthe finite size scaling plot for the order parameter ∆ Tis\ncarried out. According to the scaling hypothesis, the δA\ndependence of the order parameter ∆ Tof the finite size\nsystems near the critical point should obey the finite size\nscaling law39)\n∆TNβ/ν=f(N(δA−δc\nA)ν), (8)\nintermsofthescaledvariables∆ TNβ/νandN(δA−δc\nA)ν\nand the scaling function f(x). In Figs. 9(a) and 9(b),\n∆TNβ/νis plotted against N(δA−δc\nA)νaround the\nHDC1-UH and HDC2-UH phase boundaries assuming\nthe Ising and 3-clock universality classes, respectively.\nThe critical points δc\nA= 0.0307 (Fig. 9(a)) and 0.008248\n(Fig. 9(b)) are chosen so that all data fall on a single\nuniversal scaling curve as well as possible. These plots\nare consistent with the expected universality class.\nThe critical behavior at the HDC1-UH transition in6 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\n0 0.101∆TN1/8\n(δA−δAc)N3N=144\n216\n288(a)\n−0.2 0 0.201∆TN2/15\n(δA−δAc)5/6N3N=138\n210\n282(b)\nFig. 9. Finite-size scaling plot of ∆ Taround the critical points.\n(a) Plot around the HDC1-UH phase boundary at λ= 2.85.\nThe Ising critical exponents ν= 1 and β= 1/8 are assumed.\nThe critical point is set at δc\nA= 0.0307. (b) Plot around the\nHDC2-UHphase boundary at λ= 2.62. The3-state Potts critical\nexponents ν= 5/6 andβ= 1/938)are assumed. The critical\npoint is set at δc\nA= 0.008248.\n(a)\n(b)\nFig. 10. Valence bond structures of the ground states of spin -1\nbilinear-biquadraticchaininthe (a)Haldanephase and (b) dimer\nphase. The spins with a magnitude of unity represented by ope n\ncircles are decomposed into two spin-1/2 degrees of freedom rep-\nresented by smallfilled circles.The valence bonds are repre sented\nby dashed ovals. The spins belonging to disconnected cluste rs in\nthe dimer phase are connected by the valence bonds in the Hal-\ndane phase.\nour model should be compared with that of the S=\n1 bilinear-biquadratic chain at the Takhtajan-Babujian\npoint.40,41)Both transitions are accompanied by Z2-\nSTSB which contributes to the conformal charge by 1/2.\nFortheHDC1-UHtransitioninourmodel,therearrange-\nment of valence bonds take place only within each dia-\nmond unit, as shown in Fig. 5(a) and 5(b). In contrast,\nin theS= 1 bilinear-biquadratic chain, the spins be-\nlonging to disconnected clusters in the dimer phase are0 1 2 300.51\nλδB 3N=18\nUHFDC0\nm=1\nFDC1\nm=1/2\nFDC2\nm=1/3\nFig. 11. Phase diagram of the MDC with type B distortion with\n3N= 18. The triangles indicate the position of the phase bound-\nariesλc(n,n+1) for δB= 0.12)\nconnected by the valence bonds in the Haldane phase, as\nshown in Fig. 10. Apart from Z2-STSB, this is similar\nto the Gaussian criticality of the Haldane-dimer transi-\ntion in the spin-1 alternatingbond Heisenbergchain that\ncontribute to the conformal charge by 1.42–44)Therefore,\ntheS= 1 bilinear-biquadratic chain at the Takhtajan-\nBabujian point is described by the conformal field theory\nwithc= 1/2+1 = 3 /2, while the HDC1-UH transition\nin our model is described by the c= 1/2 Ising conformal\nfield theory.\n4. Ground-State Properties of the MDC with\nType B Distortion\nIn the case of type B distortion, the effective inter-\naction between the spins of two cluster- n’s separated by\nthedimerconsistingof τ(1)\nlandτ(2)\nlisferromagnetic,be-\ncause both SlandSl+1tend to be antiparallel to τ(1)\nl.\nTherefore,we expect the ferrimagneticgroundstate with\nspontaneous magnetization quantized as m= 1/(n+1)\nper unit cell for small δBin the range λc(n,n+ 1)<\nλ < λc(n−1,n). We call this phase a ferrimagnetic DC n\nphase (FDC nphase). In contrast, the ground state for\nλ < λ c(3,∞) will remain in the Haldane phase, since a\nnonmagnetic gapped phase is generally robust against a\nweakdistortion.Forfinite δB,we determined the ground-\nstate phase diagramby the numerical diagonalizationfor\nthe system size 3 N= 18, as shown in Fig. 11. Among\nsystem sizes tractable by numerical diagonalization, only\nthis size of 3 N= 18 is compatible with all the ground-\nstate structures with n= 0,1, and 2. As expected, the\nFDCnphases with m= 1/(n+ 1) are found for these\nvalues of n.\nBy inspecting numerical data for the 3 N= 18 system,\nwe also find other narrow ferrimagnetic phases between\nthe FDC nand FDC( n+ 1) phases with n= 0, 1, and\n2, although they are too narrow to be shown in Fig. 11.\nIn order to investigate these phases in detail, we employJ. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki 7\n2.4 2.6 2.800.51\nλm\n3N=18(ED)\n3N=72(DMRG)δB=0.2(a)\n1.7 1.800.51\nλm\nλm\n3N=18(ED)\n3N=72(DMRG)δB=0.6(b)\nFig. 12. Spontaneous magnetization for (a) δB= 0.2 and (b)\nδB= 0.6. The triangles on the vertical axes indicate the values\nof the spontaneous magnetization m= 1/(n+ 1) in the FDC n\nphases.\nthe DMRG calculation for 3 N= 72 keeping 120 states in\neach subsystem. Typical examples of the λdependence\nof spontaneous magnetization are shown in Fig. 12(a)\nforδB= 0.2 and Fig. 12(b) for δB= 0.6. Between the\nFDCnand FDC( n+1) phases with n= 0,1,2, we find\nthe partial ferrimagnetic phase in which the spontaneous\nmagnetization varies continuously with λ. The ferrimag-\nnetic phase of this kind has been found in various frus-\ntrated one-dimensional quantum spin systems.45–51)In\ncontrast, between the nonmagnetic phase and the FDC3\nphase, we find no partial ferrimagnetic phase for small\nδB.\nThis can be understood as follows: At λ=λc(n,n+1),\nthe cluster- nand cluster-( n+1) can coexist. As stated\nabove, it is physically evident that the effective magnetic\ninteraction between the clusters is ferromagnetic. There-\nfore, wecan restrict the statesof eachcluster to the max-\nimally polarized ground state with ˆSz\ni= 1. Hence, the\ngroundstateofthe wholechainisdescribedbyspecifying\nthe arrangement of cluster- n’s and cluster-( n+1)’s. We\nmap the two possible values of the length of i-th cluster,\nni=nandni=n+ 1, to two possible values of the\nspin-1/2 pseudospin, σz\ni= 1/2 andσz\ni=−1/2, respec-\ntively. Then, the total magnetization Mis equal to the\nnumber of clusters Nc. The total number of unit cells,N, is related to the pseudospins σz\nias\nN=Nc/summationdisplay\ni=1/parenleftbigg\nn+1+1\n2−σz\ni/parenrightbigg\n=Nc/parenleftbigg\nn+3\n2/parenrightbigg\n−Nc/summationdisplay\ni=1σz\ni.\n(9)\nTherefore, the ground-state magnetization per unit cell\nmis given by\nm=Nc\n/an}bracketle{tN/an}bracketri}ht=1\nn+3\n2−σ(10)\nwithσ≡/summationtextNc\ni=1/an}bracketle{tσz\ni/an}bracketri}ht/Nc.Thebracket /an}bracketle{t···/an}bracketri}htrepresentsthe\nground-state expectation value. In the presence of δB,\nthe length of neighboring clusters can exchange through\na second order process in δB. This corresponds to the\nspin exchange in terms of pseudospins. In this case, the\ninteraction between the pseudospins is approximated by\nthe spin-1 /2 XXZ Hamiltonian\nHXXZ=Nc/summationdisplay\ni=1HXXZ(i,i+1), (11)\nHXXZ(i,i+1) =Jeff\nzσz\niσz\ni+1+Jeff\n⊥(σx\niσx\ni+1+σy\niσy\ni+1)\n(12)\nuptothesecondorderin δB.Here,furtherneighborinter-\nactions are neglected. We estimate the effective exchange\nconstants by comparing the energy spectrum of the pair\nHamiltonian HXXZ(i,i+1) with that of the correspond-\ning pair of clusters as follows:\n(i)λ=λc(0,1)\nJeff\nz≃ −0.039δB2,\nJeff\n⊥≃0.087δB2. (13)\n(ii)λ=λc(1,2)\nJeff\nz≃ −0.0082δB2,\nJeff\n⊥≃0.069δB2. (14)\n(iii)λ=λc(2,3)\nJeff\nz≃ −0.0029δB2,\nJeff\n⊥≃0.018δB2. (15)\nThe details of the calculation are explained in Appendix.\nIn all cases (i) ∼(iii), we find that the effective cou-\npling constants satisfy the inequality −|Jeff\n⊥|< Jeff\nz≤\n|Jeff\n⊥|. As is well known, the ground state of the spin-\n1/2 XXZ chain in this parameter regime is nonmagnetic\nand gapless in the absence of a magnetic field. Roughly\nspeaking, ∆ λ≡λ−λc(n,n+1) corresponds to the effec-\ntive magnetic field heffconjugate to the total pseudospin/summationtext\niσz\ni, because the increase in λfavors cluster- nover\ncluster-(n+1); however, this correspondence should not\nbe taken literally. A more precise argument is also given\nin Appendix. When ∆ λtakes a large negative value,\nthe pseudospins are fully polarized downward to give\n/an}bracketle{tσz\nl/an}bracketri}ht=−1/2. This state corresponds to the FDC( n+1)\nstate with m= 1/(n+2). When heffreaches the critical\nvaluehc1≡ −(|Jeff\n⊥|+Jeff\nz), the magnetization starts to\nincrease continuously until all spins are fully polarized8 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\nupward at the critical effective field hc2≡ |Jeff\n⊥|+Jeff\nz.\nThis corresponds to the FDC nstate with m= 1/(n+1).\nOn the other hand, the magnetization jumps from 0 to\n1/4 at the phase boundary between the Haldane phase\nand the FDC3 phase for small δB. At this phase bound-\nary, no finite size clusters coexist with cluster-3. There-\nfore, no pseudospin degrees of freedom can be defined.\nConsequently, no partial ferrimagnetic phase can be re-\nalized. In contrast,for largervalues of δB, we numerically\nfind a partialferrimagnetic phasebetween the FDC3 and\nUHphases. This wouldbe ascribed tothe contributionof\nother finite-length clusters with low lying energies which\ncome into play through higher-order processes in δB.\n5. Summary and Discussion\nWeintroducedtwotypesofdistortion,typeAandtype\nB, into the MDC with spins 1 and 1/2, and investigated\nthe ground-state phases. The phase diagrams are charac-\nteristicofthetypeAandtypeBdistortions,respectively.\nFor the type A distortion, the effective interaction be-\ntween the cluster spins is antiferromagnetic with bilinear\nand biquadratic terms. The numerically estimated val-\nues of the effective couplings show that the DC nground\nstatesaretransformedintotheHDC ngroundstates.The\norder parameters characterizing the HDC nphases are\ndefined and the UH-HDC nphase boundaries are deter-\nmined using the DMRG data. From the valence bond\nstructure of each phase, we expect that the UH-HDC n\nphase transition belongs to the universality class of the\n2-dimensional ( n+1)-clock model. The finite size scaling\nplot of the order parameter is consistent with this iden-\ntification. For the type B distortion, the effective inter-\naction between the cluster spins is ferromagnetic. In ad-\ndition to the FDC nphases with quantized spontaneous\nmagnetization m= 1/(n+1), the partial ferrimagnetic\nphasesarealsofoundnumericallybetweentheFDC nand\nFDC(n+1) phases. A physical interpretation of the par-\ntial ferrimagnetic phase is given for small δBby mapping\nonto an effective pseudospin-1/2 XXZ chain.\nGenerally, the introduction of lattice distortion into\na physical model increases the possibility that a corre-\nsponding material is realized. In the MDC, there are\nthree types of distortion modes affecting the exchange\ninteractions. Among them, the two types investigated in\nthe present paper are of generic nature, because the lo-\ncal conservation laws that hold in the undistorted MDC\nare broken. This suggests that the observation of the ex-\notic phenomena predicted in the present paper is pos-\nsible even if the corresponding material is not exactly\ndescribed by the model Hamiltonians (1) and (2).\nIf a distorted MDC material is synthesized, the dis-\ntortion may be controlled by, e.g., applying pressure. If\nthe distortion is of type A, the Curie constant vanishes\nas the DC nground state turns into one of the HDC n\nground states. The magnetic susceptibility and magnetic\nspecific heat will have an activation-type temperature\ndependence with activation energy proportional to the\neffective coupling between the cluster spins, which is of\nthe order of δA2. These HDC nphases are not realized\nif the distortion δAexceeds∼0.03 even in the most ro-\nbust case of n= 1. In a real material, the STSB in thevalence bond structure manifests itself as a magnetic su-\nperstructure. It is also possible that it is accompanied\nby a lattice superstructure of corresponding periodicity\nif the spin-lattice coupling is present. Therefore careful\nmeasurements of magnetic and lattice superstructures\nwould help with the observation of HDC nphases with\n1≤n≤3.\nOn the other hand, if the distortion of the material\nis of type B, the ground sate is ferrimagnetic. At low\nbut finite temperatures, however, the spontaneous mag-\nnetization vanishes owing to the one-dimensionality. As\na precursor of ordering at T= 0, the low-temperature\nmagnetic susceptibility should diverge as T−2with a co-\nefficient proportional to the effective coupling ∼δB2be-\ntween the cluster spins.52–54)This means that even a\nweak magnetic field of the order of H∼T2/δB2derives\nthe finite-temperature magnetization up to the value of\nthe ground-state spontaneous magnetization. This en-\nables the experimental estimation of the spontaneous\nmagnetization in real materials. The quantized ferrimag-\nnetic behavior should be observed for wide ranges of the\nparameters λandδBas shown in Fig. 11, and should\nbe easily observed if an appropriate material is synthe-\nsized. The partial ferrimagnetic phases are limited to\nnarrow intervals of the parameters δBandλ. Therefore,\nthesecanonlybeobservedasatemperature-independent\ncrossoverbetween two quantized ferrimagnetic behaviors\nwith careful exclusion of the thermal effect.\nWe have demonstrated that various exotic ground\nstates and phase transitions between them are realized\nin the distorted MDC with spins 1 and 1/2, which has\na strong frustration. The physical pictures of these phe-\nnomenahavebecomeclear.Thisismadepossiblebecause\nthe ground state of the undistorted MDC is known ex-\nactly. Therefore, we expect that our model may provide\na means of understanding the similar exotic phenomena\nrealized owing to the interplay of spin ordering,quantum\nfluctuation, and strong frustration in more general frus-\ntrated quantum chains on a firm ground. For example,\npartial ferrimagnetic phases are found in various one-\ndimensional frustrated quantum spin models.45–51)How-\never, some of them are only numerically confirmed and\nno physical explanation has been given so far. We hope\nthat the present study paves the way to the general un-\nderstanding of these partial ferrimagnetic states.\nWe thank J. Richter for drawing our attention to ref.\n28andrelatedworks.Thenumericaldiagonalizationpro-\ngram is based on the package TITPACK ver.2 coded by\nH. Nishimori. The numerical computation in this work\nhas been carried out using the facilities of the Super-\ncomputer Center, Institute for Solid State Physics, Uni-\nversity of Tokyo and Supercomputing Division, Informa-\ntion Technology Center, University of Tokyo. KH is sup-\nported by a Grant-in-Aid for Scientific Research on Pri-\nority Areas, ”Novel States of Matter Induced by Frustra-\ntion” (20048003) from the Ministry of Education, Cul-\nture, Sports, Science and Technology of Japan and a\nGrant-in-Aidfor Scientific Research(C) (21540379)from\nthe Japan Society for the Promotion of Science. KT and\nHS are supported by a Fund for Project Research from\nToyota Technological Institute.J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki 9\nAppendix\nThe Hamiltonian HBwith the type B distortion is\nrewritten as\nHB=H0+δH, (A·1)\nwhere\nδH=δBN/summationdisplay\nl=1/parenleftbig\nSl+Sl+1/parenrightbig/parenleftbig\nτ(1)\nl−τ(2)\nl/parenrightbig\n.(A·2)\nFor small δB, the ground state around λ=λc(n,n+1)\nconsists almost entirely of cluster- n’s and cluster-( n+\n1)’s. Hence, as a good approximation, we consider HB\nonly in the restricted Hilbert space where each state\ninvolves no clusters except for cluster- n’s and cluster-\n(n+ 1)’s. Under the fixed cluster number Ncin this\nHilbert space, H0is equivalent to the following effective\nHamiltonian expressed in terms of pseudospin operators:\nH0\neff=E0\nG(n+1;λ)Nc/summationdisplay\ni=1/parenleftbigg1\n2−σz\ni/parenrightbigg\n+E0\nG(n;λ)Nc/summationdisplay\ni=1/parenleftbigg1\n2+σz\ni/parenrightbigg\n, (A·3)\nwhereσz\ni= 1/2 andσz\ni=−1/2 correspond to ni=n\nandni=n+ 1, respectively. E0\nG(n;λ) is the ground-\nstate energy of a cluster- nand a dimer in the absence of\ndistortion, and is given by\nE0\nG(n;λ) =˜E(2n+1)+λn\n4−3λ\n4,(A·4)\nwhere˜E(2n+1) is the ground-state energy of the spin-1\nantiferromagneticHeisenbergchainwith length2 n+1.12)\nTheapplicationof δHtotheunperturbed groundstate\ntransforms one of the Tl= 0 bonds to a Tl= 1 bond or\nviceversa.Thentheresultingstatescontainclusterswith\nlengths less than nor greater than 2 n. Since these states\nare outside the restricted Hilbert space, no correction to\nthe ground-state energy is present within the first order\ninδB. Hence, the lowest-order correction is of the order\nofδB2. Up to the second order in δB, the effective pseu-\ndospin Hamiltonian is given by\nHeff=EG(n+1;λ,δB)Nc/summationdisplay\ni=1/parenleftbigg1\n2−σz\ni/parenrightbigg\n+EG(n;λ,δB)Nc/summationdisplay\ni=1/parenleftbigg1\n2+σz\ni/parenrightbigg\n+HXXZ,(A·5)\nwhereEG(n;λ,δB) is the ground-state energy of a\ncluster-nand a dimer including the second order cor-\nrection in δB. This is simply expressed as\nHeff=Nc¯EG+∆EGNc/summationdisplay\ni=1σz\ni+HXXZ,(A·6)\nwith\n¯EG=1\n2(EG(n+1;λ,δB)+EG(n;λ,δB)),(A·7)\n∆EG=EG(n;λ,δB)−EG(n+1;λ,δB).(A·8)The effective coupling constants Jeff\nzandJeff\n⊥inHXXZ\nare also of the second order in δB. We determine Jeff\nzand\nJeff\n⊥soastoreproducethelow-lyingenergyspectrumofa\npairofcluster- n’sbythatoftwo-pseudospinHamiltonian\nHeff(i,i+1) = 2 ¯EG+∆EG(σz\ni+σz\ni+1)+HXXZ(i,i+1)\n(A·9)\nIn each of the subspaces σz\ni+σz\ni+1=±1,σz\ni=\nσz\ni+1=σ=±1/2. Therefore, the Hilbert space is one-\ndimensional and the eigenvalue of Heff(i,i+ 1) is sim-\nplyEσσ= 2¯EG+ 2∆EGσ+Jeff\nz/4 withσ=±1/2.\nIn the subspace σz\ni+σz\ni+1= 0, the Hilbert space is\ntwo-dimensional and the eigenvalues of Heff(i,i+1) are\nE±= 2¯EG−Jeff\nz/4±Jeff\n⊥/2.\nThe original Hamiltonian of the cluster consisting of a\ncluster-nand a cluster- n′is the distorted diamond chain\nwith length n+n′.\nH(n+n′) =n+n′+1/summationdisplay\nl=1/bracketleftBig\n(1+δB)Slτ(1)\nl+(1+δB)τ(1)\nlSl+1\n+(1−δB)Slτ(2)\nl+(1−δB)τ(2)\nlSl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,\n(A·10)\nWe denote the α-th eigenvalue of H(n+n′) asE(n+\nn′;α). Comparing the corresponding expression for the\neigenvalues, we find\nE(2n;0) =E1\n2,1\n2= 2¯EG+∆EG+Jeff\nz\n4(A·11)\nE(2n+2;0) = E−1\n2,−1\n2= 2¯EG−∆EG+Jeff\nz\n4(A·12)\nE(2n+1;0) = E−= 2¯EG−Jeff\nz\n4−Jeff\n⊥\n2(A·13)\nE(2n+1;1) = E+= 2¯EG−Jeff\nz\n4+Jeff\n⊥\n2.(A·14)\nSolving these sets of equations, with respect to Jeff\nzand\nJeff\n⊥, we find\nJeff\n⊥=E(2n+1;1)−E(2n+1;0), (A·15)\nJeff\nz= 2[E(2n+2;0)+ E(2n;0)\n−E(2n+1;1)−E(2n+1;0)].(A·16)\nNote that the rhs’s of (A ·15) and (A ·16) vanish for\nδB= 0. We numerically evaluated E(2n;0),E(2n+1;0),\nE(2n+1;1), and E(2n+2;0) at λ=λc(n,n+1) (n=\n0,1,2) for small δB. Using these values in eqs. (A ·15) and\n(A·16), we determined Jeff\n⊥andJeff\nzas eqs. (13)-(15).\nForthewholeMDC,theground-stateenergyiswritten\nas\nE0=Nc¯EG+Nc∆EGσ+NcǫXXZ(σ) (A ·17)\nwhereǫXXZ(σ) is the ground-state energy per site of a\nmagnetized spin-1/2 XXZ chain with /an}bracketle{tσz\ni/an}bracketri}ht=σ. The\nnumber of unit cells, N, of the original MDC is given by\nthe expectation value of eq. (9) as N=Nc(n+3\n2−σ).\nTherefore, we have\nE0=N\nn+3\n2−σ/parenleftbig¯EG+∆EGσ+ǫXXZ(σ)/parenrightbig\n.(A·18)10 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\n−4 −2 0 2−0.500.5\n∆λ/Jeff\n⊥σn=0n=1\nn=2\nFig. A·1. Relationship between σand ∆λforn= 0,1, and 2.\nMinimizing this with respect to σwith fixed N, we find\n∆λ=/parenleftbigg\nn+3\n2−σ/parenrightbigg∂ǫXXZ(σ)\n∂σ+ǫXXZ(σ),(A·19)\nwhere ∆ λ=λ−λc(n,n+ 1;δB) andλc(n,n+ 1;δB) is\ndefined by\n(n+2)EG(n;λc,δB)−(n+1)EG(n+1;λc,δB) = 0.\n(A·20)\nTo simplify the calculation, we replace ǫXXZ(σ) by the\nground-state energy of the spin-1/2 XY chain ǫXY=\n−(Jeff\n⊥/π)cosπσ, because |Jeff\n⊥|is substantially larger\nthan|Jeff\nz|in all cases. Then we find\n∆λ\nJeff\n⊥=/parenleftbigg\nn+3\n2−σ/parenrightbigg\nsinπσ−1\nπcosπσ. (A·21)\nThis relation is plotted in Fig. A ·1 forn= 0,1 and 2. 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Montenegro-Filho∗and M. D. Coutinho-Filho†\nLaborat´ orio de F´ ısica Te´ orica e Computacional, Departam ento de F´ ısica,\nUniversidade Federal de Pernambuco, 50670-901, Recife-PE , Brazil\nAbstract\nWe provide evidence of a superfluid-insulator transition (SIT) of ma gnons in a quasi-one-dimensional\nquantum ferrimagnetwith isotropic competing antiferromagneticspin interactions. This SIT occurs be tween\ntwo distinct ferrimagneticphasesdue to the frustration-induced closing ofthe gapto amagnon excitation. It\nthuscausesacoherentsuperpositionofsinglet andtripletstates at latticeunit cellsandapower-lawdecayon\nthe staggered spin correlation function along the transverse dire ction to the spontaneous magnetization. A\nhard-core boson map suggests that asymptotically close to the SI T the magnons attain the Tonks-Girardeau\nlimit. The quantized nature of the condensed singlets is observed be fore a first-order transition to a singlet\nmagnetic spiral phase accompanied by critical antiferromagnetic o rdering. In the limit of strong frustration,\nthe system undergoes a decoupling transition to an isolated gapped two-leg ladder and a critical single linear\nchain.\nPACS numbers: 75.10.Pq,75.10.Jm,75.40.Mg,75.30.Kz,75. 50.Gg\n∗Electronic address: rene@df.ufpe.br\n†Electronic address: mdcf@ufpe.br\n1I. INTRODUCTION\nRecently, several experimental and theoretical studies in dicate that, under very special condi-\ntions, magnons [1, 2, 3] and polaritons [4] undergo Bose-Ein stein condensation (BEC) in two- and\nthree-dimensional materials. In magnetic systems, BEC of m agnons can be driven by an applied\nmagnetic field ( h) (Ref. [1]), by varying the external pressure [2], or by micr owave pumping [3] In\n1D gapped antiferromagnets, e. g., spin-1 chains [5] and sin gle spin-1/2 two-leg ladders [6], the gap\nto the magnon excitation closes at a critical value ( hc) of the field and the magnetization increases\nas (h−hc)1/2. Although, stricto sensu , there is no BEC of magnons in these 1D systems, it is very\nappealing to describe the transition in terms of the condens ation of the uniform component of the\nmagnetization along the applied field [5]. In fact, rigorous results [7] on low dimensional ( D≤2)\nuniform interacting boson systems preclude the occurrence of BEC in finite temperature ( T). In\n2D systems phase fluctuations have mainly a thermal origin, s o that only the T= 0 condensate\nsurvives, with superfluid behavior persisting up to the Kost erlitz-Thouless temperature. In con-\ntrast, in 1D boson systems phase fluctuations have a quantum o rigin and there is no BEC, even at\nT= 0, but superfluidity is expected [7]. However, in finitesystems the scenario is more complex,\nsince in real confined systems [7, 8] one may be dealing with me tastable states.\nIn this work we introduce an isotropic Heisenberg spin Hamiltonian with two competing anti-\nferromagnetic (AF) exchange couplings [ J1(≡1) andJ] exhibiting a continuous quantum phase\ntransition at a critical value Jc1which, we argue, is a superfluid-insulator transition (SIT) of\nmagnons associated with the creation of a coherent superpos ition of singlet and triplet states at\nlattice unit cells. For J= 0, the model shares its phenomenology and unit cell topolog y with\nquasi-one-dimensional ferrimagnetic compounds [9], such as the line of trimer clusters present in\ncopper phosphates [10], and the organic ferrimagnet PNNBNO (Ref. [11]). On the theoretical side,\nseveral features of the ferrimagnetic phase have been studi ed through Hubbard [12], t−J(Ref.\n[13]) and Heisenberg [14] models, including magnetic excit ations [15, 16] and the occurrence of\nnew phases induced by hole doping of the electronic band [17] . Also, the physical properties of the\ncompoundCu 3(CO3)2(OH)2were successfully explained [18] by the distorted diamond c hain model\n[19], which is a system with three spin 1/2 magnetic sites per unit cell and coupling parameters\nsuch that the ferrimagnetic state is frustrated.\nNumericalresultshavebeenobtainedforfiniteclustersthr oughDensity MatrixRenormalization\nGroup (DMRG) (Refs. [20, 21]) using open boundary condition s and exact diagonalization (ED)\nusing periodic boundary conditions and Lanczos algorithm.\n2(a)B1\nB2A(b)\nJc1= 0.342 Jt= 0.445 0.5\nJ00.20.40.60.81Sg / SLMDMRG (Nc = 33)\nED (Nc = 10)\nHCB Model(c)F1 PhaseF2 Phase\nFIG. 1: (a) Illustration of the A and B sublattices (circles) and AF sp in couplings which favor (full lines)\nand destabilize (dashed lines) the LM ferrimagnetic GS: J1(≡1) andJ, respectively. (b) Illustration of the\nLM ferrimagnetic GS. (c) Results (see text) for Sg/SLM; dashed and dotted lines are guides to the eye.\nThe paper is organized as follows: in Sec. II we introduce the model Hamiltonian and analyze\nthe magnetic correlations of the competing phases close to J=Jc1. In Sec. III we define a\nhard-core boson model (HCB model), which is used to describe the main characteristics of the\nmagnon SIT at J=Jc1, in particular, the Tonks-Girardeau limit. Further, in Sec . IV we discuss\nthe singlet magnetic spiral phase accompanied by critical a ntiferromagnetic ordering, which sets\nin after a first-order transition at J=Jt, as well as the decoupling transition, at J=Jc2, to an\nisolated gapped two-leg ladder and a critical single linear chain. Finally, a summary of the results\nis presented in Sec. V.\nII. MODEL HAMILTONIAN AND ORDERED PHASES\nThe model Hamiltonian reads:\nH=Nc/summationdisplay\nl=1/summationdisplay\nα=1,2Al·(Bαl+Bα,l−1)+J(/summationdisplay\nlAl·Al+1\n+B1l·B2l+/summationdisplay\nα=1,2Bα,l·Bα,l+1), (1)\n30.34 0.36 0.38 0.4\nJ12FA(q=0)\nFA(q=π)(a)\nJc = 0.342\n0.34 0.36 0.38 0.4\nJ1234FB(q=0)\nFB(q=π)(b)\nJc= 0.342\n(c)\nFIG. 2: DMRG results for the magnetic structure factor, FX(q), atq= 0 and q=πfor (a)X= A and (b)\nX= B (B 1or B2) spins in a chain with Nc= 33; dashed lines are guides to the eye. (c) Illustration of the\nF2 phase.\nas sketched in Fig. 1(a). In Eq. (1), Al,B1landB2ldenote spin 1/2 operators at sites A l, B1l\nand B2lof the unit cell l, respectively, and Ncis the number of unit cells. For J= 0 the model\n(namedAB2chainordiagonal ladder ) is bipartite and the Lieb-Mattis (LM) theorem [22] predict s\na ground state (GS) total spin\nSg=|NA−NB|\n2=Nc\n2≡SLM, (2)\nwhereNA(NB) is the number of A (B 1and B2) sites. The GS spin pattern is represented in Fig.\n1(b). In Fig. 1(c) we report data for Sg/SLMas a function of Jusing DMRG ( Nc= 33) and ED\n(Nc= 10). Although the LM theorem is not applicable for J/negationslash= 0, the ferrimagnetic phase ( F1\nphase) is robust up to J≈0.342≡Jc1(Ref. [23]), beyond which Sgsteadily decreases ( F2 phase )\nbefore a first order transition to a phase with Sg= 0 (apart from finite size effects) at J≈0.445.\nIn order to characterize the F2 phase, we have calculated the magnetic structure factor,\nFX(q) =Nc/summationdisplay\nlCX(l)eiql, (3)\nwithq= 2πn/(Nc−1),n= 0,1,...,Nc−1, where CX(l) is the two-point correlation function\nbetween spins separated by lunit cells at sites X=A,B1andB2. We first noticed that the A\nspins remain ferromagnetically ordered as the critical poi ntJc1= 0.342 is crossed, although the\n40 0.01 0.02 0.03 0.04 0.05 0.06\n1 / (Nc+1)00.010.020.03\nm2\nAL(0)\nm2\nAL(π)\nm2\nAT(0)\nm2\nAT(π)(a)\n0 0.01 0.02 0.03 0.04 0.05 0.06\n1 / Nc00.020.030.045\nm2\nBL(0)\nm2\nBL(π)\nm2\nBT(0)\nm2\nBT(π)(b)\n1 10 100\nl10-610-510-410-310-2(-1)lCBT(l)\nDMRG (Nc=121)\na0 / l0.93(c)\nFIG. 3: DMRG results for the square of the longitudinal (L) and tra nsverse (T) order parameters at the\nspin se ctor Sz=SgandJ= 0.395 for (a) A and (b) B (B 1or B2) spins (full lines are polynomial fittings).\n(c) DMRG results for the transverse staggered correlation func tionCBTπ(l).\nmagnitude of the peak at q= 0 decreases for J > Jc1, as displayed in Fig. 2(a), while no peak\nis observed at q=π. The B i(i= 1 or 2) spins also remain ferromagnetically ordered (peak a t\nq= 0), with similar J-dependence, as shown in Fig. 2(b). However, an extra peak at q=π\ndevelops after the transition, which indicates the occurre nce of a period-2 modulation in the spin\npattern for J/greaterorsimilarJc1. Further, the average value of the correlation function /angbracketleftB1l·B2l/angbracketright, which\namounts to ≈0.25 (triplet state) in the F1 phase, steadily decreases after the transition at Jc1.\nThesefindingssuggest that theF2 phasewould display a cante d configuration, as illustrated in Fig.\n2(c). However, to check whether these features are robust in the thermodynamic limit, we have\nstudied the finite size scaling behavior of the transverse (T ) and longitudinal (L) order parameters\nin the F2 phase:\nm2\nX(L,T)(q) =FX(L,T)(q)\nNc, (4)\nforq= 0 (uniform component) and q=π(staggered component), in the subspace of maximum to-\ntal spinz-component ( Sz=Sg). The correlations are studied at J= 0.395, for which Sg=SLM/2,\nand the results are shown in Fig. 3. We confirmed that in the (ex trapolated) thermodynamic\n5limit the spins at sites A and B are ferromagnetically ordere d, as indicated by m2\nXL(q= 0)/negationslash= 0 in\nFigs. 3(a) and (b). Further, since the A and B net magnetizati ons are oppositely oriented, the F2\nphase is ferrimagnetic. The values of m2\nAL(q=π),m2\nBL(q=π) andm2\nBT(q= 0) nullifies linearly\nwith system size, which evidences short-range correlation s. On the other hand, the best fitting to\nthe data for m2\nBT(q=π) presents a nonlinear dependence with the inverse of the sys tem size and\nalso nullifies in the thermodynamic limit. This behavior ind icates that the staggered correlation\nfunction of the spins at sites B along the transverse directi on to the spontaneous magnetization,\nCBTπ(l), exhibits a power-law decay, as explicitly confirmed in Fig . 3(c). We thus conclude that\nforJc1< J <0.445 the GS is also ferrimagnetic but with critical correlati ons along the transverse\ndirection to the spontaneous magnetization ( F2 phase ).\nNext we focus on the effect of Jon the magnetic excitations. For J= 0 the Hamiltonian\nexhibits three magnon modes [15, 16]. One is AF, i. e., the spi n is raised by one unit with respect\nto the GS total spin, while the other two are ferromagnetic, a ssociated with the lowering of the\nGS total spin by one unit. The AF gapped dispersive mode is res ponsible for a quantized plateau\nin the magnetization curve as function of hand should also exhibit condensation, as suggested\nby the numerical data in Ref. [16]. One of the ferromagnetic m agnons is the gapless dispersive\nGoldstone mode, while the other is a flat mode and is the releva nt excitation for the transition\natJ=Jc1. To understand some nontrivial features of this excitation , we must comment on the\nsymmetry properties of the model. For J= 0 the Hamiltonian is invariant under the exchange\nof the B sites at the samecell. This symmetry implies that spins at these B sites can be found\nonly in singlet or triplet states (mutually exclusive possi bilities); in the GS only triplets are found.\nThe relevant magnon is a localized gapped mode which induces the formation of a singlet pair in\none cell, as illustrated in Fig. 4(a). For J/negationslash= 0, this local symmetry is explicitly broken and the\nspins at these B sites can be found in a coherent superpositio n of singlet and triplet states. In Fig.\n4(b), data using ED for the magnon band ( q= 2πn/Nc,n= 0,1,...,Nc−1) is displayed for various\nvalues of Jbefore the transition point. For J= 0 the band is flat with a gap ∆ 0≈1.0004. By\nincreasing J, the bandwidth increases and the gap to the GS lowers, closin g at the wave vector\nq=πat the transition point.\n6(a)\n00.25 0.5 0.75 1\nq / π00.51Magnon Band\nED (Nc=10)\nHCB Model(b)\nJc0.445 0.6 0.7 0.8\nJ05101520\nNS\nDMRG (Nc= 33)\nHCB Model(c)\n1 10 20 30 33 40\nl00.10.2<ηl> = 0.25 - < B1l. B2l>\n0.340\n0.345\n0.350\nHCB (NS = 2)\nHCB (NS = 4)(d)\nFIG. 4: (a) Illustration of the relevant magnon excitation for J= 0: ellipse indicates a local singlet state.\n(b) Magnon band for J= 0.00,0.05,0.10,0.15,0.20,0.25,0.30 and 0 .34, from top to bottom. (c) NSas a\nfunction of J. Full lines are the HCB model predictions in the TG limit and dashed lines a re guides to the\neye.\n7III. HARD-CORE BOSON MODEL, SUPERFLUID-INSULATOR TRANSIT ION AND\nTHE TONKS-GIRARDEAU LIMIT\nThe GS total number of singlets is given by\nNS=Nc/summationdisplay\nl=1/angbracketleftηl/angbracketright, (5)\nwith singlet density /angbracketleftηl/angbracketright=/angbracketlefts†\nlsl/angbracketright, where\ns†\nl≡1√\n2(B†\n1l,↑B†\n2l,↓−B†\n1l,↓B†\n2l,↑) (6)\nis the creation operator of a singlet pair at cell landB†\nil,σis the creation operator of an electron\nwith spin σat the B i(i= 1,2) site of cell l. In fact, it is easy to show that\n/angbracketleftηl/angbracketright=1\n4−/angbracketleftB1l·B2l/angbracketright, (7)\nsoNS= 0 forJ= 0. In Fig. 4(c) we observe that NSstarts to increase in steps of unity after\nJ=Jc1, indicating the quantized nature of the condensing singlet s.\nWe now examine the nature of the quantum critical point at J=Jc1. For this purpose we split\nthe Hamiltonian of Eq. (1) in three terms: the first favors fer rimagnetism,\nHAB=/summationdisplay\nlAl·(Sl+Sl−1), (8)\nwhereSl=B1l+B2l; the second one favors AF ordering between A spins, i. e.,\nHA=J/summationdisplay\nlAl·Al+1, (9)\nand shall play no significant role in our analysis; the last te rm, also unfavorable to ferrimagnetism,\nis a two-leg ladder Hamiltonian connecting spins at sites B1andB2(discarding a constant factor)\n[24, 25]:\nHB=J\n2/parenleftBigg/summationdisplay\nlS2\nl+/summationdisplay\nlSl·Sl+1+/summationdisplay\nlDl·Dl+1/parenrightBigg\n, (10)\nwhereDl=B1l−B2l. We represent the Hamiltonian in a basis with two states for e ach pair\nB1landB2l: the singlet and the triplet component in the magnetization direction. In addition,\nwe define the vacuum of the HCB model as the state with this trip let component in each cell.\nWe now study the GS energy when a number NSof singlet pairs is added to the vacuum. For\nJ= 0, the energy cost of a singlet pair is ∆ 0(the gap to the flat mode); thus, for NSsinglets, the\n8contribution from HABisNS∆0. The first term in HBis diagonal and will add a factor of −JNS;\nthe second causes a repulsion between singlets and adds also an extra factor of −JNS; finally,\nthe last term in HBintroduces the singlet itinerancy. Grouping these contrib utions, we arise to a\nmodel of hard-core bosons with nearest-neighbor repulsion :\nHS= (∆0−2J)NS+J\n2/summationdisplay\nlηlηl+1+J\n2/summationdisplay\nl(s†\nlsl+1+h.c.). (11)\nWeremarkthatthehard-corebosoninteraction isimpliedby thealgebraofthesingletoperators:\n[sl,s†\nl]+= 1; and (12)\n[sl,sm]−= 0 forl/negationslash=m. (13)\nBefore the transition, the single magnon dispersion relati on,\nωq(J) = ∆0−2J+Jcosq, (14)\nagrees well with the numerical data for q≈π, as can be seen in Fig. 4(b). The resulting critical\npoint:ωq=π(Jc1,S) = 0, i. e.,\nJc1,S=∆0\n3≈0.333, (15)\nis in excellent agreement with the numerical prediction Jc1= 0.342. Moreover, the closing of the\nmagnon gap is also in excellent agreement with the predictio n\n∆J=ω(q=π) = 3(Jc1,S−J) (16)\nand with the expected linear vanishing of the Mott gap [26]: zν= 1, where z= 2 and ν= 1/2 are\nthe correlation length and dynamic critical exponents, res pectively (see below).\nAfter the transition and in the highly diluted limit (NS\nNc≡η→0), the energy of NShard-core\nbosons in 1D is well approximated by the energy of NSfree spinless fermions [5]. Through this\nmap, the energy density reads:\nEGS(J) =EGS(J)\nNc=/integraldisplaykF\n−kFdk\n2π[ǫk(J)−µF] (17)\n≈3(Jc1,S−J)η+Jπ2η3\n6, (18)\nwherekF=πηand\nǫk(J)−µF=ωk+π(J)≈ −3(J−Jc1,S)+Jk2\n2. (19)\n90.36 0.38Jc\nJ-10EGS\nDMRG (Nc= 33)\nHCB Model(a)\n00.050.10.150.20.25η0.50.751\nKDMRG (Nc = 33)\nDMRG (Nc = 65)\nHCB Model\n1− 4η(b)\nFIG. 5: (a) Groundstate energy, EGS, relativeto the energyofthe LM state. (b) LuttingerLiquid expon ent,\nK, as function of η. Full lines are the HCB model predictions in the TG limit and dashed lines a re guides\nto the eye.\nNotice that the Fermi chemical potential satisfies the Tonks -Girardeau (TG) limit [7, 27, 28]\n(1D Bose gas of impenetrable particles), corresponding to a n infinitely high repulsive potential in\nthe Lieb-Liniger solution [29] of the δ-function 1D Bose gas:\nµF=ǫF(J) =π2Jη2\n2, (20)\nwhereJ−1is the fermion mass, /planckover2pi1≡1, andηis the density of singlets for J≥Jc1,Sderived from\nthe equilibrium condition ∂ηEGS(J) = 0:\nη=/radicalbig\n6(J−Jc1,S)\nπ√\nJ,η→0, (21)\nmuch in analogy with the 1D field-induced transition. Furthe r, in Fig. 4(d) we display the good\nagreement between thenumerical estimate for the density /angbracketleftηl/angbracketrightof a two (four) particle state, NS= 2\n(NS= 4), in an open system and the HCB model in a continuum space gi ven by [24]\n/angbracketleftη(l)/angbracketright=2\nNc−1NS/summationdisplay\nn=1sin2(knl), (22)\nwithkn= 1,...,πNS\nNc−1. Also, as shown in Figs. 1(c), 4(c) and 5(a), the HCB model pre dictions for\nSg\nSLM= 1−2η, (23)\nηandEGS(J), respectively, are very close to the numerical data for J/greaterorsimilarJc1≈Jc1,S.\nOn the other hand, using the Luttinger liquid description [3 0, 31] for our highly diluted HCB\n10model we have the following general relations for the sound v elocitycand the compressibility κ:\nc=πJη\nK; (24)\n1\nη2κ=πc\nK, (25)\nwhereKis the Luttinger parameter governing the decay of the correl ation functions. However,\nsince\n1\nη2κ=d2EGS\ndη2=π2Jη, (26)\nit implies that K= 1; thus c=πJη=JkF, in accord with the TG limit [7, 28]. Further, taking\nηas the order parameter of the SIT, Eq. (21) implies β= 1/2, while η2κdiverges with a critical\nexponent α=γ= 1/2, in agreement with the scaling and hyperscaling relations [26]:\nα+2β+γ= 2, (27)\n2−α=ν(d+z), (28)\nrespectively, assuring that the SIT is in the free spinless g as universality class [32].\nIn an interacting Bose gas [33], K= 1/2 is the separatrix between systems dominated by\nsuperfluid fluctuations, K >1/2, from those dominated by charge density fluctuations, K <1/2,\n(in our magnetic model spin fluctuations prevail). Affleck and collaborators [34] have succeeded in\ntaking into account corrections from interactions between pairs of dilute magnons parametrized by\na scattering length, a, thus implying that\nK= 1−2am+O(m2), (29)\nwheremis the field-induced magnetization for the S= 1 chain, with a≈ −2. The predicted\nincrease of Kwithmwas confirmed by numerical calculations [34]. This parametr ization can also\nbe implemented in our problem. In fact, in Fig. 5(b) we show th atK= 1−4η, witha≈2,\nfits quite well the data for the Luttinger liquid parameter in the highly diluted regime. Kwas\ncalculated using DMRG and assuming\nCBTπ∼a0\nl1\n2K. (30)\nIV. SPIRAL CORRELATIONS, WEAKLY COUPLED AF CHAINS, AND LADD ER-\nCHAIN DECOUPLING\nWe now turn our attention to the transition point Jt≈0.445, which marks the onset of a singlet\nphase, as can be seen in Fig. 1(c), characterized by non-quan tized values of NS, as shown in Fig.\n114(c). On the other hand, from the Hamiltonians in Eqs. (8)-(1 0), we can infer that for J >>1 the\nsystem should decompose into a linear chain (A sites) and an i sotropic two-leg ladder system (B 1\nand B2sites); see Fig. 1(b). The linear chain is known to be gapless with critical spin correlations\n(power-law decay), while the two-leg ladder is gapped with e xponentially decaying correlations. In\nwhat follows we discuss the complex phase diagram in the regi onJ > Jt.\nInitially, we display in Fig. 6 the magnetic structure facto rsFA(q) andFBi(q), withi= 1 or 2,\nas well as FS(q), which is associated to the magnetic structureof the compo site spin Sl=B1l+B2l.\nIn Fig. 6(a) we see that FA(q) peaks at q= 0 forJ= 0.44, i. e., the system remains in the F2\nphase and the A spins are ferromagnetically ordered. For J= 0.45, a sharp peak in a spiral wave-\nvectorqmaxis observed. The peak broadens and qmaxincreases with increasing J. ForJ= 0.56 we\nnotice the emergence of a commensurate AF peak, coexisting w ith the spiral one, particularly for\nJ≥0.60, as seen in Figs. 6(a) and 6(b). On the other hand, we observ e in Fig. 6(c) the presence of\ntwo peaks in FBi(q) forJ= 0.44: theq= 0 peak associated with the ferromagnetic ordering of the\nBisites in the F2 phase, and the q=πpeak related to the critical staggered transverse correlat ion\nat the same phase. Likewise, for J= 0.45, a spiral peak is observed at the same wave-vector qmax\nofFA(q). Further, notice in Fig. 6(d) that the magnitude of the AF pe ak drops in the interval\n0.96≤J <1.00.\nIn order to develop a physical meaning of the above referred d ata, we first point out that the\ncoupling between spins at A and B sites occurs through the com position Sl=B1l+B2l, as can\nbe seen in Eq. (8). Further, as the singlet component of Slis magnetically inert, only its triplet\ncomponents affect the magnetic ordering at the A sites. In fact , as shown in Figs. 6(e) and 6(f),\nshort-range spiral ordering is observed in the magnetic str ucture of Slup toJ≈1.00. However,\nsince the peak is weak and broad for J/greaterorsimilar0.6, its feature is overcomed by the AF one in the data\nof Figs. 6 [(a)-(d)]. In the sequence, we focus on the AF order ing observed for J/greaterorsimilar0.6 and study\nhow the system approaches the ladder-chain decoupling.\nIn Fig. 7(a), we present the staggered AF correlation functi on between A-spins as J→1. As\nobserved, itsbehavioriswelldescribedbythatfoundinasi nglelinearchain, whichisasymptotically\ngiven by\nC(l)∼(−1)l\nl, (31)\napartfromlogarithmiccorrections[30]. Asimilarbehavio risobservedinFig. 7(b)forthestaggered\nAF correlation between Bi-spins up to J= 0.88, a value beyond which the shape of the curve is\nvisibly changed. In order to understand this dramatic behav ior, we recall that in a two-leg ladder\n120 1 2 3 q0.70.80.91FA(q)J = 0.44\nJ = 0.45\nJ = 0.48\nJ = 0.52\nJ = 0.56\nJ = 0.60\nJ = 0.64(a)\n0 1 2 3 q12\nFA(q)J = 0.68\nJ = 0.80\nJ = 0.92\nJ = 0.96\nJ = 1.00(b)\n0 1 2 3 q0.60.811.2FBi(q)J = 0.44\nJ = 0.45\nJ = 0.48\nJ = 0.52\nJ = 0.56\nJ = 0.60\nJ = 0.64(c)\n2.6 2.8 3 q0.81.21.6\nFBi(q)J = 0.68\nJ = 0.80\nJ = 0.92\nJ = 0.96\nJ = 1.00(d)\n0 1 2 3 q00.511.52FS(q)(e)\n0.440.60 0.96\nJ00.501\n0.250.75\nqmax / π(f)\nFIG. 6: Magnetic structure factor FX(q) for the Aspins [(a) and (b)], Nc= 32, and for the Bispins\n[(c) and (d) with i= 1 or 2], Nc= 33, for the indicated values of J. (e) Magnetic structure factor for\nthe composition Sl=B1l+B2landJ= 0.44,0.45,0.48,0.52,0.56,0.60,0.64,0.68,0.80,0.92,0.96 and 1 .00,\nfrom top to bottom at q= 0. (f) The value of the wave-vector for which the peak at the mag netic structure\nfactor exhibited in (e) is observed. Dashed lines are guides to the ey e.\nsystem the asymptotic form of the correlation is given by [35 ]\nC(l)∼(−1)le−l/ξ\nl1/2, (32)\nwhereξ(≈3.2,see Ref.[36]) defines the correlation length, associated wi th the gapped spin liquid\n131 10 30\n l10-410-2(-1)lCA(l)\n~ 1 / l\nJ = 1.00\nJ = 0.96\nJ = 0.88\nJ = 0.80\nJ = 0.72\nJ = 0.64(a)\n1 10 30\n l10-410-310-210-1\n(-1)lCBi(l)\n~ 1 / l\nJ = 0.64\nJ = 0.72\nJ = 0.80\nJ = 0.88\nJ = 0.92\nJ = 0.96\nJ = 1.00(b)\n1 10 20 30\n l10-610-410-2(-1)lCBi(l)\n~ 1 / l\nLadder(c)\n1 1.2 1.4 1.61.8 2\nJ3.2681012\nξNc = 33\nNc = 65\nFitting \nLadder (d)\nFIG. 7: Staggered correlation functions between (a) A spins, Nc= 32, and (b) Bi(withi= 1 or 2)\nspins,Nc= 33, for the indicated values of J. In (a) and (b) solid lines indicate the asymptotic be-\nhavior for a single chain. (c) Staggered correlation functions betw eenBi(withi= 1 or 2) spins for\nJ= 0.88,0.92,0.96,1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9 and 2.0, circles-dash from top to bottom and\nNc= 33. For comparison we plot the behavior for a single chain (solid line) a nd for a two-leg ladder with\n32 rungs. (d) Correlation length ξas a function of J: solid line indicates the fitting of the data for Nc= 65\nto Eq. (33) with ggiven by Eq. (40); dashed line indicates the value of ξ(≈3.2) for a two-leg ladder.\nstate of this system. Indeed, as displayed in Fig. 7(c) the st aggered correlations CBiasymptotically\napproaches the correlation in a two-leg ladder system. In Fi g. 7(d) we present the behavior of ξ\nas a function of JforNc= 33 and Nc= 65. These data were obtained by a proper fitting of CBi\nin the interval l0< l <(Nc/2): starting from J= 2 and taking l0≈6 (about twice the value of\nξof a two-leg ladder), we find ξ; twice this value of ξwas used as input ( l0= 2ξ) for the next\nchosen value of J, and so on. Moreover, we have obtained a good fitting to these d ata by using the\ntwo-loop analytic form of the O(3) non-linear sigma model (N LSM) correlation length in (1+1)\ndimension [37]:\nξ=ae2π\ng/parenleftbigg\n1+2π\ng/parenrightbigg−1\n, (33)\nwhereais a constant and gis the NLSM coupling. Further, we assume (see below) that the\n14coupling gis the one suitable to the anisotropic quantum Heisenberg tw o-leg ladder to the NLSM\n[38]:\ng= 2κ/radicalBigg\n1+J⊥\n2J/bardbl, (34)\nwhereJ⊥(J/bardbl) is the exchange coupling between spins at the same rung (leg ) andκis a constant\nthat depends on the choice of the lattice regularization.\nIn order to justify Eq. (34) for g, we consider a mapping of the model Hamiltonian, Eq. (1), to\nthe Hamiltonian of an isolated two-leg ladder by eliminatin g the spin degrees of freedom associated\nwiththeA sites. Themappingis performed, inasemiclassica l manner, by thefollowing assumption\nonHAB[Eq. (8)]:\nHAB→HAB=γ/summationdisplay\nlAl·Sl, (35)\nwhereγis an effective coupling constant. This amounts to reduce the A -B coupling to spins within\nthe same unit cell, and cell-cell interactions are taken int o account through the effective coupling\nγ. We now write: ( Al+Sl)2=A2\nl+S2\nl+2Al·Sl, withS2\nl=B2\n1l+B2\n2l+2B1l·B2l; since within\na unit cell ( Al+Sl)2≈(1/2)2in an AF phase, and dropping constant terms, HABcan be written\nas\nHAB=−γ/summationdisplay\nlB1l·B2l. (36)\nSince correlations between spins at Asites does not play a significant role close to the transition ,\nwe discard the term HA, and, finally, obtain the following anisotropic two-leg ladder Hamiltonian:\nH→HaL=HAB+HB, (37)\nwhere the exchange couplings are given by\nJ⊥=J−γ, (38)\nJ/bardbl=J. (39)\nSubstituting Eqs. (38) and (39) into Eq. (34), we find the effect ive NLSM coupling:\ng=κ/radicalbigg\n6(J−Jc2)\nJ, (40)\nwhereJc2=γ/3. We have fitted the data in Fig. 7(d) to Eq. (33), with ggiven by Eq. (40), and\na,κandJc2as fitting parameters. The obtained value of a(=2.7) is such that ξ→3.1 asJ→ ∞,\n150.3420 1 2\nJ00.20.40.6η\nDMRG (Nc= 33)\nHCB Model\nFIG. 8: Density of singlets as function of J.\nwhich agrees with the expected value for an isolated isotropic two-leg ladder ( ≈3.2), while κ= 4.5\nandJc2= 0.91, in agreement with the correlation function behavior sho wn in Fig. 7(b).\nFinally, inFig. 8wedisplaytheveryinteresting behavioro fthedensity ofsinglets, η, as function\nofJ. It is clear that the effect of the A-spins and singlet-singlet interaction is relevant only for\nJt/lessorsimilarJ/lessorsimilarJc2, otherwise the solution, Eq. (21), for low density of single ts can be extended to the\nregion of low density of triplets above Jc2(strongly coupling limit), where correlations between\nB-spins are exponentially small [see Eq. (32)]. In fact, the asymptotic value predicted by Eq. (21),\ni. e.,η=√\n6/π≈0.78, compares well with the numerical one: ≈0.71.\nV. SUMMARY AND CONCLUSIONS\nJc1JtLieb−Mattis \nFerrimagnetism\nJc2+\nAntiferromagnetic\nCorrelationsSpiralFerrimagnetism\n+\nCritical\nAntiferromagnetic\nTransverse Correlations\nJ0DecouplingLadder−Chain\nFIG. 9: Schematic representation of the phase diagram.\nIn this work we have derived the rich phase diagram of a three- leg spin Hamiltonian related\nto quasi-one-dimensional ferrimagnets, as function of a fr ustration parameter Jwhich destabilizes\n16the ferrimagnetic phase. In Fig. 9 we present an illustratio n of the obtained phase diagram,\nwhich displays two critical points, Jc1≈0.342 and Jc2≈0.91, and a first order transition point\natJt≈0.445. Through DMRG, exact diagonalization and a hard-core bo son model, we have\ncharacterized the transition at Jc1as an insulator-superfluid transition of magnons (built fro m\nthe coherent superposition of singlet and triplet states be tween B sites at lattice unit cells), with\na well defined Tonks-Girardeau limit in the high diluted regi me. Ferrimagnetism with critical\nstaggered correlations in a direction transverse to the spo ntaneous magnetization is observed for\nJc1< J < J c2. Further, for Jc1< J < J tthe number of singlets in the lattice is quantized,\nwhile above the first order transition at J=Jtthis quantity is a continuous one. Also, in the\nintervalJt< J < J c2the magnetic structure factor displays a singlet phase with incommensurate\n(q/negationslash= 0 and π) spiral and AF peaks. However, the spiral peak broads and the AF peak is the salient\nfeature as Jincreases within this phase. At J=Jc2a remarkable gapped two-leg ladder / critical\nsingle-linear chain decoupling transition occurs, charac terized by an essential singularity in the\ncorrelation length as predicted by the NLSM through a mappin g of our model onto an anisotropic\nquantum Heisenberg two-leg ladder. For J≫Jc2the ladder approaches the isotropic limit (full\ndecoupling), while the linear chain remains critical.\nIn summary, our reported results clearly reveal that frustr ated quasi-one-dimensional magnets\nare quite remarkable systems to study magnon condensation, including the crossover to coupled\nladder systems of higher dimensionality [39] and related ch allenging phenomena [40], as well as\nfrustration-driven quantum decoupling transition in ladd er systems.\nVI. ACKNOWLEDGMENTS\nWe acknowledge useful discussions with A. S. F. Ten´ orio and E. P. Raposo. This work was\nsupported by CNPq, Finep, FACEPE and CAPES (Brazilian agenc ies).\n[1] M. Jaime et al., Phys.Rev. Lett 93, 087203(2004); T. 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Fisher, Natur e\n(London) 441, 617 (2006).\n19"    },    {        "title": "2309.01965v1.Strong_and_nearly_100_____spin_polarized_second_harmonic_generation_from_ferrimagnet_Mn___2__RuGa.pdf",        "content": "arXiv:2309.01965v1  [cond-mat.mtrl-sci]  5 Sep 2023Strong and nearly 100 %spin-polarized second-harmonic generation from ferrimag net\nMn2RuGa\nY. Q. Liu,1M. S. Si∗,1and G. P. Zhang†2\n1School of Materials and Energy, Lanzhou University, Lanzho u 730000, China\n2Department of Physics, Indiana State University, Terre Hau te, IN 47809, USA\n(Dated: September 6, 2023)\nSecond-harmonic generation (SHG) has emerged as a promisin g tool for detecting electronic and\nmagnetic structures in noncentrosymmetric materials, but 100% spin-polarized SHG has not been\nreported. In this work, we demonstrate nearly 100% spin-pol arized SHG from half-metallic ferri-\nmagnet Mn 2RuGa. A band gap in the spin-down channel suppresses SHG, so t he spin-up channel\ncontributes nearly all the signal, as large as 3614 pm/V abou t 10 times larger than that of GaAs.\nIn the spin-up channel, χ(2)\nxyzis dominated by the large intraband current in three highly d ispersed\nbands near the Fermi level. With the spin-orbit coupling (SO C), the reduced magnetic point group\nallows additional SHG components, where the interband cont ribution is enhanced. Our finding is\nimportant as it predicts a large and complete spin-polarize d SHG in a all-optical spin switching\nferrimagnet. This opens the door for future applications.\nPACS numbers:\nI. INTRODUCTION\nThe interaction between an intense optical field and a\nmaterial is always fascinating. This gave birth to non-\nlinear optics [1, 2]. Second-harmonic generation (SHG),\na special case of sum frequency generation, has received\nenormous attention worldwide. SHG only exists in non-\ncentrosymmetricmaterialswith brokeninversionsymme-\ntryI[3], while it is absent in centrosymmetric systems.\nIn general, impurities and surfaces introduced in a mate-\nrial can break I. For instance, Iof the NV center is bro-\nken by introducing nitrogen-vacancies in diamond [4–7].\nMoreover, a few layers of crystal, created by mechanical\nexfoliation, exhibit different symmetry properties. Odd\nlayers of MoS 2and h-BN belong to the noncentrosym-\nmetric space group, different from their bulk, can also\ngenerate SHG [8–10]. So far, most of the materials stud-\nied are nonmagnetic. For magnetic materials, magnetic\norder can break time reversal symmetry T. The sizable\nSHG appears in the antiferromagnetic (AFM) CrI 3and\nthe even septuple layers of MnBi 2Te4[11–13], where the\nAFM orderingbreaks I. In these two cases, SOC destroys\nthe symmetry of band structure thereby enhancing SHG.\nAlthough nearly 100% spin polarization at the Fermi\nlevel is observed in materials such as half-metal Cr 2O3\n[14], they possess I, where only the odd-order harmon-\nics are observed. Until now, little is known about 100%\nspin-polarized SHG in a half-metallic ferrimagnet.\nIn this work, we predict a strong SHG signal from the\nhalf-metallic Heusler Mn 2RuGa. We show that a single\nspin channel mainly contributes to SHG in Mn 2RuGa.\nThe band gap in the spin-down channel is open and lim-\nits SHG, resulting in SHG mainly from the spin-up chan-\nnel. Surprisingly, χ(2)\nxyzreaches as large as 3614 pm/V for\nthe spin-upchannel, atleastanorderofmagnitudelarger\nthan that ofGaAs. It is found that the intrabandcurrent\ndominates this large χ(2)\nxyz, which originates from threehighly dispersive bands near the Fermi level. To con-\nfirm our conclusion, we remove the Ru atoms to obtain\nMn2Ga, where both the spin-up and spin-down channels\nare metallic. It is found that a highly dispersive band\nnear the Fermi level appears in the spin-down channel,\nwhich does enhance the SHG spectrum χ(2)\nxyz. With SOC,\nthe spin-up and spin-down channels arecoupled. As a re-\nsult, the restricted transitions in the spin-polarized case\nare now SOC-allowed between the flat valence and con-\nduction bands. The underlying physics stems from the\nreduced magnetic point group induced by SOC, where\nthe magnetization field is applied along the z-axis. This\ndirectly leads to the appearance of additional SHG com-\nponents such as χ(2)\nxxz, where the allowed interband tran-\nsitions play a role. Our study demonstrates that nearly\n100% spin-polarized SHG can detect the half-metallicity\nin Heusler alloy Mn 2RuGa.\nThe rest of the paper is arrangedas follows. In Sec. II,\nwe show our theoretical methods. Then, the results and\ndiscussions are given in Sec. III. Finally, we conclude our\nwork in Sec. IV.\nII. COMPUTATIONAL METHODS\nA. First-principle electronic structure calculations\nThe electronic structures of Mn 2RuGa are calculated\nwithin the first-principle density functional theory using\ntheprojector-augmentedwave(PAW)[15,16]method, as\nimplemented in the Vienna Ab intio Simulation Package\n(VASP) [17–20]. The generalized gradient approxima-\ntion (GGA) [21] is employed within the Perdew-Burke-\nErnzerhof (PBE) scheme as the exchange-correlation\nfunctional. We self-consistently solve the Kohn-Sham2\nequation\n/bracketleftbigg\n−¯h2\n2me∇2+Vne(/vector r)+e2\n4πǫ0/integraldisplayn(/vector r)\n|/vector r−/vector r′|d3/vector r′+Vxc(/vector r)/bracketrightbigg\n×ψn/vectork(/vector r) =εn/vectorkψn/vectork(/vector r).\n(1)\nThe first term is the kinetic energy and next three terms\nare the potential energy, the Coulomb, and the exchange\ninteractions, respectively. meis the electron mass and\nn(/vector r) is the electron density. ψn/vectork(/vector r) denotes the Bloch\nwave function of band nat crystal momentum /vectork, and\nεn/vectorkis the band energy. The cutoff energy is set to 500\neV. The structural optimizations and self-consistent are\ncarried out by Γ-centered k-point mesh of 15 ×15×15.\nThe density of states (DOS) is calculated using a denser\nk-point mesh of 21 ×21×21. In order to obtain accurate\nresults, we set the energy convergence less than 10−6eV.\nB. First-principle nonlinear optical response\ncalculations\nWe use the length gauge to compute SHG [22–\n24]. The nonlinear polarization is given by Pa(2ω) =\nχabc(2ω;ω,ω)Eb(ω)Ec(ω), whereχabcdenotes the SHG\nsusceptibility, and Eb(ω) is thebcomponent of the op-\ntical electric field at frequency ω. In general, χabccon-\ntains three major contributions: the interband transi-\ntionsχabc\ninter, the intraband transitions χabc\nintra, and the\nmodulation of interband terms by intraband terms χabc\nmod,\nand can be expressed as\nχabc(2ω;ω,ω) =χabc\ninter(ω)+χabc\nintra(ω)+χabc\nmod(ω),\n=χabc\n2ph,inter(ω)+χabc\n1ph,inter(ω)\n+χabc\n2ph,intra(ω)+χabc\n1ph,intra(ω)\n+χabc\nmod(ω),(2)\nwherethe subscripts2 phand1phrepresenttwo-andone-\nphoton transitions, respectively. The interband and in-\ntraband transitions are\nχabc\ninter(ω) =χabc\n2ph,inter(ω)+χabc\n1ph,inter(ω),\nχabc\nintra(ω) =χabc\n2ph,intra(ω)+χabc\n1ph,intra(ω).(3)\nThedetailedexpressions[25]ofthesefourtermsaregiven\nby\nχabc\n2ph,inter(ω) =e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlra\nnm(k)/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\nωln(k)−ωml(k)\n×2fnm\nωmn(k)−2ω−2iη,\n(4)\nχabc\n1ph,inter(ω) =e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlra\nnm(k)/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\nωln(k)−ωml(k)\n×/braceleftbiggfml\nωml(k)−ω−iη+fln\nωln(k)−ω−iη/bracerightbigg\n,\n(5)χabc\n2ph,intra(ω) =e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmra\nnm(k)/braceleftbig\n∆b\nmn(k)rc\nmn(k)/bracerightbig\nω2mn(k)\n×−8ifnm\nωmn(k)−2ω−2iη\n−e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlra\nnm(k)/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\nω2mn(k)\n×2fnm(ωln(k)−ωml(k))\nωmn(k)−2ω−2iη,\n(6)\nχabc\n1ph,intra(ω) =e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlra\nnm(k)/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\n×ωmn(k)/braceleftbiggfnl\nω2\nln(k)(ωln(k)−ω−iη)\n−flm\nω2\nml(k)(ωml(k)−ω−iη)/bracerightbigg\n.\n(7)\nWe know that χabc\nmodin Eq. (2) contributes little to SHG\nin comparison with χabc\ninterandχabc\nintra, and it has the form\nas\ni\n2ωχabc\nmod(2ω;ω,ω) =\nie3\n2¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlωnl(k)ra\nlm(k)/braceleftbig\nrb\nmn(k)rc\nnl(k)/bracerightbig\nfnm\nω2mn(k)(ωmn(k)−ω−iη)\n−ie3\n2¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlωlm(k)ra\nnl(k)/braceleftbig\nrb\nlm(k)rc\nmn(k)/bracerightbig\nfnm\nω2mn(k)(ωmn(k)−ω−iη)\n+ie3\n2¯h2/integraldisplaydk\n4π3/summationdisplay\nnmfnm∆a\nnm(k)/braceleftbig\nrb\nmn(k)rc\nnm(k)/bracerightbig\nω2mn(k)(ωmn(k)−ω−iη).\n(8)\nIn Eqs. (4)-(8), ωmn(k) =ωm(k)−ωn(k), and the\nenergy of band nis ¯hωn. ∆a\nmn(k) =υa\nmm(k)−\nυa\nnn(k), whereυa\nnm(k) is theacomponent of the ve-\nlocity matrix elements, and fmn=f(¯hωm)−f(¯hωn),\nwheref(¯hωn) is the Fermi Dirac function. The ma-\ntrix elements of position operator rmn(k) is given by\nra\nmn(k) =−iυa\nnm(k)/ωmn(k), and/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\n=\n1\n2/bracketleftbig\nrb\nml(k)rc\nln(k)+rc\nml(k)rb\nln(k)/bracketrightbig\n. The matrix elements of\nposition operator rmn(k) are directly computed by the\nfirst-principle calculations, which accounts for the effect\nof non-local potentials [26]. The damping parameter η\nis set to 0.003 Hartree. In the realistic calculations, the\nnumber ofkpoints and energy bands affect the accuracy\nofχabc. A large number of kpoints are required to ob-\ntain an accurate NLO response, so a very dense k-point\nmesh of32 ×32×32is used. The number of energybands\nis set to 32 to converge the spin-polarized SHG spectra,\nand 64 for the SHG spectra with SOC.3\nIII. RESULTS AND DISCUSSIONS\nA. Crystal, electronic structures and\nspin-polarized SHG of Mn 2RuGa\n(a)\nHalf-metallic ferrimagnet0.0 0.5 1.0 1.5 2.0 2.5 3.030006000900012000\n0Spin-up Spin-dn(b)\n|\n\u0001xyz| (pm/V)(2)\n0 1 2 3 4 5 6100200300\n0(c)\nE (eV) GaAsMn2RuGa2 \u0002\n7 8zLaser\nDetector\nEF\nSpin-up Spin-dngap2\u0000 2\u0003\n\u0004\u0005\n\u0006400\n|\n\u0007xyz| (pm/V)(2)\nFIG. 1: (a) A sketch of the nearly 100% spin-polarized SHG\nin half-metallic ferrimagnets, where the contribution fro m the\nspin-up channel is significant, while that from the spin-dow n\nchannel is largely suppressed. (b) The absolute value of the\nSHG susceptibility χ(2)\nxyzfor Mn 2RuGa without SOC. The red\nsolid andblack dashed lines denote the spin-upand spin-dow n\nchannels, respectively. The damping parameter η= 0.003\nHartree is taken. (c) χ(2)\nxyzof GaAs is also given for compari-\nson.\nSHG appears in nonmagnetic materials with broken I,\nbut it does not distinguish spin. For ferrimagnetic ma-\nterials, both spin-up and spin-down channels contribute\nto SHG. However, SHG from two spin channels is differ-\nent, which is closely related to the spin-polarized band\nstructures near the Fermi level. For a half-metallic ferri-\nmagnet with nearly 100% spin polarization at the Fermi\nlevel, SHG mainly comes from one spin channel show-\ning metallicity, as schematically displayed in Fig. 1(a).\nMn2RuGa is such a half-metallic ferrimagnet with many\ndifferent structures [27–29]. We choose the most stable\nXAHeusler structure, which belongs to the space group\nF-43m [30], as shown in Appendix A. This structure is\nmore consistent with the experimental results [31]. The\nexperimental lattice parameters are a=b=c= 5.97\n˚A [32]. In Mn 2RuGa, there are two Mn atoms, Mn 1\nand Mn 2, with different Wyckoff positions 4a(0, 0, 0)\nand 4c(1/4, 1/4, 1/4). The positions of Ru and Ga are\n4d(3/4, 3/4, 3/4) and 4b(1/2, 1/2, 1/2), respectively.\nThe magnetic moments of two Mn atoms are M4a= 3.13\nµB, andM4c=−2.29µB, consistent with the prior\nstudy [30], and they are antiferromagnetically coupled.\nHere, the magnetic moment direction is along the zdi-\nrection. It is found that the total magnetic moment is\nMtot= 1.03µB, which satisfies the Slater-Paulingrule as\nreported in the literature [33]. This rule provides a sim-\nple relationshipbetween the totalmagneticmoment Mtot\nand the valence electron Z. For Mn 2RuGa, they satisfy\nMtot=Z−24, and the number of valence electron is 25,TABLE I: The SHG susceptibilities for different materials.\n|χ(2)|and|χ(2)|SPrepresent the absolute values of SHG sus-\nceptibilities without and with spin polarization, respect ively,\nin units of pm/V. The unit of the photon energy is eV.\ncMQWs is the abbreviation of the coupled metallic quantum\nwells.\nMaterial |χ(2)| |χ(2)|SPPhoton energy Reference\nMn2RuGa 3614 0.38 This work\nGaAs 358 1.80 This work\nGaAs 350 1.53 Ref.[35]\nTaAs 3600 1.55 Ref.[36]\nCo3Sn2S21050.05 Ref.[37]\ncMQWs 1500 1.35 Ref.[38]\nBiFeO 315-19 0.80 Ref.[39]\nCaCoSO 6.9 1.17 Ref.[40]\nsoMtotis about 1µB[34].\nIn orderto understand the nonlinear optical properties\nof Mn 2RuGa, we have calculated the second-order non-\nlinear optical susceptibilities. Nonmagnetic Mn 2RuGa\nbelongs to the point group T d. There are six equiva-\nlent nonvanishing SHG susceptibilities χ(2)\nxyz=χ(2)\nxzy=\nχ(2)\nyxz=χ(2)\nyzx=χ(2)\nzxy=χ(2)\nzyx. When the antiferromag-\nnetic coupling appears along the zdirection, the symme-\ntry is reduced from 24 to 8, belonging to the magnetic\npoint group -42m. This changes SHG. The system con-\ntains three independent nonvanishing elements, namely,\nχ(2)\nxyz=χ(2)\nyxz,χ(2)\nxzy=χ(2)\nyzx, andχ(2)\nzxy=χ(2)\nzyx. The SHG\nsusceptibility satisfies the intrinsic permutation symme-\ntry, that is χ(2)\nabc=χ(2)\nacb. Thus, there are only two inde-\npendent nonvanishing elements, χ(2)\nxyz=χ(2)\nxzy=χ(2)\nyxz=\nχ(2)\nyzx, andχ(2)\nzxy=χ(2)\nzyx.\nFor the SHG susceptibility χ(2)\nxyz, the absolute value of\nthe spin-up channel is given by the red solid line in Fig.\n1(b). It is shown that |χ(2)\nxyz|is at the maximum value\nof 11670.76 pm/V when the photon energy approaches 0\neV. Due to the metallic nature for the spin-up channel\nin Mn 2RuGa, the intensity of |χ(2)\nxyz|at 0 eV is not accu-\nrate, which also depends on the damping parameter [see\nAppendix A for more details]. When the photon energy\nis between 0 and 0.22 eV, |χ(2)\nxyz|decreasesmonotonically.\nAt 0.22 eV, the value of |χ(2)\nxyz|is close to zero. As the\nphoton energyincreases, a dramaticpeak appearsat 0.38\neV, as shown in Table I. The intensity of this peak is as\nlarge as 3614.37 pm/V. We also note that this peak is in-\nsensitive to the damping parameter, as discussed in Ap-\npendix A. As the energy further increases, the spectrum\noscillates and gradually decreases, and the final intensity\nis close to zero. By contrast, the spin-down |χ(2)\nxyz|[see\nthe black dashed line in Fig. 1(b)] is much smaller, with\na maximum of about 652.46 pm/V in the entire energy\nrange. This is consistent with the experimental finding\n[41], where only the majority spin channel is optically4\nexcited highly. When the energy is less than 0.2 eV, the\nintensity difference of |χ(2)\nxyz|between the spin-down and\nspin-up channels is largest. In the energy range of 0.2\nto 1.6 eV, the spin-down |χ(2)\nxyz|change is relatively sta-\nble, but the spin-up decreases, so the difference between\nthe two spins decreases. When the energy is larger than\n1.6 eV, the spin-down |χ(2)\nxyz|is very close to the spin-up,\nand the intensity is almost zero. Therefore, we can con-\nclude that SHG in Mn 2RuGa is mainly contributed by\nthe spin-up channel [41], while the contribution of spin-\ndown channel is negligible. As a result, we obtain nearly\n100% spin-polarized SHG in Mn 2RuGa. We also notice\nthat GaAs and Mn 2RuGa share the same point group.\nHowever, as shown in Fig. 1(c), the maximum intensity\nof|χ(2)\nxyz|in GaAs is only about 358.19 pm/V at 1.8 eV,\nwhich is about 10 times smaller than the spin-polarized\nSHG of Mn 2RuGa. The SHG susceptibilities for other\nmaterials are also presented in Table I. We believe that\nsuch a large spin-polarized SHG in Mn 2RuGa has poten-\ntial applications in spin-filter devices [42].\nB. Intraband and interband contributions\nTo reveal insights into this nearly 100% spin-polarized\nSHG, we resort to the band structure of Mn 2RuGa, as\nshown in Fig. 2(a). The red line indicates the spin-up\nchannel, and the black line indicates the spin-down chan-\nnel. The band of spin-up channel crosses the Fermi level\nat multiple kpoints, indicating a metallic state. This re-\nsultcanbeverifiedfromthetotaldensityofstates(DOS),\nas shown in Fig. 2(b). Bands in the energy range from\n−0.08 to 0.12 eV are mainly occupied by dorbitals, in\nwhich Mn-3 dorbitals are dominant, Ru-4 dorbitals con-\ntribute less, and the contribution from Ga-3 dorbitals\nis negligible. However, the spin-down band only touches\nthe Fermi level near the Γ point. The total DOS near the\nFermi level is close to 0. Therefore, Mn 2RuGa is a half\nmetal, which is consistent with previous report [29, 30].\nThe band gap disappears in the spin-up channel, but ap-\npears in the spin-down channel. Therefore, the appear-\nance of band gap hinders the transition of electrons from\nthe valence band to the conduction band.\nNext, we further analyze the difference between the\nspin-up and the spin-down χ(2)\nxyzof Mn 2RuGa by exam-\nining the real and imaginary parts separately. For the\nspin-up channel, the real and imaginary parts of χ(2)\nxyzare\ngiven in red solid and black dashed lines, respectively, as\nshown in Fig. 2(c). The real part of χ(2)\nxyz(Re(χ(2)\nxyz))\ndecreases monotonously as the photon energy increases\nfrom 0 to 0.16 eV. When the photon energy reaches 0.16\neV,itsvaluereachesaminimum of-1099.71pm/V.When\nthe energy is larger than 0.16 eV, the spectrum oscillates\nand finally approaches zero. For the imaginary part of\nχ(2)\nxyz, that is Im( χ(2)\nxyz), a Lorentzian-like resonance ap-\npears in the energy range between 0 and 0.4 eV. Com-\npared with the real part, its first negative peak shiftsΓ L K0.01.0\n-0.5\n-1.0W \b X0.5\n0.03.06.09.0\n-3.0\n-6.0\n-9.0\n0.0 0.5 1.0 1.5 2.0 -0.5-1.0-1.5-2.0DOS (States/eV)\nE (eV) E (eV)(a) (b)Total Mn-3d Ru-4d\nSpin-up Spin-dnW1\nW2\nW3\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(c)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) \n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (d)\n(e) (f)Spin-up Spin-up\nSpin-dn Spin-dnInter (\t) Intra (\n)\nInter (2\u000b) Intra (2\f)\nInter (\r) Intra (\u000e)\nInter (2\u000f) Intra (2\u0010)\n05001000\n-500\n-10001500\n-150004000800012000\n-4000\n-8000\n-12000\nIm(χxyz) (pm/V)(2)Im(χxyz) (pm/V)(2)χxyz (pm/V)(2)χxyz (pm/V)(2)04000800012000\n-4000\n-8000\n-12000\n05001000\n-500\n-10001500\n-1500\nFIG.2: (a)BandstructurewithoutSOCforMn 2RuGa, where\nred line represents the spin-up channel, and black line repr e-\nsents the spin-down channel. The blue circles represent the\ndxy,dyzanddxzorbitals of Mn atoms for the bands W 1, W2\nand W 3. The dashed line denotes the Fermi level. (b) To-\ntal DOS of Mn 2RuGa, where the partial DOS of Mn-3 dand\nRu-4dstates are also given and represented by red and green\nlines, respectively. The upward arrow indicates the spin-u p\nchannel, and the downward arrow indicates the spin-down\nchannel. The vertical dashed line denotes the Fermi level. ( c)\nReal and imaginary parts of the SHG susceptibility χ(2)\nxyzfrom\nthe spin-up channel in Mn 2RuGa without SOC. (d) Calcu-\nlated Im( χ(2)\nxyz) from inter( ω)/(2ω) (black solid, black dashed\ncurve)andintra( ω)/(2ω)(red solid, reddashed-dottedcurve)\nparts. (e) and (f) are similar to (c) and (d), but from the spin -\ndown channel.\nto the lower energy by about 0.06 eV, and the intensity\nis decreased to 7768.47 pm/V. When the photon energy\nfurther increases, the spectrum oscillates.\nIn general, the imaginary part of χ(2)\nxyzreflects the op-\ntical absorption in NLO experiments. From the band\nstructure, the optical absorption involves in the intra-\nand interband currents. Thus, we decompose Im( χ(2)\nxyz)\ninto the inter- and intraband parts for the spin-up chan-\nnel, asshowninFig. 2(d). It clearlyshowsthattheintra-\nband contribution dominates the spectrum in the lower\nphoton energy window from 0 to 0.4 eV. By contrast,\nthe interband contribution from both single- and two-\nphoton resonances is much smaller. In the energy range\nof around 0.4 −2.0 eV, these four spectra are comparable\nand have opposite signs for the single- and two-photon\nresonances, leading to the oscillation of Im( χ(2)\nxyz) in this5\nenergy range. Thus, the negative characteristic peak of\nIm(χ(2)\nxyz) in the lower energy is determined by the in-\ntraband current. In other words, the interband current\ncontributes little to the negative characteristic peak in\nthe lower energy. We know that the negative character-\nistic peak locates in the energy range of 0 −0.4 eV. The\ntwo-photon resonance would correspond to the energy\nrange of around 0 −0.8 eV. In this perspective, the bands\nrelated to the intraband current would locate in the en-\nergy range from −0.4 to 0.4 eV near the Fermi level. As\nshown in Fig. 2(a), we can see that only three bands W 1,\nW2and W 3appear in this energy range. The bands W 1\nand W 2are degenerate along the L-Γ direction, while\nthe bands W 2and W 3are degenerate along the Γ- Xdi-\nrection. More importantly, these three bands disperse\nquadratically along the L-Γ-Xdirection. This means\nthat they highly disperse along this high-symmetry line,\ncontributingalargenormalvelocitytothe intrabandcur-\nrent [43]. This is the reason why the intraband current\ndominates the negative characteristic peak in the lower\nenergy range.\nWealsonoticethatthereexisttworegionsfortheinter-\nbandtransitionsamongthesethreebands. Oneislocated\nnear theLpoint along the L-Γ direction, where the dou-\nble degenerate bands of W 1and W 2form the conduction\nbands while the band W 3is the valence band. The other\ninterband transition appears near the Xpoint along the\nΓ-Xdirection,wherethebandW 1istheconductionband\nwhile the two-fold degenerate bands of W 2and W 3are\nthe valence bands. According to the selection rules, the\ninterband transitions from these two regions are largely\nlimited. This is because these three bands are mainly\nformed by the dxy,dyzanddxzorbitals of Mn atoms\n[see Fig. 2(a) for more details]. The interband transi-\ntions between the same dorbitals are not allowed. As\na result, the interband current contributes little to the\nnegative characteristic peak in the lower energy range.\nThis is generic for metallic ferro- or ferrimagnets as the\nspin-splitting states near the Fermi level are dominated\nby thedorbitals.\nIn the case of spin-down channel, the real and imag-\ninary parts of χ(2)\nxyzare comparable, as shown in Fig.\n2(e). Both oscillate around zero as the photon energy\nincreases. However, they are largely suppressed in com-\nparison with those of spin-up channel. Similarly, we also\ndecompose Im( χ(2)\nxyz) into the inter- and intraband con-\ntributions, as shown in Fig. 2(f). It is found that both\nthe inter- and intraband currents are comparable and os-\ncillate around zero, which are much smaller than those\nof the spin-up channel. This is because only few elec-\ntrons are allowed to transit from the valence band to the\nconduction band near the Γ point. However, the large\nband gap of the spin-down channel limits both the inter-\nand intraband currents. This explains why SHG of the\nspin-down channel is much smaller.C. SHG in metallic Mn 2Ga\nTo further confirm the contribution of SHG from the\nspin-up quadratic bands in Mn 2RuGa, we artificially re-\nmove the Ru atoms in Mn 2RuGa and obtain the crystal\nstructure of Mn 2Ga [30]. In Mn 2Ga, two Mn atoms are\nantiferromagnetic coupled and their magnetic moments\nare close to 3 µB, which can compensate each other. The\nmagnetic moments of Ga atoms are very small, so the to-\ntal magnetic moment of the unit cell is nearly zero. This\ncoincides with the previous report [34]. However, the\nantiferromagnetic coupling in Mn 2Ga has a huge effect\non the spin-polarized band structures, as shown in Fig.\n3(a). It is found that both the spin-up and spin-down\nbands cross the Fermi level, indicating that Mn 2Ga is a\nmetal. This is also confirmed from the PDOS, as shown\nin Fig. 3(b). We can see that obvious DOS exits both\nfor the spin-up and spin-down channels near the Fermi\nlevel, where the Mn-3 dorbitals dominates.\nΓ L K0.02.0\n-1.00.5\n-2.0\nW W\u0011-1.5-0.51.01.5\nSpin-up Spin-dnE (eV)(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.050001000015000\n0\nE (eV) (c)0.03.0\n6 \u0012 \u00139 \u0014 \u0015-3.0- \u0016 \u0017 \u0018\u0019 \u001a \u001b \u001c\n0.0 0.5 1.0 1.5 2.0 -0.5-1.0-1.5-2.0DOS (States/eV)\nE (eV) (b)\n(d)Total Mn-3d\nSpin-dn\nInter (\u001d) Intra (\u001e)\nInter (2\u001f) Intra (2 )\nx ! \" # $ % & ' ( ) * + , . /\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) 03000\n-30000 1 2 3 45 7 8 : ;Im(\n<xyz) (pm/V)(2)|= abc\n>?@ABCD(2)Spin-up\nSpin-dn\n-12000\nFIG. 3: (a) Band structures for Mn 2Ga, where the red and\nblack lines represent the spin-up and spin-down channels, r e-\nspectively. The dashed line denotes the Fermi level. (b) Tot al\nDOS of Mn 2Ga, where the partial DOS of Mn-3 dorbitals is\nalso given and represented by red line. The upward arrow\nindicates the spin-up channel, and the downward arrow indi-\ncates the spin-down channel. The vertical dashed line denot es\nthe Fermi level. (c) The absolute value of the SHG suscep-\ntibilityχ(2)\nabcfor Mn 2Ga, where abcrefers to xyz,xzy,yxz,\nandyzx. The red solid and black dashed lines denote the\nspin-up and spin-down channels, respectively. (d) Calcula ted\nIm(χ(2)\nxyz) from inter( ω)/(2ω)(black solid, black dashed curve)\nand intra ( ω)/(2ω) (red solid, red dashed-dotted curve) parts\nfor the spin-down channel.\nFor Mn 2Ga, SHG has two independent nonvanishing\nelements, namely, χ(2)\nxyz=χ(2)\nxzy=χ(2)\nyxz=χ(2)\nyzx, and\nχ(2)\nzxy=χ(2)\nzyx. The absolute value of χ(2)\nxyzis displayed\nin Fig. 3(c). For the spin-up channel, the first peak\nappears at the photon energy 0.14 eV and its intensity\nis 5602.48 pm/V. Then, the intensity sharply decreases\nand finally approaches zero as the photon energy further6\nincreases. In the case of the spin-down channel, χ(2)\nxyz\nis similar to that of the spin-up channel, but the inten-\nsity is much larger. However, this is contrast to that of\nthe spin-down channel in Mn 2RuGa, where χ(2)\nxyzis much\nsmaller. This is because no band gapappears in the spin-\ndown channel of Mn 2Ga and three metallic bands cross\nthe Fermi level. More importantly, one of them disperses\nquadratically with knear the Γ point, which largely con-\ntributes to the intraband part of χ(2)\nxyz. This implies that\nthe presence of band gap in the spin-down channel of\nMn2RuGa limits χ(2)\nxyz, while the absence of band gap or\nthe quadratic band near the Fermi level enhances χ(2)\nxyz.\nTo this end, we decompose Im( χ(2)\nxyz) of the spin-down\nchannel into the inter- and intraband contributions, as\nshown in Fig. 3(d). It clearly shows that the intraband\ncurrent dominates, while the interband contribution is\nnearly neglected. In addition, the two-photon resonance\nis obviously larger than that of one-photon resonance.\nThis would be easily detected by SHG in experiment.\nD. Role of SOC in SHG in Mn 2RuGa and the\ngroup symmetry\nWith SOC, the spin-up and the spin-down channels\nmix together. The band structure of Mn 2RuGa is shown\nin Fig. 4(a), which almost coincides with the spin-\npolarized band structures. Figure 4(b) shows that DOS\nhas a peak between −0.1 and 0.1 eV, and mainly comes\nfrom the Mn-3 dorbitals, which is the same as the spin-\nup channel. This is because the spin-down DOS is close\nto zero in this energy range. However, in the energy\nrange below −0.1 eV and above 0.1 eV, DOS under SOC\nchanges compared to the spin-up channel, which is due\nto the contribution of the spin-down channel. This will\naffect SHG.\nIn fact, if we only include SOC, but ignore the mag-\nnetic field direction, the symmetries of the system re-\nmain unchanged. In real calculations, SOC is considered\nthrough a tiny magnetic field applied along the z-axis.\nThe symmetry is reduced and belongs to the magnetic\npoint group of -4. The remaining four symmetry op-\nerations are the identity operation E, the twofold rota-\ntional symmetry C2zwith the binary axis as the zaxis,\nand two combination operations IC4zandIC−1\n4z.IC4z\ndenotes the rotation of π/2 around the z-axis, followed\nby a mirror symmetry σxy.IC−1\n4zis similar to IC4z,\nbut with a rotation −π/2 around the z-axis. As a re-\nsult, there are six independent nonvanishing elements,\nχ(2)\nxyz=χ(2)\nyxz,χ(2)\nxzy=χ(2)\nyzx,χ(2)\nzxy=χ(2)\nzyx,χ(2)\nxxz=−χ(2)\nyyz,\nχ(2)\nxzx=−χ(2)\nyzy, andχ(2)\nzxx=−χ(2)\nzyy. Based on the in-\ntrinsic permutation symmetry, only four components in-\ndeed appear, χ(2)\nxyz=χ(2)\nxzy=χ(2)\nyxz=χ(2)\nyzx,χ(2)\nzxy=χ(2)\nzyx,\nχ(2)\nxxz=χ(2)\nxzx=−χ(2)\nyyz=−χ(2)\nyzy, andχ(2)\nzxx=−χ(2)\nzyy. It\nshould be noted that χ(2)\nxxzandχ(2)\nzxxare induced from the\nreduced magnetic point group.Γ L K0.01.0\n-0.5\n-1.0\nW W X0.5E (eV)(a)\n0.03.0\nE F GH I J0.0 0.5 1.0 1.5 2.0 -0.5-1.0-1.5-2.0DOS (States/eV)\nE (eV) (b)\nTotal Mn-3d Ru-4d\n0.0 0.5 1.0 1.5 2.0 2.5 3.03000\nK L M NO P Q R0\nE (eV) (c)S T U V Y Z [ \\ ] ^ _ ` a b c\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (d)d e f g h i j k l m n o p q r s tz u v w y { } ~       \n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(e)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (f)\nInter ( ) Intra ( )\nInter (2 ) Intra (2 )020004000\n   \n-2000\n-4000    020004000\n   \n-2000\n-4000    \n abc\n¡¢£¤(2)¥¦ abc\n§¨©ª«¬(2) Im(\n®xxz) (pm/V)(2)¯xxz (pm/V)(2)3000\n° ± ² ³´ µ ¶ ·0\nFIG. 4: (a) Band structure for Mn 2RuGa under SOC. The\ndashed line denotes the Fermi level. The rectangles denote\nthe flat valence and conduction bands and the arrow labels\nthe transitions between those flat bands. (b) Total DOS of\nMn2RuGa, where the partial DOS of Mn-3 dand Ru-4 dstates\nare also given and represented by red and green lines, re-\nspectively. The vertical dashed line denotes the Fermi leve l.\n(c) and (d) The absolute values of the SHG susceptibility\nχ(2)\nabcfor Mn 2RuGa under SOC. (e) Real and imaginary parts\nofχ(2)\nxxz. (f) Calculated Im( χ(2)\nxxz) from inter( ω)/(2ω) (black\nsolid, black dashed curve) and intra ( ω)/(2ω) (red solid, red\ndashed-dotted curve) parts.\nFigure 4(c) shows the absolute values of SHG suscep-\ntibilitiesχ(2)\nxyzandχ(2)\nzxy. It clearly shows that those two\nSHG spectra are nearly the same as that of the spin-\nup channel [see Fig. 1(d)]. This is because the contri-\nbution from the spin-down channel is negligible. Thus,\nSOC has little effect on these six SHG spectra, which\nalso appear in the spin-polarized case. It is also found\nthat the vanishing SHG spectra χ(2)\nxxzandχ(2)\nzxxin the\nspin-polarized case are recovered now, as shown in Fig.\n4(d). The intensity of χ(2)\nzxxis obviously larger than that\nofχ(2)\nxxz. The appearance ofthem directly comes from the\nreduced magnetic point group. In the following, we use\nthe four remaining symmetries to understand the nonva-\nnishing SHG spectrum χ(2)\nxxz. The matrix representations\nofEandC2zare diag{1,1,1}and diag {−1,−1,1}. The7\nother twoIC4zandIC−1\n4zare\nIC4z=\n0 1 0\n−1 0 0\n0 0 −1\n,IC−1\n4z=\n0−1 0\n1 0 0\n0 0 −1\n.\n(9)\nThe transformations of position operator under these\nfour symmetry operations are as follows. E: (x,y,z)→\n(x,y,z),C2z: (x,y,z)→(−x,−y,z),IC4z: (x,y,z)→\n(y,−x,−z), andIC−1\n4z: (x,y,z)→(−y,x,−z). As a\nresult, we can get:\nE:χ(2)\nxxz→χ(2)\nxxz,\nC2z:χ(2)\nxxz→χ(2)\n(−x)(−x)z=χ(2)\nxxz,\nIC4z:χ(2)\nxxz→χ(2)\nyy(−z)=−χ(2)\nyyz,\nIC−1\n4z:χ(2)\nxxz→χ(2)\n(−y)(−y)(−z)=−χ(2)\nyyz.(10)\nIt clearly shows that these four SHG susceptibilities do\nnot cancel out each other. We can obtain these induced\nSHG spectra via the above symmetry analysis, where\nχ(2)\nxxzandχ(2)\n−yyzare protected by C2zandIC4z, respec-\ntively.\nThe reduced magnetic point group is closely related\nto SOC and the applied magnetic filed direction. This\nmeans that the induced SHG spectra must have a deep\nrelation to them. Here, we take χ(2)\nxxzto reveal the un-\nderlying physics. The decomposed real and imaginary\nparts ofχ(2)\nxxzare displayed in Fig. 4(e). We can see\nthat both have a similar manner. Sizable intensities are\nmainly located in the photon energy range of 0 −0.75 eV.\nTo understand the contributions of inter- and intraband\ncurrents, we decompose Im( χ(2)\nxxz) into the inter- and in-\ntraband parts, as shown in Fig. 4(f). There also exists a\nlarge contribution from the intraband current, where the\nhighly dispersed bands near the Fermi level play a role.\nIt should be noted that the interband contributions are\nlargely enhanced, which are nearly limited in the spin-\npolarized case. We find that many transitions are now\nallowed between the flat valence and conduction bands,\nas shown in rectangles of Fig. 4(a). However, those tran-\nsitions are not allowed in the spin-polarized case. This is\nbecause the flat valence bands are originally spin-up po-\nlarized, while those flat conduction bands are spin-down\npolarized. The direct transition from the spin-up band\nto the spin-down band is forbidden. But, this does occur\nunder SOC as the conservation of spin is not needed. To\ncheck it, we calculate some matrix elements of the posi-\ntionoperatorbetweenthoseflatbandsatseveral kpoints,\nas shown in Table II. We can see that the y-componentof\nmatrix elements are small but not zero. However, the x-\nandz-components are much larger. The maximum abso-\nlute value reaches as large as 8.02 a 0with a 0being the\nBohrradius. Thesenonzeromatrixelementsconfirmthat\nthe transitions are not allowed in the spin-polarized case,\nbut did occur with SOC. This tells us that the reduced\nmagnetic point group or the remaining four symmetriesprotect the induced SHG spectra, which is similar to our\nprevious study [44].\nTABLE II: The matrix elements of position operator between\nthose flat bands near the Fermi level under SOC in Mn 2RuGa\nfor three kpoints, where the atomic unit is used.\nkpointx y z\nRe Im Re Im Re Im\n(0,0.139,0) -0.83 0.32 0.05 0.12 -0.68 -1.74\n(0,0.209,0) -1.07 -2.34 0.01 -0.01 -2.49 1.13\n(0,0.278,0) 1.93 2.45 -0.09 0.07 6.29 -4.96\nIV. CONCLUSIONS\nIn conclusion, we have demonstrated that large nearly\n100% spin-polarized SHG carries rich information about\nthe electronic structures of the half-metal Mn 2RuGa.\nThe band gap in the spin-down channel limits SHG.\nIn contrast, the spin-up channel is metallic and gives\nrise toχ(2)\nxyzas large as 3614 pm/V, which is about 10\ntimes largerthan that of typical nonlinear materials such\nas GaAs. For the spin-up channel, the intraband cur-\nrent mainly contributes to χ(2)\nxyz, which stems from three\nhighly dispersed bands near the Fermi level. In addition,\nSOC under the z-axis magnetization field induces addi-\ntionalSHG susceptibilities suchas χ(2)\nxxzfrom the reduced\nmagnetic point group, where the interband transitions\ndominate. Our study would provide a good guide in fu-\nture application of largespin-polarized SHG in spin-filter\ndevices.\nACKNOWLEDGMENTS\nThis work was supported by the National Science\nFoundationofChinaunderGrantNo. 11874189. Wealso\nacknowledge the Fermi cluster at Lanzhou University for\nproviding computational resources. GPZ was supported\nby the U.S. Department of Energy under Contract No.\nDE-FG02-06ER46304. The research used resources of\nthe National Energy Research Scientific Computing Cen-\nter,whichissupportedbytheOfficeofScienceoftheU.S.\nDepartment of Energy under Contract No. DE-AC02-\n05CH11231.\n∗sims@lzu.edu.cn\n†guo-ping.zhang@outlook.com\nAPPENDIX A: The crystal structure of Mn 2RuGa\nand the effect of damping parameters on SHG\nHeusler alloyshavethree different structures belonging\nto different space groups [45]. The normal full-Heusler\nX2YZalloys belong to group symmetry L2 1(No. 225).8\n0.0 0.5 1.0 1.5 2.0 2.5 3.050001000015000\n0\nE (eV) Spin-up(b)\n|χxyz| (pm/V)(2)η¸ ¹ º » ¼ ½ ¾¿ À Á  à Ä\nηÅ Æ Ç È É Ê ËÌ Í Î Ï Ð Ñ\nη Ò Ó Ô Õ Ö × ØÙ Ú Û Ü Ý Þ200002500030000\nz\nxyMn1\nMn2Ruß à(a)\nFIG. 5: (a) Crystal structure of Mn 2RuGa. Purple, or-\nange, grey, and green spheres represent Mn 1, Mn2, Ru, and\nGa atoms, respectively. The arrows marked on the Mn atoms\nindicate the local magnetic moments, forming the antiferro -\nmagnetic configuration. (b) The absolute value of the spin-u p\nχ(2)\nxyzof Mn 2RuGa without SOC for different damping param-\netersη.\nThe Half-Heusler XYZcompounds have group symme-\ntryC1b(No. 216), and the inverse-Heusler X2YZalloys\nwith group symmetry XA(No. 216) [28]. We select\na stable one, that is XAstructure, as our example, as\nshown in Fig. 5(a), where the sublattice Mn 2RuGa are\nferrimagnetic ordering. The magnetic moments of these\nsublattice Mn atoms are 3.13 and -2.29 µB, respectively.\nAs aresult, the net magneticmoment ofunit cellisabout\n1µB, which agrees well with the Slater-Pauling rule.\nDue to the metallic nature for the spin-up channel in\nMn2RuGa, the nonlinear optical response at zero fre-\nquency must be estimated, because the energy difference\n¯hωnm=En−Emis zero for those metallic states near\nthe Fermi level. As a result, ωnm−ωorωnm−2ωin\nthe denominators of Eqs. (4)-(7) in the main text di-\nverges. To avoid this, the damping parameter ηis in-\ntroduced to estimate |χ(2)|at zero frequency. However,\nthe obtained results are still not accurate. As shown in\nFig. 5(b), we can see that the intensity of the spin-up\n|χ(2)\nxyz|at 0 eV largely depends on η. Whenη= 0.004\nHartree, the intensity of spin-up |χ(2)\nxyz|at 0 eV is about\n6150.37 pm/V. When ηis decreased to 0.003 Hartree,\nthe value is about twice larger than that for η= 0.004\nHartree. When ηis further decreased to 0.002 Hartree,\nthe intensity of spin-up |χ(2)\nxyz|is dramatically increased.\nThis means that |χ(2)|usually diverges at zero frequency.\nIn other words, it is a challenge to accurately compute\n|χ(2)|at zero frequency. This is also the case of linear\nresponse in Hall effect. According to the Drude model,\nthe frequency-dependent conductivity σ(ω) is given by,\nσ(ω) =σ0\n1−iωτ, (11)\nwhereσ0is the DC Drude conductivity without the ex-\nternal magnetic field and τis the relaxation time. At\nzero frequency, we can see σ(ω) reduces to σ0=ne2τ/m\nwithn,e, andmbeing the electron density, the electron\ncharge, and the electron mass, respectively. This shows\nthe failure of zero-frequency response already exists in\nthe linear response, which would be our future researchfocus of SHG.\nWhen the photon energy is further increased to about\n0.38 eV, a stable peak of spin-up |χ(2)\nxyz|with respect to\nηappears. The intensity of this peak increases as η\ndecreases. But the change is small. This implies that\nthe intensity of this second peak is insensitive to η. We\nalso note that the intensity of this peak corresponding to\nη= 0.003Hartree is as large as3614.37pm/V, which can\nbe compared with SHG spectra of other materials such\nas GaAs, TaAs, and CaCoSO [see Table I].\nAPPENDIX B: The SHG susceptibility for\nMn2RuGa without SOC\n0.0 0.5 1.0 1.5 2.0 2.5 3.030006000900012000\n0\nE (eV)Spin-up\nSpin-dn\ná â ã ä å æ çèχabc\néêëìíîï(2)\nFIG. 6: The absolute value of the SHG susceptibilities χ(2)\nabc\nfor Mn 2RuGa without SOC. The red solid and black dashed\nlines denote the spin-up and spin-down channels, respectiv ely.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) \n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (b)\n(c) (d)Spin-up Spin-up\nSpin-dn Spin-dnInter (ω) Intra (ω)\nInter (2ω) Intra (2ω)\nInter (ω) Intra (ω)\nInter (2ω) Intra (2ω)\n05001000\n-500\n-10001500\n-150004000800012000\n-4000\n-8000\n-12000\nIm(χzxy) (pm/V)(2)Im(χzxy) (pm/V)(2)χzxy (pm/V)(2)χzxy (pm/V)(2)04000800012000\n-4000\n-8000\n-12000\n05001000\n-500\n-10001500\n-1500\nFIG. 7: (a) The real and imaginary parts of the SHG sus-\nceptibility χ(2)\nzxyfrom the spin-up channel in Mn 2RuGa with-\nout SOC. (b) Calculated Im( χ(2)\nzxy) from inter( ω)/(2ω) (black\nsolid, black dashed curve) and intra ( ω)/(2ω) (red solid, red\ndashed-dotted curve) parts. (c) and (d) are similar to (a) an d\n(b), but from the spin-down channel.\nNext, theabsolutevalueofthe SHGsusceptibility χ(2)\nzxy\nfor Mn 2RuGa without SOC is given in Fig. 6. For the\nspin-upchannel,theintensityatzerofrequencyis9936.41\npm/V, which is smallerthan that of χ(2)\nxyz. The secondary9\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(c)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (d)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(e)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (f)0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (b)Inter (ω) Intra (ω)\nInter (2ω) Intra (2ω)\nInter (ω) Intra (ω)\nInter (2ω) Intra (2ω)\nInter (ω) Intra (ω)\nInter (2ω) Intra (2ω)4000800012000\n0\n-8000\n-12000-4000χxyz (pm/V)(2)χzxy (pm/V)(2)χzxx (pm/V)(2)\nIm(χxyz) (pm/V)(2)Im(χzxy) (pm/V)(2)Im(χzxx) (pm/V)(2)4000800012000\n0\n-8000\n-12000-4000\n4000800012000\n0\n-8000\n-12000-4000\n4000800012000\n0\n-8000\n-12000-40004000800012000\n0\n-8000\n-12000-4000\n4000800012000\n0\n-8000\n-12000-4000\nFIG. 8: (a) The real and imaginary parts of χ(2)\nxyz. (b) Calcu-\nlated Im( χ(2)\nxyz) from inter( ω)/(2ω) (black solid, black dashed\ncurve)andintra( ω)/(2ω)(redsolid, reddashed-dottedcurve)\nparts. (c) and (d) are similar to (a) and (b), but for χ(2)\nzxy. (e)\nand (f) are similar to (a) and (b), but for χ(2)\nzxx.\npeak appears at 0.41 eV with an intensity of 3452.36\npm/V. When the photon energy is larger than 0.59 eV,the spectrum oscillates and then approaches to 0. In\ncontrast,the intensityofthespin-down χ(2)\nzxyisverysmall\nin the energy range from 0 to 3 eV. Therefore, in the low\nenergy region, the intensity of the spin-up χ(2)\nzxyis much\nstronger than that of the spin-down channel, which is\nsimilar toχ(2)\nxyz. This is because the spin-up channel has\nno band gap, while a gap in the spin-down channel limits\nSHG.\nThe real and imaginary parts of the spin-up channel\nforχ(2)\nzxyis shown in Fig. 7(a). We decompose Im χ(2)\nxyz\nof the spin-up channel into the inter- and intraband con-\ntributions, as shown in Fig. 7(b). It is found that the\nintraband contributions are dominant in the low energy\nregion. For the spin-down channel, the real and imag-\ninary parts are very small, as shown in Fig. 7(c). It\nmainly comes from the intraband and interband tran-\nsitions, as shown in Fig. 7(d). These results of both\nspin-up and spin-down channels are similar to χ(2)\nxyz.\nAPPENDIX C: The SHG susceptibility for\nMn2RuGa with SOC\nWith SOC, χ(2)\nxyzandχ(2)\nzxyare similar to the spin-up\nχ(2)\nxyzandχ(2)\nzxy, respectively, as shown in Figs. 8(a)-8(d).\nIt shows that SOC has little effect on these two compo-\nnents. On the contrary, the reduced symmetries induce\nχ(2)\nxxzandχ(2)\nzxx, showing different characteristics. The\nreal and imaginary parts of χ(2)\nzxxare shown in the red\nand black lines in Fig. 8(e). It is contributed by inter-\nand intraband transitions, as shown in Fig. 8(f). Com-\npared with the spin-polarized χ(2)\nxyzandχ(2)\nzxy, the inter-\nband contribution increases, which is similar to χ(2)\nxxz.\n[1] Y. R. Shen, The Principles of Nonlinear Optics (Wiley,\nNew York, 1984).\n[2] R. W. 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Phys. 113, 17101 (2013)."    },    {        "title": "2004.10475v2.Magnetic_correlations_in_polycrystalline___mathrm_Tb__0_15_Co__0_85___.pdf",        "content": " \n \n 1  \n  Magnetic correlations in polycrystalline Tb 0.15Co0.85 \nMathias Bersweiler1, Philipp Bender1, Inma Peral1, Lucas Eichenberger2,  \nMichel  Hehn2, Vincent Polewczyk3, Sebastian Mühlbauer4, and Andreas Michels1 \n1 Department of Physics and Materials Science, University of Luxembourg, 162A Avenue de la \nFaïencerie, L -1511 Luxembourg, Grand Duchy of Luxembourg  \n2 Institut Jean Lamour (UMR CNRS 7198), Université de Lorraine, 54000 Nancy, France  \n3 Istituto Officina dei Materiali (IOM) -CNR, Laboratorio TASC, 34149, Trieste, Italy  \n4 Heinz Maier -Leibnitz Zentrum (MLZ), Technische Universität München, D -85748 Garching, \nGermany  \n \nE-mail:  mathias.bersweiler @uni.lu , andreas.michels@uni.lu  \n \nSubmitted to Journ al of Physics D: Applied Physics  \nReceived 5 February 2020  \nAccepted for publication 21 April 2020  \nAccepted Manuscript online 21 April 2020  \n \nDoi: https://doi.org/10.1088/1361 -6463/ab8b95  \nAbstract  \nWe investigated a  polycrystalline sample of the ferrimagnetic compound Tb 0.15Co0.85 by magnetometry and small -angle \nneutron scattering (SANS). The magnetization curve at 300 K is characteristic for soft ferrimagnets but at 5 K the hysteresis  \nindicates the exi stence of magnetic domains. The magnetic SANS signal suggests that at 300 K the Tb and Co moments are \ncorrelated over large volumes within the micrometer -sized grain s with correlation lengths > 100 nm. At 5 K, however, the \nmagnetic SANS analysis reveals a reduced correlation length of around 4.5 nm, which indicates the formation  of narrow \nmagnetic domains  within the ferrimagnet with one d imension being in the nm range. We attribute the observed changes of the \ndomain structure to the tempera ture-dependence o f the magnetic properties of the Tb sublattice.  \nKeywords: ferrimagnetism , magnetic domains , small -angle neutron scattering  \n \n1. Introduction  \nOver the last decades ferrimagnetic rare -earth transition -\nmetal alloys (RE -TM) raise d a lot of attention, since they are \ninteresting materials for fundamental research  [1–4] and \npromising candidates for technological applications, such as \nmagneto -optical recording media  [5], permanent magnets  [6], \nor spintronic devices  [7–10]. Previously, all -optical switching \nhas been observed in some selected RE -TM alloys  [11–13], \nrendering them suitable candidates for optically -controlled \nmagnetic data storage devices. More recently, Mangin et \nal.  [14] have demonstrated that all -optical helicity -dependent \nswitching can be extended to more complex, multilayered RE -\nTM systems containing for example HoFeC o, DyCo, or TbCo. \nMost of the studies are focused on amorphous  RE-TM alloys, \nsince their fabrication is relatively easy and their magnetic properties can be straightforwardly controlled by changing the \nconcentrations, the nature of RE and TM, or the \ntemper ature  [15,16] . It is well established that amorphous RE -\nTM alloys exhibit a noncollinear spin structure, the so -called \nsperimagnetic structure  [17,18] , where the magnetic moments \nare frozen into random orientations. In contras t to the \namorphous alloys, in crystalline  binary intermetallic \nferrimagnetic RE -TM alloys, it is assumed that the magnetic \nmoment of RE and TM are antiparallel coupled and form a \nferrimagnetic collinear arrangement  [19–22].  \nThe goal of the  present work is to investigate the structural \nand magnetic properties of a binary intermetallic Tb -Co \nferrimagnetic alloy, one of the most promising candidate \nsystem for the next generation of magnetic memories based on \nall-optical switching  [14]. In particular, we study  the \ntemperature dependence of the magnetic properties using \nconventional magnetometry combined with magnetic field - \n 2  \n dependent unpolarized small -angle neutron scattering \n(SANS). Magnetic SANS is a very powerful technique which \nprovides volume -averaged information about variations of the \nmagnetization vector field on a mesoscopic length scale of ~ \n1-300 nm  [23,24] . This method has previously been applied to \nstudy the structures of magnetic nanoparticles  [25–32], soft \nmagnetic nanocomposites  [33,34] , proton domains  [35–37], \nmagnetic steels  [38–42], or Heusler -type alloys  [43–46]. \nHere, we aim to estimate the temperature de pendence of the \nmagnetic correlation length in a polycrystalline  bulk Tb-Co \nferrimagnetic alloy . \n2. Methods  \nThe polycrystalline Tb 0.15Co0.85 sample has been prepared \nby arc melting under a high -purity argon atmosphere starting \nfrom stoichiometric quantitie s of the two high -purity elements \n(> 99.9% wt. % from Alfa Caesar). The mixture was melted \nin a water -cooled copper crucible and was not annealed after \nmelting. The sample was then ground manually, compacted to \na pellet, and enclosed in a silica tube under  purified argon to \nprevent oxidation. The structural properties were determined \nby X -ray wide -angle diffraction of the powder using a Bruker \nD8 DISCOVER diffractometer with a Co -Kα radiation source. \nThe magnetic analysis was performed on a pellet using a \nPhysical Property Measurement System (PPMS) from \nQuantum Design (from 350 K to 5 K in applied magnetic \nfields up to 4 T). Thermomagnetization M(T) curves and \nhysteresis loops M(H) were recorded after cooling down the \nsample under a constant magnetic field of 4 T. The SANS \nexperiments were also performed on a circular pellet, in this \ncase with a diameter of 8 mm and a thickness of 1.2 ± 0.1 mm. \nThe neutron experiments were performed at the instrument \nSANS -1 at the Heinz -Maier -Leibnitz Zentrum (MLZ), \nGarching , Germany  [47]. The measurements were done using \nan unpolarized incident neutron beam with a mean wavelength \nof λ = 4.51 Å and a wavelength broadening of Δλ/λ = 10 % \n(FWHM). The measurement s were conducted at room (300 K) \nand low temperature (5 K) and within a q-range of 0.06 nm-1 \n≤ q ≤ 3.0 nm-1. A magnetic field H0 was applied perpendicular \nto the incident neutron beam ( H0 ⊥ k0). Neutron data were \nrecorded at the maximum field available (4 T) and then in the \nremanent state (0 T). The neutron -data reduction (correction \nfor background scattering, sample transmission, and detector \nefficiency) was performed using the GRASP software \npackage  [48]. \nIn the neutron data analysis (see below), the magnetic  \nSANS cross section is discussed when the total (nuclear + \nmagnetic) SANS cross section at the highest field (near to \nsaturation) is subtracted from the total cr oss section at a lower \nfield. This procedure assumes that the nuclear SANS cross \nsection is independent of the applied magnetic field. \nTherefore, it is useful to explicitly display the total and the \npurely magnetic (difference) SANS cross sections.  When t he applied magnetic field is perpendicular to the \nincident neutron beam ( H0 ⊥ k0), the elastic total (nuclear + \nmagnetic) unpolarized SANS cross section d Σ/dΩ and the \npurely magnetic SANS cross section d ΣM/dΩ are given as:  \n \n𝑑𝛴\n𝑑𝛺(𝒒)=8𝜋3\n𝑉𝑏H2(𝑏H−2|𝑁̃|2+|𝑀̃𝑥|2+|𝑀̃𝑦|2cos2(𝜃)\n+|𝑀̃𝑧|2sin2(𝜃)\n−(𝑀̃𝑦𝑀̃𝑧∗+𝑀̃𝑦∗𝑀̃𝑧)sin(𝜃)cos(𝜃))      (1) \n \n𝑑𝛴𝑀\n𝑑𝛺(𝒒)=8𝜋3\n𝑉𝑏H2(𝛥|𝑀̃𝑥|2+𝛥|𝑀̃𝑦|2cos2(𝜃)\n+𝛥|𝑀̃𝑧|2sin2(𝜃)\n−𝛥(𝑀̃𝑦𝑀̃𝑧∗+𝑀̃𝑦∗𝑀̃𝑧)sin(𝜃)cos(𝜃))   (2) \n \nwhere V is the scattering volume, bH = 2.91 x 108 A-1m-1 relates \nthe atomic magnetic moment to the atomic magnetic scattering \nlength, 𝑁̃(𝒒) and 𝑴̃(𝒒)=[𝑀̃𝒙(𝒒),𝑀̃𝒚(𝒒),𝑀̃𝒛(𝒒)] represent \nthe Fourier transforms of the nuclear scattering length density \nN(r) and of the magnetization vector field M(r), respectively,  \nθ specifies the angle between H0 and q  q{0, sin ( θ), cos ( θ)} \nin the small -angle approximation, and the asterisks “*” denote \nthe complex conjugated quantities. For small -angle scattering \nthe component of the scattering vector along the incident \nneutron beam, here qx, is smaller than the other two \ncomponents, so that only correlations in the plane \nperpendicular to the incoming neutron beam are probed. The \n’s in equation  (2) represent the difference between the \nFourier components at a certain applied fiel d and the highest \nfield of 4 T, which is subtracted in the data analysis. More \ndetails about the magnetic SANS technique can be found in \nRefs.  [23,49] . \n3. Results  \n3.1 XRD and SEM  \nFigure 1 shows the X -ray diffraction results and displays a \nscanning electron microscopy (SEM) image s of the powder. \nThe SEM image s show  that the primary particles (i.e., grains) \nare several μm in size. The XRD analysis confirms that \nTb0.15Co0.85 crystallizes in the hexagonal CaCu 5 structure -type \nwith the space group P6/mmm, indicat ing a pure single phase \nTbCo 5. This is expected from the hypothetical phase diagram \nof Tb xCo1-x and for a composition of x = 0.15  [50]. Moreover,  \nthe XRD pattern exhibits  no impurity peaks , which confirms \nthe high -quality synthesis of the Tb -Co alloy by arc melting. \nThe lattice -parameter values a and c were determined by the \nLe Bail fit method (LBF) implemented in the Fullprof \nsoftware  [51].The values obtained from the XRD refinement \n(a ≈ 0.49 3 nm and c ≈ 0.40 1 nm) are consistent with the values \ntypically obtained in TbCo 5 alloy  [50,52] . Furthermore, no \nadditional broadening of the diffraction peaks (apart from \ninstrumental broadening) are observed which verifies that the \ncrystallites are at least 100 nm in si ze.  \n 3  \n  \n3.2 Magnetometry  \nFigure 2(a) shows the magnetization curves at 300 K and 5 \nK. At 300 K, the measured hysteresis loop is similar to that \nexpected for a soft polycrystalline ferrimagnet with randomly \ndistributed anisotropy axis. By cooling down to 5 K, the \nmagnetization curve significantly changes, namely , the \nmagnetization is strongly reduced over the whole field range \nas compared to 300 K, and the shape of the hysteresis is \ndistinctly different.  \nThe characteristic shape of the hysteresis measured at 5 K \n(zoom in figure  2(b)) indicates that the reversal becomes \ndominated by the nucleation (i.e., the jump of M at small \nreversal fields) and propagation of magnetic domains (i.e., the \nshearing of the h ysteresis at intermediate fields). In fact, the \nmagnetization curve is qualitatively similar to that obtained in \nsynthetic antiferromagnetic magnetic systems whose field \nreversal behavior has been correlated to the collective \npropagation of magnetic stripe  domains (see figure  3(a) in \nRef. [53]).  \nThe temperature dependence of the total magnetization , \nmeasured under a cooling -field of 4 T, is displayed in figure  \n2(c). By decreasing the temperature the total magnetization \ndecreases, as expected by considering the negative exchange \ncoupling  between the Co and Tb sublattices (ferrimagnetic) \nand the temperature dependences of the Co and Tb magnetic \nmoments within the TbCo 5 crystal structure. As shown in \nRef.  [54], in case of Tb -Co alloys having a TbCo 5 structure, \nthe magnetization of the Co sublatt ice remains roughly \nconstant over the temperature range 2 -400 K, whereas the \nmagnetization of the Tb sublattice increases significantly with \ndecreasing temperature, which consequently results in a \nreduction of the total magnetization.  \n3.3 Magnetic Small -angle neutron scattering  \nFigure s 3(a) and (b) display the two dimensional (2D) total \n(i.e., nuclear + magnetic) SANS cross sections at 300 K and at \n5 K, respectively , while figure  4 features the corresponding \n(over 2) azimuthally -averaged 1D SANS cross sections. As \ncan be seen, the total 2D SANS cross sections d Σ/dΩ are only \nweakly field -dependent and isotropic, which suggests the \ndominance of the isotropic nuclear scattering contribution. \nAccording to magnetometry (see figure  2(a)), the sample is \nnearly  magnetically saturated at a field of 4 T for both \ntemperatures. Therefore, assuming a field -independent and \nisotropic nuclear SANS cross section d Σnuc/dΩ, the 1D sector \naverage of the total SANS cross section parallel to the applied \nfield ( q // H0) at 4 T  is a good approximation for the nuclear \nSANS cross section d Σnuc/dΩ [compare equation  (1)]. The in \nthis way estimated 1D d Σnuc/dΩ [red filled circles in figure  5] \nexhibit an asymptotic q-4 Porod behavior at the smallest \nmomentum transfers. This indicates scattering due to large -scale structures (e.g., grains or pores), which lie outside of the \nexperimentally accessible q-range (> 2 /qmin  100 nm). This \nis expected for the studied  sample that consists of μm sized \ngrains (compare the results of the XRD anal ysis in figure  1). \nThe 2D magnetic SANS cross section d ΣM/dΩ in the \nremanent state at both temperatures [compare Eq. (2)]  is \ndetermined by subtracting the 4 T data from the measurements \nat zero field . This data reduction procedure has already been \nused to extract the purely magnetic SANS cross section of \nnanoparticle systems  [55,56] . The obtained 2D d ΣM/dΩ are \ndisplayed in figures 3(c) and (d)  for 300 K and 5 K , \nrespectively. With reference to equation  (2) it is reemphasize d \nthat the 2D magnetic cross sections d ΣM/dΩ contain in the \nsector perpendicular to the field (vertical sector) the difference \nof 𝑀̃𝑧(𝒒) at zero field and at 4 T. On the other hand, the sector \naverage parallel to the field contains the corresponding \ndifference s between the transverse magnetization Fourier \ncoefficients 𝑀̃𝑥(𝒒) and 𝑀̃𝑦(𝒒) at zero field and at 4 T. Thus, \nanalysis of this sector allows to access the transversal \nmagnetic correlation lengths in the remanent state. The 1D \nmagnetic cross  sections at 300 K and 5 K, which are obtained \nby integration along the field direction over an angular range \nof ± 20°, are displayed in figure  5. At 300 K, the q-dependence \nof dΣM/dΩ is similar to that obtained for d Σnuc/dΩ ∝ q-4 (at the \nsmallest momentum transfers). This suggests the presenc e of \nlarge magnetic spin -correlation lengths ( lC > 100 nm), lying \noutside of the measured q-range. By contrast, at 5 K, a \ndeviation from the q-4 dependence can be discerned below 0.2 \nnm-1. This q-dependence can be described using a Lorentzian -\nsquared funct ion (blue solid line in figure  5) from which an \nestimate for the transversal magnetic correlation length of lC = \n4.5 ± 0.3 nm  is obtained . As discussed by Hellman et al.  [57], \na Lorentzian -squared term in magnetic SANS data may be \nattributed to meandering domain walls with lC being a measure \nfor the domain size. The magnetization data at 5 K ( figure  \n2(a)) together with the estimated nanoscale transversal \ncorrelation length are compatible with this result.  \n4. Discussion  \nWe surmise that  the formation of narrow magnetic domain s \nobserved at 5 K is connected to the temperature dependence \nof the magnetic anisotropy  in TbC o5. The magnetic anisotropy \nof RE -TM systems is determined by  the magnetic anisotropy \nof both sublattices  (here the Tb and Co sublattice s). In \nRef. [54], it could be shown that the magnetic properties of the \nCo sublattice barely change within the temperature range 300 –\n5 K. Therefore, it can be assumed that the temperature -\ndependenc y of the magnetic  properties of Tb Co5 is dominated  \nby the one of the Tb sublattice . The magnetic anisotropy of the \nTb sublattice  depends of the inter -sublattice exchange energy \nand on the easy-magnetization direction  [58], which are both \nvery sensitive regarding temperature . The magnetization of \nthe Tb sublattice significantly  increases with decreasing \ntemperature  [54], so that  it can  be assumed that also the  \n 4  \n magnetic anisotropy constant K increases accordingly  [59]. \nAn increase of K favors the formation of narrow domain walls , \nsince the domain -wall width 𝛿w is proportional to  1√𝐾⁄  [60]. \nAs reported for Tb2Co17 [61], narrow domain walls are \ndifficult to move and thus may qualitatively explain the \nobservation of narrow domains at low temperature in TbCo 5. \nAt room temperature, on the other hand, the magnetic \nanisotropy of Tb is expected  to be weak due to the increased \nthermal fluctuation of the magnetic moments of the Tb \nsublattice and , thus, 𝛿w is expected to become larger than at 5 \nK (the temperature dependence of the magnetic anisotropy in \nTbCo 5.1 suggests that K can vanish  around 300  K [62]). \nTherefore, at 300 K, the system may rather favor a correlated \nsingle -domain  structure  within the grains. This feature \nqualitatively explains the soft ferrimagnet ic behavior of the \nmagnetization and the observation of a large spin -correlation \nlength by magnetic SANS , lying outside of the measured q-\nrange at room temperature.  Further neutron  studies, for \ninstance, magnetic -field-dependent polarized SANS and very \nsmall -angle neutron scattering (providing access to lower \nmomentum transfers) are required to shed light on the precise \nnature of the observed correlation lengths : In agreement with \nprevious neutron work  [57], we interpreted the origin of the \ncorrelation lengths with the domain size, although it has to be \nconsidered that the correlation length could  also be attributed \nto the domain walls.  \n \n5. Conclusion  \nTo summari ze, we employed  magnetometry and \nunpolarized SANS to investigate the structural and magnetic \nproperties of polycrystalline samples of the ferrimagnetic \nalloy Tb 0.15Co0.85. The XRD analysis confirms the high quality \nof the synthesis with a single phase TbCo 5 as expected for this \ncomposition. The magnetometry results suggest a reversal of \nthe magnetization by rotation at 300 K, whereas at 5 K the \ncharacteristic shape of the hysteresis indicates the nucleation \nand propagation of magnetic domains. From the unpolarized \nSANS measurements, the purely m agnetic SANS cross \nsections in the remanent state were determined by subtracting \nthe scattering patterns measured at a large magnetic field of 4 \nT. The 1D magnetic SANS cross section parallel to the applied \nfield suggests that at 300 K both the Co and Tb m oments are \ncorrelated over large distances with correlation lengths of at \nleast 100 nm. At 5 K, on the other hand, analysis of the \nmagnetic SANS signal in terms of a Lorentzian -squared \nscattering function reveals a reduced correlation length of \naround 4.5 nm. This result in combination with the \nmagnetization curve indicates the formation of domains \nwithin the ferrimagnet with one dimension being in the nm \nrange.  Finally, we relate our results  to the temperature \ndependence of the magnetic  anisotropy  of TbCo 5, which is \ndominated by the Tb sublattice for temperatures below 300 K.  \nAcknowledgements  The authors acknowledge the Heinz Maier -Leibnitz \nZentrum for provision of neutron beamtime. We also thank the \nC.C. Magnetism and C.C. X -Gamma of the Institut Jean \nLamo ur (Université de Lorraine) for technical support \nregarding the magnetometry and XRD experiments. P.B. and \nA.M. acknowledge financial support from the National \nResearch Fund of Luxembourg (CORE SANS4NCC grant).  \n \nReferences  \n[1] D. H. Kim, T. Okuno, S. K. Kim, S. H. Oh, T. Nishimura, \nY. Hirata, Y. Futakawa, H. Yoshikawa, A. Tsukamoto, Y. \nTserkovnyak, Y. Shiota, T. Moriyama, K. J. Kim, K. J. Lee, \nand T. Ono, Phys. Rev. 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For the analysis, the Le \nBail fit method (implemented in the Fullprof software) was \nused, considering the space group P/6mmm. The “*” indicate \nthe diffraction peaks coming from the Kβ radiation of the \nCobalt source. The bottom black solid line  represents the \ndifference between the calculated and observed intensities. (b) \nSecondary electron scanning electron microscopy images of \nthe grain microstructure of our sample. Here the black color \ncorresponds to the carbon tape used for the discharging.  \n \nFigure  2. (a) Magnetization curves measured in a field \nrange of ± 4 T at 300 K (black solid line) and 5 K (blue solid \nline). (b) Zoom of the magnetization curve measured at 5 K in \na field range of ± 1 T. The onset of nucleation and propagation \nof the magn etic domains are sketched by the arrows (1) and \n(2), respectively. (c) Temperature dependence of the total \nmagnetization under a fixed field of 4 T.  \n \nFigure  3. (a) and (b) E xperimental two -dimensional  (2D)  \ntotal  (nuclear + magnetic) unpolarized SANS cross sections \ndΣ/dΩ measured at 300  K and 5  K, respectively . (c) and (d) \nPurely magnetic 2D SANS cross sections d ΣM/dΩ measured \nat 300 K and 5 K, respectively . The purely magnetic 2D SANS \ncross section s in the  remanent state were obtained by \nsubtracting the total scattering at the (near) saturation field of \n4 T from the data at H = 0 T. The applied magnetic field H0 is \nhorizontal in the plane of the detector ( H0 ⊥ k0). Note that the \ndΣ/dΩ and d ΣM/dΩ scales are plotted in polar coordinates ( q \nin nm-1, θ in degree, and the intensity in cts/exposure time).  \n \nFigure  4. (a) and (b) Azimuthally -averaged 1D total SANS \ncross sections d Σ/dΩ as a function of the momentum transfer \nq and at selected applied -field values (see insets) (log -log \nscale) at 300 K and 5 K , respectively. The error bars of d Σ/dΩ \nare smaller than the data point size.  \n \nFigure  5. Red filled circles: nuclear 1D SANS cross section \ndΣnuc/dΩ as a function of momentum transfer q. Color ed filled \nsquares:  radially -averaged 1D magnetic SANS cross sections \ndΣM/dΩ along the field direction at 300 K (white filled \nsquares) and at 5 K (blue fi lled squares). Red dashed  line: \npower law d Σnuc/dΩ ∝ q-4. Blue solid line: Lorentzian -squared \nfit of the transverse scattering contribution at 5 K to determin e \nthe magnetic transverse correlation length lC. The d Σnuc/dΩ \nwas determined by  10° horizontal sector averages ( q // H0) \nof the total d Σ/dΩ at an applied magnetic field of μ 0H0 = 4T \nand T = 300 K. The radially -averaged 1D magnetic SANS \ncross section was determined by  20° horizontal sector \naverages ( q // H0) of the 2D magnetic SANS cross section at the remanent state taken from figure  3. Note: the magnetic \nSANS cross section intensities at 300 K and 5 K have been \nrescale d to the nuclear 2D SANS cross section intensity  for \nbetter comparison. (log -log scale)Figure 1  \n \n \n  \n \n 8  \n Figure 2  \n \n \n  \n \n 9  \n Figure 3  \n  \n \n 10  \n Figure 4  \n \n  \n \n 11  \n Figure 5  \n \n \n"    },    {        "title": "2102.13502v2.Direct_imaging_of_chiral_domain_walls_and_Néel_type_skyrmionium_in_ferrimagnetic_alloys.pdf",        "content": " 1 Direct imaging of chiral domain walls and Néel -\ntype skyrmionium in ferrimagnetic alloys  \nBoris Seng1,2,3,4, Daniel Schönke1, Javi er Yeste1, Robert M. Reeve1,3, Nico Kerber1,3,4, Daniel \nLacour2, Jean -Loïs Bello2, Nicolas Bergeard5, Fabian Kammerbauer1, Mona  Bhukta1, Tom \nFerté5, Christine Boeglin5, Florin Radu6, Radu Abrudan6, Torsten Kachel6, Stéphane Mangin2, \nMichel Hehn2 and Mathias Kläui1,3,4,  † \n1. Institut für Physik, Johannes Gutenberg -Universität Mainz, Staudingerweg 7, 55128 Mainz, \nGermany  \n2. Institut  Jean Lamour, UMR CNRS 7198, Université de Lorraine, 2 allée André Guinier, 54011 \nNancy, France  \n3. Graduate School of Excellence Materials Science in Mainz, Staudingerweg 9, 55128 Mainz, \nGermany  \n4. Max Planck Graduate Center mit der Johannes Gutenberg -Universität, Staudingerweg 9, \n55128 Mainz, Germany  \n5. Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, \nUMR 7504, F -67000 Strasbourg, France  \n6. Institut für Methoden und Instrumentierung der Forschung mit  Synchrotronstrahlung \nHelmholtz -Zentrum Berlin für Materialien und Energie GmbH, Albert -Einstein -Str. 15, 12489 \nBerlin, Germany  \n† Corresponding author:  klaeui@uni -mainz.de  \n   2 Abstract  \nThe evolution of chiral spin s tructures is studied in ferrimagne t Ta/Ir/Fe/GdFeCo/Pt multilayers  as \na function of temperature using scanning electron microscopy with polarization analysis \n(SEMPA).  The GdFeCo ferrimagnet exhibit s pure right -hand Néel -type domain wall (DW) spin \ntextures over a large temperature range. This indicates the presence of a negative Dzyaloshinskii -\nMoriya interaction (DMI) that can originate from both the top Fe/Pt and the Co/Pt interfaces . From \nmeasurement s of the DW width , as well as complementary magnetic char acterization, the \nexchange stiffness  as a function of temperature is ascertained. The exchange stiffness is \nsurprisingly mostly constant, which is explained by theoretical predictions . Beyond single \nskyrmions, we find by direct imaging a pure Néel -type skyrmionium , which due to the absence of \na skyrmion Hall angle is a promising topological spin structure to enable high impact potential \napplications in the next generation of spintronic devices.  \nIntroduction  \nThe competition between different interactions in magnetic materials leads to a wide variety of \ndifferent magnetic spin structures from single -domain states to chiral spin textures. Among them, \nmagnetic skyrmions in thin -film multilayer systems  [1-7] are nowadays widely studied due to their \nattractive pro perties for potential applications including their room temperature (RT) stability  [8-\n10]. Skyrmions are topological spin structures that can be stabilized through an antisymmetric \nexchange interaction that arises in a system with a broken inversion symmet ry: the Dzyaloshinskii -\nMoriya interaction (DMI)  [11-12]. Recent studies confirmed the current -driven dynamics of \nskyrmions in ultrathin ferromagnet s via spin -orbit torques (SOTs)  that act efficiently on Néel -type \nspin textures  [1,8,13] . For deterministic d ynamics, sufficient DMI is then required to yield \nhomochiral domain walls (DW). Therefore, skyrmions stabilized with DMI are particularly \nsuitable for next generation spintronics devices, such as the skyrmion -based racetrack memory  [8]. \nHowever, the dynami cs of ferromagnetic skyrmions via SOTs exhibits a transverse motion due to \ntheir non -zero topological charge  [14-16]. This behavior is especially unwanted in many \napplications since skyrmions could be annihilated at an edge of a device with the information  \ncarried by the skyrmion lost as a result. Materials with antiferromagnetically exchange -coupled \nmagnetic sublattices have been proposed to reduce or annihilate this skyrmion Hall effect owing \nto the overall zero topological charge  [17-18].  3 Ferrimagnetic m aterials such as rare earth (RE) – transition metal (TM) alloys are made of two \nantiferromagnetically exchange -coupled magnetic sublattices. For a given stoichiometry, at a \ntemperature called the magnetization compensation temperature, the magnetization of  the two \nsublattices is equal and opposite and therefore the net magnetization is zero  [19]. Similarly, an \nangular momentum compensation temperature can be defined where a zero s kyrmion Hall angle \nis predicted  [20]. GdFeCo ferrimagnetic alloys have attract ed significant attention since the \ndiscovery of all -optical s witching (AOS) in these alloys  [21]. Since then two types of AOS has \nbeen observed  [22-23] and the possibility to reverse the magnetization with a single electron pulse \nhas been demonstrated  [24]. In particular, to use these systems for topological spin structure \ndevices, one needs deterministic behavior in devices and for that it is necessary that the DMI is \nsufficient to generate a single chirality of skyrmions with a Néel -type spin texture. Thi s then \nenables efficient SOT driven DW or skyrmion motion. So to realize this, one requires a detailed \nanalysis and high resolution magnetic imaging on the internal spin structures of such ferrimagnetic \nskyrmions, which is still missing to date. In particu lar, as ferrimagnet can exhibit strong \ntemperature dependence of the properties, one needs to ascertain the temperature range where \nrobust properties are found.  \nIn this study, we investigate the chirality of spin textures in a ferrimagnet namely \nTa/Ir/Fe/G dFeCo/Pt by imaging the spin structure of the domain walls using SEMPA  [25-27]. This \nsurface -sensitive imaging technique has already been successfully used to determine the chiral \ncharacter of out -of-plane (OOP) magnetized spin text ures in ferromagnetic materials  [28-29] and \nhere we demonstrate that we can determine the chiral character of spin textures also for \nferrimagnetic materials. From the SEMPA images, we are also able to extract the domain wall \nwidth across a wide range of temperatures, which allo ws us to determine the exchange stiffness \nevolution with temperature as a crucial parameter that governs the stability and operation \ntemperature range.  \nExperimental details  \nA multilayer thin film of Si//Ta(5)/Ir(5)/Fe(0.3)/Gd 26.1Fe65.5Co8.3(8)/Pt(5)  is dep osited using \nmagnetron sputtering in a chamber with a base pressure of 2.4x10-8 Torr, with the thickness of \neach individual layer given in nanometers in parentheses. The ferrimagnetic layer was grown by \nco-sputtering and the atomic compositions were estima ted from the deposition rates of each target.  4 The Gd26.1Fe65.5Co8.3(8) alloy has been chosen for its low coercivity that makes the stabilization of \nOOP spin textures by magnetic fields easier. The CoFe dominant alloy phase has been selected to \nprovide a lo wer compensation temperature and lower resistivity to avoid sample heating effects. \nInterfacial DMI comes from a strong spin -orbit coupling with broken inversion symmetry between \na heavy metal (HM) and ferromagnetic materials (FM) that arises mainly from t he first atomic \nlayer  [30]. An ultrathin Fe(0.3) has been inserted between Ir and the ferrimagnetic layer to enhance \nthe interfacial DMI which is known to be particularly strong and positive at the Ir/Fe interface. \nHowever, we note that the DMI at the Ir/F e interface has also been calculated to be negative  [31]. \nAt the other interface, the DMI a t the Co/Pt or Fe/Pt interfaces is expected to be strong and negative  \n[2,30,32] .  \nResults and discussion  \nThe sample exhibits perpendicular magnetic anisotropy (PMA) with a small switching field after \nmilling ( see Supplementary S 1). The stabilization of spin textures is achieved  by cycling an in -\nplane (IP) magnetic field at RT where structures start to nucleate randomly. Figures 1(a) and 1(b) \nshow the direction of the in -plane magnetization, imaged with SEMPA  (see S upplementary S2 ), \nunder zero magnetic field after the nucleation process. Contrast in the DWs can be observed where \nthe in -plane magnetization is expected with a brighter contrast present for the top (left) of each \nspin textures indicating an up (left) tendency for the local direction of the magnetization.  \nConversely da rk contrast is seen at the bottom and right edges of the domains as shown in figures \n1(a,b), the absolute in -plane magnetization image has been generated in figure 1(c) where the \nbrighter contrast indicates the in -plane saturated magnetization. This contra st clearly indicates the \nposition of the domain walls where in -plane components are then present. From the analysis, the \nposition of the skeleton of the domain walls, i.e. the line at the center of the DWs, can then be \ndefined. The determination of the ske leton allows us to establish the DW intensity profile shows \nin figure 1(d) by averaging the measured local DW profile at each position of the skeleton (see \nSupplementary S3 ). However, the imaged DW profile needs to be analyzed to obtain the true \nprofile, p rincipally due to the broadening of the features due to the finite size of the beam profile. \nThe experimental profile can be approximated as the convolution of the theoretical DW profile \nwith a Gaussian function describing the electron beam distribution as  follows:   5 𝐷𝑊𝑝𝑟𝑜𝑓𝑖𝑙𝑒∝ 𝑐𝑜𝑠ℎ−1(𝑥\n∆)⊗𝑒−𝑥2\n2𝜎2 (1) \nwhere the hyperbolic function represents the real DW profile with ∆ the Bloch parameter. In our \ncase, we use for the domain wall width 𝛿, defined by Lilley [33] where 𝛿=𝜋∆. To determine the \nGaussian function, we considered a sharp defect modeled as a step function (see S upplementary \nS4). The Gaussian is found to be narrower than the imaged DW and here it has a rather small but \nnon-negligible influence on the domain wall widt h measurements. From this analysis, we finally \ndetermine the real DW width to be 𝛿𝑟𝑒𝑎𝑙=175±5 𝑛𝑚 at RT.  \n \n  \nFigure 1 . Determination of the average DW width of s pin textures in Ta/Ir/Fe/GdFeCo/Pt taken \nwith SEMPA  at RT . (a) Vertical and (b) horizontal  in-plane component s of the magnetization . The \n 6 direction of magnetization is indicat ed by the grayscale contrast as displayed on the  double arrows.  \nScale bar in (a): 1 µm.  (c) Reconstruction of the absolute in -plane magnetization intensity. The \nwhite contr ast indicates a saturated in-plane magnetization. (d) Distribution of the average domain \nwall intensity profile for all the spin textures present in the image.  \n \nAfter having defined the DW width, we analyze the spin distribution within the DWs. Figure 2(a)  \ndisplays the direction of the in -plane magnetization in the DWs by way of the color wheel shown \nin the inset. Qualitatively, we first see that all the magnetic structures present the same chirality, \nnamely they are homochiral clockwise Néel -type spin text ures. For a precise analysis, we then \ncompare the spin direction of each position inside the DW with the direction of the local tangent \nof the DW (figure 2(b)) (see S upplementary S3 ). The histogram indicates that the magnetization \ndirection in the DW struc tures forms a distribution centered around -90°. We conclude that our \nmaterial presents a pure clockwise Néel -type homochiral character, which can be explained by the \nnegative sign of the interfacial DMI in typical Fe/Pt and Co/Pt interface when the Pt lay er is on \ntop of the Fe or Co layer. Since SEMPA is a surface sensitive technique, we only probe the \ndirection of the magnetization close to the surface. To ascertain that this chiral character is not a \nlocal effect at the surface due to flux closure but is  indeed supported through the whole thickness \nof the film, asymmetric bubble expansion has been performed that confirms our clockwise Néel \nhomochirality in the full film  [34]. \nAs explained above, an IP magnetic field is used to nucleate spin textures that start to propagate at \nrandom positions. Remarkably, using an IP oscillating magnetic field with a certain damping ratio \nallows for the nucleation of a spin texture inside another spin texture that has been nucleated \npreviously. In figures 2(c) and (d), one  can observe a small 160 nm diameter skyrmion stabilized \nin a larger skyrmion bubble, forming a magnetic skyrmionium, topological spin texture that is \nespecially attractive due t o its zero topological charge , which leads to a vanishing skyrmion Hall \nangle that is promising for application  [35-36].  7  \n \nFigure 2. Determination of the magnetization direction inside the DWs of different ferrimagnetic \nchiral spin textures.  (a) The direction of the in -plane magnetization in the domain walls is displayed \nas defined  by the color wheel in the bottom left corner of the image. Scale bar in (a): 1 µm. (b) \nDistribution of the direction of the in -plane magnetization 𝑀⃗⃗  in the domain wall with respect to \nthe local tangent 𝑡  at RT : angle ψ (see inset). A Gaussian fit indica tes a central value around -90°. \n(c) Absolute in -plane magnetization intensity with skeleton displayed in white and (d) direction of \nthe in -plane magnetization in the domain walls of a ferrimagnetic skyrmionium at 320 K. Scale \nbars in (c) and (d): 500 nm.  \n \nTo extract the keys magnetic parameters of the system, we analyze the measured domain wall \nprofiles by considering the model put forward in I. Lemesh et al.  [37]. In that work, the authors \ndemonstrated that the DW width ∆  is given by:  \n∆(𝑑,𝛹) = ∆0−1\n2𝜋(𝑄−1)\n𝑑+1\n∆0−∆∞(𝛹) (2) \n 8 where d is the thickness of the magnetic material and Ψ is the domain wall angle; ∆0=√𝐴\n𝐾𝑒𝑓𝑓, ∆∞=\n√𝐴\n𝐾𝑢+𝜇0𝑀𝑆2\n2𝑠𝑖𝑛(𝛹)2 , 𝑄=2𝐾𝑢\n𝜇0𝑀𝑆2, 𝐾𝑒𝑓𝑓=𝐾𝑢−𝜇0𝑀𝑆2\n2 depending on the exchange stiffness 𝐴, the \nuniaxial anisotro py 𝐾𝑢 and the saturation magnetization 𝑀𝑠.  \nThe DMI dependence of the DW width enters via the angle 𝛹. In the case of pure -Néel type domain \nwalls, above a certain DMI threshold value, the exact value of the DMI does not affect the DW \nwidth sin ce |𝛹|=90°. In this case, by measuring the saturation magnetization 𝑀𝑠 and the \neffective anisotropy 𝐾𝑒𝑓𝑓 (defined as the difference in the areas of the IP and OOP hysteresis loops) \nusing a Superconducting Quantum Interference Device  (SQUID) , the excha nge stiffness can be \nevaluated from the DW profile to be 𝐴=8.0±0.5 𝑝𝐽.𝑚−1 at RT. Therefore, through the \nmeasurement of the domain wall width via SEMPA, it is possible to determine the exchange \nstiffness of a material, a parameter that is not easily acc essible using other simple technics.  \nOne crucial point for the use of magnetic skyrmions in spintronic devices is that the homochiral \ncharacter of the DWs is preserved over a large temperature range. Therefore, to assess this, the \nprevious analysis has be en carried out from 150 K to 315 K. We see in figure 3(a) that the pure \nNéel character rema ins for this temperature range.  Next, we analyze the DW width for these \ntemperatures where the different values are reported in figure 3(b). We find that the DWs are  \nnarrower when the temperature is decreased due to the increase of the effective anisotropy that is \nconfirmed by SQUID. The results can be analyzed with the help of measurements of the thermal \nvariation of 𝐾𝑒𝑓𝑓 and 𝑀𝑠 at different temperat ures (figur e 3(c)). As expected  for the CoFe transition \nmetal dominant ferrimagnet, lower temperatures lead to a reduction in the saturation magnetization \nsince the rare earth sub -lattice magnetization increase faster than the transition metal one. On the \nother hand,  𝐾𝑒𝑓𝑓 increases when the temperature decreases. Since the chiral character is kept over \nthe whole temperature range and |𝛹|=90°, eq. 2 can be used to extract the thermal variation of \nthe exchange stiffness of the ferrimagnetic alloy over the whole temperature range. The exchange \nstiffness is found to be mostly constant with a slight increase when the temperature increases as \nshown in fig ure 3(c) . \nThis dependence of the effective exchange on temperature is counterintuitive since in \nferromagnetic materials, 𝐴(𝑇)/𝐴(0) is expected to vary as (𝑀(𝑇)/𝑀(0))𝛾 with 𝛾 around 2  [38-\n39]. Considering that in ferrimagnets the magnon spectrum consi sts of acoustical and optical  9 branches, Nakamura et al.  have shown that the temperature dependence of A(T) is rather weak up \nto a certain high temperature  [40]. This is due to competing effects of thermal -acoustical and \noptical magnons, A(T) then decreases  when T decreases in case of acoustic branch magnons. This \nprediction has been compared to experimental data by Srivastava et al.  in case of YIG and \nmagnetite and a satisfactory agreement have been obtained  [41]. In their paper, they proposed a \ncomprehensi ve description by a formula:  \n𝐴(𝑇)\n𝐴(0)=(𝑀𝑎(𝑇)/𝑀𝑎(0))(𝑀𝑏(𝑇)/𝑀𝑏(0))\n(𝑀𝑎(𝑇)−𝑀𝑏(𝑇))/(𝑀𝑎(0)−𝑀𝑏(0)) (3) \nwhere 𝑀𝑎(𝑇) and 𝑀𝑏(𝑇) are the thermal variation of magnetization of the subnetworks a and b, \nwhich we now want to use to analyze our data.  \nWhile very few experimental data on the variation of 𝐴(𝑇) for ferrimagnetic systems are available, \nand to our knowledge none on the CoFeGd system, we can use experimental data sets acquired by \nmeans of  element selective X -ray Magnetic Circular Dichroism (XMCD) spectroscopy  (see \nSupplementary S5)  a 20 nm thick Co 84Gd16 alloy (N. Bergeard et al. , in preparation) to extract \n𝑀𝐶𝑜(𝑇) and 𝑀𝐺𝑑(𝑇) and then to derive 𝐴(𝑇) using eq.  3. The compensation temperature of this \nalloy is the same as our 8 nm thic k Gd 26.1Fe65.5Co8.3 alloy. The result of 𝐴(𝑇) is shown in figure \n3c. We find that 𝐴(𝑇) is roughly constant between 80 K and 350 K in line with the experimental \ndata of our GdFeCo ferrimagnet. The good agreement allows us to conclude that the key properti es \ndo not vary strongly as a function of temperature showing robust behaviors over the full \ntemperature range.   10  \nFigure 3 . Measurements and determinations of magnetic parameters of the ferrimagnet multilayer.  \n(a) Average value of the angle ψ at different temperatures for the TM dominant GdFeCo \nferrimagnet. (b) Domain wall width measurements as a function of temperature. (c) Display of the \nexchange stiffness (blue) for different temperatures  as well as the saturation magnetization M s \n(black) a nd the effective anisotropy K eff (red). Blue fil led circles: exchange stiffness deduced from \ndomain wall width  measurements and magnetic characterizations of our GdFeCo ferrimagnet \nusing eq. 2.  The exchange stiffness at 26 K has been calculated assuming an  expected pure Néel -\ntype character of the domain wall ( |ψ|=90°) . Blue open  circles: exchange stiffness deduced from  \nXMCD  measurements of Co 84Gd16(20) using  eq. 3 for comparison.  \nConclusion  \nIn this study, we demonstrate imaging the internal spin structure of domains and domain walls in \nGdFeCo ferrimagne tic alloys across a 1.5 nm Pt c apping layer using SEMPA. This approach is a \n 11 new path to characterize chiral spin textu res in ferrimagnetic multilayers . In the studied GdFeCo -\nbased ferrimagnet, we find that th e domain wall spin textures exhibit a pure Néel -type \nhomochirality that is preserved over a large temperature range, even far away from the \ncompensation temperature. 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Temperature scaling of the Dzyaloshinsky -Moriya \ninteraction in the spin wave spectrum . Phys. Rev. B  96, 094436  (2017) .  \n \n[40] T. Nakamura, M. Bloch . Temperature Dependence of the Exchange Stiffness in \nFerrimagnets. Phys. Rev.  132, 2528 -2539  (1963) . \n \n[41] C. M. Srivastava, R. Aiyar . Spin wave stiffness constants in some ferrimagnetics.  J. Phys. \nC: Solid State Phys.  20, 1119 -1128  (1987) .   18 Acknowledgment  \nThe authors acknowledge funding from TopDyn, SFB TRR 146, SFB TRR 173 S pin+X (projects \nA01 & B02). The experimental part of the project was additionally funded by the Deutsche \nForschungsgemeinschaft (DFG, German Research Foundation) project No. 403502522 (SPP 2137 \nSkyrmionics) and the EU (3D MAGIC ERC -2019 -SyG 856538. We acknowledge financial \nsupport from the Horizon 2020 Framework Programme of the European Commission under FET -\nOpen  Grant No. 863155 (s -Nebula). This work was supported by the Institut Carnot ICEEL, by \nthe impact project LUE -N4S, part of the French PIA proje ct “Lorraine Université d’Excellence”, \nreference ANR -15-IDEX -04-LUE, and by the “FEDER -FSE Lorraine et Massif Vosges 2014 -\n2020”, a European Union Program. N.K., B.S. and M.K. gratefully acknowledge financial support \nby the Graduate School of Excellence Mat erials Science in Mainz (MAINZ, GSC266) and the \nMax Planck Graduate Center (MPGC).  We thank HZB for the allocation of synchrotron radiation \nbeamtime. We acknowledge funding from the French “Agence National de la Recherche” via \nproject No. ANR -11-LABX -0058_ NIE, the project EQUIPEX UNION No. ANR -10-EQPX -52 \nand the CNRS -PICS program.  \nAuthor c ontributions  \nThe M.K., M.H. and S.M. proposed the study. M.K., M.H., S.M. and R.M.R. supervised the study. \nB.S, M.H. and J. -L.B grew the samples and B.S and M.H. optimized  the samples. D.S., R.M.R., \nB.S, J.Y. and M.B. performed the SEMPA imaging. B.S, D.S, J.Y. and D.L. performed the analysis \nof the SEMPA images. B.S., F.K. and N.K. performed the magnetometry. F.R and R. A are the \nHZB’s local contact that operated t he ALICE reflectometer while T. K. is in charge of the PM3 \nbeamline at BESSYII. C.B, T.F. and N. B performed the XMCD measurements and took care of  19 the analysis. B.S. drafted the manuscript with the help of M.K., M.H., R.M.R., S.M. and N.B. All \nthe authors commented on the manuscript.  \nCompeting interests  \nThe authors declare no competing interests.  \n   20 Supplementary  \nS1: Kerr m icroscopy at RT \nThe sample exhibit s a perpendicular magnetic anisotropy (PMA) with a smaller switching field \nafter etching (see figure S1(a)) as expected from the presence of nucleation sites introduced by the \nAr ion beam etcher where 3.5 nm of Pt was removed.   \nIn Figure S1 (b), we see the spin structures that have been nucleated using a damped oscillating IP \nmagnetic field. We see that micr ometer -sized up spin textures have been nucleated that we imaged \nafter with SEMPA to determine the chirality (see main text).  \n \n \nFigure S1. (a) H ysteresis loop s performed with a polar magneto -optical Kerr effect (MOKE) \nmicroscope on Si//Ta( 5)/Ir(5)/Fe(0.3)/Gd 26.1Fe65.5Co8.3(8)/Pt( 5) before (Black) and after milling \n(red). (b) Image taken with a MOKE microscope at RT after nucleation using an IP magnetic field. \nScale bar: 10 µm.  \nS2: SEMPA methodology  \n 21 To enable surface sensitive SEMPA imaging,  Ar ion milling is performed to remove 3.5 nm of the \n5 nm Pt capping layer. In SEMPA, a primary beam of 3 nA at 7.5 keV is used to image the DW \nmagnetization orientation. In the case of RE -TM ferrimagnets, the SEMPA acquisition is primarly \nsensitive to the  magnetization of the TM sublattice which then allow to determine the system \nchirality even in a presence of a reduced net magnetization of the alloy. Since SEMPA is a very \nsurface sensitive technique with a relat ively low -efficiency detector [42] and the imaging is \nperformed through a cap l ayer of 1.5 nm Pt  [43], very long acquisition times are needed (about 20 \nhours per image). The top Pt interface has only been partially removed to retain a sizable negative \nDMI interaction at the top interface as well as  keeping an out -of-plane anisotropy while making \nthe SEMPA imaging possible. Since thermally induced drift of the image during a long acquisition \nwould be detrimental for the required high resolution, we scan the field of view with a fast scan \nfrequency of  1000 pixels/s multiple times subsequently resulting in a total dwell time of 1 ms per \npixel. The secondary electrons emitted from the sample surface scatter at the W(001) crystal via \nlow energy electron diffraction (LEED) in different directions depending  on their spin. Calculating \nthe normalized asymmetries of the electron flux at the opposite (2,0) LEED spots gives the x and \ny SEMPA asymmetry images that correspond to the in -plane magne tization of the sample surface  \n[44]. Before further analysis a linear  gradient correction is applied to the asymmetry images, which \nresemble the x and y component of the magnetization vector. From these images, a third image of \nthe absolute magnetization vector is calculated showing the position of the domain walls. Outside  \nthe DWs , the measured absolute mag netization value is around zero  since in the conventional \ngeometry the system detects the in -plane magnetization component. However, within the DW, a \nmeasurable in -plane component exists and is detected. The spatial resol ution is determined by  22 imaging the edge of a particle in the SEMPA sum image, where all four detector channels are \nadded up.  \nS3: DW width and magnetization angle into the DWs  \nFrom the intensity image of th e IP magnetization (see figure S2 (a)), we first bin arize this image in \norder to exactly localize the domain walls of each spin textures (see figure S2(b)). In figure S2 (c), \nwe shrink the binarized image to get the skeleton of the domain walls, i.e. the one -line at the center \nof the domain walls. Finally, w e cut into several parts the skeleton in order to be able to fit each \nsection by a polynomial function  (see figure S2(d)) . This will be crucial in order to get the local \ntangents of the domain walls that will allow us to:  \n1. Extract the DW width (figure 1(d)  in the main text) by measuring and averaging a radial \nprofile at each position of the DWs.  \n2. Determine the chiral character by comparing the local magnetization direction into the DWs \nwith its local tangent extracted from the polynomial functions (see figur e 2(b) in the main \ntext).   23  \nFigure S2. (a) IP intensity image taken with SEMPA at RT of our GdFeCo -based ferrimagnetic \nalloy. (b) Binarization  of the image (a). (c) Skeleton of the image (b). (d) Decomposition of the \nskeleton into several parts allowing a fit of each section of the skeleton with a polynomial function.  \n \nS4: Determination of the SEMPA resolution  \nThe spatial resolution is determined using an edge of a particle in the SEMPA sum image. The \nedge is assumed to be perfectly sharped even if it maximize our resolution. Therefore the edge \ndefect is modeled as a Heaviside step function. Since the microscope  can not be  perfectly aligned \nand that the measured electron beam has a spatial resolution, we consider the imaging technique \nhaving an influence on the observed objects that is modeled as a Gaussian function. Finally, the \nmeasured edge defect profile is d efined as:   \n𝐸𝑑𝑔𝑒𝑝𝑟𝑜𝑓𝑖𝑙𝑒∝ 𝐻(𝑥)⊗𝑒−𝑥2\n2𝜎2 \nwhere H(x) represents the Heaviside step function and the Gaussian function represents the \ninfluence of the imaging technique on the real edge defect  (Figure S3 a). On the Figure 1b, we fit \nthe edge prof ile with our convolution function . The resolution is commonly measured as the spatial \ninterval corresponding to a variation of the intensity between 20% and 80% of a sharp edge  [45]. \nWe finally f ound that the resolution is about 28 nm  at RT . \n 24  \nFigure S3. x: distance in nm. y: reduced intensity between 0 and 1. (a) Simulated edge profile \n(yellow) calculated from the convolution between a Heaviside step function (blue) with a Gaussian  \nfunction (orange) (see eq.  1 in the main text) using arbitrary parameters. (b) Real edge profile \nmeasured across a defect on a SEMPA image (red crosses) with the fitting edge profile function.  \nS5: XMCD spectroscopy methodology  \nThe measurements on CoGd have been performed by using the ALICE reflectometer installed on \nthe PM3 beaml ine at the BESSY II synchrotron radiation source operated by the Helmholtz -\nZentrum Berlin  [46]. The X -ray Absorption Spectra (XAS) at the Co L3 and Gd M5 edges were \nacquired by monitoring the transmission of circularly polarized X -ray under a magnetic fiel d of \n+/-1 kOe. The magnetic field was applied along the X -ray propagation while the sample was tilted \nby 30° according to the in -plane magnetic anisotropy of the alloys.  \nSupplement ary references  \n[42] R. Frömter, S. Hankemeier, H. P.  Oepen, J. Kirschner . Optimizing a low -energy electron \ndiffraction spin -polarization analyzer for imaging of magnetic surface structures.  Rev. Sci. \nInstrum.  82, 033704  (2011) . \n \n 25 [43] S. Kuhrau, F.  Kloodt -Twesten, C. Heyn, H. P.  Oepen, R. Frömter. Cap-layer -dependent \noxidation of ultr athin cobalt films and its effect on the magnetic contrast in scanning \nelectron microscopy with polarization analysis.  Appl. Phys. Lett.  113, 172403  (2018) . \n \n[44] H. P. Oepen, R. Frömter . Scanning Electron Microscopy with Polarisation Analysis . \nHandbook of Magnetism and Advanced Magnetic Materials . Wiley, 1442 -1463  (2007) . \n \n[45] K. Koike. Spin -polarized scanning electron microscopy. Microscopy  62, 191  (2013) . \n \n[46] R. Abrudan, F. Brüssing, R. Salikhov,  J. Meermann, I. Radu,  H. Ryll,  F. Radu , H. Zabel . \nALICE —An advance d reflectometer for static and dynamic experiments in magnetism at \nsynchrotron radiation facilities. Rev. Sci. Instrum.  86, 063902  (2015) . "    },    {        "title": "1401.2747v1.Room_Temperature_Ferrimagnet_with_Frustrated_Antiferroelectricity__Promising_Candidate_Toward_Multiple_State_Memory.pdf",        "content": " \nRoom -Temperature Ferrimagnet with Frustrated Antiferroelectricity : \nPromising Candidate Toward  Multiple State Memory  \n \nP. S. Wang and H. J. Xiang* \nKey Laboratory of Computational Physical  Sciences (Ministry of Education), State Key \nLaboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 200433, \nP. R. China  \n \ne-mail: hxiang@fudan.edu.cn  \nOn the basis of first-principle s calculation s we show that the M -type hexaferrite BaFe 12O19 \nexhibit s frustrated antiferroelectricity  associated with  its trigonal bipyramidal Fe3+ sites. The \nferroelectric (FE) state of BaFe 12O19, reachable by applying an external electric field to the \nantiferroelectric  (AFE)  state, can be made stable at room temperature by  appropriate  element \nsubstitution or strain engineering. Thus M-type hexaferrite, as a new type of multiferoic with \ncoexistence of antiferroelectricity  and ferrimagnetism, provide a basis for studying the \nphenomenon of frustrated antiferroelectricity  and realizing multiple state memory devices . \n \nPACS numbers:  75.85.+t, 71.20.- b, 75.30.Et, 75.50.Gg  \n \n \n \n1 \n  \n The phenomenon of frustration, typically observed in the field  of magnetism, is found in  \nsolids [ 1] and soft materials  [2]. Prototype example s of geometrical spin frustration are provided \nby magnetic systems consisting of triangular or pyrochlore spin lattice  with nearest -neighbor \nantiferromagnetic  spin exchang e. Frustrated magnetic systems can give rise to exotic phenomena \nsuch as quantum spin liquid [3] and spin -order induced ferroelectricity  [4,5]. An important  \nquestion in the field of ferroelectricity is whether geometrical ly frustrat ed antiferroelectricity \nexists  or not. Currently, all well- known multiferroics (e.g., TbMnO 3, BiFeO 3) possess \nsimultaneously ferroelectricity and antiferromagnetism. It is not clear whether antiferroelectricity \ncan coexist with ferromagnetism.  \n Hexaferrite contains  triangular lattice s of Fe3+ ions, which has been found to display \nintriguing magnetoelectric (ME) effects at room temperature and low magnetic fields (~ 0.01 T)  \n[6,7,8,9]. The room -temperature insulating ferrimagnetic  undoped M-type hexaferrite, AFe 12O19 \n(A = Ca , Sr, Ba, Pb, etc.) , was widely believed  [10] to crystallize in the magnetoplumbite -type \ncentrosymmetric structure (space group P6 3/mmc) with high -spin Fe3+ ions in the octahedral \n(OCT) (12k, 4f and 2a), tetrahedral ( TET) ( 4f) and trigonal bipyramidal (TBP) (2b) sites  [Fig. \n1(a)], see Fig. S 1 of the supplementary material  (SM) for details . The Fe3+ ion position at the \nTBP site has been controvers ial. In the centrosymmetric structure, the  TBP Fe3+ ion lies on the \nequatorial plane  (i.e., the local mirror plane) of the TBP FeO 5. The x-ray-diffraction study at \nroom temperature  [11] suggested that the Fe3+ ion is displaced out of  the equatorial mirror plane  \nby about 0.16 Å , which was supported by a Mössbauer study  [12]. Collomb et al.  [13] carried \nout neutron- diffraction studies  for a number of hexagonal ferrites  at room temperature and 4.2 K. \nAt room  temperature, they found an even greater displacement  (i.e., 0.26 Å)  of the TBP site ions. \nHowever, their structur e refinement  at 4.2 K  suggest ed a freezing of the Fe3+ ion at the mirror -\n2 \n  \nplane site , and this  was supported by e mpirical rigid -ion model calculations  [14]. In this work , \nwe carry  out a comprehensive first -principles study  to resolve this controversy and reveal  that the \nM-type hexaferrite BaFe 12O19 exhibits  frustrated antiferroelectricity , giving rise to both room -\ntemperature polar order and strong ferrimagnetic order . Our work predicts that BaFe 12O19 \nrepresents a first example of a new type of multiferroic possessing both antiferroelectricity  and \nferrimagnetism. \n The presence of a structural  instability  in BaFe 12O19 can be examined by comput ing its \nphonon dispersion within the density functional theory (see [15] for details) . In BaFe 12O19, the \nhigh-spin Fe3+ ions (S = 5/2) order ferrimagnetically below 450 °C with  16 up-spin and eight \ndown- spin Fe3+ ions per unit cell , resulting in a net magnetization 20 μ B per unit cell  [10] (see \nFig. S1) . We first show that this ferrimagnetic state is the magnetic ground state by calculat ing \nthe spin exchange parameters  of BaFe 12O19 (see SM), and then carry out  phonon calculations for  \nthe ferrimagnetic spin ground state. Contra ry to the conclusion of  the empirical rigid -ion model  \n[14], our calculat ions show the presence of  two unstable modes in the whole B rillouin zone  of \nthe phonon dispersion [Fig. 1(b)] , providing a clear evidence for  the structural instability in \nBaFe 12O19. Both unstable phonon modes at Γ are contributed mainly by the displacement of  the \nFe3+ ions at the TBP sites along the c axis  [Fig. 1(c)] . The lower (higher) frequency mode is \nassociated with the in -phase (out -of-phase) vibration of the two TBP Fe3+ ions in the unit cell.  \nThe eigenvectors of the two modes can be used to generate one FE  and one AFE  structure along \nthe c axis. After performing structural relaxations, the FE and AFE  structures become more \nstable than the centrosymmetric paraelectric (PE) structure by 4.3 meV/f.u. and 0.1 meV/f.u., \nrespectively . This  further evidences  the structural instability of the TBP Fe3+ ions at the local \n3 \n  \nmirror- plane sites . In the FE structure, the TBP Fe3+ ion move s out of  the mirror plane by 0.19 Å, \nin good agreement with the experimental result (0.16 Å)  [11].  \n We now investigate the interaction between the local dipoles caused by the displacements of \nthe TBP Fe3+ ions by considering the five different dipole arrangements  (Fig. 2) : (I) The 11× FE \nstate in which all dipole moments aligned along the c  axis (i.e., FE ab-FEc state) ; (II) The 11× \nFEab-AFEc state with the same ab- plane FE arrangement as the FE ab-FEc state but with  an \nantiparallel dipole moments between adjacent lattices along the c axis; (III) The 21× AFE ab-FEc \nstate with the dipole moments aligned ferroelectrically along the c  axis and a chain -like AFE  \narrangement in the ab -plane; (IV) The 33×  FIab-FEc state with the dipole moments aligned \nferroelectrically along the c  axis and a two -up-one-down ferri electric arrangement in the ab -\nplane;  (V) The 33×  FIab-AFE c state with the same ab -plane ferrielectric arrangement as the \nFIab-FEc state but with an antiparallel dipole moments between adjacent lattices along the c axis. \nAfter structural relaxation s, we obtain the relative energies of these five states summarized in \nFig. 2, which shows that if two states have the same in- plane dipole arrangement, the state with a \nFE alignment of the dipoles along the c  axis has the lower energy [i.e., E(FE ab-FEc) < E(FE ab-\nAFE c), and E(FI ab-FEc) < E(FI ab-AFE c)]. On the other hand, the dipole moments prefer an AFE \narrangement in the ab -plane [i.e., E(AFE ab-FEc) < E(FI ab-FEc) < E(FE ab-FEc)]. This can be \nexplained in terms of the dipole -dipole interaction ( DDI), \n3[ 3( )( )]ij\nDDI i j i ij j ij\nijCE p p pe p er= ⋅− ⋅ ⋅  ,  \nwhere ije is a unit vector parallel to the line joining the centers of the two dipoles , r is the \ndistance between two dipoles, ip and jp, and C is a constant related to the dielectric constant. \n4 \n  \nIt can be easily seen that t he two ferroelectrically aligned dipoles along the c axis has a lower \nDDI energy than that of an antiparallel dipole pair , while  two dipoles with the dipole direction \nperpendicular to the distance vector tend to be antiparallel to each other  (see the inset of Fig. 2) . \nThe energies (DFTE ) of the five states from the density functional theory ( DFT ) calculations are \ncompared with their total DDI energies \n,ij\nDDI DDI\nijEE\n<>=∑  in Fig. 2, which reveals that the DFT \nresults are very well described by the DDI model. This is  due probably to  the fact that the dipoles \nassociated with the displacement of the TBP Fe3+ ions are well separated from each other so that \nthe short -range interactions between the dipoles are not important, unlike the case of traditional \nFE systems ( e.g. BaTiO 3) [16]. \n We now examine the ground state and thermodynamic properties of BaFe 12O19 using the \nDDI model  since it closely reproduces the DFT results . Note that the dipoles form a hexagonal \nlattice and are always perpendicular to the ab- plane [ i.e., along the c  or –c direction  (Ising -like)] . \nTo find out the ground state configuration of the dipole arrangement, we adopt two approach es. \nOne is to  enumerate all the symmetrically nonequivalent  configurations with the total number of \ndipoles no more than 12 in each supercell  [17]. We find  that the 21× AFE ab-FEc state has the \nlowest DDI energy. The other approach is to perform parallel tempering Monte Carlo (MC)  [18] \nsimulations , which confirm that the 21× AFE ab-FEc state (space group: Pnma) is the ground \nstate and is  consistent with the DFT result that it has  the lowest energy among all five  considered \nstates.  The ground state of the NN antiferromagnetic  (AFM)  Ising model on a triangular lattice is \nknown to have a m acroscopic degeneracy. The 6-fold degenerate 21× AFE ab-FEc state has the \nlowest energy due to the long range nature of the DDI. As a matter of fact, a similar 21× chain -\nlike AFM state is the ground state of the Ising model on a triangular lattice with AFM NN and \n5 \n  \nAFM next NN interactions  [19]. Although the AFE ab-FEc state has the lowest energy, there are \nmany low -lying excited states , and this affects the thermodynamic properties of BaFe 12O19. \n We perform parallel tempering MC  simulations on a 10 10 4×× lattice using the DDI \nmodel  to determine the thermodynamic properties of BaFe 12O19. The effect of vibrational free \nenergy is discussed in SM. Our calculations reveal  that there is a sharp  peak at around 3 K in the \nspecific heat curve, indicating a long -range order of the dipoles  (see Fig. 3 ). To characterize the \nphase transition, we compute the local correlation  i j abpp〈⋅〉 between the NN dipole pair in the \nab-plane and that i jcpp〈⋅〉 between the NN dipole pair along the c axis . The i jcpp〈⋅〉 becomes \nnonzero when the temperature is lowered below 15 K, and gradually increase s to the maximum  \nvalue of 1.0 when the transition temperature (3.0 K) is approached. The in -plane local correlation \ni j abpp〈⋅〉 becomes nonzero at much high temperature (about 300 K, not shown here) . This is so \nbecause the distance (5.8 Å ) between the NN dipoles in the plane is much shorter than that (11.5 \nÅ) between the NN dipoles along the c axis, thus the in -plane DDI is much s tronger. The in -\nplane correlation i j abpp〈⋅〉 saturates to  1\n3− at the transition temperature, which is the smallest \nvalue that can be achieved in a 2D Ising triangular system. However, this does not mean that the \nsystem fully orders in  the ground state  below the transition temperature because there are many \nlow-lying excited states with the same i j abpp〈⋅〉 and i jcpp〈⋅〉 as does the ground state. For \nexample, the 33×  FIab-FEc is less stable than the ground state only by 0.15 m eV per formula \nunit (FU) . The existence of  significant local correlation s above the phase transition temperature \nis a typical signature of a frustrated system . The 21× AFE ab-FEc ground state can be described \nas a modulated structure with a dipole modulation ve ctor q. There are three symmetrically \n6 \n  \nequivalent modulation vectors: 1(0.5,0,0)q=, 2(0,0.5,0)q=, and 3( 0.5,0.5,0)q= −. The order \nparameter can be chosen as the dipole structure factor \n123()\n2\n,,1ijiq r r\nij\nq q q q ijO p peN⋅−\n== ⋅∑∑\n, where ir \nis the position of the i -th dipole , and N is the number of dipoles in the supercell . For the AFE ab-\nFEc ground state, the order parameter is  1. Fig. 3 shows  that this parameter starts to become non -\nzero only when the temperatur e is below the transition point, suggesting that the low temperature \nphase is  the AFE ab-FEc state . \n We now compare our theoretical results with previous experiments. We find that the TBP \nFe3+ ion is displaced out of the equatorial mirror plane, which agrees with the x- ray-diffraction \nstudy [11] and the Mössbauer study [12]. The neutron- diffraction study by Collomb et al. [13] \nfound a large displacement of the TBP Fe3+ ion at room temperature, but suggested that the TBP \nFe3+ ion freezes at the mirror -plane site at 4.2 K. This is contrary to the usual phenomenon that \nsymmetry lowers  when temperature decreas es as predicted by Landau ’s theory . The puzzling \nneutron- diffraction results may be due to the frustrated nature of the TBP Fe3+ related dipoles and \nthe fact that the TBP Fe3+ ion is at the mirror -plane site on average in the AFE state. A future \nneutron- diffraction study at very low temperature (e.g., 1 K) may confirm the AFE ground state \npredicted in this work.  \n The above  discussion shows the ground state of BaFe 12O19 to be an AFE ferr imagnetic \nstate.  Our phonon calculations reveal that the FE ferrimagnetic state is metastable (see SM) \nbecause flipping a dipole needs  to go through the high energy PE -like state. The metastability of \nthe FE ferrimagnetic state suggests that BaFe 12O19 is a promising candidate for realizing multiple \nstate memory devices . We find that the FE BaFe 12O19 has the same ferrimagnetic ground state \nand similar magnetic Curie temperature as PE BaFe 12O19 (see SM), indicating that the spin -\n7 \n  \nphonon coupling in this system is not very important. Our calculations show  that the electric \npolarization (3.23 μC/cm2) and magnetization ( 10 μ B/FU) of the FE ferrimagnetic state in \nBaFe 12O19 are both large.  This is different from the usual single -phase multiferroics, for which \neither the electric polarizatio n or the magnetization is small: For example, the polarization in \nBa(Fe,Sc,Mg) 12O19 is almost three -order of magnitude smaller  [9]. \n However, t he FE state of freestanding BaFe 12O19 can be locally stable only at low \ntemperature because the energy barrier for the dip ole flip is about 2.1 meV/dipole. Therefore, we \nconsider two ways of mak ing this FE state stable at room temperature . One way is to replace \nsome of the TBP Fe3+ ions by other +3 ions . We examine the stability of the FE state by \nreplacing the TBP Fe3+ ions with nonmagnetic ions such as Al3+, Ga3+, Sc3+, and In3+. Fig. 4(a) \nshows that the FE state becomes more stable by replacing the TBP Fe3+ ions with Al3+ or Ga3+ \nions, but becomes less stable when the TBP Fe3+ ions are replaced with Sc3+ and In3+ ions . This is \nexplained  by considering the ionic radii of the +3 ions. The interaction between an inert M3+ \ncation  and an O2- anion consists of  the attractive Coulomb electrostatic interaction and the short -\nrange Pauli repulsion between core electrons [ 20]. For a linear O -M-O arrangement, the \nCoulomb interaction favors a FE -like asymmetric arrangement with two different M -O bond \nlengths, while the Pauli repulsion favors a PE -like centrosymmetric arrangement. If the size of \nthe metal ion is small, the Coulom b interaction will stabilize the FE -like arrangement . Otherwise, \nthe PE -like arrangement becomes more stable. The ionic radii  [21] increase in the order Al3+ < \nGa3+ < Fe3+ < Sc3+ < In3+, suggesting  that the  stability of the FE state follows the relationship  \nAl3+>Ga3+>Fe3+>Sc3+>In3+, in good agreement wit h our first -principles results  [see Fig. 4(a)] . \nThus, t he FE state of BaFe 12O19 can be made more stable at a higher temperature if the TBP Fe3+ \nions can be selectively replaced with smaller cations Al3+ and Ga3+.  \n8 \n  \n The other way of stabiliz ing the FE state at room temperature is to apply compressive \nepitaxial strain [Fig. 4(b)]. An in -plane compressive epitaxial strain makes the TBP FeO 5 \nelongated along the c axis, so the distance between the Fe ion and the apical  O ion becomes \nlonger  than the sum of the ionic radius, which enhances the FE distortion of the TBP Fe3+ site. \nThe stability of the FE state increases with decreasing the in -plane lattice constant.  For example, \nif BaFe 12O19 is grown on CaFe 12O19 (which introduces a 2% compressive strain), the FE state is \nmore stable than the P E state by about 26 meV/dipole, close to the room temperature energy \nscale. Our molecular dynamics  [22] simulation shows that the FE state at a 5% compressive \nstrain is stable at least up to room temperature: The TBP Fe3+ ions stay at the original positions \nof an initial FE state after a  5 ps simulation  at 300 K . As can also be seen from Fig. S7 of SM, \nthe FE state becomes even more stable than the AFE state when the compressive strain is larger \nthan 4% possibility due to the strain -polarization coupling. We also calculate the energy barrier \nfrom the FE state to the AFE state by using the climbing image nudg ed elastic band method [23]. \nAs shown in Fig. 5, the barrier is 1.26 meV/f.u. in the case of zero strain, while the  barrier is \nincreased to 116.14  meV/f.u. at 5% compressive strain. Thus, the FE state could become stable \nboth thermodynamically and kinetically at large compressive strain.  \n The above discussion suggests the possibility of realiz ing room temperature four -state \nmemory devices  [24] by selectively doping BaFe 12O19 with smaller +3 ions at the TBP Fe3+ sites \nor by growing BaFe 12O19 on the hexagonal substrate with a small lattice constant.  Gajek et al.  \nsuccessfully demonstrated  [25] that a multiferroic tunnelling junction made by the FE and \nferromagnetic La 0.1Bi0.9MnO 3 can act as a four- state resistive memory  system  although its \nmagnetic Curie temperature (105 K) is well below  the room temperature. Our work may provide \nnew recipes toward realizing a room -temperature four -state memory  device. \n9 \n  \n The precise definition of antiferroelectricity  is in general more subtle than for \nantiferromagnets  and has not reach consensus . In this work, antiferroelectricity refers to the case \nwhere the ground state contains anti -parallel aligned dipole moments and there is a ferroelectric \n(FE) low -lying excited state. Recentl y, Rabe proposed the following definition [26]: “an \nantiferroelectric is like  a ferroelectric in that its structure is obtained through distortion of a \nnonpolar high- symmetry ref erence phase; for ferroelectrics, the distortion is polar, while for \nantiferroelectrics it is nonpolar. However, not all nonpolar phases thus obtained are \nantiferroelectric : in addition, there must be an alternative low -energy ferroelectric phase \nobtained by a polar distortion of the same high -symmetry  reference structure, and an applied \nelectric field must induce a first- order transition from the anti ferroelectric phase to this \nferroelectric phase, producing a characteristic P -E double -hysteresis loop.” For the system \nBaFe 12O19 studied in this work, it satisfies almost all the above conditions. Currently , it is not \nclear whether a double hysteresis loop exist s in BaFe 12O19. The exact shape of the P -E curve \ndepends on the temperature and the barrier between the FE state and the AFE state. If the FE \nstate is metastable and the barrier is higher than the thermal energy (k BT) at a given temperature \nT, then the zero -field polarization after a pooling process may be non- zero. Otherwise, there may \nbe a P -E double -hysteresis loop. Some  other  factors such as d omain pinning , and/or defects may \nalso change the shape of the loop. \n In summary, the M -type hexaferrite BaFe 12O19 exhibit s frustrated antiferroelectricity due \nto the local dipole moments  arising from  its TBP Fe3+ ion sites, and is a novel multiferroic  with \nboth ferrimagnetic order and antiferroelectric order.  The M-type hexaferrites a re expected to \nprovide a basis  for realizing room -temperature multiple state memory devices.  \nWe thank Professor M.- H. Whangbo for invaluable discussions. Work was supported by \n10 \n  \nNSFC, FANEDD, NCET -10-0351, Research Program of Shanghai M unicipality and MOE, the \nSpecial Funds for Major State Basic Research, and Program for Professor of Special \nAppointment (Eastern Scholar).  \n \n[1] A. P.  Ramirez, C. L.  Broholm, R. J. Cava,  and G. R.  Kowach, Geometrical frustration, spin \nice and negative thermal expansion — the physics of underconstraint. Physica B 280, 290 \n(2000).  \n[2] M. Kléman, O. D. Lavrentovich, and J. Friedel, Soft Matter Physics: An Introduction \n(Springer, 2003).  \n[3] L. Balents, Spin liquids in frustrated magnets.  Nature 464, 199 (2010).  \n[4] T. Kimura , T. Goto, H. Shintani,  K. Ishizaka , T. Arima,  and Y . Tokura, Magnetic control of \nferroelectric polarization. Nature (London) 426, 55 (2003).  \n[5] H. J.  Xiang , E. J. Kan, Y. Zhang, M.-H. Whangbo, and X. G.  Gong , General Theory for the \nFerroelectric Polarization Induced Spin -Spiral Order.  Phys . Rev. Lett. 107, 157202 (2011); H. 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B 44, 619 \n(1991).  \n[15] Our total energy calculations are based on the DFT plus the on- site repulsion (U) method \n[27] within the generalized gradient approximation  [28] on the basis of the projector augmented \nwave method [29,30] encoded in the Vienna ab initio simulation package  [31,32]. Unless \notherwise noted, the plane -wave cutoff energy is set to 400 eV . For optimizing the lattice vectors, \na 500 eV plane -wave cutoff energy is adopted. For the calculation of elect ric polarization, the \nBerry phase method  [33,34] is employed. We use U = 5 eV and J = 1 eV  [see Jacek C. Wojdeł \nand Jorge Íñiguez, Phys. Rev. Lett. 103, 267205 (2009)]  for Fe 3d states which reproduces rather \nwell the ferrimagnetic critical temperature (se e SM).  The summation of the DDIs is carried out \nusing the Ewald technique  [16,35] . \n12 \n   \n[16] W. Zhong, D. Vanderbilt, and K. M.  Rabe, First-principles theory of ferroelectric phase \ntransitions for perovskites: The case of BaTiO 3. Phys. Rev. B 52, 6301 (1995). \n[17] G. Hart  and R. W. Forcade, Generating derivative structures from multilattices: Algorithm \nand application to hcp alloys. Phys. Rev. B 80, 014120 (2009).  \n[18] K. Hukushima  and K. Nemoto, Exchange Monte Carlo Method and Application to Spin \nGlass Simulations . J. Phys. Soc. Jpn. 65, 1604 (1996). \n[19] K. Takasaki,  I. Harada and  T. Tonegawa, Magnetic Phase Diagram of the Ising Model on a \nTriangular Lattice with Antiferromagnetic Nearest -Neighbor and Next -Nearest -Neighbor \nInteractions. J. Phys. Soc. Jpn. 55 (1986). \n[20] M. Born  and J. E. Mayer, Zur Gittertheorie der Ionenkristalle.  Z. Physik 75, 1 (1932).  \n[21] R. D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances \nin halides and chalcogenides. Acta Cryst A  32, 751 (1976).  \n[22] In order to check this stability of the FE state, a supercell with 128 atoms is built and first-\nprinciples molecular dynamic simulations are performed at 300 K.  \n[23] G. Henkelman, B.P. Uberuaga, and H. Jónsson, A climbing image nudged elastic band \nmetho d for finding saddle points and minimum energy paths. J. Chem. Phys. 113 , 9901 (2000).  \n[24] J. F. Scott, Data storage: Multiferroic memories. Nat . Mater. 6, 256 (2007). \n[25] M. Gajek, M . Bibes, S . Fusil, K . Bouzehouane, J . Fontcuberta,  A . Barthélémy  and A. Fert, \nTunnel junctions with multiferroic barriers. Nat . Mater. 6, 296 (2007).  \n13 \n   \n[26] K. Rabe, \"Antiferroelectricity in oxides: a reexamination,\" in Functional Metal Oxides: New \nScience and Novel Applications, ed. by Satish Ogale and V . Venkateshan, to be published by \nWiley (2012).  \n[27] A. I.  Liechtenstein, V.  I. Anisimov, and J. Zaanen, Density -functional theory and strong \ninteractions: Orbital ordering in Mott- Hubbard insulators. Phys. Rev. B 52, R5467 (1995). \n[28] J. P. Perdew, K. Burke, and M. Ernzerhot, Generalized Gradient Approximation Made \nSimple.  Phys. Rev. Lett. 77, 3865 (1996).  \n[29] P. E. Blöchl, Projector augmented -wave method. Phys. Rev. B 50, 17953 (1994).  \n[30] G. Kresse  and D. Joubert, From ultrasoft pseudopotentials to the projector augmented -wave \nmethod. Phys. Rev. B 59, 1758 (1999).  \n[31] G. Kresse  and J. Furthmüller, Efficiency of ab -initio total energy calculations for metals and \nsemiconductors using a plane -wave basis set.  Comput. Mater. Sci. 6, 15 (1996). \n[32] G. Kresse  and J. Furthmüller, Efficient iterative schemes for ab initio total -energy \ncalculations using a plane- wave basis set.  Phys. Rev. B 54, 11169 (1996).  \n[33] R. D. King -Smith  and D. Vanderbilt, Theory of polarization of crystalline solids. Phys. Rev. \nB 47, 1651 (1993).  \n[34] R. Resta, Macroscopic polarization in crystalline dielectrics: the geometric phase approach.  \nRev. Mod. Phys. 66, 899 (1994).  \n[35] P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 369, \n253 (1921).  \n14 \n   \n[36] S. R. Gawali, G. Kishor, Rewatkar , and V.  M .  Nanoti, Structural and Electrical properties of \nM-type Nanocrystalline Aluminium substituted Calcium Hexaferrites.  Adv. Appl. Sci. Res. 3, \n2672 (2012) . \n \n \n \n \nFIG. 1 (a) A polyhedral representation of the crystal structure of BaFe 12O19. (b) The phonon \ndispersion relations  calculated for BaFe 12O19 by PBE+U calculations.  For clarity , we only show \nthe phonon branches below 3 THz . (c) The ionic displacements of the FE and AFE phonon \nmodes at Γ  calculated for BaFe 12O19 by PBE+U calculations . For clarity, only the TBP Fe3+ ions, \ntheir neighboring O2- ions, and the Ba2+ ions are displayed.  \n \n15 \n   \n \n  \nFIG. 2 Comparison of the results from the DDI model with those from the DFT \ncalculations. The top panel plots the energies from the DDI model against the DFT energies for \nfive different configurations (i.e., FE, AFE ab-FEc, FE ab-AFE c, FI ab-FEc, FI ab-AFE c) of the local \ndipoles. The zero energy reference is taken to be that of the FE state. A straight line from the \nlinear fitting ( 18.5DDI DFT E Ec= + ) is also shown. The inset in the top panel schematically \nillustrates the idea that the two in -plane dipoles tend to be antiparallel to each other, while two \ndipoles along the c axis tend to be parallel to each other. The five dipole configurations are \nshown in the bottom panel. For the side view, we only indicate that whether the dipole \n16 \n   \narrangement along c is AFE or FE. For the top view, ⊗ (⊙) represents the dipole along c ( -c). \nThe NN in -plane dipoles are connected by dashed lines. The solid lines denote the in -plane unit \ncell of the dipole configurations. \n \n \n \n \nFIG. 3 Thermodynamic properties of the dipoles in BaFe 12O19 from the parallel tempering \nMonte Carlo simulations. The specific heat curve shown in the top panel indicates a long range \norder at around 3.0 K. The bottom panel shows the spin structure factor (i.e, the order parameter \nO defined in the main text) and local dipole correlations (i.e., in -plane correlationi j abpp〈⋅〉 and \nout-of-plane correlation  i jcpp〈⋅〉) as a function of temperature.  \n \n \n17 \n   \n \nFIG. 4 Stability of the FE distortion.  (a) The energy difference between the PE and FE state as \na function of the radius of the +3 ion at the TBP Fe3+ ion sites. (b) The energy difference between \nthe PE and FE state as a function of the in- plane lattice constant. The out -of-plane lattice vecto r \nis fully optimized. The experimental in -plane lattice constants  [10,36] of AFe 12O19 with A = Ca, \nSr, Ba are denoted by arrows. \n \n \n18 \n   \n \nFIG. 5 Energy barrier from the FE state to AFE (i.e., AFE ab-FE c) state.  Upper panel shows \nthe case for the zero -strain case, while lower panel  shows the case when the compressive \nepitaxial strain is 5%.  \n19 \n  "    },    {        "title": "2101.05831v1.Anomalous_Hall_effect_in_weak_itinerant_ferrimagnet_FeCr__2_Te__4_.pdf",        "content": "arXiv:2101.05831v1  [cond-mat.str-el]  14 Jan 2021Anomalous Hall effect in weak-itinerant ferrimagnet FeCr 2Te4\nYu Liu,1,∗Hengxin Tan,2Zhixiang Hu,1,3Binghai Yan,2and C. Petrovic1,3\n1Department of Condensed Matter Physics and Materials Scien ce,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n2Department of Condensed Matter Physics, Weizmann Institut e of Science, Rehovot 7610001, Israel\n3Department of Materials Science and Chemical Engineering,\nStony Brook University, Stony Brook, NY 11790, USA\n(Dated: January 18, 2021)\nWe carried out a comprehensive study of electronic transpor t, thermal and thermodynamic prop-\nerties inFeCr 2Te4single crystals. Itexhibits bad-metallic behavior andano malous Hall effect(AHE)\nbelow a weak-itinerant paramagentic-to-ferrimagnetic tr ansition Tc∼123 K. The linear scaling be-\ntween the anomalous Hall resistivity ρxyand the longitudinal resistivity ρxximplies that the AHE\nin FeCr 2Te4is most likely dominated by extrinsic skew-scattering mech anism rather than intrinsic\nKL or extrinsic side-jump mechanism, which is supported by o ur Berry phase calculations.\nINTRODUCTION\nThe anomalous Hall effect (AHE) in metals is linked\nto an asymmetry in carrier paths and the effects of spin-\norbit interaction. This is typically observed in ferromag-\nnets since an electric current induces a transversevoltage\ndrop in zero magnetic field which is proportional to mag-\nnetization[1,2]. Spin-orbitcouplinginthe ferromagnetic\nbands leads to anomalous carrier velocities and intrinsic\nAHE [3]. The intrinsic Kaplus-Luttinger (KL) mecha-\nnism can be reinterpreted as a manifestation of Berry-\nphase effects on occupied electronic Bloch states [4, 5].\nThe extrinsic mechanisms involving skew-scattering and\nside-jump mechanisms can also give rise to the AHE and\nare induced by asymmetric scattering of conduction elec-\ntrons [6, 7]. In recent years it has been shown that the\nAHEvelocitiesarisefromthetopologicalBerrycurvature\nwhich generate an effective magnetic field in momentum\nspace in varieties of Dirac materials with noncollinear\nspin configuration [8–12].\nFeCr2Ch4(Ch = O, S, Se, Te) materials show rich cor-\nrelatedelectronphysics. FeCr 2O4spinelshowsacomplex\nmagnetic phase diagram with a ferrimagnetic (FIM) and\nmultiferroic order below 80 K, a strong spin-lattice cou-\npling and orbital order due to the Jahn-Teller distortion\n[13–17]. FeCr 2S4isa multiferroicferrimagnetbelow Tc=\n165 K with large changes of resistivity in magnetic field\n[18–22]. FeCr 2Se4orders antiferromagnetically with TN\n= 218 K in an insulating state despite with larger ligand\nchalcogenatom[23–25]. FeCr 2S4andFeCr 2Se4havesim-\nilar electronic structure with nearly trivalent Cr3+and\ndivalent Fe2+states, and there is a strong hybridization\nbetween Fe 3 d- and Ch p-states [26]. FeCr 2Te4shows no\nsemiconducting gap and a FIM order below Tc= 123 K\n[27, 28].\nIn this work, we performed a comprehensive study of\nelectronic and thermal transport properties in FeCr 2Te4\nsinglecrystals. TheAHEobservedbelow Tcisdominated\nby the skew-scattering mechanism, i.e., by the Bloch\nstate transport lifetime arising from electron scatteringby impurities or defects in the presence of spin-orbit ef-\nfects, and is smaller than the intrinsic AHE revealed by\ndensity functional calculations.\nEXPERIMENTAL AND COMPUTATIONAL\nDETAILS\nSingle crystals growth and crystal structure details\nare described in ref.[28]. Electrical and thermal trans-\nport were measured in quantum design PPMS-9. The\nlongitudinal and Hall resistivity were measured using\na standard four-probe method. In order to effectively\neliminate the longitudinal resistivity contribution due to\nvoltage probe misalignment, the Hall resistivity was ob-\ntained by the difference of transverse resistance mea-\nsured at positive and negative fields, i.e., ρxy(µ0H) =\n[ρ(+µ0H)−ρ(−µ0H)]/2. Isothermal magnetization was\nmeasured in quantum design MPMS-XL5.\nWe performed density functional theory (DFT) cal-\nculations with the Perdew-Burke-Ernzerhof (PBE) [29]\nexchange-correlation functional that is implemented in\nthe Vienna ab initio simulation package(VASP) [30]. We\nadoptedtheexperimentalcrystalstructurewiththe ferri-\nmagnetism (parallelto the lattice vector c) [28]. The cut-\noff energy for the plane wavebasis is 300eV. A k-mesh of\n10×10×10wasused in the Brillouin zone sampling. The\nspin-orbit coupling was included. The intrinsic anoma-\nlousHallconductivity(AHC) andSeebeckcoefficientwas\ncalculated in a tight-binding scheme based on the maxi-\nmally localized Wannier functions [31].\nRESULTS AND DISCUSSIONS\nFigure 1(a) shows the temperature-dependent heat ca-\npacityCp(T) for FeCr 2Te4. A clear anomaly around 123\nK corresponds well to the paramagnetic (PM)-FIM tran-\nsition. The high temperature Cp(T) approaches the Du-\nlong Petit value of 3 NR≈172 J mol−1K−1, whereR2\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s52/s54/s56/s49/s48/s48 /s49/s50/s48 /s50/s52/s48 /s51/s54/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s52/s54\n/s48 /s53/s48 /s49/s48/s48/s52/s46/s48/s52/s46/s53/s53/s46/s48/s67\n/s112/s32/s40/s74/s47/s109/s111/s108/s45/s75/s41\n/s84/s32/s40/s75/s41/s40/s97/s41\n/s51/s78/s82/s49/s50/s51/s32/s75/s83/s32/s40 /s86/s47/s75/s41\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s67\n/s112/s47/s84/s32/s40/s74/s47/s109/s111/s108/s45/s75/s50\n/s41\n/s84/s50\n/s32/s40/s75/s50\n/s41\n/s40/s99/s41/s120/s120/s32/s40/s49/s48/s45/s52\n/s32/s99/s109/s41\n/s84/s32/s40/s75/s41/s120/s120/s40/s49/s48/s45/s52\n/s99/s109/s41\n/s84/s32/s40/s75/s41\nFIG. 1. (Color online) (a) Temperature-dependent heat ca-\npacityCp(T) for FeCr 2Te4. Inset shows the low temperature\nCp(T)/TvsT2curve fitted by Cp(T)/T=γ+βT2. (b) See-\nbeck coefficient S(T) and (c) in-plane resistivity ρxx(T) for\nFeCr2Te4single crystal. Inset in (c) shows data below 100 K\nfitted by ρ(T) =ρ0+aT3/2+bT2(solid line) in comparison\nwithρ(T) =ρ0+cT2(dashed line).\n= 8.314 J mol−1K−1is the molar gas constant. The\nlow temperature data from 2 to 18 K are featureless and\ncould be fitted by using Cp(T)/T=γ+βT2, where the\nfirst term is the Sommerfeld electronic specific heat co-\nefficient and the second term is low-temperature limit of\nlattice heat capacity[inset in Fig. 1(a)]. The fitting gives\nγ= 61(2) mJ mol−1K−2andβ= 1.7(1) mJ mol−1K−4.\nThe Debye temperature Θ D= 199(1) K can be calcu-\nlated by using Θ D= (12π4NR/5β)1/3, whereN= 7 is\nthe number of atoms per formula unit.\nThe Seebeck coefficient S(T) of FeCr 2Te4is positive in\nthe whole temperature range, indicating dominant hole-\ntype carriers [Fig. 1(b)]. The S(T) changes slope around\nTcand gradually decreases with decreasing temperature.\nAs we know, the S(T) depends sensitively on the Fermi\nsurface. The slope change of S(T) reflects the possible\nreconstruction of Fermi surface passing through the PM-\nFIM transition. At low temperature, the diffusive See-\nbeck response of Fermi liquid dominates and is expected\nto be linear in T. In a metal with dominant single-band\ntransport, the Seebeck coefficient could be described by\nthe Mott relationship,\nS=π2\n3k2\nBT\neN(εF)\nn, (1)\nwhereN(εF) is the density of states (DOS), εFis the\nFermi energy, nis carrier concentration, kBis the Boltz-\nman constant and eis the absolute value of electronic\ncharge [32]. The derived dS/dTbelow 26 K is ∼0.074(1)\nµV K−2. TheS(T) curve is consistent with our calcu-/s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s49/s50/s51/s52\n/s114\n/s120/s121/s32/s40 /s109/s87 /s32/s99/s109/s41\n/s109\n/s48/s72/s32/s40/s84/s41/s32/s56/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s49/s50/s48/s32/s75/s40/s98/s41\n/s109\n/s48/s72/s32/s47/s47/s32/s99\n/s32/s50/s48/s32/s75\n/s32/s52/s48/s32/s75\n/s32/s54/s48/s32/s75/s77/s32/s40 /s109\n/s66/s47/s102/s46/s117/s46/s41\n/s109\n/s48/s72/s40/s84/s41/s32/s50/s48/s32/s75\n/s32/s52/s48/s32/s75\n/s32/s54/s48/s32/s75\n/s32/s56/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s49/s50/s48/s32/s75/s40/s97/s41\n/s109\n/s48/s72/s32/s47/s47/s32/s99\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s84/s32/s40/s75/s41/s82\n/s48/s32/s40/s49/s48/s45/s50\n/s32/s99/s109/s51\n/s47/s67/s41/s40/s99/s41\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s82\n/s115/s32/s40/s99/s109/s51\n/s47/s67/s41\nFIG. 2. (Color online) Out-of-plane field dependence of (a) d c\nmagnetization M(µ0H) and (b) Hall resistivity ρxy(µ0H) for\nFeCr2Te4at indicated temperatures. (c) Temperature depen-\ndence of ordinary Hall coefficient R0(left axis) and anoma-\nlous Hall coefficient Rs(right axis) fitted from the ρxyvsµ0H\ncurves using ρxy=R0µ0H+RsM.\nlations based on Boltzmann equations and DFT band\nstructure [see below in Fig. 4(b)]. The electronic specific\nheat is:\nCe=π2\n3k2\nBTN(εF). (2)\nFrom Eq. (1), thermopower probes the specific heat per\nelectron: S=Ce/ne. The units are V K−1forS, J K−1\nm−3forCe, and m−3forn, respectively. It is common to\nexpressγ=Ce/TinJK−2mol−1units. Inordertofocus\non theS/Ceratio, we define a dimensionless quantity\nq=S\nTNAe\nγ, (3)\nwhereNAis the Avogadro number. This gives the num-\nber of carriers per formula unit (proportional to 1 /n)\n[33]. The obtained q= 0.10(1) indicates about 0.1 hole\nper formula unit within the Boltzmann framework [33].\nFigure 1(c) shows the temperature-dependent in-plane\nresistivity ρxx(T) of FeCr 2Te4, indicating a metallic be-\nhavior with a relatively low residual resistivity ratio\n[RRR = ρ(300K)/ ρ(2K) =1.7]. Aclearkinkis observed\natTc, correspondingwell to the PM-FIM transition. The\nrenormalized spin fluctuation theory suggests that the\nelectrical resistivity shows a T2dependence for itinerant\nferromagnetic system [34]. In FeCr 2Te4, the low temper-\nature resistivity fitting gives a better result by adding an3\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s50/s52/s54/s56/s49/s48/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s51/s46/s52/s48 /s45/s51/s46/s51/s53 /s45/s51/s46/s51/s48/s45/s53/s46/s53/s53/s45/s53/s46/s53/s48/s45/s53/s46/s52/s53/s45/s53/s46/s52/s48\n/s65 /s120/s121/s32/s40\n/s99/s109/s41/s45/s49\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s40/s100/s41/s40/s97/s41\n/s83\n/s72/s32/s40/s49/s48/s45/s50\n/s32/s86/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s110/s32/s40/s49/s48/s50/s49\n/s32/s99/s109/s45/s51\n/s41\n/s84/s32/s40/s75/s41/s108/s111/s103/s32/s65 /s120/s121\n/s32/s40 /s99/s109/s41\n/s108/s111/s103/s32\n/s120/s120/s32/s40 /s32/s99/s109/s41/s32/s61/s32/s49/s46/s49/s40/s50/s41/s40/s99/s41\nFIG. 3. (Color online) Temperature dependence of the carrie r\nconcentration (a) and the anomalous Hall conductivity σA\nxy=\nρA\nxy/(ρ2\nxx+ρ2\nxy) (b). Scaling behavior of the anomalous Hall\nresistivity (c) and the coefficient SH=µ0Rs/ρ2\nxx(d).\nadditional T3/2term that describes the contribution of\nspin fluctuation scattering [35].\nρ(T) =ρ0+aT3\n2+bT2, (4)\nwhereρ0is the residual resistivity, aandbare constants.\nThe fitting yields ρ0= 366(1) µΩ cm,a= 1.00(3) ×10−1\nµΩ cm K−1, andb= 2.8(3)×10−3µΩ cm K−2, indicating\ntheT3/2term predominates. This means the interaction\nbetween conduction electrons and localized spins could\nnot be simply treated asa small perturbation to a system\nof free electrons, i.e., strong electron correlation should\nbe considered in FeCr 2Te4.\nFigure 2(a) shows the isothermal magnetization mea-\nsured at varioustemperatures below Tc. All the M(µ0H)\ncurves rapidly increase in low field and change slowly in\nhigh field. Field dependence of Hall resistivity ρxy(µ0H)\nfor FeCr 2Te4at the corresponding temperatures are de-\npicted in Fig. 2(b). All the ρxy(µ0H) curves jump in low\nfield and then become linear-in-field in high field, indi-\ncating an AHE in FeCr 2Te4crystal. In general, the Hall\nresistivity ρxyin ferromagnets is made up of two parts,\nρxy=ρO\nxy+ρA\nxy=R0µ0H+RsM, (5)\nwhereρO\nxyandρA\nxyare the ordinary and anomalous Hall\nresistivity, respectively [36–39]. R0is the ordinary Hall\ncoefficientfromwhichapparentcarrierconcentrationand\ntype can be determined ( R0= 1/nq).Rsis the anoma-\nlous Hall coefficient. With a linear fit of ρxy(µ0H) in\nFIG. 4. (Color online) (a) Crystal structure, Brillouin zon e\n(BZ),andelectronic structureofFeCr 2Te4. Theredvectorsin\nthe crystal structure represent the directions of the magne tic\nmoments on Fe and Cr. The high symmetric k-paths in the\nBZ are shown. The x, y, and z directions of the Cartesian co-\nordinate are along the lattice vectors a, b, and c, respectiv ely.\nThe Fermi energy is set tozero. Calculated (b)Seebeck coeffi-\ncientSand (c) anomalous Hall conductivity σxyof FeCr 2Te4.\nThe calculated Sin low temperature shows good agreement\nwith the experiment. The σxyat the Fermi level (zero) is ∼\n127 (Ω cm)−1, much larger than the measured value of 22 (Ω\ncm)−1.\nhigh field, the slope and intercept corresponds to R0and\nρA\nxy, respectively. Rscan be obtained from ρA\nxy=RsMs\nwithMstaken from linear fit of M(µ0H) curves in high\nfield. The temperature dependence of derived R0andRs\nis plotted in Fig. 2(c). The value of R0is positive, in line\nwith the positive S(T), confirming the hole-type carries.\nThe derived Rsgradually decreases with decreasing tem-\nperature. Its magnitude is about two orders larger than\nthat ofR0.\nThe derived carrier concentration nis shown in Fig.\n3(a). The n∼0.5×1021cm−3at 20 K corresponds to ∼\n0.04 holes per formula unit, comparable to the value es-\ntimated from q. Taken into account a weak temperature-\ndependent ρ(T) [Fig. 1(c)], the estimated n∼1.11×1021\ncm−3from 484 µΩ cm near 100 K points to a mean free\npathλ∼0.44 nm. This is comparable to the lattice pa-\nrameters and is close to the Mott-Ioffe-Regel limit [40].\nThe AHC σA\nxy(≈ρA\nxy/ρ2\nxx) is plotted in Fig. 3(b). Theo-\nretically, intrinsic contribution of σA\nxy,inis of the order of\ne2/(hd), where eis the electronic charge, his the Plank\nconstant, and dis the lattice parameter [41]. Taking\nd≈V1/3∼4.3˚A,σA\nxy,inis estimated ∼900 (Ω cm)−1,4\nmuch larger than the obtained values in Fig. 3(b). Ex-\ntrinsic side-jump contribution of σA\nxy,sjis usually of the\norder of e2/(hd)(εSO/EF), where εSOandEFis spin-\norbital interaction energy and Fermi energy, respectively\n[42]. The value of εSO/EFis generally less than 10−2for\nmetallic ferromagnets. As we can see, the σA\nxyis about\n22 (Ω cm)−1at 20 K and exhibits a moderate tempera-\nture dependence. This value is much smaller than σA\nxy,in\n∼900 (Ω cm)−1, which precludes the possibility of in-\ntrinsic KL mechanism. Based on the band structure, as\nshown in Fig. 4, we obtained the intrinsic AHC as127(Ω\ncm)−1,whichismuchlargerthanthemeasuredvaluetoo.\nThe extrinsic side-jump mechanism, where the potential\nfield induced by impurities contributes to the anomalous\ngroup velocity, follows a scaling behavior of ρA\nxy=βρ2\nxx,\nthe same with intrinsic KL mechanism. The scaling be-\nhaviorof ρA\nxyvsρxxgivesα∼1.1(2) by using ρA\nxy=βρα\nxx\n[Fig. 3(c)], which also precludes the possibility of side-\njump and KL mechanism with α= 2. It points to that\nthe skew-scattering possibly dominates, which describes\nasymmetric scattering induced by impurities or defects\nand contributes to AHE with α= 1. Furthermore, the\nscaling coefficient SH=µ0Rs/ρ2\nxx=σA\nxy/Ms[Fig. 3(d)]\nis weaklytemperature-dependent and iscomparablewith\nthose in traditional itinerant ferromagnets, such as Fe\nand Ni (SH∼0.01−0.2V−1) [43, 44]. It is proposedthat\nthe FIM in FeCr 2Te4is itinerant ferromagnetism among\nantiferromagnetically coupled Cr-Fe-Cr trimers [28]. In\na noncomplanar spin trimer structures the topologically\nnontrivial Berry phase is induced by spin chirality rather\nthan spin-orbit effect, resulting in chirality-induced in-\ntrinsic AHE [45–48]. Our result excludes such scenario\nin Cr-Fe-Cr trimers in FeCr 2Te4[28].\nCONCLUSIONS\nIn summary, we studied the electronic transport prop-\nerties and AHE in FeCr 2Te4single crystal. The AHE be-\nlowTc= 123 K is dominated by extrinsic skew-scattering\nmechanism rather than the intrinsic KL or extrinsic side-\njump mechanism, which is confirmed by our DFT calcu-\nlations. 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Applied Physics Department, University of Salamanca. \n2. Dahlem Center for Complex Quantum Systems and Fachbereich Physik . \n*,+: Corresponding author s: victor@usal.es , edumartinez@usal.es   \n \nAbstract  \nBoth helicity -independent and helicity -dependent all -optical  switching  processes  driven \nby single ultrashort laser pulse ha ve been experimentally  demonstrated in ferrimagnetic \nalloys  as GdFeCo. Although the switching has been previously reproduced by atomistic \nsimulations, the lack of a robust micromagnetic  framework  for ferrimagnets  limits the \npredictions to small nano -systems, whereas the experiments are usually performed  with \nlasers and samples of tens  of micrometers. Here we develop a micromagnetic model based \non the extended Landau- Lifshitz -Bloch equation, which is  firstly validated by directly \nreproducing atomistic results for small samples  and uniform laser heating. After that, the \nmodel is used to study ultrafast single  shot all -optical switching in ferrimagnetic alloys  \nunder realistic conditions. We find that the helicity -independent switching under a \nlinearly polarized laser pulse is a pure thermal phenomenon, in which the size  of inverted \narea directly correlates with the maximum electron  temperature  in the sample . On the \nother hand, the  analysis of the helicity -dependent processes  under circular polarized \npulses in ferrimagnetic alloys with different composition indicates qualitative differences \nbetween the results predicted by  the magnetic circular dichroism and the ones from \ninverse Faraday effect . Based on these predictions, we propose experiments that would \nallow to resolve the controversy over the physical phenomenon that underlies these \nhelicity -dependent all optical processes . \n  2 \n I. INTRODUCTION \nAll optical switching (AOS) refers to  the manipulation of the magnetic state of a \nsample through the application of short laser pulses. The discovery of subpicosecond \ndemagnetization of a nickel sample   [1] upon application of  a short laser pulse, ranging \nfrom tens of femtosecond to several picoseconds , opened the path for other experiments \nto manipulate the magnetization using ultrashort laser pulses in ferromagnetic  [2,3] , \nsynthetic antiferromagnetic   [4–6] and ferrimagnetic materials  [7 –9]. While the AOS in \nferromagnetic materials is usually described by the Magnetic Circular Dichrois m \n(MCD)  [10,11]  or the Inverse Faraday Effect (IFE)  [12–14] , and it requires multiple \nshots of circularly polarized laser pulses  [15] , the inversion of the magnetization of \nferrimagnetic materials can be achieved by single- shot pulse  [16] , even with linear \npolarization. In these Helicity -Independent AOS  (HI-AOS)  processes, the reversal takes \nplace as the two antiferromagnetically coupled sublattices demagnetize at different rates \nwhen submitted to a laser pulse of adequate duration and energy. Since exchange \nprocesses conserve total angular momentum, the system transits through a ferromagnetic -\nlike state despite being ferrimagnetic at the ground state   [17] . The switching of the \nmagnetization is completed when the sublattices relax back to their thermodynamic \nequilibrium  [18] . On the other hand, several experimental studies  [7,9]  have also \nobserved that the magnetic state of ferrimagnetic alloys can be also reversed under \ncircular polarized laser pulses within a narrow range of laser  energies, resulting in a \nHelicity -Dependent AOS (HD -AOS ) which could be useful to develop ultrafast magnetic \nrecording devices purely controlled by optical means. While the single -shot HI -AOS can \nbe caused by the strong non- equilibrium due to the heating induced by the laser pulse, the \nphysical mechanisms behind the HD -AOS are still not completely understood, and \nseveral works participated by  the same authors ascribe it either to the IFE  [9]  or the \nMCD  [8]. Although several attempts have been performed to explain such AOS \nprocesses, a realistic numerical description of experimental observations is still missing . \nIndeed, some  theoretical studies usually adopt an atomistic description which is limited \nto small samples   [16,19] , with dimensions at the nanoscale, well below the size of the \nexperimentally studie d samples, with lateral sizes of several hundreds of microns . Such \natomistic approach cannot describe the non -uniform heating caused by laser beams of \nseveral microns, so it does not predict some multidomain patterns typically observed in \nthe experiments   [9]. On the other hand, other numerical attempts  have been carried out 3 \n by describing the ferrimagnetic alloy as an effective ferromagnetic sample, without \nconsidering  the individual nature of the two sublattices forming the ferrimagnet  [9] . \nAlthough these micromagnetic studies predict some features of the AOS processes, so  \nfar, the structure of the ferrimagnetic alloys has not been taken into account to investigate \nthe reversal of magnetic samples of micrometer size under realistic excitation conditions . \nAs the switching happens due to angular momentum transfer between subl attices , \nsomething impossible to account for within an effective ferromagnetic description, it is needed  to develop studies considering the two sublattice nature of the such alloys to \nnaturally evaluate their role on the reversal processes.  \nHere we present  a micromagnetic framework that is able to reproduce accurately \nthe atomistic results  of the laser -induced switching  by the extension of the conventional \nLandau -Lifshitz -Bloch (LLB) model  for ferrimagnets   [20,21] . Note that the  conventional \nLLB model does not allow to accurately describe AOS as indicated in  [21] , and here we \nextend it to solve this limitation . However, and differently from  atomistic simulations, \nwhich are limited to small samples at the nanoscale submitted to uniform laser heating, \nour micromagnetic formalism allows us to realistically describe AOS experimental \nobservations  by directly evaluating  extended samples at the microscale and non- uniform \nenergy absorption from the laser  pulse . The procedures here developed are essential to \nunderstand the physical aspects underlying these experiments , and will be useful for the \nfuture development of novel ultra -fast devices based on  these AOS processes.  After \npresenting and validating both the atomistic and the extended micromagnetic models for \nsample size of ten s of nanometers, the upper size limit of the atomistic spin models , we \ndescribe the results for HI -AOS processes in realisti c samples at the microscale for a \ntypically ferrimagnetic alloy  (GdFeCo) . Later on, we focus our attention to the description \nof the HD -AOS processes by exploring the role of the IFE and MCD separately for two \ndifferent ferrimagnetic alloys where the relative composition is slightly varied. Our \nresults allow us to suggest future  experiments which could be useful to infer the \ndominance of the IFE or the MCD  in single -shot HD-AOS  in ferrimagnetic alloys . \n \nII. ATOMISTIC AND MICROMAGNETIC MODELS  \nTypical ferrimagnetic (FiM) samples formed by a Transition Metal (TM:Co, CoFe) \nand a Rare Earth (RE:Gd) are considered here. Square samples in the 𝑥𝑥𝑥𝑥  plane with side 4 \n length ℓ and with thickness  𝑡𝑡𝐹𝐹𝐹𝐹𝐹𝐹 =5.6 nm are studied. At atomistic level the FiM sample \nis formed by a set of coupled spins, and the magnetization dynamics is described the \nLangevin -Landau -Lifshitz -Gilbert eq uation \n𝜕𝜕𝑆𝑆⃗𝐹𝐹\n𝜕𝜕𝑡𝑡=−𝛾𝛾0\n(1+𝜆𝜆2)�𝑆𝑆⃗𝐹𝐹×�𝐻𝐻��⃗𝐹𝐹+𝐻𝐻��⃗𝑡𝑡ℎ,𝐹𝐹 �+𝜆𝜆𝑆𝑆⃗𝐹𝐹×�𝑆𝑆⃗𝐹𝐹×�𝐻𝐻��⃗𝐹𝐹+𝐻𝐻��⃗𝑡𝑡ℎ,𝐹𝐹 ��� (1) \nwhere  𝑆𝑆⃗𝐹𝐹 is the localized magnetic moment and 𝐻𝐻��⃗𝐹𝐹 is the local effective field including \nintra- and inter -lattice exchange and anisotropy contributions . 𝐻𝐻��⃗𝑡𝑡ℎ,𝐹𝐹 is the local stochastic \nthermal field .  𝛾𝛾0 and 𝜆𝜆 are the gyromagnetic ratio and the damping parameter \nrespectively   [22] . Except the contrary is indicated, t ypical parameters  of Gdx(CoFe )1-x \nwith relative composition 𝑥𝑥 =0.25 were considered  [22] . See Supplemental Material \nNote  SN1  [23]  for further details , including material and numerical parameters . \nStarting from an initial uniform state of the FiM  with the spins  of the two sublattices  \nantiparallelly aligned each other along the easy axis 𝑧𝑧, a laser pulse is applied , and the \nirradiated sample absorbs energy from the laser pulse. The laser spot is assumed to have \na spat ial Gaussian profile ( 𝜂𝜂(𝑟𝑟)), with 𝑟𝑟0 being the radius  spot (𝑑𝑑0=2𝑟𝑟0 is the full width \nat half maximun, FWHM). Its temporal profile ( 𝜉𝜉(𝑡𝑡)) is also Gaussian, with 𝜏𝜏𝐿𝐿 \nrepresenting the pulse duration (FWHM). The  absorbed power density can be expressed \nas 𝑃𝑃(𝑟𝑟,𝑡𝑡)=𝑄𝑄𝜂𝜂(𝑟𝑟)𝜉𝜉(𝑡𝑡) where 𝜂𝜂(𝑟𝑟)=exp[−4ln(2)𝑟𝑟2/ (2𝑟𝑟0)2 ]  is the spatial profile \nwith 𝑟𝑟=�𝑥𝑥2+𝑥𝑥2 being the distance from the center of the laser spot, and 𝜉𝜉 (𝑡𝑡)=\nexp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/ 𝜏𝜏𝐿𝐿2] the temporal profile. 𝑄𝑄 is the maximum value of the \nabsorbed power density reached at 𝑡𝑡=𝑡𝑡0 just below  the center of the laser spot. \nLaser pulse heats the FiM  sample, and consequently, it is transiently dragged into a \nnon-equilibrium thermodynamic state, where its magnetization changes according to the \ntemperature dynamics. The temperature evolution is described by the Two Temperatures Model (TTM)  [9,24]  in terms of two subsystems: the electron ( 𝑇𝑇\n𝑒𝑒=𝑇𝑇𝑒𝑒(𝑟𝑟⃗,𝑡𝑡)) and the \nlattice (𝑇𝑇𝑙𝑙=𝑇𝑇𝑙𝑙(𝑟𝑟⃗,𝑡𝑡)), \n𝐶𝐶𝑒𝑒𝜕𝜕𝑇𝑇𝑒𝑒\n𝜕𝜕𝑡𝑡=−𝑘𝑘𝑒𝑒∇2𝑇𝑇𝑒𝑒−𝑔𝑔𝑒𝑒𝑙𝑙(𝑇𝑇𝑒𝑒−𝑇𝑇𝑙𝑙)+𝑃𝑃(𝑟𝑟,𝑡𝑡)−𝐶𝐶𝑒𝑒(𝑇𝑇𝑒𝑒−𝑇𝑇𝑅𝑅)\n𝜏𝜏𝐷𝐷 (2) \n𝐶𝐶𝑙𝑙𝜕𝜕𝑇𝑇𝑙𝑙\n𝜕𝜕𝑡𝑡=−𝑔𝑔𝑒𝑒𝑙𝑙(𝑇𝑇𝑙𝑙−𝑇𝑇𝑒𝑒) (3) \nwhere 𝐶𝐶𝑒𝑒 and 𝐶𝐶𝑙𝑙 denote the specific heat of electrons and  lattice subsystems, respectively. \n𝑘𝑘𝑒𝑒 is the electronic thermal conductivit y. 𝑔𝑔𝑒𝑒𝑙𝑙 is a coupling parameter between the electron 5 \n and lattice subsystems, and 𝜏𝜏𝐷𝐷 is the characteristic heat diffusion time to the substrate and \nthe surrounding medi a  [25] . Conventional values were adopted  (see  [1,16,26] and \nSupplemental Material Note SN1  [23]). \nThe approach that consists on solving Eq. (1 ) coupled to Eqs. ( 2)-(3) is named as \nAtomistic Spin Dynamics  (ASD), and due to computational restrictions,  its numerical \nsolution is limited to small samples at the nanoscale ( ℓ ≲ 100  nm, see Supplemental \nMaterial Note  SN2  [23]) . While A SD predicts the single -shot switching in small FiM \nnano- samples  [16,22,27] , the lack of a realistic micromagnetic framework for micro -size \nsamples and non- unifom laser spot limits the description of many experimental \nworks  [9] . In particular, the appearance of central regions with a multi -domain \ndemagnetized pattern s  [28] , or the observation of rings  of switched magnetization  under \nirradiation with lasers of tens of micrometers   [29] cannot be reproduced by AS D due to \nsuch computing limitations. In order to overcome the ASD limitations , here we develop \nan extended continuous micromagnetic model that describes the temporal evolution of \nthe reduced local magnetization  𝑚𝑚��⃗𝐹𝐹(𝑟𝑟 ⃗,𝑡𝑡) of each sublattice 𝑖𝑖 :RE,TM based on the \nconventional ferrimagnetic Landau -Lifshitz -Bloch (LLB) Eq  [21,30], \n𝜕𝜕𝑚𝑚��⃗𝐹𝐹\n𝜕𝜕𝑡𝑡=−𝛾𝛾0𝐹𝐹′�𝑚𝑚��⃗𝐹𝐹×𝐻𝐻��⃗𝐹𝐹�+ \n−𝛾𝛾0𝐹𝐹′𝛼𝛼𝐹𝐹⊥\n𝑚𝑚𝐹𝐹2𝑚𝑚��⃗𝐹𝐹×�𝑚𝑚��⃗𝐹𝐹×�𝐻𝐻��⃗𝐹𝐹+𝜉𝜉⃗𝐹𝐹⊥��+ \n+𝛾𝛾0𝐹𝐹′𝛼𝛼𝐹𝐹∥\n𝑚𝑚𝐹𝐹2(𝑚𝑚��⃗𝐹𝐹·𝐻𝐻��⃗𝐹𝐹)𝑚𝑚��⃗𝐹𝐹+𝜉𝜉⃗𝐹𝐹∥ (4) \nwhere 𝐻𝐻��⃗𝐹𝐹=𝐻𝐻��⃗𝐹𝐹(𝑟𝑟 ⃗,𝑡𝑡) is the local effective field on sublattice magnetic moment 𝑖𝑖 at location \n𝑟𝑟 ⃗ of the FiM sample , 𝛼𝛼𝐹𝐹∥ and 𝛼𝛼𝐹𝐹⊥ are the longitudinal and perpendicular damping \nparameters, and 𝜉𝜉⃗𝐹𝐹∥ and 𝜉𝜉⃗𝐹𝐹⊥ are the longitudinal and perpendicular stochastic thermal fields. \nDetails of the LLB model can be found in [21,30] and in Supplemental Material Note  \nSN2  [23]. C ontrary to the ASD, where the spatial discretization is imposed by the \natomistic scale ( 𝑎𝑎= 0. 35 nm ), within the micromagnetic model the sample is discretized \nin elementary cells with dimensions of  Δ𝑥𝑥=Δ𝑥𝑥~1nm a\nnd Δ𝑧𝑧=𝑡𝑡𝐹𝐹𝐹𝐹𝐹𝐹. Therefore, it is \npossible to numerically  evaluate, with manageable  computing effort, extended samples at \nthe microscale  (ℓ~100  μm), three order s of magnitude larger than the  ones which can be \ndealt with  the ASD model. 6 \n Numerically solving Eq. (4) coupled to Eqs . (2)- (3) under ultra -short laser pulses \nprovides a micromagnetic description of several AOS processes in ferromagnetic \nsystems  [13] . However, when dealing with ferr imagnetic  samples we checked that some \ndisagreement with the predictions of the ASD mode l were observed (see Supplemental \nMaterial Note  SN3  [23] ), which are related to the lack of a proper description of the \nangular moment exchange between sublattices during the non- equilibrium transient state \npromoted by the laser pulse. Indeed, magnetization dynamics in FiMs  is driven by \ndissipative processes of relativistic and exchange nature. The relativistic ones allow \nexchange of angular momentum between the magnetization and the lattice degree of \nfreedom due to the spin- orbit coupling between them, and are phenomenologically \ndescribed by the usual damping terms in the LLB Eq. (4). Additiona lly, in multisublattice \nmagnets as FiMs, another different pathway opens local exchange of angular momentum \nbetween both sublattices of the FiM, and to account for  it, the LLB Eq . (4) has to be \nenhanced by an additional exchange relaxation torque  [18,31–34] . The simplest model \nto describe the sublattice -specific magnetization dynamics in FiMs, was derived from \nOnsager’s relations  [31]  within a macrospin approach based on a microscopic spin \nmodel. In this simplified description, the magnetization dynamics of sublattice 𝑖𝑖  can be \nexpressed as 1\n𝛾𝛾0𝑖𝑖𝑑𝑑𝑚𝑚𝑖𝑖\n𝑑𝑑𝑡𝑡=𝛼𝛼𝐹𝐹𝐻𝐻𝐹𝐹+𝛼𝛼𝑒𝑒𝑒𝑒�𝜇𝜇𝑖𝑖\n𝜇𝜇𝑗𝑗𝐻𝐻𝐹𝐹−𝐻𝐻𝑗𝑗�, where 𝑖𝑖,𝑗𝑗:RE,TM, 𝜇𝜇𝐹𝐹 and 𝐻𝐻𝐹𝐹 are the \nmagnitude of the magnetic moment and the e ffective field acting on macrospin of \nsublattice 𝑖𝑖 respectively . The relativistic relaxation parameter in this model, 𝛼𝛼𝐹𝐹, \ncorresponds  to the longitudinal damping parameter in the LLB E q. and it depends on the \ntemperature of the thermal bath  to which ang ular momentum and energy is dissipated. \nDifferently, it is assumed that the exchange relaxation parameter 𝛼𝛼 𝑒𝑒𝑒𝑒, only depends on \nthe non- equilibrium sublattice magnetizations, 𝛼𝛼𝑒𝑒𝑒𝑒=𝛼𝛼𝑒𝑒𝑒𝑒(𝑚𝑚𝐹𝐹,𝑚𝑚𝑗𝑗). Considering that \nexchange relation rate should be symmetric with respect to the sublattice index, \n𝛼𝛼𝑒𝑒𝑒𝑒(𝑚𝑚𝐹𝐹,𝑚𝑚𝑗𝑗)=𝛼𝛼𝑒𝑒𝑒𝑒(𝑚𝑚𝑗𝑗,𝑚𝑚𝐹𝐹), a simple functional fulfilling these heuristic conditions \nyields to 𝛼𝛼𝑒𝑒𝑒𝑒(𝑚𝑚𝐹𝐹,𝑚𝑚𝑗𝑗)=𝜆𝜆𝑒𝑒𝑒𝑒𝑚𝑚𝑖𝑖+(𝑒𝑒𝑗𝑗𝜇𝜇𝑗𝑗/𝑒𝑒𝑖𝑖𝜇𝜇𝑖𝑖 )𝑚𝑚𝑗𝑗\n𝑚𝑚𝑖𝑖𝑚𝑚𝑗𝑗 where 𝜆𝜆𝑒𝑒𝑒𝑒 is a phenomenological parameter  \nrepresenting the exchange relaxation rate and 𝑥𝑥𝐹𝐹 the concentration of each specimen 𝑖𝑖. \nInspired by this two sublattice phenome nological model based on Onsager’s relations, \nhere we add an additional torque 𝜏𝜏⃗𝐹𝐹𝑁𝑁𝑁𝑁 to the micromagnetic LLB Eq. (4) that accounts for \nnon-equilibrium magnetic moment exchange  between sublattices , and becomes crucial to \ndescribe AOS ultra -fast switching in FiMs  under realistic conditions . The torque reads as  7 \n 𝜏𝜏⃗𝐹𝐹𝑁𝑁𝑁𝑁=𝛾𝛾0𝐹𝐹′𝜆𝜆𝑒𝑒𝑒𝑒𝛼𝛼𝐹𝐹∥𝑥𝑥𝐹𝐹𝜇𝜇𝐹𝐹𝑚𝑚𝐹𝐹+𝑥𝑥𝑗𝑗𝜇𝜇𝑗𝑗𝑚𝑚𝑗𝑗\n𝜇𝜇𝐹𝐹𝑚𝑚𝐹𝐹𝜇𝜇𝑗𝑗𝑚𝑚𝑗𝑗�𝜇𝜇𝐹𝐹𝐻𝐻��⃗𝐹𝐹∥−𝜇𝜇𝑗𝑗𝐻𝐻��⃗𝑗𝑗∥� (5) \nwhere and 𝐻𝐻��⃗𝐹𝐹∥ and 𝐻𝐻��⃗𝑗𝑗∥ are the longitudinal effective field s for each lattice 𝑖𝑖:RE,TM  [21] , \n𝑥𝑥𝐹𝐹≡𝑥𝑥 and 𝑥𝑥𝑗𝑗=1−𝑥𝑥𝐹𝐹=1−𝑥𝑥 are the concentration s of each specimen, and 𝜆𝜆𝑒𝑒𝑒𝑒 is a \nparameter representing the exchange relaxation rate  [18] . By including Eq. (5) in the \nRHS of Eq. (4), and numerically solving it coupled to TTM Eqs. (2) -(3), w e can provide \na realistic description of the magnetization dynamics in FiM systems  under ultra -short \nlaser pulses. In what follows, we refer to this formalism as the extended micromagnetic \nLLB model  (eLLB) , to distinguish it from the conventional  LLB model  (LLB) when \n𝜏𝜏⃗𝐹𝐹𝑁𝑁𝑁𝑁=0. See Supplemental Material Note  SN2  [23]  for the rest of details.  \n \nIII. RESULTS  AND DISCUSSION  \nBefore presenting the predictions of the extended micromagnetic model for realistic \nFiM samples and laser  beam s at the microscale, here we firstly compare the results \nobtained from the extended LLB model ( eLLB) to the ones resulting from the atomistic \nspin dynamics simulations (ASD) for a small FiM dot at the nano -scale (ℓ≈25 nm ). As \ntypical laser spots have radius  of 𝑟𝑟0~1 μm−10 μm or even larger , we assume here that \nthe power absorbed by the FiM dot the from the laser pulse is uniform, that is, 𝜂𝜂(𝑟𝑟)=1. \nThe pulse duration is 𝜏𝜏 𝐿𝐿=50 fs. Typical results showing  the temporal evolution of the \nout-of-plane averaged mag netization ( 𝑚𝑚𝑧𝑧𝐹𝐹) for each sublattice ( 𝑖𝑖:TM (red),  RE (blue)) are \nshown in FIG.  1 for two different values of 𝑄𝑄. 8 \n  \nFIG. 1 . Comparison between atomistic simulation (ASD, solid lines) and micromagnetic model \n(eLLB , dashed lines) results for the temporal evolution of the out -of-plane magnetization for the \ntransition metal (TM, red) and the rare earth (RE, blue) under two different laser power densities: \n(a) 𝑄𝑄=6×1021 W/m3 and (b) 𝑄𝑄=12×1021 W/m3. Four consecutive laser pulses with 𝜏𝜏𝐿𝐿=\n50 fs are applied every 20  ps. (c) shows a detailed view of first pulse switching event as in ( a), \nwhile (d) shows temporal evolution of the electron ( 𝑇𝑇𝑒𝑒) and lattice ( 𝑇𝑇𝑙𝑙) temperatures for 𝑄𝑄=\n6×1021 W/m3. (e) and  (f) corresponds to 𝑄𝑄=12×1021 W/m3. Shaded interval  in (e)  shows \nthe transient ferromagnetic state. The pulse length  is 𝜏𝜏𝐿𝐿=50 fs. The eLLB results were obtained \nwith 𝜆𝜆𝑒𝑒𝑒𝑒=0.013. \n \nA remarkable agreement between both ASD and e LLB models with similar \ndynamics  for both sublattices  is observed in FIG. 1 (a) and (b) . For low 𝑄𝑄 values ( FIG. \n1(a) and (c) ) there is no switching, but when 𝑄𝑄 increases above a threshold, which \ndepends on the pulse length ( 𝜏𝜏𝐿𝐿), the deterministic AOS  is predicted by both ASD and \neLLB models ( FIG.1(b)  and (e) ). It is important to note that similar switching was  also \nobtained within the deterministic e LLB framework, that is, in the absence of thermal \n9 \n fluctuations  (𝜉𝜉⃗𝐹𝐹⊥=𝜉𝜉⃗𝐹𝐹∥=0 in Eq. (4) , see FIG. S4(b) in Supplemental Material Note  \nSN3  [23] ). On the contrary, the conventional  LLB model ( LLB, 𝜏𝜏⃗𝐹𝐹𝑁𝑁𝑁𝑁=0) fails to \nreproduce the switching of FIG. 1(b)  (see FIG. S4(c) -(d) in Supplemental Note \nSN3  [23] ). FIG. 1(c)  show s the details of the temporal evolution 𝑚𝑚𝑧𝑧𝐹𝐹 for 𝑄𝑄=\n6×1021 W/m3 during the first laser pulse , while the corresponding evolutions  of 𝑇𝑇𝑒𝑒 and \n𝑇𝑇𝑙𝑙 are depicted in FIG. 1(d) , which also shows the laser pulse . Corresponding results for \n𝑄𝑄=12×1021 W/m3 are shown in FIG.  1(e) and (f) respectively . For 𝑄𝑄=\n6×1021 W/m3, the electron temperature reaches a peak maximum value of  𝑇𝑇𝑒𝑒≈850  K \nat the end of the laser pulse, but this is not enough to achieve the switching . For 𝑄𝑄=\n12×1021 W/m3, 𝑇𝑇𝑒𝑒 reaches  a peak  of 𝑇𝑇𝑒𝑒≈1150  K and switching takes place . Not ice \nthat this value is well above the Curie temperature ( 𝑇𝑇𝐶𝐶≈600  K), and therefore the system \nneeds to be significantly heated above the Curie threshold to achieve the deterministic \nAOS  in FiM . These processes  are explained by the different demagnetization rates of the \nRE and TM sublattices, that lead to a transient ferromagnetic aligment. Such transient \nferromagnetic state is observed during a  short transient  (see shaded interval in FIG. 1(e)), \nand it is only present when the system is far away from the thermodynamic equilibrium, \nas caused by the ultrafast laser heating.  Except the contrary is indicated, all eLLB results \nwere obtained wi th 𝜆𝜆𝑒𝑒𝑒𝑒=0.013, (s ee FIG. S 5 and its corresponding discussion in \nSupplemental Material Note SN3 (c)  [23]  for results with other values of 𝜆𝜆𝑒𝑒𝑒𝑒). \n \nHelicity -Independent A ll Optical Switching (H I-AOS) . Once validated the e LLB \nformalism by reproducing the ASD results for small nano- sample s under uniform \nlinearly -polarized laser pulses, we can now use it to explore the influence of the laser \nduration ( 𝜏𝜏𝐿𝐿) and maximum absorbed power  density  (𝑄𝑄) in realistic extended samples at \nthe microscale (ℓ ~10 μ m). This is illustrated  in the phase diagram of FIG.  2(a), which \nshows the final state under a single  linearly -polarized laser  pulse starting from a uniform \nstate of the FiM . White color indicates the combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) where the sample \nreturns to the original state after the pulse  (no-switching) . The blue region corresponds to \ncombinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) presenting  deterministic HI-AOS  after each pulse, and red \ncorresponds to combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) resulting in a final demagnetized multidomain \nconfiguration. It is noted that there is a correlation b etween the final state and the \nmaximum electron temperature reached in the sample, which is shown by the overlapping 10 \n solid black lines in FIG.  2(a). As it is clearly observed, s olid lines coincide with \nboundaries between the three possible behaviors  already discussed. Indeed, the transition \nbetween no -switching (white) to the deterministic switching range (red) is limited by the \n~1000  K curve , whereas the transition to the thermal demagnetization (blue) occurs when \n𝑇𝑇𝑒𝑒≳1400  K, as shown in FIG.  2(a). Instead of 𝑄𝑄 , the information collected in the phase \ndiagram of FIG. 2(a)  could be also presented in terms  of the laser fluence ( 𝐹𝐹≡𝑄𝑄 𝜏𝜏𝐿𝐿 𝑡𝑡𝐹𝐹𝐹𝐹𝐹𝐹), \nas it is done in FIG.  S6 of Supplemental Material Note SN 4  [23] . Note, that such phase \ndiagram is also in good qualitative agreement with recent experimental \nobservations  [35] .  \n \nFIG. 2 . (a) Phase diagram of the final state as a function of 𝑄𝑄  and 𝜏𝜏𝐿𝐿 for a small nano -sample ℓ=\n25 nm under uniform laser heating ( 𝜂𝜂(𝑟𝑟)=1). White, red and blue colors represent no -\nswitching, deterministic switching and thermal demagnetization behaviors respectively. Solid \nlines are isothermal curves showing the maximum electron temperature ( 𝑇𝑇𝑒𝑒) reached due to the \npulse. (b) Typical mi cromagnetic snapshots of the initial and final magnetization of RE and TM \nfor three combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿). I: (8×1021 W/m3,20 fs), II: (15×1021 W/m3,30 fs) and \nIII: (20×1021 W/m3,60 fs). Here extended samples ( ℓ=20 μm) with a laser spot of 𝑑𝑑0=ℓ/2 \nwere considered. Dashed purple lines in the images of the initial state indicate the FWHM of  the \nlaser spot . The results of the phase diagram  (a) coincide with (b) for magnetization at the center \nof the laser spot . \n \nThe main advantage of the extended e LLB model over ASD simulations is that it  \nallows us  to explore  realistic samples and laser beams with dimensions that are not \naccessible with ASD models. e LLB model ( Eqs. (4) and (2) -(3)) has been used to simulate \n11 \n samples with lateral size of ℓ=20 μ m. From no w on , the spatial Gaussian dependence \nof the laser beam is considered  (𝜂𝜂(𝑟𝑟)=exp[−4ln(2)𝑟𝑟2/ (2𝑟𝑟0)2 ]), with a laser spot \ndiameter  of 𝑑𝑑0=2𝑟𝑟0=ℓ/2. Typical initial and final states corresponding to three  \nrepresentative combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) are shown in FIG. 2(b). Our micromagnetic \nsimulations point out again that the three types of behaviors observed experimentally (see  \nfor example Fig. 4(a) in   [9] or  [6,28] ) are also achieved under these realistic conditions, \nwith samples and laser spots at the microscale. Not ice that now the final magnetic state \ndepends on the local position because the power absorption from laser pulse does . The \nfinal states depict a radial symmetry  around the center of the laser spot, which coincides \nwith the center of the  FiM  sample at (𝑥𝑥𝑐𝑐,𝑥𝑥𝑐𝑐)=(0,0). \nIn order to further describe such spatial dependence, FIG.  3 plots the final state of \n𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹 as a function of 𝑥𝑥 along the central line of the FiM sample ( 𝑥𝑥=0) for the same \nthree representative combinations  of (𝑄𝑄,𝜏𝜏𝐿𝐿) as in FIG.  2(b). The maximum electron \ntemperature 𝑇𝑇𝑒𝑒=𝑇𝑇𝑒𝑒(𝑥𝑥) is also plotted in top graphs by blue curves. Bottom graphs in \nFIG. 3 show the final state over the sample plane (𝑥𝑥,𝑥𝑥). These graphs clearly correlate \nthe local final magnetic state ( 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹(𝑥𝑥,𝑥𝑥)) with the maximum electron temperature \n𝑇𝑇𝑒𝑒(𝑥𝑥,𝑥𝑥). In the no- switching regime (I), the electron temperature does not reach 1000  K \nat any point . For combinations (𝑄𝑄,𝜏𝜏𝐿𝐿) as II, 𝑇𝑇𝑒𝑒(𝑥𝑥)≳1000  K is only reached in the cent ral \nregion , whose dimensions  fit the local part of the sample that switches its magnetization. \nNote that 𝑇𝑇 𝑒𝑒 remains below 𝑇𝑇 𝑒𝑒(𝑥𝑥)≲1400  K. Finally, the demagnetiz ed case (multi -\ndomain pattern, III) occurs in the part of the sample where 𝑇𝑇𝑒𝑒≳1400  K, but deterministic \nswitching is s till obtained in the ring region, where 1000  K≲𝑇𝑇𝑒𝑒(𝑥𝑥)≲1400  K. \nMicromagnetic images  (bottom graphs in FIG. 3 ) are in good agreement with typical \nexperimental HI -AOS observations   [6,28] . 12 \n  \nFIG. 3 . Final out -of-plane magnetization ( 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹(𝑥𝑥)) and maximum electron temperature ( 𝑇𝑇𝑒𝑒(𝑥𝑥)) \nas function of 𝑥𝑥  for 𝑥𝑥=0 and for the t hree representative combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) as in FIG. 2(b)  \n(top graphs). The corresponding final states over the sample plane (𝑥𝑥,𝑥𝑥) are shown in bottom \ngraphs.  \n \nThe inferred  correlation between the maximum electron temperature and the final \nmagnetic state allows us to predict the size of the inverted region by studying the \nmaximum electron temperature  reached in the sample by using the TTM (Eqs. (2)-(3)) in \ncombination with t he switching diagram of FIG. 2(a) . The radius of the switched area \n(𝑅𝑅𝑠𝑠) calculated from micromagnetic simulations (dots, e LLB), and the one predicted by \nthe TTM (lines, TTM) is shown in FIG. 4  as function of 𝜏𝜏 𝐿𝐿 for two different values of 𝑄𝑄 \nwithin the deterministic switching range  (1000  K≲𝑇𝑇𝑒𝑒(𝑥𝑥)≲1400  K). Again, good \nagreement is obtained , a fact that points out that the origin of these HI -AOS processes \nunder linearly polarized laser pulse is a purely thermal  phenomenon. Indeed, as the local \nmaximum electron temperature  reached in the sample only depends on the absorbed \npower from the laser ( 𝑄𝑄) and the laser pulse length ( 𝜏𝜏𝐿𝐿), the size of the inverted region \ncan be directly obtained from the TTM by the condition of 𝑇𝑇 𝑒𝑒≳1000  K. \n13 \n  \nFIG. 4 . Radius of the switched area ( 𝑅𝑅𝑠𝑠) as a function of laser duration ( 𝜏𝜏𝐿𝐿) for two values of the \nmaximum absorbed power ( 𝑄𝑄). Dots are micromagnetic results from the e xtended e LLB model . \nLines are predictions from the TTM, where 𝑅𝑅𝑠𝑠 is inferred from the condition that the local \nmaximum electron temperature reaches 𝑇𝑇𝑒𝑒≳1000  K. \n \nHelicity -Dependent All Optical Switching (H D-AOS) . Previous results were \ncarried out by applying laser pulses with linear polarization  (𝜎𝜎=0), and show that the \nHI-AOS can be achieved in a controlled manner with an adequate election of the laser \npower (𝑄𝑄) and duration ( 𝜏𝜏𝐿𝐿): the magnetization switches its direction in the p icoseconds  \nrange independently on the initial state. While this is interesting for toogle memory \napplications, the procedure to store and record a bit  using linearly  polarized  laser pulses \nwould still require two steps: ( 𝑖𝑖) a pre liminary  reading operation of the magnetic state, \nand after that, ( 𝑖𝑖𝑖𝑖) deciding or not to apply the laser pulse depending on the  preceeding \nstate. This two -step procedure can be avoided by using circularly polarized laser pulses, \nresulting in  Helicity -Dependent AOS  processes  (HD- AOS) . However, as it was already \ncommented the physics behind these HD -AOS observations still remains un clear, and \nboth the Magnetic Circular Dichroism (MCD)  [8,10]  and the Inverse Faraday Effect \n(IFE)  [12,14]  have been suggested as responsible of the experimental observations. In \nwhat follows, we explore both mechanisms in a separated manner by including them in \nthe extended micromagnetic model.  \nLet firstly consider the Magnetic Circular Dichroism . It has been suggested to be \nplay a dominant  a role on these HD- AOS  processes  in GdFeCo ferrimagnetic samples, \nwhich are known for its strong magneto- optical effect   [8]. According to the MDC \nformalism,  right -handed (𝜎𝜎+) and  left-handed (𝜎𝜎−) circular ly polarized laser  pulses \n14 \n experience different refractive indices, and consequently a differen ce in energy \nabsorption of the FiM sample for 𝜎𝜎+ and 𝜎𝜎− pulses is expected. The MCD coefficient \ncan be calculated from the total absorption for each polarization, resulting in MCD≡𝑘𝑘=\n(𝐴𝐴−−𝐴𝐴+)/(1\n2(𝐴𝐴++𝐴𝐴−)), where 𝐴𝐴± represent the total absorption  for each polarization, \n(±≡𝜎𝜎±). Indeed, the MCD makes the power absorbed by the sample ( 𝑃𝑃(𝑟𝑟,𝑡𝑡)) to depend \non the laser helicity ( 𝜎𝜎±=±1 for right -handed and left -handed circular helicities ) and \non the initial net magnetic state ( 𝑚𝑚𝑁𝑁(0)=𝑀𝑀𝑆𝑆𝑇𝑇𝐹𝐹𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹(0)+𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁(0)), up (↑: \n𝑚𝑚𝑁𝑁(0)>0) or down (↓:𝑚𝑚𝑁𝑁(0)<0). Note that 𝑚𝑚 𝑧𝑧𝑇𝑇𝐹𝐹(0)=±1 and 𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁(0)=∓1, \nwhereas 𝑀𝑀𝑆𝑆𝑇𝑇𝐹𝐹 and 𝑀𝑀𝑆𝑆𝑅𝑅𝑁𝑁 are both positive. Under a right -handed laser pulse  (𝜎𝜎+), an \ninitially up (down)  magnetic state is expected to absorb more (less)  energy than the \ninitially down  (up) state . Therefore, 𝑃𝑃(𝑟𝑟,𝑡𝑡) in Eq. (2) is replaced by 𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)𝑃𝑃(𝑟𝑟,𝑡𝑡) \nwith 𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)=�1+12𝑘𝑘𝜎𝜎±sign (𝑚𝑚𝑁𝑁)� describing  the different absorption power  for \nup and down magnetization states as depending on the laser helicity.  See further details \non the implementation of the MCD in Supplemental Material Note SN 5  [23] . \nWe have evaluated the role of the MCD in the eLLB model with several values of \nthe MCD coefficient  (𝑘𝑘). The isothermal curve delimiting the border between the no-\nswitching and switching  regimes  now depends on the combination of helicity and initial \nnet magnetic state (see such isothermal threshold curves for different values of the MCD \ncoefficient in of SN 5). Considering  a realistic value of  MCD≡𝑘𝑘~2%, as estimated \nin  [8], the electron temperature variation is quite small, typically  a few  units of K, and \ntherefore small variations in the phase diagram are obtained with respect to the one for \nlinearly -polarized pulses  FIG. 2(a) (see also FIG. S 7 in Supplemental Material Note \nSN5  [23] ). However, when exciting with circular polarized pulses close to the no-\nswitching /switching  boundary, the FiM  switches or not depending on the helicity and \ninitial net state, only within a narrow  interval  of 𝑄𝑄. This is represented in FIG.  5(a), where \nthe H D-AOS is shown for 𝜏𝜏 𝐿𝐿=50 fs pulses with different 𝑄𝑄 in a sample with  ℓ=20 μm. \nNote that these results correspond to a F iM alloy Gdx(FeCo) 1-x with 𝑥𝑥=0.25, and that for \nthis relative composition the RE is the dominant sublattice at room temperature, 𝑀𝑀 𝑠𝑠𝑅𝑅𝑁𝑁>\n𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹 at 𝑇𝑇=300  K (see inset of FIG. S3 of Supplemental Material  [23] or FIG. 6(a) ). No \nswitching is achieved for low energy values (see left colum in FIG. 5(a) for 𝑄𝑄=\n5.7×1021 W/m3). However, if 𝑄𝑄 increases to 𝑄𝑄 =5.8×1021 W/m3, the system shows  \nthe so -called H D-AOS: if 𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁 is initially down (up), the reversal is only achieved for  15 \n left-handed helicity, 𝜎𝜎−=−1 (right -handed helicity, 𝜎𝜎+=+1). Consequently, the final \nstate can be selected by chosing the laser helicity, which is relevant  for ultrafast memory \napplications. It is important to note that this H elicity Dependent AOS is only obtained in \na very narrow range of 𝑄𝑄 around the H elicity Independent AOS boundary . Indeed, a small \nincrease of the absorbed power results again in H I-AOS as the linear polarized case (see \n3rd and 4th columns in FIG. 5(a)  for 𝑄𝑄 =5.9×1021 W/m3 and 𝑄𝑄=9.0×1021 W/m3). \nFor high values of 𝑄𝑄, the final state depicts a ring around a central demagnetized state, \nsimilar to HI -AOS case (see right column in FIG. 5(a)  for 𝑄𝑄=18×1021 W/m3). In this \ncase, the maximum electron temperature overcomes the  𝑇𝑇𝑒𝑒≃1400  K threshold in the \ncentral region below the laser spot , resulting in a central  demagnetized or multidomain \nstate. However, the maximum 𝑇𝑇𝑒𝑒 remains with in the range of HD -AOS deterministic \nswitching ( 1000  K≲𝑇𝑇𝑒𝑒(𝑥𝑥)≲1400  K) in the ring  around the central part . These \nmicromagnetic predictions , including the narrow range of HD -AOS,  are in good \nagreement with several experimental observations  (see, for instance,  Fig. 4 and 7(a) \nin  [9] , Fig. 1(b) in  [4] , or Fig. 3 in  [36] ).  \n \n16 \n FIG. 5. Helicity -Dependent AOS  predicted by the MCD . (a) Snapshots of the final RE  magnetic \nstate (𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁) after a laser pulse of 𝜏𝜏𝐿𝐿=50 fs for five  different values of the absorbed power density \n(𝑄𝑄). Results are shown for four combinations of the initial state ( ↑,↓) and helicities ( 𝜎𝜎±) as \nindicated at the left side.  The HD- AOS is shown in panel corresponding to 𝑄𝑄=\n5.8×1021 W/m3. (b) RE magnetic state after every pulse for 𝑄𝑄=5.9×1021 W/m3, showing \nthe appearance of a ring due to the MCD and the switching of the central  part. Here pulses with \nleft-handed chirality are applied ( 𝜎𝜎+). The sample side is ℓ=20 μm and the laser spot diameter \nis 𝑑𝑑0=ℓ/2. \n \nMoreover, the inclusion of the MCD in our e LLB  model allows us to explain the \nexperimental observation of rings  [28,36,37]  which appear after the application of a \nsecond laser pulse. This is illustrated  in FIG. 5(b)  for pulses with 𝑄𝑄=5.9×1021 W/m3 \nand 𝜏𝜏𝐿𝐿=50 fs. The central part of the sample reaches temperatures that lead to H I-AOS, \nand ther efore, its magnetization reverses after each pulse. On the contrary, the ring around \nof the inverted region is within the H D-AOS  regime , and therefore, its local magnetic \nstate (going from up to down) is only reversed by the first pulse. For the second and \nsubsequent  pulses , the ring maintains its down state while the inner part changes again to \nup (white). This is repeated every pulse, with the inversion of the central part and the \nmaintenance of bla ck ring in the external shell, as it is clearly seen in even pulses (see \nsnapshots after pulses #2 and #4 in FIG. 5(b) ). Note that this ring structure differs from \nthe ones shown in FIG. 2(b)  and FIG.3 , as they were caused by the inversion of the \nmagnetiza tion around the central demagnetized part under high -power linear pulses  (𝜎𝜎=\n0). For circularly polarized pulses  (𝜎𝜎±=±1) the images correspond to alternative \nswitching and the H D-AOS  without the central demagnetized (multidomain) state . Again, \nthese results are in good agreement with recent experimental observations (see figures \nin  [28,36,37] ). \n \nInstead  of the MCD, several other works claim that the observations of the HD -\nAOS can be ascribed to the Inverse Faraday Effect (IFE)  [9]. Within this formalism, the \nlaser pulse generates an effective out -of-plane magneto -optical field which direction \ndepends on the laser pulse helicity , 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)=𝜎𝜎±𝐵𝐵𝐹𝐹𝑀𝑀𝜂𝜂(𝑟𝑟)𝜖𝜖(𝑡𝑡)𝑢𝑢�⃗𝑧𝑧, where 𝜂𝜂(𝑟𝑟)=\nexp[−4ln(2)𝑟𝑟2/ (2𝑟𝑟0)2 ]  is the spatial field profile, and 𝜖𝜖(𝑡𝑡) is its temporal profile. \nNote that the spatial dependence of 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) is the same as the one of the  absorbed power \ndensity. However, a ccording to the literature   [9], the so- called magneto -optical field \n𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) has some temporal persistence with respect to the laser pulse, and therefore  its 17 \n temporal profile is different for 𝑡𝑡 <𝑡𝑡0 and 𝑡𝑡 >𝑡𝑡0: 𝜖𝜖(𝑡𝑡<𝑡𝑡0)=exp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/\n 𝜏𝜏𝐿𝐿2], and 𝜖𝜖 (𝑡𝑡≥𝑡𝑡0)=exp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/(𝜏𝜏𝐿𝐿+𝜏𝜏𝐷𝐷)2], where 𝜏𝜏 𝐷𝐷 is the delay time of \nthe 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) with respect to the laser pulse. We have evaluated this IFE scenario by \nincluding this field 𝐵𝐵 �⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) in the effective field of Eq. (4). The  results  for the same \nFiM alloy considered up to here ( Gdx(FeCo) 1-x, with x=0.25, see SN 5), are similar to the \nones already presented in FIG. 5  for the MCD considering a maximum magneto -optical \nfield of 𝐵𝐵𝐹𝐹𝑀𝑀=20 T  with a delay time of 𝜏𝜏𝐷𝐷=𝜏𝜏𝐿𝐿. These IFE results can be seen in FIG. \nS8 in Supplemental Material Note  SN6  [23] . Therefore, we could conclude from this \nanalysis that, from the  micromagnetic  modeling point of view, both the MCD and the IFE \nare compatible with experimental observations of the HD -AOS . At this point, it is worth \nto mention here that in real experiments there is not a clear distinction between MCD and \nIFE phenomena. Indeed, the modeling of the IFE for absorbing materials can account for \nabsorption phenomena as MCD (see for instance  [38,39] ). These works suggested that \nin micromagnetic simulations the IFE could induce  a change of the magnetic moment \n(Δ𝑚𝑚��⃗𝐹𝐹) modifying the initial magnetic moments in the two sublattices of the FiM  when \nsubmitted to circular polarized laser pulses. We have evaluated in our modeling this alternative manner of studying the role of the IFE by adding such an induced magnetic \nmoment in the eLLB Eq. (4), and compared the results to the case where the IFE is \nsimulated by the magneto -optical field 𝐵𝐵\n�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) as discussed above. As presented and \ndiscussed in Supplemental Material Note S N7  [23] , both alternatives ( 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) or Δ𝑚𝑚��⃗𝐹𝐹) \nare equivalent from the simulation point of view. Therefore, in what follows we will \nsimulate the IFE as an effective out -of-plane magneto- optical field.  \nIn order to get a further understanding on the physics of these two mechanisms, \neither the MCD or th e IFE, we have explored the switching probability as a function of \n𝑄𝑄 for laser pulses with  fixed duration ( 𝜏𝜏𝐿𝐿=50 fs ) in two FiM alloys with slightly \ndifferent composition: Gd x(FeCo) 1-x with x=0.25 and x=0.24 respectively . The \ncorresponding parameters to numerically evaluate these two alloys are given in SN8, and  \nthe temperature dependence of the saturation magnetization of each sublattice (RE: Gd; \nTM: CoFe) are shown in FIG. 6  (a) and (c) respectively . Note that magnetization \ncompensation temperature at which the net magnetization of the sample vanishes ( 𝑇𝑇𝐹𝐹) is \nabove and below room temperature for x=0.25 and x=0.24 respectively. In other words, \nthe FiM sample is dominated by the RE (TM) at 𝑇𝑇 =300  K for x=0.25 (x=0.24) \ncompositions . The MCD and IFE parameters remain fixed as indicated above (MCD: 18 \n 𝑘𝑘~2%,; IFE: 𝐵𝐵 𝐹𝐹𝑀𝑀=20 T, 𝜏𝜏𝐷𝐷=𝜏𝜏𝐿𝐿). The two  possible initial states prior the laser pulse  \n(𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹:(↑,↓)), and the three laser polarizations (linear: 𝜎𝜎 =0, and circular 𝜎𝜎±=±1) were \nevaluated. The switching probability was computed by evaluating ten different stochastic \nrealization s for each for each 𝑄𝑄, and t he results  are presented in FIG. 6(b) and  (d) for \nx=0.25 and x =0.24 respectively . As in experimental observations   [9], the HD -AOS takes \nplace o nly in a narrow range of 𝑄𝑄 around the threshold value of 𝑄𝑄  at which the switching \nprobability abruptly changes from 0 to 1 under linear polarized pulses ( black dots in FIG. \n6(b) and (d)) . \n \n \nFIG. 6. Temperature dependence of the spontaneous magnetization of each sublattice (RE:Gd; \nTM:CoFe) of the FiM alloy (Gd x(CoFe) 1-x) for two different compositions: (a) x=0.25 and (c) \nx=0.24. The vertical grey line indicates the initial room temperature prior the laser pulse ( 𝑇𝑇=\n300  K). Probability of switching as a function of the absorbed power density ( 𝑄𝑄) for a laser pulse \nof 𝜏𝜏𝐿𝐿=50 fs for different combinations of the initial state  (𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹:(↑,↓)) and the polarization  \n(linear: 𝜎𝜎=0 (black dots), and circ ular 𝜎𝜎±=±1) of the laser pulse as indicated  in the legend \nand in the main text: (b) corresponds to 𝑥𝑥 =0.25 and (d) to x =0.24.  MCD results are shown by \nsolid dots, whereas IFE results are presented by open symbols. Lines are guide to the eyes.  \n \nFor 𝑥𝑥=0.25, as all result s presented up to here, the FiM is dominated by the RE:Gd \nat room temperature: 𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹<𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁 at 𝑇𝑇=300K , see FIG. 6(a) . In this case, the switching \nrequires  less 𝑄𝑄 with circular polarization ( 𝜎𝜎±) with respect to the linearly polarized case \n19 \n (𝜎𝜎=0) for two different combinations of the circular laser polarization and the initial \nstate of the FiM: ( 𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↑) and (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↓). This happens for both MCD (solid \nsymbols) and IFE scenarios (open symbols) as it is shown in FIG. 6(b) . Note that the \ninitial state in the TM is the opposite to the RE: 𝑚𝑚 𝑧𝑧𝑇𝑇𝐹𝐹↑ (up) corresponds to 𝑚𝑚 𝑧𝑧𝑅𝑅𝑁𝑁↓ (down) \nand vice, and the AOS is independent on the initial state for linear polarization (HI -AOS), \nwhereas under circular polarized laser pulses the switching is helicity -dependent (HD -\nAOS). For the rest of combinations, either (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↑) or (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↓), a higher 𝑄𝑄 is \nneed ed to achieve 100% of switching probability with respect to the linear polarized laser \npulse, and again for x =0.25 both MCD and IFE scenarios result in similar behavior of the \nswitching probability ( FIG. 6(b) ).  \nRemarkably, the MCD and IFE r esults are qualitatively different w hen the \ncomposition is slightly modified to x =0.24, where the FiM becomes dominat ed by the \nTM:CoFe at room temperature: 𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹>𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁 at 𝑇𝑇=300K , see FIG. 6(c) . In this case  \n(x=0.24) , the results in the IFE scenario  are qualitatively  similar to the ones already \nobtained for 𝑥𝑥=0.25: the HD -AOS occurs with small 𝑄𝑄  with respect to the linearly \npolarization case for (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↑) and (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↓) (open blue symbols in FIG. 6(d) ), and \nit requires high  𝑄𝑄 for the two other combinations ( (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↑), (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↓), open red \nsymbols in FIG. 6(d) ). However, for this concentration ( x=0.24), the results in the MCD \nscenario (see solid symbols in FIG. 6(d) ) are opposite as for x =0.25, and also opposite to \nthe ones obtained in the IFE scenario.  \nThese results can be understood as follows. In the MCD scenario, the HD -AOS \ndepends on the net initial magnetization at room temperature ( 𝑚𝑚𝑁𝑁=𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹+\n𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁, with 𝑚𝑚 𝑧𝑧𝑇𝑇𝐹𝐹=±1 and 𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁=∓1) and on the laser helicity ( 𝜎𝜎±=±1): if \ninitially 𝑚𝑚𝑁𝑁>0, a laser pulse with 𝜎𝜎+=+1 promotes the reversal, and this happens \neither for 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹=−1 (𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁=+1) when  x=0.25 because 𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁>𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹 at 𝑇𝑇=300K , or \nfor 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹=+1 (𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁=−1) if x=0.24 because  now 𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹>𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁 at 𝑇𝑇=300K . On the \nother hand, the HD -AOS within the IFE scenario is essentially determined by the \ndominant sublattice magnetization just below the Curie threshold  (𝑇𝑇≲𝑇𝑇𝐶𝐶), due to the \npersistence of the magneto -optical field 𝐵𝐵 �⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) when the laser puls e has already \nfinished. Note that for both concentrations 𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹>𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁 for 𝑇𝑇≲𝑇𝑇𝐶𝐶. Indeed, during the \ncooling down after the pulse, the magneto- optical field 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)∝𝜎𝜎±𝑢𝑢�⃗𝑧𝑧 promotes up \nor down magnetic state for the TM for 𝜎𝜎+ and 𝜎𝜎− respectively, and therefore, if 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹 is 20 \n initially up (𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹=+1), 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) promotes the reversal for 𝜎𝜎− and vice. Our analysis \nsuggests a set of experiments which could help to elucidate the  physical mechanism \nbehind these HD -AOS , just by evalu ating the switching probability as function of the \ninitial state and o f the laser pulse helicity for two different concentrations x , one resulting \nin a TM -dominated and other in a RE -dominated FiM alloy at room temperature.  Another \nalternative could be to use a single FiM with a given composition, and working with a \ncryostat to fix different initial temperatures below and above the magnetization \ncompensation temperature. Similar results to ones obtained by changing the composition \nx for a fixed temperature of the thermal bath are also  predicted by our simulations  when \nit is the temperature of the thermal bath what is varied for a given composition  (see Fig. \nS10 in Supplemental Material Note SN 9  [23] ): both IFE and MCD scenarios give similar \nresults below compensation and opposite above it . These theoretical predictions on the \nHD-AOS could be checked by experiments, which all together would allow us to shed \nlight on the real scope of these two mechanisms . \n \nIV. CONCLUSIONS  \nAs summary, the extension of the two sublattice Landau- Lifshitz -Bloch equation \nwith the angular momentum non- equilibrium exchange is proven to be a powerful tool to \nstudy ultrafast AOS switching in ferrimagnetic alloys . The formalism here developed \nreproduces  the atomistic spin dynamics  results for small samples at the nanoscale , while \nit opens the possibility to  numerically study realistic extended micro -size systems, with \ndimensions  comparable to the experiment al one s. The deterministic single -shot switching  \nand the demagnetization at high power regime are found to be in very good agreement with the experimental observations of Helicity -Independent AOS  under linearly polarized \nlaser pulses . The phase diagram combined with  the thermal analysis allow ed us to \ndetermine and predict the size of the inverted regions as depending the absorbed power \nand duration of the laser pulse . Moreover, w e have also explored and reproduced \nexperimental observations for the Helicity -Dependent AOS within the two physical \nmechanism s suggested in the literature: M agnetic Circular Dichroism  and Inverse \nFaraday Effect. According to the  Magnetic Circular Dichroism, the absorbed power by \nthe FiM depends on the laser helicity under  circularly polarized pulses , and our model \nalso predicts the main features of the Helicity -Dependent AOS measurements. Indeed, 21 \n both the Helicity -Dependent AOS  and the appearance of rings around the circularly \npolarized laser beam appear naturally  in our simulations . Additionally, similar r esults of \nthe HD -AOS switching were also obtained in Inverse Faraday Effect scenario, where the \ncircular polarization has been suggested to generate a persistent magneto -optical field \npromoting the switching for proper combinations of initial magnetic stat e and laser pulse \nhelicity . By exploring FiM samples with different compositions resulting in TM -\ndominated or in a RE -dominated FiM alloy at room temperature, we have found a \ndifference between the predictions of the IFE and the MCD scenarios. These result s could \nbe tested by performing the corresponding experiments, and consequently helping \ntogether to elucidate the  true basis of such  HD-AOS processes . Therefore, our methods \nwill be useful to understand recent and future experiments on AOS , and a lso to the \ndevelop novel recording devices where the information can be manipulated by optical \nmeans in an ultra -fast fashion. \n \nACKNOWLEDGMENTS  \nThis work was supported by projects MAT2017- 87072- C4-1-P funded by \nMinisterio de Educacion y Ciencia and PID2020117024GB -C41 funded by Ministerio de \nCiencia e Innovacion , both from the Spanish government , projects No. SA299P18 and \nSA114P20 from Consejer ia de Educaci on of Junta de Castilla y León, and project \nMagnEFi, Grant Agreement 860060, (H2020- MSCA -ITN-2019) funded by the European \nCommission.  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Li, All -Optical \nHelicity- Dependent Magnetic Switching by First -Order Azimuthally \nPolarized Vortex Beams , Applied Physics Letters 113, 171108 (2018).  \n[37] C. Banerjee, K. Rode, G. Atcheson, S. Lenne, P. Stamenov, J. M. D. Coey, and J. Besbas, Ultrafast Double Pulse All -Optical Reswitching of a \nFerrimagnet , Physical Review Letters 126 , 177202 (2021).  \n[38] M. Battiato, G. Barbalinardo, and P. M. Oppeneer, Quantum Theory of the Inverse F araday Effect , Physical Review B 89, 014413 (2014).  \n[39] M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer, Ab Initio Theory of \nCoherent Laser -Induced Magnetization in Metals , Physical Review Letters \n117, 137203 (2016).  \n \n 1 \n Supplementa l Material: Realistic micromagnetic description of all -optical ultrafast \nswitching  processes in ferrimagnetic alloys  \n \nV. Raposo1,*, F. García- Sánchez1, U. Atxitia2, and E. Martínez1,+  \n \n1. Applied Physics Department, University of Salamanca. \n2. Dahlem Center for Complex Quantum Systems and Fachbereich Physik . \n*,+ Corresponding authors : victor@usal.es , edumartinez@usal.es   \n \n \nCONTENT:  \n \n Supplemental Note SN1. Atomistic Spin Dynamics (ASD) model  \n Supplemental Note SN2. Micromagnetic LLB model s: conventional (LLB) and \nextended (eLLB) cases  \n Supplemental Note SN3. Comparison between atomistic, conventional -LLB \nand extended- LLB models  \n Supplemental Note SN4. Phase diagram in terms of the fluence and the pulse \nduration  \n Supplemental Note SN5. Helicity -Dependent AO S (HD-AOS) and M agnetic \nCircular Dichroism (MCD)  \n Supplemental Note SN 6. Helicity -Dependent AO S (HD-AOS) and Inverse \nFaraday Effect ( IFE) \n Supplemental Note SN7. I nverse Faraday Effect: magneto-optical field or \ninduced  magnetic moment  \n Supplemental note SN 8. Material inputs for two different compositions  \n Supplemental note SN 9: Helicity -Dependent All Optical Switching: MCD & \nIFE for different compositions and initial temperatures  \n \n \n  2 \n SN1. Atomistic S pin Dynamics  (ASD) model  \nWith the growing power of modern computers, numerical modeling has become an \nessential tool for scientific research, especially when handling systems as complex as \nmagnetic devices. For magnetic modeling, there are several  methodological choices in \nterms o f dimensions and time scales. In this section  we review the bases of the Atomistic \nSpin Dynamic  (ASD) model used to study the magnetization dynamics  in small nano -size \nferrimagnetic FiM samples submitted to  ultra -short of laser pulses. The physical basis o f \nthe ASD model is the  localization of unpaired electrons to atomic sites, leading  to an \neffective local atomistic magnetic moment at site 𝑖𝑖 , (𝜇𝜇⃗𝑖𝑖), which is  treated as a classical \nspin of fi xed length. The FiM alloys are composed of  two sublattices: rare -earth (RE) and \ntransition -metal (TM). Typical examples of these FiM alloys are GdCo, GdFe , GdFe Co, \nTbFeCo , etc. We consider  that the ordered TM alloy is represented by the fcc -type lattice , \nand t o simulate the amorphous character of t he TM -RE alloy, 𝑥𝑥𝑅𝑅𝑅𝑅⋅100% lattice sites are \nsubstituted randomly with RE  magnetic moments.  An example of a typical atomistic \narrangement is shown in Fig. S1  for a Gd x(FeCo) 1-x with 𝑥𝑥=𝑥𝑥𝑅𝑅𝑅𝑅=0.25 being the \nrelative composition of the sublattices  (𝑥𝑥=𝑥𝑥𝑅𝑅𝑅𝑅, and 𝑥𝑥 𝑇𝑇𝑇𝑇=1−𝑥𝑥𝑅𝑅𝑅𝑅). Each atom, either \nRE (Gd) or TM (FeCo), has its local atomistic magnetic moment 𝜇𝜇⃗𝑖𝑖 at site 𝑖𝑖 . \n \nFIG. S1. Atomistic scheme showing a FiM Gd x(FeCo) 1-x alloy consisting on two sublattices of \nRE (Gd, red) and TM (CoFe , grey) magnetic moments, with 𝑥𝑥 representing the relative \ncomposition of each sublattice. The sample dimensions along the Cartesian coordinate’s  \ndirections  are (ℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇) with 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇=5.6 nm. Atomistic simulations are restricted to samples \nat the na noscale (ℓ≈25 nm). The dimensions of an elementary computational cell ( Δ𝑥𝑥=Δ𝑦𝑦=\n3 nm) within the micromagnetic approach in shown at the bottom, along with the indication of \nthe lattice constant ( 𝑎𝑎=0.32 nm) for comparison.  \n \n3 \n The basis of ASD model  (see for instance  [1,2]  and references  therein ) is a classical \nspin Hamilt onian based on the Heisenberg  exchange formalism. The spin Hamiltonian ℋ \ntypically  has the form:  \n ℋ=ℋ𝑒𝑒𝑒𝑒𝑒𝑒+ℋ𝑎𝑎𝑎𝑎𝑖𝑖+ℋ𝑑𝑑𝑖𝑖𝑑𝑑+ℋ𝑎𝑎𝑑𝑑𝑑𝑑 (eS1) \nwith the terms on the RHS representing respectively exchange, anisotropy, dipolar and \nZeeman terms. The exchange energy for a system of interacting atomic moments is given by the expression  \n ℋ𝑒𝑒𝑒𝑒𝑒𝑒=−�𝐽𝐽𝑖𝑖𝑖𝑖𝑆𝑆⃗𝑖𝑖·𝑆𝑆⃗𝑖𝑖\n𝑖𝑖≠𝑖𝑖 (eS2)  \nwhere 𝐽𝐽𝑖𝑖𝑖𝑖 is the exchange interaction betwe en atomic sites 𝑖𝑖 and 𝑗𝑗, 𝑆𝑆⃗𝑖𝑖 is a unit vector \ndenoting the local spin moment direction ( 𝑆𝑆⃗𝑖𝑖=𝜇𝜇⃗𝑖𝑖/𝜇𝜇𝑠𝑠𝑖𝑖 with 𝜇𝜇𝑠𝑠𝑖𝑖=|𝜇𝜇⃗𝑠𝑠𝑖𝑖|) and 𝑆𝑆⃗𝑖𝑖 is the spin \nmoment direction of neighboring atoms. The sum in Eq. (eS2) is truncated to nearest neighbors only. As the FiM alloy is formed by two sublattices of magnetic moments, we \ncan split the exchange interaction ( 𝐽𝐽\n𝑖𝑖𝑖𝑖) between ferromagnetic intra- lattice  (𝐽𝐽𝑖𝑖𝑖𝑖>0) and \nantiferromagnetic inter -lattice (𝐽𝐽𝑖𝑖𝑖𝑖<0) exchange interactions. In what follows, we adopt \nthe notation of 𝐽𝐽 𝑇𝑇𝑇𝑇−𝑇𝑇𝑇𝑇>0, 𝐽𝐽𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅>0 for intra -lattice interactions, and 𝐽𝐽𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅<0 for \nthe inter -lattice interactions.  \nThe second term in Eq. (eS2) is the magnetic anisotropy. Here, a standard uniaxial \nanisotropy along the easy -axis (𝑧𝑧, out-of-plane  direction ) is considered ,  \n ℋ𝑎𝑎𝑎𝑎𝑖𝑖=−𝑑𝑑𝑢𝑢��𝑆𝑆�⃗𝑖𝑖·𝑢𝑢�⃗𝐾𝐾�2\n𝑖𝑖 (eS3) \nwhere 𝑑𝑑𝑢𝑢 is the anisotropy energy per atom and 𝑢𝑢�⃗𝐾𝐾=𝑢𝑢�⃗𝑧𝑧 is the  unit vector denoting the \npreferred direction . The last two terms in Eq. ( eS1) account for the dipolar  (ℋ𝑑𝑑𝑖𝑖𝑑𝑑) and \nexternal applied field (ℋ𝑎𝑎𝑑𝑑𝑑𝑑) contributions . Since the demagnetizing field is usually \nrelatively small in FiMs samples , this term is generally ignored  here. No external field is \napplied in the present work. \nThe equilibrium state of the FiM  sample can be obtained by minimiz ing of the total \nenergy. Here, it consists o f the two antiferromagnetic coupled sublattices aligned along \nthe easy -axis in anti -parallelly . Its dynamics response is governed by the  Langevin -\nLaudau -Lifshitz -Gilbert equation  for each atomistic moment ( 𝑆𝑆⃗𝑖𝑖)  [1,2] : \n𝜕𝜕𝑆𝑆⃗𝑖𝑖\n𝜕𝜕𝑡𝑡=−𝛾𝛾0\n(1+𝜆𝜆2)�𝑆𝑆⃗𝑖𝑖×�𝐻𝐻��⃗𝑖𝑖+𝐻𝐻��⃗𝑡𝑡ℎ,𝑖𝑖 �+𝜆𝜆𝑆𝑆⃗𝑖𝑖×�𝑆𝑆⃗𝑖𝑖×�𝐻𝐻��⃗𝑖𝑖+𝐻𝐻��⃗𝑡𝑡ℎ,𝑖𝑖 ��� (eS4) \nwhere 𝜆𝜆=0.02 is the Gilbert damping parameter  and 𝛾𝛾0=2.21×105 m/(A⋅s) the \ngyromagnetic ratio . 𝐻𝐻��⃗𝑖𝑖 is the local  effective magnetic field obtained from the spin \nHamiltonian  as \n𝜇𝜇0𝐻𝐻��⃗𝑖𝑖=−1\n𝜇𝜇𝑠𝑠𝑖𝑖𝜕𝜕ℋ\n𝜕𝜕𝑆𝑆⃗𝑖𝑖 (eS5) 4 \n and 𝐻𝐻��⃗𝑡𝑡ℎ,𝑖𝑖 is the stochastic thermal field is given by:  \n𝜇𝜇0𝐻𝐻��⃗𝑡𝑡ℎ,𝑖𝑖 =Γ⃗𝑖𝑖(𝑡𝑡)�2𝜆𝜆𝑘𝑘𝐵𝐵𝑇𝑇\n𝛾𝛾0𝜇𝜇𝑠𝑠𝑖𝑖Δ𝑡𝑡 (eS6) \nwith 𝑘𝑘𝐵𝐵 is the Boltzmann constant, 𝑇𝑇  the temperature, Δ 𝑡𝑡 the integration time step . Γ⃗𝑖𝑖(𝑡𝑡) \nis a local vector wh ose components a re Gaussian -distributed white -noise random \nnumbers  with zero mean value . \nUnder a laser pulse, the FiM  sample absorbs energy and its temperature changes in time. \nThe temperature 𝑇𝑇 that enters in Eq. (eS6) is the temperature of the electronic subsystem \n(𝑇𝑇≡𝑇𝑇𝑒𝑒), which is coupled to the lattice subsystem temperature ( 𝑇𝑇𝑙𝑙) and to the laser pulse \nas given b y the Two Temperature Model ( TTM ). Details of the power absorbed by sample \nas due to the laser pulse and the Eqs. (2)-(3) of the TTM were already discussed in the \nmain text. Conventional TTM values were adopted  [3 –5]: 𝐶𝐶𝑒𝑒=1.8×105 J/(m3K) at \n𝑇𝑇𝑅𝑅= 300  K, 𝐶𝐶𝑙𝑙=3.8×106 J/(m3K), 𝑘𝑘𝑒𝑒=91 W/(m K), 𝑔𝑔𝑒𝑒𝑙𝑙=7×1017 W/(m3K) \nand 𝜏𝜏𝐷𝐷=10 ps. \nEq. (eS4)  [or Eq. (1) in the main text] is numerically solved coupled to the T TM \nEqs. ((2) -(3) in the main text) with a home -made solver using a 4th-order Runge -Kutta  \nscheme with Δ𝑡𝑡=0.1 fs. Typical Gd 0.25(CoFe )0.75 parameters were considered   [2]: intra-\nlattice exchange energies:  intra-lattice exchange energies 𝐽𝐽 𝑇𝑇𝑇𝑇−𝑇𝑇𝑇𝑇=3.58×10−21 J, \n𝐽𝐽𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅=1.44×10−21 J, inter -lattice energy 𝐽𝐽𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅=−1.25×10−21 J, magnitude of \nthe local magnetic moments: 𝜇𝜇𝑇𝑇𝑇𝑇=1.92 𝜇𝜇𝐵𝐵, 𝜇𝜇𝑅𝑅𝑅𝑅=7.63 𝜇𝜇𝐵𝐵, and anisotropy energy of \n𝑑𝑑𝑢𝑢=8.07×10−24 J. The FiM sample dimensions for atomistic simulations are \nℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 (see FIG. S1) with ℓ ≃25 nm  and 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 =5.6 nm, and magnetic moment  \nof the two sublattices are randomly generated with 25% and 75% of RE:Gd and \nTM:(FeCo) respectively (FIG. S1). The lattice constant is 𝑎𝑎 =0.35 nm . TTM parameter \nThe power absorbed by the laser pulse is assumed to be uniform over the nano- size FiM  \nsample considered in the main text for atomistic simulations  (ASD) .  \n \nSN2. Micromagnetic LLB models: conventional (LLB) and extended (eLLB) cases  \nAs it was already stated, due to memory and computation limitations , atomistic \nsimulations are restricted to small samples at the nano -scale (ℓ≲100  nm ). However, \nexperimental studies on All Optical Switching are typical carried out in extended samples \nwith several micro meters  in length ( ℓ≃100  μm ). Also,  the size of conventional laser \nspot (FWHM) are usually at the micro -scale (𝑑𝑑0~1−10 μm). It is worthy  to provide an \nestimation  of the maximum size that c ould be numerically manage d in atomistic \nsimulations  (ASD) . For instance, in the present work we used one of the most power ful \nGraphics Processing Units (GPUs) , model RTX3090 (24GB) , and performing atomistic \nsimulations of FiM samples of ℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 with ℓ≃150  nm and 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇=5.6 nm \n(𝑎𝑎~0.35 nm ) will be already out of the computational power of such GPU. Therefore, in \norder to explain experimental observations another coarse- grained formalism is needed. \nHere, we adopt the mesoscopic description ( micromagnetic model)  which assumes that \nthe magnetization of each sublattice is a continuous vector field: 𝑚𝑚��⃗𝑖𝑖=𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡), where 5 \n here the sub- index 𝑖𝑖 refers to the local magnetic magnetization of each sublattice, \n𝑖𝑖:RE,TM. The FiM sample, with dimension ℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇, is discretized using a 2D finite \ndifferences scheme, using computational cells of volume Δ 𝑉𝑉=Δ𝑥𝑥2 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇. The continuous \ndescription ( Δ𝑥𝑥≫𝑎𝑎) is justified by the short- range of the exchange interaction, which \npromote s the ferromagnetic and the antiferromagnetic orders for intra - and inter -lattices \ninteractions respectively. At the same time, in ord er to resolve magnetic patterns such as \ndomain walls, the cell size Δ𝑥𝑥 must be smaller than the characteristic length scale along \nwhich the magnetization varies significantly (for instance, the so -called domain wall \nwidth , 𝛿𝛿≫Δ𝑥𝑥). Therefore, within the m icromagnetic approach, at each cell location ( 𝑟𝑟⃗) \nthere are two magnetic spin s representing the magnetization of the two sublattices of the \nFiM, 𝑚𝑚��⃗𝑖𝑖=𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)=𝑀𝑀��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)/𝑀𝑀𝑠𝑠𝑖𝑖(𝑇𝑇) where 𝑀𝑀��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡) is the local magnetization of \nsublattice 𝑖𝑖  in units of A/m, and 𝑀𝑀 𝑠𝑠𝑖𝑖(𝑇𝑇) is the corresponding spontaneous magnetization  \nat temperature 𝑇𝑇. FIG. S2 shows these ideas, along with different spatial scales of the \nASD and micromagnetic model.  \n \n \nFIG. S2 . (a) Arrange of atoms for atomistic simulations , limited to nanoscale samples, ℓ≈\n25 nm. (b) Micromagnetic discretization scheme for extended samples at the microscale, ℓ ≈\n20 μm. The FiM sample is discretized using a 2D finite differences scheme using computational  \ncells of volume 𝑉𝑉 =Δ𝑥𝑥2 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇. Each cell contains two micro -magnetic moments, one for each \nsublattice , and its size would include thousands of atomistic magnetic moments  (𝑆𝑆⃗𝑖𝑖). The \nmagnetization of  each sublattice i s a continuous  vector function (𝑚𝑚��⃗𝑖𝑖=𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)) over the FiM \nsample.  \n \nIn order to overcome the ASD limitations here we  adopt  an extended continuous \nmicromagnetic model that describes the temporal evolution of the reduced local \nmagnetization  𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡) of each sublattice 𝑖𝑖 :RE,TM based on the stochastic  Landau -\nLifshitz -Bloch (LLB) equation   [1,6]  \n \n6 \n 𝜕𝜕𝑚𝑚��⃗𝑖𝑖\n𝜕𝜕𝑡𝑡=−𝛾𝛾0𝑖𝑖′�𝑚𝑚��⃗𝑖𝑖×𝐻𝐻��⃗𝑖𝑖�+ \n−𝛾𝛾0𝑖𝑖′𝛼𝛼𝑖𝑖⊥\n𝑚𝑚𝑖𝑖2𝑚𝑚��⃗𝑖𝑖×�𝑚𝑚��⃗𝑖𝑖×�𝐻𝐻��⃗𝑖𝑖+𝜉𝜉⃗𝑖𝑖⊥��− \n+𝛾𝛾0𝑖𝑖′𝛼𝛼𝑖𝑖∥\n𝑚𝑚𝑖𝑖2(𝑚𝑚��⃗𝑖𝑖·𝐻𝐻��⃗𝑖𝑖)𝑚𝑚��⃗𝑖𝑖+𝜉𝜉⃗𝑖𝑖∥ (eS7) \nwhere 𝛾𝛾0𝑖𝑖′=𝛾𝛾0𝑖𝑖/(1 +𝜆𝜆𝑖𝑖2) is the reduced gyromagnetic ratio, which is defined via the \ncoupling parameter 𝜆𝜆𝑖𝑖 of sublattice 𝑖𝑖  to the heat bath . 𝛼𝛼𝑖𝑖∥ and 𝛼𝛼𝑖𝑖⊥ are the longitudinal and \nperpendicular damping parameters . 𝐻𝐻��⃗𝑖𝑖=𝐻𝐻��⃗𝑖𝑖(𝑟𝑟 ⃗,𝑡𝑡) is the local effective field at location 𝑟𝑟 ⃗ \nacting  on sublattice magnetic moment 𝑖𝑖, and 𝜉𝜉⃗𝑖𝑖∥ and 𝜉𝜉⃗𝑖𝑖⊥ are the longitudinal and \nperpendicular stochastic thermal fields.  \nFor 𝑇𝑇<𝑇𝑇𝐶𝐶, the damping constants for sublattice 𝑖𝑖 are \n𝛼𝛼𝑖𝑖∥= 2𝜆𝜆𝑖𝑖𝑘𝑘𝐵𝐵𝑇𝑇𝑚𝑚𝑒𝑒,𝑖𝑖\n𝐽𝐽0,𝑖𝑖𝑖𝑖𝑚𝑚𝑒𝑒,𝑖𝑖+�𝐽𝐽0,𝑖𝑖𝑖𝑖�𝑚𝑚𝑒𝑒,𝑖𝑖 (eS8) \n𝛼𝛼𝑖𝑖⊥=𝜆𝜆𝑖𝑖�1−𝑘𝑘𝐵𝐵𝑇𝑇𝑚𝑚𝑒𝑒,𝑖𝑖\n𝐽𝐽0,𝑖𝑖𝑖𝑖𝑚𝑚𝑒𝑒,𝑖𝑖+�𝐽𝐽0,𝑖𝑖𝑖𝑖�𝑚𝑚𝑒𝑒,𝑖𝑖� (eS9) \nwhere 𝑚𝑚𝑒𝑒,𝑖𝑖 and 𝑚𝑚𝑒𝑒,𝑖𝑖 are the equilibrium magnetization in sublattices 𝑇𝑇 𝑀𝑀 a nd 𝑅𝑅𝑅𝑅 \nrespectively . These values are obtained from the mean field approximation  (see \nRef.  [1,6]) . The exchange energies became 𝐽𝐽0,𝑖𝑖𝑖𝑖=𝑧𝑧𝑥𝑥𝑖𝑖𝐽𝐽𝑖𝑖−𝑖𝑖, 𝐽𝐽0,𝑖𝑖𝑖𝑖=𝑧𝑧𝑥𝑥𝑖𝑖𝐽𝐽𝑖𝑖−𝑖𝑖, with 𝑧𝑧  the \nnumber of nearest neighbors and 𝐽𝐽𝑖𝑖−𝑖𝑖 and 𝐽𝐽𝑖𝑖−𝑖𝑖 the atomistic exchange energy between \nspecies. 𝑥𝑥𝑖𝑖 is the concentration  of sublattice 𝑖𝑖 . Above the Curie temperature,  𝑇𝑇>𝑇𝑇𝐶𝐶, the \ndamping parameters are  \n𝛼𝛼𝑖𝑖∥=𝛼𝛼𝑖𝑖⊥=2𝜆𝜆𝑖𝑖𝑇𝑇\n3𝑇𝑇𝐶𝐶 (eS10 ) \nThe effective field consists on several contributions: \n𝐻𝐻��⃗𝑖𝑖=𝐻𝐻��⃗𝑎𝑎𝑎𝑎𝑖𝑖,𝑖𝑖+𝐻𝐻��⃗𝑒𝑒𝑒𝑒,𝑖𝑖+𝐻𝐻��⃗𝑖𝑖∥ (eS11 ) \nwhere 𝐻𝐻��⃗𝑎𝑎𝑎𝑎𝑖𝑖,𝑖𝑖 and 𝐻𝐻��⃗𝑒𝑒𝑒𝑒,𝑖𝑖 are the anisotropy and exchange fields given by  \n𝐻𝐻��⃗𝑎𝑎𝑎𝑎𝑖𝑖,𝑖𝑖=2𝐾𝐾𝑖𝑖\n𝜇𝜇0𝜇𝜇𝑖𝑖𝑚𝑚��⃗𝑖𝑖·𝑢𝑢�⃗𝐾𝐾 (eS12 ) \n𝐻𝐻��⃗𝑒𝑒𝑒𝑒,𝑖𝑖=2𝐴𝐴𝑒𝑒𝑒𝑒,𝑖𝑖\n𝜇𝜇0𝑀𝑀𝑠𝑠𝑖𝑖(∇2𝑚𝑚��⃗𝑖𝑖)−𝐽𝐽0,𝑖𝑖𝑖𝑖\n𝜇𝜇0𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖2�𝑚𝑚��⃗𝑖𝑖× (𝑚𝑚��⃗𝑖𝑖×𝑚𝑚��⃗𝑖𝑖)� (eS13 ) \nwhere the first term in 𝐻𝐻��⃗𝑒𝑒𝑒𝑒,𝑖𝑖 is the continuous exchange between neighbors’ cells, and the \nsecond one accounts for the inter -lattice contribution. Finally, the term 𝐻𝐻��⃗𝑖𝑖∥ is a f ield \ncomputed differently below 𝑇𝑇 𝐶𝐶, \n𝐻𝐻��⃗𝑖𝑖∥(𝑇𝑇<𝑇𝑇𝐶𝐶) =−1\n𝜇𝜇0�1\n𝜒𝜒 �𝑖𝑖∥+�𝐽𝐽0,𝑖𝑖𝑖𝑖�\n𝜇𝜇𝑖𝑖𝜒𝜒 �𝑖𝑖∥\n𝜒𝜒 �𝑖𝑖∥�𝛿𝛿𝑚𝑚𝑖𝑖\n𝑚𝑚𝑒𝑒,𝑖𝑖 𝑚𝑚��⃗𝑖𝑖+1\n𝜇𝜇0�𝐽𝐽0,𝑖𝑖𝑖𝑖�\n𝜇𝜇𝑖𝑖𝛿𝛿𝜏𝜏𝐵𝐵,𝑖𝑖\n𝑚𝑚𝑒𝑒,𝑖𝑖𝑚𝑚��⃗𝑖𝑖 (eS14 ) 7 \n and above 𝑇𝑇𝐶𝐶 \n𝐻𝐻��⃗𝑖𝑖∥(𝑇𝑇 ≥ 𝑇𝑇𝐶𝐶) =−1\n𝜇𝜇0�1\n𝜒𝜒 �𝑖𝑖∥+�𝐽𝐽0,𝑖𝑖𝑖𝑖�\n𝜇𝜇𝑖𝑖𝜒𝜒�𝑖𝑖∥\n𝜒𝜒 �𝑖𝑖∥�𝑚𝑚��⃗𝑖𝑖+1\n𝜇𝜇0�𝐽𝐽0,𝑖𝑖𝑖𝑖�\n𝜇𝜇𝑖𝑖𝜏𝜏𝐵𝐵,𝑖𝑖\n𝑚𝑚𝑖𝑖𝑚𝑚��⃗𝑖𝑖 (eS15 ) \nwith 𝜏𝜏𝐵𝐵,𝑖𝑖=�𝑚𝑚��⃗𝑖𝑖·𝑚𝑚��⃗𝑖𝑖�/𝑚𝑚𝑖𝑖, 𝛿𝛿𝑚𝑚𝑖𝑖=𝑚𝑚𝑖𝑖−𝑚𝑚𝑒𝑒,𝑖𝑖 and 𝛿𝛿𝜏𝜏𝐵𝐵,𝑖𝑖=𝜏𝜏𝐵𝐵,𝑖𝑖−𝜏𝜏𝑒𝑒,𝐵𝐵,𝑖𝑖. The longitudinal \nsusceptibilities  𝜒𝜒 �𝑖𝑖∥ are calculated from mean field approach  [1] . Below the Curie \ntemperature ( 𝑇𝑇<𝑇𝑇𝐶𝐶): \n𝜒𝜒 �𝑖𝑖∥(𝑇𝑇<𝑇𝑇𝐶𝐶)\n=𝜇𝜇𝑖𝑖ℒ𝑖𝑖′(𝜁𝜁𝑖𝑖)�𝐽𝐽0,𝑖𝑖𝑖𝑖�ℒ𝑖𝑖′�𝜁𝜁𝑖𝑖�+𝜇𝜇𝑖𝑖ℒ𝑖𝑖′(𝜁𝜁𝑖𝑖)�𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖ℒ𝑖𝑖′�𝜁𝜁𝑖𝑖��\n�𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖ℒ𝑖𝑖′(𝜁𝜁𝑖𝑖)��𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖ℒ𝑖𝑖′�𝜁𝜁𝑖𝑖��−�𝐽𝐽0,𝑖𝑖𝑖𝑖�ℒ𝑖𝑖′(𝜁𝜁𝑖𝑖)�𝐽𝐽0,𝑖𝑖𝑖𝑖�ℒ𝑖𝑖′�𝜁𝜁𝑖𝑖� (eS16 ) \nwhere ℒ𝑖𝑖 is the Langevin function with argument 𝜁𝜁𝑖𝑖=𝐽𝐽0,𝑖𝑖𝑖𝑖𝑚𝑚𝑖𝑖+�𝐽𝐽0,𝑖𝑖𝑖𝑖�𝑚𝑚𝑖𝑖\n𝑘𝑘𝐵𝐵𝑇𝑇 and ℒ𝑖𝑖′ is the \nderivative respect 𝜁𝜁𝑖𝑖. Above the Curie temperature ( 𝑇𝑇>𝑇𝑇𝐶𝐶):  [7] \n𝜒𝜒 �𝑖𝑖∥(𝑇𝑇 ≥ 𝑇𝑇𝐶𝐶) =𝜇𝜇𝑖𝑖�𝐽𝐽0,𝑖𝑖𝑖𝑖�+𝜇𝜇𝑖𝑖�3𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖�\n��3𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖��3𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖�−�𝐽𝐽0,𝑖𝑖𝑖𝑖��𝐽𝐽0,𝑖𝑖𝑖𝑖�� (eS17) \n \nStochastic fluctuations due to temperature are introduced through thermal fields 𝜉𝜉 ⃗𝑖𝑖∥ \nand 𝜉𝜉⃗𝑖𝑖⊥. These fields  are time and space uncorrelated, and they are  generated from white \nnoise random numbers with zero mean field and variance given by  [6]: \n〈𝜉𝜉𝑖𝑖,𝛼𝛼𝜂𝜂(𝑟𝑟⃗,𝑡𝑡)𝜉𝜉𝑖𝑖,𝛽𝛽𝜂𝜂(𝑟𝑟⃗′,𝑡𝑡′)〉=2𝐷𝐷𝑖𝑖𝜂𝜂𝛿𝛿𝛼𝛼𝛽𝛽𝛿𝛿(𝑟𝑟⃗−𝑟𝑟⃗′)𝛿𝛿(𝑡𝑡−𝑡𝑡′) (eS18) \nwhere 𝛼𝛼,𝛽𝛽 are the cartesian components of the stochastic thermal fields , 𝑖𝑖 denotes \nsublattice 𝑇𝑇\n𝑀𝑀 o\nr 𝑅𝑅𝑅𝑅 and 𝜂𝜂:∥,⊥ represents parallel or perpendicular field components. \nThe diffusions constants are given by: \n𝐷𝐷𝑖𝑖⊥=�𝛼𝛼𝑖𝑖⊥−𝛼𝛼𝑖𝑖∥�𝑎𝑎3𝑘𝑘𝐵𝐵𝑇𝑇\n(𝛼𝛼𝑖𝑖⊥)2𝛾𝛾0𝑖𝑖′𝜇𝜇0𝑛𝑛𝑎𝑎𝑡𝑡𝑥𝑥𝑖𝑖𝜇𝜇𝑖𝑖Δ𝑉𝑉 (eS19) \n𝐷𝐷𝑖𝑖∥=𝛼𝛼𝑖𝑖∥𝛾𝛾0𝑖𝑖′𝑎𝑎3𝑘𝑘𝐵𝐵𝑇𝑇\n𝑛𝑛𝑎𝑎𝑡𝑡𝑥𝑥𝑖𝑖𝜇𝜇0𝜇𝜇𝑖𝑖Δ𝑉𝑉 (eS20) \nwith Δ𝑉𝑉=Δ𝑥𝑥2𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 the discretization volume, 𝑎𝑎 is the lattice constant and 𝑛𝑛𝑎𝑎𝑡𝑡 is the \nnumber of atoms per unit cell.  \n \nA\ns it will be shown in Supplemental Note N 3(b), we have confirmed that the \nconventional LLB Eq. (eS7) (or Eq. (4) in the main text)  [1,6,8] is not able to fully \nreproduce the atomistic model predictions of some all optical switching processes in FiM  \nsystems. Indeed, atomistic simulations show that the sudden change in the temperature \ndue to the laser pulse results in transient non-equilibrium states  where  a transfer of angular \nmomentum between sublattices  takes place. This transference is not fully reproduced by \nthe conventional LLB Eq. (eS7). In order to overcome this limitation, in the present work 8 \n we exten d the LLB E q. (eS7) by add ing an additional term  to the RHS  that account s of \nthe angular momentum transfer between lattices in non -equilibrium states . This non-\nequilibrium torque is  given  \n𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅=𝛾𝛾0𝑖𝑖′𝜆𝜆𝑒𝑒𝑒𝑒𝛼𝛼𝑖𝑖∥𝑥𝑥𝑖𝑖𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖+𝑥𝑥𝑖𝑖𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖\n𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖�𝜇𝜇𝑖𝑖𝐻𝐻��⃗𝑖𝑖∥−𝜇𝜇𝑖𝑖𝐻𝐻��⃗𝑖𝑖∥� (eS21) \nwhere and 𝐻𝐻��⃗𝑖𝑖∥ and 𝐻𝐻��⃗𝑖𝑖∥ are the longitudinal effective fields for each lattice 𝑖𝑖 :RE,TM  [6], \n𝑥𝑥𝑖𝑖≡𝑥𝑥 and 𝑥𝑥𝑖𝑖=1−𝑥𝑥𝑖𝑖≡1−𝑥𝑥 are the concentrations of each specimen, and  𝜆𝜆𝑒𝑒𝑒𝑒 is a \nparameter representing the exchange relaxation rate  [9]. By including Eq. (eS21) in the \nRHS of Eq. (eS7), and numerically solving it coupled to TTM Eqs. (2) -(3), we can provide \na realistic description of the magnetization dynamics in FiM systems  under ultra -short \nlaser pulses. In what follows, and in order to distinguish it from the conventional  LLB \nmodel  (LLB, 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅=0), we refer to this formalism as the extende d micromagnetic LLB \nmodel  (eLLB, 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅≠0), Note that the  additional torque has not been previously included \nin the LLB model, and therefore, here we name it  here as the extended LLB model  \n(eLLB) . A comparison between atomistic (ASD) predictions and the results without  \n(conventional LLB)  and with (extended LLB, eLLB)  the additional t orque 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅 is shown \nin FIG. S4, discussed in the next section.  \nIn the present work , FiM samples of ℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 are discretized using a 2D finite \ndifferences scheme, using computational cells of volume Δ 𝑉𝑉=Δ𝑥𝑥2 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇. Typical \nmaterial parameters for Gd 0.25(FeCo) 0.75 were adopted:  𝐽𝐽0,𝑇𝑇𝑇𝑇−𝑇𝑇𝑇𝑇=2.59×10−21 J, \n 𝐽𝐽0,𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅=1.35×10−21 J, 𝐽𝐽0,𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅=−1.13×10−21 J. 𝜆𝜆𝑇𝑇𝑇𝑇=𝜆𝜆𝑅𝑅𝑅𝑅=0.02. \nMicromagnetic parameters needed to solve the LLB Eq.  can be obtained from atomistic \ninputs as follows   [2]: the spontaneous magnetization at zero temperature of each \nsublattice is 𝑀𝑀 𝑠𝑠𝑖𝑖(0)=𝜇𝜇𝑖𝑖𝑥𝑥𝑖𝑖/(𝑝𝑝𝑓𝑓⋅𝑎𝑎3), where 𝑝𝑝𝑓𝑓 is a packing factor which depends on the \ncrystalline structure ( 𝑝𝑝𝑓𝑓=0.74 for fcc,  [2]): 𝑀𝑀𝑆𝑆,𝑇𝑇𝑇𝑇(0)=0.41 MA /m, 𝑀𝑀𝑆𝑆,𝑅𝑅𝑅𝑅(0)=\n0.55 MA /m. The continuous exchange stiffness constant   [10]  is 𝐴𝐴𝑒𝑒𝑒𝑒,𝑖𝑖=𝑛𝑛𝑛𝑛𝑥𝑥𝑖𝑖2𝐽𝐽𝑖𝑖/(2𝑎𝑎)  \nwith 𝑛𝑛=2 being the number of atoms pe r unit cell for fcc, 𝑛𝑛=0.79 the spin wave mean \nfield correction value also for FCC, and 𝑥𝑥𝑖𝑖 the concentration of sublattice 𝑖𝑖  (𝑥𝑥𝑖𝑖=1−𝑥𝑥𝑖𝑖): \n𝐴𝐴𝑒𝑒𝑒𝑒,𝑇𝑇𝑇𝑇=3.2 pJ/m, 𝐴𝐴𝑒𝑒𝑒𝑒,𝑅𝑅𝑅𝑅=0.19 pJ /m. The perpendicular anisotropy parameter is \n𝐾𝐾𝑖𝑖=𝑑𝑑𝑢𝑢𝑥𝑥𝑖𝑖/(𝑝𝑝𝑓𝑓⋅𝑎𝑎3), see  [2] , so 𝐾𝐾𝑢𝑢,𝑇𝑇𝑇𝑇=1.87 MJ /m3, 𝐾𝐾𝑢𝑢,𝑅𝑅𝑅𝑅=0.62 MJ /m3. For the \npresented results we take 𝜆𝜆𝑒𝑒𝑒𝑒=0.013 for the exchange relaxation rate. Within the \nmicromagnetic  model, the  laser -induced magnetization dynamics is evaluated by \nnumerically solving Eq. (eS7) coupled to the TTM Eqs. (2) -(3) using the Heun algorithm \nwith Δ𝑡𝑡=1 fs with cell sizes of Δ𝑥𝑥=3 nm. This was done  by implementing the models \nin a home -made CUDA -based code, which was run a RTX3090 GPU . It was also checked \nin several tests that decreasing the cell size to Δ 𝑥𝑥=1 nm does not change the results. \n \nSN3. Comparison between atomistic, conventional LLB and extended LLB models  \n(a) In order to validate the extended LLB model  (eLLB, 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅≠0), we have firstly \ncompared the predictions of the conventional  LLB model (LLB , 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅=0) with the \natomistic results  (ASD)  by computing for the equilibrium magnetization of each 9 \n sublattice ( 𝑖𝑖:RE,TM ) as a function of the temperature of the thermal bath (w hich coincides \nwith the electronic temperature). This study was carried out in a small FiM  sample  with \nℓ≈25 nm, and the results are shown in F IG. S3. Both ASD and eLLB models lead to \nsimilar results . \n \nFIG. S3. Temperature dependence of the reduced equilibrium magnetization  (𝑚𝑚) of the two \nsublattices . Solid (dashed ) lines corresponds to ASD ( eLLB) results. Blue and red color \ncorrespond to the RE and TM sublattices respectively.  The inset shows the corresponding \ndependence of the spontaneous magnetiz ation value of each sublattice ( 𝑀𝑀𝑠𝑠𝑖𝑖(𝑇𝑇) vs 𝑇𝑇). \n \n(b) In the main text and also in S N2, we claimed that the conventional LLB mode l \n(LLB , 𝜏𝜏⃗𝑁𝑁𝑅𝑅,𝑖𝑖=0) is not able to fully reproduce the atomistic results  (ASD)  for some all -\noptical switching processes in FiM samples. On the other hand, when such term is \nconsidered ( 𝜏𝜏⃗𝑁𝑁𝑅𝑅,𝑖𝑖≠0), the extended LLB model (eLLB) naturally reproduces the \natomistic temporal variation of the magnetization in small FiM samples under ultra -short \nlaser pulses. This is shown in F IG. S4 for the same study as in Fig. 1(b) of the main text.  \nContrary to the ASD and the eLLB models, the conventional LLB (𝜏𝜏⃗𝑁𝑁𝑅𝑅,𝑖𝑖=0) does not \npredicts switching for 𝑄𝑄=12×1021 W/m3 pulses ( Fig. S4(c) -(d)). Moreover, as shown \nin FIG. S4(a) -(b), t he extended LLB model  (eLLB) , which includes 𝜏𝜏⃗𝑁𝑁𝑅𝑅,𝑖𝑖≠0, repro duces \nsimilar results  even in the absence of therma l fluctuations ( 𝜉𝜉⃗𝑖𝑖∥=𝜉𝜉⃗𝑖𝑖⊥=0). \n10 \n  \nFIG. S4. Temporal evolution  of the out -of-plane component  (𝑚𝑚𝑧𝑧 vs 𝑡𝑡) calculated by three different \nmodels : ASD, conventional LLB and extended eLLB models for the case studied in Fig. 1(b) of \nthe main text. Solid lines in all  graphs correspond to the ASD results. Red and blue c urves \ncorrespond to the TM and RE respectively . In top graphs, dashed lines correspo nd to extended \neLLB results  (𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅≠0): (a) including thermal fluctuations 𝜉𝜉⃗𝑖𝑖∥,𝜉𝜉⃗𝑖𝑖⊥≠0, and (b) without thermal \nfluctuations ( 𝜉𝜉⃗𝑖𝑖∥=𝜉𝜉⃗𝑖𝑖⊥=0, deterministic case) . In bottom  graphs, dashed lines correspond to LLB \nresults (𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅=0): (a) including thermal fluctuations 𝜉𝜉⃗𝑖𝑖∥,𝜉𝜉⃗𝑖𝑖⊥≠0, and (b) without thermal \nfluctuations ( 𝜉𝜉⃗𝑖𝑖∥=𝜉𝜉⃗𝑖𝑖⊥=0, deterministic case) . \n \n(c) Except the contrary is indicated, all presented results with the extended eLLB \nmodel were computed with 𝜆𝜆𝑒𝑒𝑒𝑒=0.013. FIG. S5 show the comparison of ASD results \nto eLLB  results for three different values of 𝜆𝜆𝑒𝑒𝑒𝑒. A good quanti tative agreement  in the \ntime traces of the out -of-plane components of each sublattice  predicted by the ASD model \nis achieved time when 𝜆𝜆𝑒𝑒𝑒𝑒=0.013 is considered  (central graph in FIG. S5) . For a laser \npulse of 𝜏𝜏 𝐿𝐿=50 fs, and using 𝜆𝜆 𝑒𝑒𝑒𝑒=0.013, the threshold laser power density to achieve \nthe switching is 𝑄𝑄=7.9×1021 W/m3 within the ASD, whereas the threshold is 𝑄𝑄=\n9.9×1021 W/m3 with the eLLB model. However, if 𝜆𝜆𝑒𝑒𝑒𝑒=0.024 both models give the \nsame threshold  (𝑄𝑄=9.9×1021 W/m3). \n \n11 \n  \nFIG. S5 . Temporal evolution of the out -of-plane component ( 𝑚𝑚𝑧𝑧 vs 𝑡𝑡) calculated with  three \ndifferent  values of 𝜆𝜆𝑒𝑒𝑒𝑒 as indicated within each graph.  Here 𝑄𝑄 =10×10−21 W/m3. The rest of \ninputs are the same as in  FIG. S4 .  \n \nSN4. Phase diagram in terms of the f luence and the pulse duration  \nFIG. 2(a) of the main text presents the phase diagram of the three possible final \nstates as function of the pulse length ( 𝜏𝜏𝐿𝐿) and the absorbed energy from the laser pulse \n(𝑄𝑄). Same results can be also presented in terms of the absorbed fluence, which is given \nby 𝐹𝐹=𝑄𝑄𝜏𝜏𝐿𝐿𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇, where 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 is the thickness of the ferrimagnetic alloy. This is shown in \nFIG. S 6 for an extended range of laser pulse  durations  (𝜏𝜏𝐿𝐿). These results are also in \nagreement with recent experimental studies   [11]  \n \n12 \n  \nFIG. S 6. Same r esults as FIG. 2(a) of the main text showing the three possible final states as a \nfunction of the fluence, 𝐹𝐹=𝑄𝑄𝜏𝜏𝐿𝐿𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 for an extended range of pulse durations ( 𝜏𝜏𝐿𝐿). White, red \nand blue colors represent no -switching, deterministic switching  (HI-AOS)  and thermal \ndemagn etization behaviors respectively . \n \nSN5. Helicity -Dependent AOS (HD -AOS) and Magnetic Circular Dichroism (MCD)  \nConsidering the Magnetic Circular Dichroism (MCD)  scenario , the absorbed power \nby the FiM  under circular polarized laser pulses , 𝑃𝑃(𝑟𝑟,𝑡𝑡), also depends on the magnetic \nstate of the system  and the helicity of the pulse . Therefore , 𝑃𝑃(𝑟𝑟,𝑡𝑡) in Eq. (2) of the TTM \nis replaced by  𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)𝑃𝑃(𝑟𝑟,𝑡𝑡), where 𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁) describes  the different absorption  \npower for up (𝑚𝑚𝑁𝑁> 0,↑) and down (𝑚𝑚𝑁𝑁< 0,↓) magnetization states as depending on \nthe laser helicity  (𝜎𝜎±) of the laser pulse, \n𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)=�1 +1\n2𝑘𝑘𝜎𝜎±sign(𝑚𝑚𝑁𝑁)� (eS22) \nwhere 𝑚𝑚𝑁𝑁 is the local net out of plane magnetization, ( 𝑚𝑚𝑁𝑁=𝑀𝑀𝑠𝑠𝑇𝑇𝑇𝑇𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇+𝑀𝑀𝑆𝑆𝑅𝑅𝑅𝑅𝑚𝑚𝑧𝑧𝑅𝑅𝑅𝑅), \n𝜎𝜎±= ±1  is the helicity  of the laser pulse ( 𝜎𝜎+= +1  for right -handed helicity and 𝜎𝜎−=\n−1 left-handed helicity) , and  sign (𝑚𝑚𝑁𝑁) is the sign of the initial net magnetization.  Note that \n𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇= ±1  corresponds to 𝑚𝑚𝑧𝑧𝑅𝑅𝑅𝑅=∓1, but 𝑀𝑀𝑠𝑠𝑇𝑇𝑇𝑇 and 𝑀𝑀 𝑠𝑠𝑅𝑅𝑅𝑅 are both positive . 𝑘𝑘 is a f actor  \nthat determines the difference in the power absorption with respect to the linearly \npolarized case ( 𝑘𝑘= 0). According to this criterium, a state with local net magnetization \n𝑚𝑚𝑁𝑁> 0 (↑: up) absorb s more energy for a right -handed laser 𝜎𝜎+ and less for left -handed \nhelicity (𝜎𝜎−) than in the linearly polarization case  (𝜎𝜎= 0). The opposite happens starting \nfrom  a state with local down net magnetization 𝑚𝑚𝑁𝑁< 0 (↓: down).  Considering \n𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)𝑃𝑃(𝑟𝑟,𝑡𝑡) in the TTM Eqs (2) -(3) of the main text, we obtain isothermal curves  \nof 𝑇𝑇=𝑇𝑇𝑒𝑒=1000  K showing the transition from no-s witching to helicity -dependent \nswitching as a f unction of 𝑄𝑄  and 𝜏𝜏𝐿𝐿 for different combinations of the laser helicity and \nthe initial magnetic state under uniform illumination.  \nThe results are shown  in FIG. S 7(a) and (b) for 𝑘𝑘= 0.1  and 𝑘𝑘= 0. 02 respectively , \nwhich correspond to differences in the power absorption of 10% and 2% between up  and \ndown states . Solid black line corresponds to the  case of linear polarization ( 𝜎𝜎= 0) already \n13 \n plotted in Fig. 2(a) of the main text. Cases with (𝜎𝜎+,↑) and (𝜎𝜎−,↓) are represented by the \nsolid -blue curve, whereas solid -red curve corresponds to cases with (𝜎𝜎+,↓) and (𝜎𝜎−,↑). \nAs it can be observed  for 𝑘𝑘=0.1 (FIG. S7(a)), the isothermal curves  𝑇𝑇=𝑇𝑇𝑒𝑒=1000  K \nfor combinations (𝜎𝜎+,↑) and (𝜎𝜎−,↓) slightly reduce the values of 𝑄𝑄  and 𝜏𝜏𝐿𝐿 needed to \nachieve switching with respect to the linear polarization case ( 𝜎𝜎=0). On the contrary, \nthe isothermal curve for combinations (𝜎𝜎+,↓) and (𝜎𝜎−,↑) slightly moves towards high \nvalues of 𝑄𝑄  and 𝜏𝜏𝐿𝐿 with respect to linear polarization case ( 𝜎𝜎=0). If 𝑘𝑘 =0.02 (FIG. \nS7(b)), the differences are even smaller, and the three curves almost overlap . This fact  \nagrees  with experimental observation s, where the Helicity -Dependent AOS is only \nachieved  in a narrow range of absorbed power ( 𝑄𝑄) for a fixed pulse length ( 𝜏𝜏𝐿𝐿). Dashed \nlines in FIG. S 7(a)-(b) correspond to the isothermal curves of 𝑇𝑇 =𝑇𝑇𝑒𝑒=1400  K, and \nindicate the transition between s witching  and demagnetized m ultidomain behaviors . \n \nFIG. S 7. Phase diagram in the MCD scenario for different values of the  MCD  coefficient 𝑘𝑘. (a) \nand ( b) show the isothermal curves of 𝑇𝑇=𝑇𝑇𝑒𝑒=1000  K (solid lines) as a  function of 𝑄𝑄 and 𝜏𝜏𝐿𝐿 \nindicating  the transition between no- switching to switching , as computed from the TTM  Eq. (2) -\n(3) for 𝑘𝑘 =0.1, and 𝑘𝑘=0.02 respectively . Dashed curves are isothermal curves indicating the \ntransition between the switching  to m ultidom ain regimes ( 𝑇𝑇𝑒𝑒=1400  K). Black curve  \ncorresponds to the case of linear  polarization  (𝜎𝜎=0) already plotted in Fig. 2(a) of the main text. \nCases with (𝜎𝜎+,↑) and (𝜎𝜎−,↓) are represented by the blue curve, whereas red curve corresponds \nto cases with (𝜎𝜎+,↓) and (𝜎𝜎−,↑). Here ↑ and ↓ represent the initial  net magnetic state (𝑚𝑚𝑁𝑁), either \nup or down  respectively.   \n \nSN6. Helicity -Dependent AOS (HD -AOS) and Inverse Fadaray Effect  (IFE)  \nAs mentioned in the main text, several works claim that the observations of the HD -\nAOS can be ascribed to the Inverse Faraday Effect (IFE)  [12] . Within this formalism, \nthe laser pulse generates an effective out -of-plane magneto -optical field whose  direct ion \ndepends on the laser pulse helicity  as 𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)=𝜎𝜎±𝐵𝐵𝑇𝑇𝑀𝑀𝜂𝜂(𝑟𝑟)𝑛𝑛(𝑡𝑡)𝑢𝑢�⃗𝑧𝑧, where 𝜂𝜂(𝑟𝑟)=\nexp[−4ln(2)𝑟𝑟2/ (2𝑟𝑟0)2 ]  is the spatial field profile, and 𝑛𝑛(𝑡𝑡) is its temporal profile. \nNote that there the spatial dependence of 𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) is the same as the one of the absorbed \npower density. A ccording to the literature  [12] , the so- called magneto -optical field \n𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) has some persistence with respect to the laser pulse, and therefore its temporal \nprofile is different for 𝑡𝑡<𝑡𝑡0 and 𝑡𝑡>𝑡𝑡0: 𝑛𝑛(𝑡𝑡<𝑡𝑡0)=exp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/ 𝜏𝜏𝐿𝐿2], and \n14 \n 𝑛𝑛(𝑡𝑡≥𝑡𝑡0)=exp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/(𝜏𝜏𝐿𝐿+𝜏𝜏𝐷𝐷)2], where 𝜏𝜏𝐷𝐷 is the delay time of the \n𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) with respect to the laser pulse. We have evaluated this scenario by including \nthis field 𝐵𝐵 �⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) in the effective field of Eq. (4) . The results for the same FiM alloy \nconsidered up to here ( Gdx(FeCo) 1-x, with x =0.25), (see Supplemental Note SN5 ), are \nshown in FIG. S 8 by considering a maximum magneto -optical field of 𝐵𝐵𝑇𝑇𝑀𝑀=20 T  with \na delay time of 𝜏𝜏𝐷𝐷=𝜏𝜏𝐿𝐿. \n \n \nFIG. S8. Micromagnetic results of the HD -AOS computed within the IFE scenario.  (a) Snapshots \nof the final RE magnetic state ( 𝑚𝑚𝑧𝑧𝑅𝑅𝑅𝑅) after a laser pulse of 𝜏𝜏𝐿𝐿=50 fs for four different values of \nthe absorbed power density ( 𝑄𝑄). Results are shown for four combinations of the initial state ( ↑,↓) \nand helicities ( 𝜎𝜎±) as indicated at the left side. (b) RE magnetic state after every pulse with 𝜎𝜎+ \nfor 𝑄𝑄=5.9×1021 W/m3, showing the appearance of a ring around the central part. The sample \nside is ℓ=20 μm and the laser spot diameter is 𝑑𝑑0=ℓ/2. The IFE was evaluated by considering \na maximum magneto -optical field of 𝐵𝐵 𝑇𝑇𝑀𝑀=20 T with a delay time of 𝜏𝜏𝐷𝐷=𝜏𝜏𝐿𝐿. The material \nparameters correspond to a FiM alloy Gd x(CoFe) 1-x with x =0.25 . \n \nSN7. Inverse Faraday Effect: magneto- optical  field or induced  magnetic moment  \nIn previous section SN6 and in the main text, the Inverse Faraday Effect (IFE) was \nconsidered  by adding an effective out -of-plane magneto -optical field  (𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)) whose \ndirection depends on the laser pulse helicity . Other authors   [13,14]  have alternatively \npointed out  that under circularly polarized laser pulses , the IFE c an be taken into account \nin micromagnetic simulations by adding a helicity -dependent  induced magnetic moment \n15 \n on each sublattice ( Δ𝑚𝑚��⃗𝑖𝑖). We have also explored this alternative by adding, instead of the \n𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡), an additional term  to the right -hand side of the eLLB Eq. ( eS7). This term can \nbe expressed as 𝛾𝛾0′ Δ𝑚𝑚��⃗𝑖𝑖, where Δ𝑚𝑚��⃗𝑖𝑖 represents the laser induced magnetization for \nsublattice 𝑖𝑖 :RE,TM, and i t is given by  \nΔ𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)=(𝜎𝜎±)𝐾𝐾𝐼𝐼𝐹𝐹𝑅𝑅,𝑖𝑖𝐼𝐼\n𝑐𝑐𝑢𝑢�⃗𝑧𝑧 (eS23)  \nwhere 𝐾𝐾𝐼𝐼𝐹𝐹𝑅𝑅,𝑖𝑖 is the IFE constant  for sublattice 𝑖𝑖 :RE,TM, 𝐼𝐼=𝑃𝑃(𝑟𝑟,𝑡𝑡)𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 is the laser \nintensity  with 𝑃𝑃(𝑟𝑟,𝑡𝑡)=𝑄𝑄𝜂𝜂(𝑟𝑟)𝜉𝜉(𝑡𝑡) (see main text) , and 𝑐𝑐 the speed of light. In FIG. S9 \nwe compare both options to account for the IFE: either by a magneto- optical field \n(𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)≠0 with Δ𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)=0, solid lines  in FIG. S9)  as described in SN6, or by an \ninduced magnetic moment ( 𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)=0 with Δ𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)≠0, dashed lines in F IG. S9). \nAs it can be observed, both alternatives provide very similar  results, so we conclude , as \nin  [14] , that both alternatives are essentially equivalent within the scope of our numerical \nstudy . Moreover, the  adopted value  of 𝜎𝜎±𝐾𝐾𝐼𝐼𝐹𝐹𝑅𝑅,𝑖𝑖=±0.0014  T−1 (see FIG. S9) is also in \ngood quantitative agreement with typical values deduced  in  [14]  from an ab initio  \nformalism. \n \nFIG. S9 . Comparison of two micromagnetic implementations of the IFE  for both laser helicities: \n(a) 𝜎𝜎+=+1 and (b) 𝜎𝜎−=−1. Solid lines correspond to results obtained assuming that the IFE \ngenerates a magneto -optical field ( 𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)≠0), whereas dashed lines are for the induced \nmagnetic moment alternative ( Δ𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)≠0). The values of the maximum magneto -optical field \nis 𝐵𝐵𝑇𝑇𝑀𝑀=20 T, wher eas the IFE constant  is 𝜎𝜎±𝐾𝐾𝐼𝐼𝐹𝐹𝑅𝑅 ,𝑖𝑖=±0.0014  T−1. The  rest of inputs are the \nsame as in Fig. S8.  \n \nSN8. Material inputs for two different compositions  \nIn the las t part of the manuscript we studied the probability of switching ( see FIG. \n6) for two FiMs ( Gdx(CoFe) 1-x:RE x(TM) 1-x) with two different compositions : x=0.25 and \nx=0.24. The inputs used in these simulations are collected in the following Table S 1. \n \n x=0.25: Gd x(CoFe) 1-x x=0.24: Gd x(CoFe) 1-x \n𝑀𝑀𝑠𝑠,𝑇𝑇𝑇𝑇  / 𝑀𝑀𝑠𝑠,𝑅𝑅𝑅𝑅 (MA/m) 0.412 / 0.546  0.412 / 0.52  \n𝐴𝐴𝑒𝑒𝑒𝑒,𝑇𝑇𝑇𝑇  / 𝐴𝐴𝑒𝑒𝑒𝑒,𝑅𝑅𝑅𝑅 (pJ/m) 3.27 / 0.189  3.35 / 0 .174 \n𝐽𝐽𝑇𝑇𝑇𝑇−𝑇𝑇𝑇𝑇  / 𝐽𝐽𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅 (×10−21J) 2.59 / 1.35  2.59 / 1.35  \n16 \n 𝐽𝐽𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅  (×1021J)  -1.12  \n𝐾𝐾𝑢𝑢,𝑇𝑇𝑇𝑇  / 𝐾𝐾𝑢𝑢,𝑅𝑅𝑅𝑅 (MJ/m3) 1.87 / 0.62  1.89 / 0.59  \n𝜇𝜇𝑇𝑇𝑇𝑇  / 𝜇𝜇𝑅𝑅𝑅𝑅 (𝜇𝜇𝐵𝐵) 1.92 / 7.63  1.92 / 7.63  \n𝛼𝛼𝑇𝑇𝑇𝑇  / 𝛼𝛼𝑅𝑅𝑅𝑅 (  ) 0.02 / 0.02  0.02 / 0.02  \n𝛾𝛾𝑇𝑇𝑇𝑇  / 𝛾𝛾𝑅𝑅𝑅𝑅 (1011×(T⋅s)−1) 1.847 / 1.759  1.847 / 1.759  \n𝑔𝑔𝑇𝑇𝑇𝑇  / 𝑔𝑔𝑅𝑅𝑅𝑅 (  ) 2.1 / 2.0  2.1 / 2.0  \n𝑎𝑎 (nm)   0.352  \n   \n𝑘𝑘𝑒𝑒 (W/ (K⋅m))   91 \n𝐶𝐶𝑒𝑒(300  K) (×105J/ (K⋅m3))  1.8 \n𝐶𝐶𝑙𝑙 (×106J/ (K⋅m3))  3.8 \n𝑔𝑔𝑒𝑒𝑙𝑙(300  K) (×1017 W/m3)  7 \n \nTABLE  S1. Material inputs adopted to explore the switching probability for two different FiM \nalloys, with different composition s x. These inputs were used to get the results of FIG. 6 in the \nmain text.  \n \nSN9. Helicity -Dependent All Optical Switching : MCD & IFE for different \ncompos itions and initial temperatures  \nIn FIG. 6 of the main text we explore the switching probability predicted by both \nthe MCD and IFE mechanism s for two different compositions of the FiM and considering \nthat the initial temperature of the thermal bath was room temperature. Similar results can \nbe also obtained by fixing the compos ition of the FiM alloy and chan ging the temperature \nof the thermal bath with a cryostat. These results are shown in FIG. S10 for with x=0.25 \n(left column) and with x =0.24 (right column) compositions and three different \ntemperatures of the thermal bath: 𝑇𝑇 =260  K, 𝑇𝑇=300  K, and 𝑇𝑇 =340  K. For x=0.25, \nboth the IFE and MCD predict similar behavior for 𝑇𝑇 =260  K (FIG. S10(b)) and 𝑇𝑇 =\n300  K (FIG. S10(c) ): respect to the linear  polarized laser pulse, the 100% switching \nprobability occurs with smaller  Q for circular polarization when (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇↑) and \n(𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇↓), and with larger Q  when (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇↑) and (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇↓). However, at 𝑇𝑇=\n340  K (FIG. S10(d)), the IFE scenario results in similar behavior but the MCD results are \nthe opposite. This can be easily understood as explained in the main, because the FiM \nalloy with 𝑥𝑥=0.25 is RE -dominated for 𝑇𝑇=260  K and 𝑇𝑇=300  K, whereas becomes \nTM-dominated for 𝑇𝑇=340 K. (Fig. S8(a)). Note that the magnetization compensation \ntemperature for 𝑥𝑥=0.25 is 𝑇𝑇 𝑇𝑇≈320  K. \nSimilar results are also achieved for 𝑥𝑥=0.24 (right column in F IG. S10), but now \nboth IFE and MCD only give similar results for 𝑇𝑇=260  K (FIG. S10(f)), whereas they \npredict opposite behavior for 𝑇𝑇=300  K (FIG. S10(g)) and 𝑇𝑇 =300  K (FIG. S10(h)). \nNote that for 𝑥𝑥=0.24, the FiM is only RE -dominated for temperature below the 𝑇𝑇𝑇𝑇, \nwhich now is 𝑇𝑇 𝑇𝑇≈280  K. \n 17 \n  \nFIG. S 10. Temperature dependence of the spontaneous magnetization of each sublattice (RE:Gd; \nTM:CoFe) of the FiM alloy (Gd x(CoFe) 1-x) for two different compositions: (a) x=0.25 and ( e) \nx=0.24. Probability of switching as a function of the absorbed power density ( 𝑄𝑄) for a laser pulse \nof 𝜏𝜏𝐿𝐿=50 fs for different combinations of the initial state ( 𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇:(↑,↓)) and the polarization \n(linear: 𝜎𝜎=0 (black dots), and circular 𝜎𝜎±=±1) of the laser pulse as indicated in the legend \nand in the main text: (b) , (c) and (d) corresponds for 𝑇𝑇=260  K, 𝑇𝑇=300  K, and 𝑇𝑇=340  K \nfor 𝑥𝑥=0.25, whereas  (f), (g) and (h)  to x=0.24. MCD results are shown by solid dots, whereas IFE \nresults are presented by open symbols. Lines are guide to the eyes.  \n \n \n18 \n REFERENCES  \n[1] U. Atxitia, P. Nieves, and O. Chubykalo -Fesenko, Landau- Lifshitz -Bloch Equation \nfor Ferrimagnetic Materials , Physical Review B 86 , 104414 (2012).  \n[2] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A. Ellis, and R. W. \nChantrell, Atomistic Spin Model Simulations of Magnetic Nanomaterials , Journal \nof Physics Condensed Matter , 26, 103202  (2014) . \n[3] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxitia, O. Chubykalo -\nFesenko, S. el Moussaoui, L. le Guyader, E. Mengotti, L. J. Heyderman, F. Nolting, A. Tsukamoto, A. Itoh, D. Afanasiev, B. A. Ivanov, A. M. Kalashnikova, K. \nVahaplar, J. Mentink, A. Kirilyuk, T. Rasing, and A. v. Kimel, Ultrafast Heating as a \nSufficient Stimulus for Magnetization Reversal in a Ferrimagnet , Nature \nCommunications 3, 1666 (2012).  \n[4] U. Atxitia, O. Chubykalo -Fesenko, J. Walowski, A. Mann, and M. Münzenberg, \nEvidence for Thermal Mechanisms in Laser -Induced Femtosecond Spin Dynamics , \nPhysical Review B, 81, 174401  (2010).  \n[5] E. Beaurepaire, J. -C. Merle, A. Daunois, and J. -Y. Bigot, Ultrafast Spin Dynamics \nin Ferrom agnetic Nickel , Physical Review Letters 76 , 4250 (1996).  \n[6] C. Vogler, C. Abert, F. Bruckner, and D. Suess, Stochastic Ferrimagnetic Landau-\nLifshitz -Bloch Equation for Finite Magnetic Structures , Physical Review B 100, \n054401 (2019).  \n[7] O. J. Suarez, P. Nieves, D. Laroze, D. Altbir, and O. Chubykalo -Fesenko, Ultrafast \nRelaxation Rates and Reversal Time in Disordered Ferrimagnets , Physical Review \nB. 92, 144425  (2015).  \n[8] P. Nieves, U. Atxitia, R. W. Chantrell, and O. Chubykalo -Fesenko, The Classical \nTwo -Sublattice Landau- Lifshitz -Bloch Equation for All Temperatures , Low \nTemperature Physics 41, 739 (2015).  \n[9] C. S. Davies, T. Janssen, J. H. Mentink, A. Tsukamoto, A. v. Kimel, A. F. G. van der \nMeer, A. Stupakiewicz, and A. Kirilyuk, Pathways for Single -Shot All -Optical \nSwitching of Magnetization in Ferrimagnets , Physical Review Applied 13, 024064 \n(2020).  \n[10] C. T. Ma, X. Li, and S. J. Poon, Micromagnetic Simulation of Ferrimagnetic TbFeCo Films with Exchange Coupled Nanophases , Journal of Magnetism and Magnetic \nMaterials 417, 197 (2016).  \n[11] J. Wei, B. Zhang, M. Hehn, W. Zhang, G. Malinowski, Y. Xu, W. Zhao, and S. \nMangin, All-Optical Helicity -Independent Switching State Diagram in Gd -  Fe - Co \nAlloys , Physical Review Applied 15 , 054065 (2021).  \n[12] K. Vahaplar, A. M. Kalashnikova, A. v. Kimel, S. Gerlach, D. Hinzke, U. Nowak, R. \nChantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, All -Optical \nMagnetization Reversal by Circularly Polarized Laser Pulses: Experiment and Multiscale Modeling, Physical Review B 85, 104402 (2012).  19 \n [13] M. Battiato, G. Barbalinardo, and P. M. Oppeneer, Quantum Theory of the \nInverse Faraday Effect , Physical Review B.  89, 014413 (2014).  \n[14] M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer, Ab Initio Theory of \nCoherent Laser -Induced Magnetization in Metals , Physical Review Letters 117, \n137203 (2016).  \n  "    },    {        "title": "1705.04836v2.Partial_Ferrimagnetism_in_S_1_2_Heisenberg_Ladders_with_a_Ferromagnetic_Leg__an_Antiferromagnetic_Leg__and_Antiferromagnetic_Rungs.pdf",        "content": "arXiv:1705.04836v2  [cond-mat.str-el]  21 Jul 2017Journal of the Physical Society of Japan FULL PAPERS\nPartial Ferrimagnetism in S= 1/2 Heisenberg Ladders with a\nFerromagnetic Leg, an Antiferromagnetic Leg, and Antiferr omagnetic\nRungs\nKazutaka Sekiguchi∗and Kazuo Hida†\nDivision of Material Science, Graduate School of Science an d Engineering,\nSaitama University, Saitama, Saitama 338-8570, Japan\n(Received May 13, 2017; accepted June 13, 2017; published on line July 19, 2017)\nGround-state and finite-temperature properties of S= 1/2 Heisenberg ladders with a ferromagnetic leg,\nan antiferromagnetic leg, and antiferromagnetic rungs are studied. It is shown that a partial ferrimagnetic\nphase extends over a wide parameter range in the ground state . The numerical results are supported by\nan analytical calculation based on a mapping onto the nonlin earσmodel and a perturbation calculation\nfrom the strong-rung limit. It is shown that the partial ferr imagnetic state is a spontaneously magnetized\nTomonaga–Luttinger liquid with incommensurate magnetic c orrelation, which is confirmed by a DMRG\ncalculation. The finite-temperature magnetic susceptibil ity is calculated using the thermal pure quantum\nstate method. It is suggested that the susceptibility diver ges asT−2in the ferrimagnetic phases as in the\ncase of ferromagnetic Heisenberg chains.\n1. Introduction\nFerrimagnetism in one-dimensional quantum magnets\nhas been attracting broad interest in condensed mat-\nter physics. Conventionalferrimagnetism in unfrustrated\nspin chains can be understood on the basis of the Lieb–\nMattis (LM) theorem,1)for which the spontaneous mag-\nnetization is quantized to the values expected from the\nLM theorem.2,3)This type of ferrimagnetism is called\nLM ferrimagnetism. For weak frustration, LM ferrimag-\nnetism often remains stable. Another type of quan-\ntum ferrimagnetism induced by frustration for which\nthe spontaneous magnetization varies continuously with\nthe strength of frustration is called partial ferrimag-\nnetism.4–13)In this case, the spontaneous magnetization\nis not quantized to a specific value. In many numerical\nexamples,7–12)partial ferrimagnetism is accompanied by\nan incommensurate quasi-long-range modulation of the\nmagnetization. Recently, an analytical approach using\nthe nonlinear σmodel has been proposed to understand\nthe partial ferrimagnetism of this kind.14)It is proposed\nthat this phase can be characterized as a spontaneously\nmagnetized Tomonaga–Luttinger liquid (SMTLL).\nIn the present work, we investigate the partial ferri-\nmagnetism in S= 1/2 Heisenberg ladders with a fer-\nromagnetic leg, an antiferromagnetic leg, and antiferro-\nmagnetic rungs. In the absence of rung interactions, the\nsystem decouples to a spin-1/2 antiferromagnetic chain\nand a spin-1/2 ferromagnetic chain. Hence, the ground\nstate has magnetization M=L/2, where Lis the length\nalong the legs. On the other hand, in the strong-rung\n∗Present address: Akikusa Gakuen High School, Sayama, Saita ma\n350-1312, Japan\n†E-mail: hida@mail.saitama-u.ac.jplimit,twospinsoneachrungformasingletdimerandthe\nground state is nonmagnetic with M= 0. This ground\nstate is called the rung-dimer state. Hence, it is plausible\nthat a partial ferrimagnetic ground state is realized in an\nappropriate range of the rung strength.\nThis paper is organized as follows. The Hamiltonian is\nintroduced in Sect. 2. The ground-state phase diagram\nis investigated numerically and analytically in Sect. 3.\nThe finite-temperature magnetic susceptibility is numer-\nically estimated in Sect. 4 using the canonical thermal\npure quantum state (cTPQ) method. The last section is\ndevoted to a summary and discussion.\n2. Hamiltonian\nWe consider the S= 1/2 Heisenberg ladders described\nby the Hamiltonian\nH=−J1L/summationdisplay\ni=1Si,1·Si+1,1+J2L/summationdisplay\ni=1Si,2·Si+1,2\n+RL/summationdisplay\ni=1Si,1·Si,2, (2.1)\nwhereSi,ais a spin-1/2 operator. The lattice structure\nis shown in Fig. 1. For J1=J2, the rung-dimer state is\nthe exact ground state down to a finite critical value of R\nas shown by Tsukano and Takahashi.5)Later, a similar\nmodel with a ferromagnetic J1, an antiferromagnetic J2,\nand an anisotropic ferromagnetic Rwas investigated by\nTonegawa et al.13)Among the variety of ground-state\nphases of this model, they also found a partial ferri-\nmagnetic phase. In the present work, we consider the\nwhole parameter region with a ferromagnetic J1, an an-\n1J. Phys. Soc. Jpn. FULL PAPERS\ntiferromagnetic J2, and an antiferromagnetic Rwithout\nanisotropy. In the remainder of this paper, we set the\nenergy unit by J2= 1.\n−J1\nR−J1\nJ2J2R Ri−1 i i+1a=1\na=2\nFig. 1. Lattice structure of the present model.\n3. Ground-State Phase Diagram\n3.1 Numerical analysis\nThe ground-state phase diagram is determined by\nLanczos numerical diagonalization with the periodic\nboundary condition for L= 12 as shown in Fig. 2. In\nthe LM ferrimagnetic phase, M=L/2 = 6. In the par-\ntial ferrimagnetic phase, 0 < M < L/ 2. It is found that\nthe partial ferrimagnetic phase extends over a wide pa-\nrameter range.\n0 1 2012\nR/J2J1/J2M=6 M=5 M=4M=3\nM=2\nM=1\nM=0Lieb\nMattis\nFerriPartial\nFerri\nNonlinear σ Perturbation\nFig. 2. Ground-state phase diagram of the S= 1/2 Heisenberg\nladder (2.1) with L= 12. The spontaneous magnetization is de-\nnoted by M.The solidcurves arethe boundaries of the partial ferri-\nmagnetic phase.The dotted curvesare the boundaries betwee n par-\ntial ferrimagnetic phases with different magnetization. Th e dashed\nand dash-dotted lines are the nonmagnetic-partial-ferrim agnetic\nphase boundaries calculated by the perturbation expansion from\nthe strong-rung limitand the mapping onto the nonlinear σmodel,\nrespectively.\nTheR-dependences of MforJ1= 0.5, 0.8, and 1 .5\nare presented in Figs. 3(a)-3(c), respectively. The criti-\ncal value Rcbetween the nonmagnetic phase and partial\nferrimagnetic phase is insensitive to the system size L.ForJ1= 0.5, 0.8, and 1.5, we obtain Rc= 0.898, 1.054,\nand 1.291, respectively.\n0 0.5 100.20.40.6\nM/L\nRJ1=0.5  J2=1.0\n:L=8\n:L=10\n:L=12\nRc(a)\n0 0.5 100.20.40.6\nM/L\nRJ1=0.8  J2=1.0\n:L=8\n:L=10\n:L=12\nRc(b)\n0 0.5 100.20.40.6\nM/L\nRJ1=1.5  J2=1.0\n:L=8\n:L=10\n:L=12\nRc(c)\nFig. 3. Spontaneous magnetization for (a) J1= 0.5, (b)J1=\n0.8, and (c) J1= 1.5.\nTo determine the R-dependence of Mmore precisely,\nlog-log plots of M/LagainstRc−Rare shown in Fig.\n4. The value of Rcorresponding to each value of M/Lis\nat the middlepoint of the steps in Fig. 3. The solid lines\nare fit assuming the form\nM\nL=A(Rc−R)β. (3.1)\nForJ1= 0.5, 0.8, and 1.5, we obtain β= 0.48±0.01,\n0.48±0.01, and 0 .49±0.03, respectively. For J1= 0.5\nand 0.8,we use two to five points for the fitting. For J1=\n2J. Phys. Soc. Jpn. FULL PAPERS\n1.5, we use two to four points. The errors are estimated\nfrom the variation of βfor different choices of the points.\nThese results are consistent with the estimation of β=\n1/2 obtained by a mapping onto the nonlinear σmodel\ndescribed in the following subsection.\n10−310−210−1100 10−210−1100\nM/L\nRc−RJ1=0.5  J2=1.0\n:L=8\n:L=10\n:L=12(a)\n10−310−210−1100 10−210−1100\nM/L\nRc−RJ1=0.8  J2=1.0\n:L=8\n:L=10\n:L=12(b)\n10−310−210−1100 10−210−1100\nM/L\nRc−RJ1=1.5  J2=1.0\n:L=8\n:L=10\n:L=12(c)\nFig. 4. Log-log plot of M/LagainstRc−Rfor (a)J1= 0.5, (b)\nJ1= 0.8, and (c) J1= 1.5.\nOn the other hand, the critical value RLM\ncbetween\nthe LMferrimagneticphaseandthe partialferrimagnetic\nphase depends strongly on the system size as shown in\nFig. 3. The size dependences of RLM\ncare shown in Figs.\n5(a)-5(c). The size extrapolation is carried out using the\ndata for L= 8,10, and 12. It is noteworthy that RLM\nc\ndecreases substantially with increasing L. The extrapo-\nlation suggests that the LM ferrimagnetic phase is much\nnarrowerthan that shown in Fig. 2 and might eventually\nvanish.0 0.02 0.04 0.0600.51\nRcLM\n1/LJ1=0.5  J2=1.0\nL=8\nL=10\nL=12(a)\n0 0.02 0.04 0.0600.51\nRcLM\n1/LJ1=0.8  J2=1.0\nL=8\nL=10\nL=12(b)\n0 0.02 0.04 0.0600.51\nRcLM\n1/LJ1=1.5  J2=1.0\nL=8\nL=10\nL=12(c)\nFig. 5. Size dependence of RLM\ncfor (a)J1= 0.5, (b)J1= 0.8,\nandJ1= 1.5.\n3.2 Mapping onto the nonlinear σmodel\n3.2.1 Transformation of spin variables\nTheground-statephasediagramisstudiedanalytically\nby mapping the Hamiltonian (2.1) onto the nonlinear σ\nmodel.14)For small J1, the classical ground-state spin\nconfiguration of the Hamiltonian (2.1) is given by the\nN´ eel state\nScl\ni,a= (−1)i+aSez. (3.2)\nHence, we decompose the whole ladder into two inter-\npenetrating sublattices as shown in Fig. 6. Since the unit\ncell is doubled, we take a unit cell as shownby the dotted\nsquare.\nWe introduce the low-energy modes corresponding to\nthe uniform and staggered components of spin variables\nl(xj) andn(xj) by15–17)\nSi,a=Aal(xj)+S/radicalbigg\n1−A2al(xj)2\nS2n(xj) (3.3)\n3J. Phys. Soc. Jpn. FULL PAPERS\n2j−2 2j−1 2ja=1\na=2\nFig. 6. Definition of two sublattices (open and filled circles) and\na unit cell (enclosed by a dotted square).\nfor (i,a) = (2j−1,1),(2j,2),\nSi,a=Aal(xj)−S/radicalbigg\n1−A2al(xj)2\nS2n(xj) (3.4)\nfor (i,a) = (2j−1,2),(2j,1),\nwhich satisfy the constraint\nn(xj)2= 1,l(xj)·n(xj) = 0, (3.5)\nwherexj= 2ja0is the coordinate of the center of\nthejth unit cell along the leg. The square root factor/radicalbigg\n1−A2al(xj)2\nS2is introduced to explicitly normalize S2\ni,a\nas\nS2\ni,a=S2. (3.6)\nThe coefficients Aaare normalized as\n2/summationdisplay\na=1Aa=a0, (3.7)\nso thatl(xj) corresponds to the net magnetization per\nunit cell as\nl(xj) =1\n2a0(S2j−1,1+S2j−1,2+S2j,1+S2j,2).(3.8)\n3.2.2 Stability of the nonmagnetic state\nTaking the continuum limit and within the second or-\nder inn′(xj) andl(xj), the Hamiltonian is rewritten as\nH=/integraldisplaydx\n2a0\n2/summationdisplay\na,b=1Ma,bAaAb\nl(x)2\n−2Sa0/integraldisplaydx\n2a0/parenleftBigg2/summationdisplay\na=1JaAa/parenrightBigg\n×[l(x)·n′(xa)+n′(xa)·l(x)]\n+2a2\n0S2/integraldisplaydx\n2a02/summationdisplay\na=1(−1)aJa(n′(xa))2,(3.9)\nwhere\nM=/parenleftbigg\n−4J1+R R\nR4J2+R/parenrightbigg\n.(3.10)To determine Aa, we follow Sierra16,17)to obtain\nA1=a0J2\nJ2−J1, A2=a0−J1\nJ2−J1.(3.11)\nUsing Eq. (3.10) and Eq. (3.11), we have\n2/summationdisplay\na,b=1Ma,bAaAb=a2\n0(J2−J1)R−4J1J2\nJ2−J1.(3.12)\nHence, we finally obtain\nH=/integraldisplaydx\n2a02a2\n0S2(J2−J1)n′(x)2\n+/integraldisplaydx\n2a0/bracketleftbigg\na2\n0(J2−J1)R−4J1J2\nJ2−J1/bracketrightbigg\nl(x)2.(3.13)\nLimiting ourselves to the case of J2≫J1, the state with\nl(x) = 0 is unstable for R < R c, where\nRc=4J1J2\nJ2−J1. (3.14)\nThe instability in limplies the transition to the ferri-\nmagnetic state. The critical value given by Eq. (3.14) is\nplotted in Fig. 2 by a dash-dotted line. Considering that\nthe present approximation is valid for J2≪J1, it is con-\nsistent with the phase boundary obtained by numerical\ncalculation.\n3.2.3 Higher-order correction in l(x)\nWe haveto considerthe higher-ordercorrectionin l(x)\nto fix the equilibrium value of l(x) in the unstable region\nR > R c. Hence, we expand Eq. (3.3) and Eq. (3.4) up to\nO(l4). Then, the Hamiltonian yields\nH=/integraldisplaydx\n2a02a2\n0S2(J2−J1)n′(x)2\n+/integraldisplaydx\n2a0/bracketleftbigg\na2\n0(J2−J1)R−4J1J2\nJ2−J1l(x)2/bracketrightbigg\n(3.15)\n+/integraldisplaydx\n2a0a4\n0\n4S2/parenleftbiggJ2+J1\nJ2−J1/parenrightbigg2\nR[l(x)2]2.(3.16)\nThe coefficient of [ l(x)2]2is positive definite. Hence,\nthe magnitude of the equilibrium value of lgrows con-\ntinuously from R=Rcas\n|∝angbracketleftl∝angbracketright|=√\n2J1J2S\nRc(J1+J2)a0/radicalbigg\nRc−R\nR∝S/radicalbig\nRc−R.(3.17)\nThe magnitude of the uniform magnetization per site M\nis given by\nM=|∝angbracketleftl∝angbracketright|a0\n2. (3.18)\nThis result implies β= 1/2 as estimated numerically.\nIn the higher-order terms, the terms such as\nl(x)2n′(x)2andl(x)2[l(x)·n′(x)+n′(x)·l(x)] also ap-\npear. Replacing l(x)2by∝angbracketleftl(x)∝angbracketright2, the first term is ab-\nsorbed by a slight redefinition of the coefficient of n′(x)2\n4J. Phys. Soc. Jpn. FULL PAPERS\nand the second term leads to a small but finite topologi-\ncal angle. In the magnetized sector, however, the ground\nstate is an SMTLL, as discussed below, and the topo-\nlogical angle does not play an essential role. The term\nl′(x)2also appears with a positive coefficient. This term\nsuppresses the spatial variation of l(x) and stabilizes the\nferrimagnetic long-range order. Hence, we conclude that\nasecond-ordertransitionto a partialferrimagneticphase\ntakes place for R < R c.\nFollowing the argument of Ref. 14, this ground state is\nan SMTLL with broken SU(2) symmetry down to U(1).\nHence,theincommensuratequasi-long-rangemodulation\nof magnetization is also expected in the partial ferrimag-\nnetic phase. This is confirmed by the finite-size DMRG\ncalculation of the expectation values/angbracketleftbig\nSz\ni,a/angbracketrightbig\n(a= 1,2) as\nshown in Fig. 7 for J1= 0.8,J2= 1,andR= 0.5 with\nsystem size L= 90. Similar behavior is also found for\nseveral other values of the parameters within the partial\nferrimagnetic phase. In each DMRG step, the number\nmof states kept in each subsystem is 240. The conver-\ngence with respect to mis confirmed. Although a true\nbreakdown of the translational symmetry is absent in\nthe infinite SMTLL state, the oscillatory modulation of\nmagnetization becomes visible in spin expectation values/angbracketleftbig\nSz\ni,a/angbracketrightbig\nowing to the presence of open boundaries.\n30 40 50 6000.20.4J1=0.8 J2=1 R=0.5\ni<Sz i,1>\n<Sz i,2>L=90\nFig. 7. Ground-state expectation values/angbracketleftBig\nSz\ni,a/angbracketrightBig\n(a= 1,2) for\nJ1= 0.8,J2= 1, and R= 0.5 withL= 90 near the center of\nthe whole ladder.\n3.3 Perturbation from the strong-rung limit\nIn the strong-rung limit R≫J1,J2, we divide the\nHamiltonian (2.1) as\nH=H0+H1, (3.19)\nH0=RL/summationdisplay\ni=1Si,1·Si,2, (3.20)\nH1=−J1L/summationdisplay\ni=1Si,1·Si+1,1+J2L/summationdisplay\ni=1Si,2·Si+1,2.(3.21)In this subsection, we regard H0as an unperturbed\nHamiltonian and H1as a perturbation Hamiltonian.\nEach spin state is described by eigenstates of Sz\ni,aas\n|↑i,a∝angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\ni,a=1\n2/angbracketrightbigg\n,|↓i,a∝angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\ni,a=−1\n2/angbracketrightbigg\n.(3.22)\nThe singlet and triplet states on each rung are defined\nby\n|si∝angbracketright=1√\n2(| ↑i,1∝angbracketright| ↓i,2∝angbracketright−| ↓ i,1∝angbracketright| ↑i,2∝angbracketright),(3.23)\n|t+\ni∝angbracketright=| ↑i,1∝angbracketright| ↑i,2∝angbracketright, (3.24)\n|t0\ni∝angbracketright=1√\n2(| ↑i,1∝angbracketright| ↓i,2∝angbracketright+| ↓i,1∝angbracketright| ↑i,2∝angbracketright),(3.25)\n|t−\ni∝angbracketright=| ↓i,1∝angbracketright| ↓i,2∝angbracketright. (3.26)\nIn the limit R→ ∞, the ground state is the rung\nsinglet (RS) state |RS∝angbracketrightdefined by\n|RS∝angbracketright=|s1∝angbracketright|s2∝angbracketright···|sL∝angbracketright. (3.27)\nThis is an eigenstate of H0that satisfies\nH0|RS∝angbracketright=−3\n4RL|RS∝angbracketright. (3.28)\nThe eigenvalue of this RS state is\nERS=−3\n4RL−3\n32(J1−J2)2\nRL+O(R−2) (3.29)\nup to the second order in J1andJ2.\nIn the single triplet (RT1) state, one of the rung sin-\nglets is replaced by a rung triplet. Owing to the trans-\nlational invariance, the eigenstate is a plane-wave state\nindexed by a wave number k(−π/a0≤k≤π/a0) and\nα(= 0,±) as\n|RT1;k,α∝angbracketright=1√\nLL/summationdisplay\ni=1exp(ikxi)|s1∝angbracketright|s2∝angbracketright···|tα\ni∝angbracketright···|sL∝angbracketright.\n(3.30)\nThis is an eigenstate of H0that satisfies\nH0|RT1;k,α∝angbracketright=/parenleftbigg\n−3\n4RL+R/parenrightbigg\n|RT1;k,α∝angbracketright.(3.31)\nUp to the second-order perturbation in J1andJ2, the\neigenvalue of |RT1;k,α∝angbracketrightis given by\nEα\nRT1(k) =−3\n4RL−3\n32(J1−J2)2\nRL+R−J1J2\nR\n+/bracketleftbiggJ2−J1\n2+(J1+J2)2\n4R/bracketrightbigg\ncosk\n−(J1−J2)2\n8Rcos2k+O(R−2).(3.32)\nThe minimum of ERT1located at k= 0 orπis given by\nEmin\nRT1=ERS+R−J1J2\nR−/vextendsingle/vextendsingle/vextendsingle/vextendsingleJ2−J1\n2+(J1+J2)2\n4R/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n5J. Phys. Soc. Jpn. FULL PAPERS\n−(J1−J2)2\n8R+O(R−2). (3.33)\nHence, if\nR−J1J2\nR−/vextendsingle/vextendsingle/vextendsingle/vextendsingleJ2−J1\n2+(J1+J2)2\n4R/vextendsingle/vextendsingle/vextendsingle/vextendsingle−(J1−J2)2\n8R<0\n(3.34)\nis satisfied, the RS state is unstable against the forma-\ntion of a rung-triplet excitation. This instability leads to\nferrimagnetic ordering. The critical value of Ris given\nby\nRc=1\n4(J2−J1+/radicalBig\n7J2\n1+18J1J2+7J2\n2).(3.35)\nThis is plotted in Fig. 2 by a dashed line. Considering\nthat the present approximationis valid for R≫J1,J2, it\nis qualitatively consistent with the phase boundary ob-\ntained by numerical calculation. This expression reduces\ntoRc=√\n2 obtained by Tsukano and Takahashi5)for\nJ1=J2= 1.\n4. Finite-Temperature Properties\nThe ground state of the present system is an SMTLL,\nanalogousto the ground state ofa spin chain in the effec-\ntive magnetic field. Nevertheless, the ferromagneticlong-\nrange order is destroyed at finite temperatures due to\none-dimensionality. This implies that the effective mag-\nnetic field vanishes as soon as the temperature becomes\nfinite. Hence, the finite-temperature properties of the\npresent system are not simply described as those of a\nconventional Tomonaga–Luttinger liquid (TLL) at finite\ntemperatures. This situation poses the nontrivial ques-\ntion “What is the fate of the SMTLL at finite tempera-\ntures?”.\nTo obtain insight into this question, the finite-\ntemperature susceptibility is calculated by the cTPQ\nmethod.18–21)The average is taken over 1200 initial vec-\ntors. The size extrapolation is carried out by the Shanks\ntransform22)fromL= 8,10, and 12. Motivated by the\nlow-temperature behavior of the susceptibility of the\nS= 1/2 ferromagnetic Heisenberg chain,23)we fit the\ndata by the formula\nχT2≃C0+C1T1/2+C2T. (4.1)\nThe plotof χT2againstT1/2is shownin Fig.8.This plot\nsuggeststhat C0>0inthepartialferrimagneticphaseas\nwell as in the LM ferrimagnetic phase. This means that\nthe susceptibility in these phases behaves as χ∼T−2at\nlow temperatures.\nAlthough the above result is not conclusive due to the\nlimited system size, the following physical argument sup-\nports the validity of this behavior. In addition to the\nexcitations of the conventional TLL, whose excitation\nenergy is normally proportional to the wave number k,\nthe ferromagnetic fluctuation modes coexist as low-lying\nmodes in the SMTLL. The amplitude of the ferromag-netic fluctuation mode in the long-wave-length limit is\nsimply the total magnetization, which commutes with\nthe Hamiltonian. Therefore, similarly to the ferromag-\nnetic fluctuations in the one-dimensional ferromagnets,\ntheir excitation energy is proportional to k2. Hence, the\ntimescaleoftheferromagneticfluctuationmodesismuch\nlonger than that of the excitations in the conventional\nTLL for small k. This implies that the whole system can\nbe regarded as a TLL in the background of slowly fluc-\ntuating almost uniform ferromagnetic modes. The latter\nmodes contribute to the finite-temperature susceptibil-\nity in the same way as the ferromagnetic modes do in a\nferromagnetic chain, leading to the behavior χ∼T−2at\nlow temperatures.\n0 0.2 0.4 0.600.1J1=0.8  J2=1.0\nR=0.2     R=0.6     R=1.4χT2\nT1/2Ferro Heisenberg \nchain\nFig. 8. Plot ofχT2againstT1/2. The solid curves are fit by Eq.\n(4.1).\n5. Summary and Discussion\nWe have investigated the ground-state properties of\nS= 1/2 Heisenberg ladders with a ferromagnetic leg, an\nantiferromagnetic leg, and antiferromagnetic rungs using\nLanczos diagonalization. It is shown that a partial fer-\nrimagnetic phase extends over a wide parameter range.\nThe numerical results are supported by analytical calcu-\nlations using the nonlinear σmodel and the perturbation\nexpansion from the strong-rung limit.\nThe finite-temperature magnetic susceptibility is cal-\nculated using the cTPQ method. Although the ground\nstate is an SMTLL, the finite-temperature properties are\nexpected to be different from those of a conventional\nTLL, since the spontaneous magnetization vanishes at\nfinite temperatures. Our numerical results suggest that\nthe susceptibility diverges as T−2in the ferrimagnetic\nphasesasinthecaseofaferromagneticHeisenbergchain.\nThis behavior can be understood if we regard the\npresentsystemasaTLLinaslowlyfluctuatingferromag-\nnetic background. Since the ferromagnetic spin wave has\n6J. Phys. Soc. Jpn. FULL PAPERS\nmuch lower excitation energy than the TLL excitation\nin the long-wavelength limit, this should make a domi-\nnant contribution to the susceptibility. Nevertheless, the\ndetails of the properties of the SMTLL at finite temper-\natures still remain to be investigated. It is hoped that\nextensive analyses of other models with ground states of\nthis kind will clarify their generic nature.\nThe authors are grateful to S. C. Furuya for enlighten-\ning comments and discussion on the nonlinear σmodel\nanalysis. They thank the authors of Ref. 13 for the dis-\ncussion and showing their results prior to their publica-\ntion. They also thank H. Shinaoka and K. Yoshimi for\nadvice on the cTPQ method. For the numerical diago-\nnalization, the package TITPACK ver. 2 coded by H.\nNishimori was used. Part of the numerical computation\nin this work was carried out using the facilities of the\nSupercomputer Center, Institute for Solid State Physics,\nUniversityofTokyo,and YukawaInstitute ComputerFa-\ncility in Kyoto University. This work was supported by\nJSPS KAKENHI Grant Number JP25400389.\n1) E. Lieb and D. Mattis, J. Math. Phys. 3, 749 (1962).\n2) T. Kuramoto, J. Phys. Soc. Jpn. 67, 1762 (1998).\n3) K. Maisinger, U. Schollw¨ ock, S. Brehmer, H.-J. Mikeska, and\nS. Yamamoto, Phys. Rev. B 58, R5908 (1998).\n4) S. Sachdev and T. Senthil, Ann. Phys. 251, 76 (1996).5) M. Tsukano and M. Takahashi, J. Phys. Soc. Jpn. 66, 1153\n(1996).\n6) N. B. Ivanov and J. Richter, Phys. Rev. B 69, 214420 (2004).\n7) S. Yoshikawa and S. Miyashita, J. Phys. Soc. Jpn. Suppl. 74,\n71 (2005).\n8) K. Hida, J. Phys.: Condens. Matter 19, 145225 (2007).\n9) R.R.Montenegro-Filho and M.D.Coutinho-Filho, Phys.Re v.\nB78, 014418 (2008).\n10) K. Hida and K. Takano, Phys. Rev. B 78, 064407 (2008).\n11) T. Shimokawa and H. Nakano, J. Phys. Soc. Jpn. 80, 043703\n(2011).\n12) T. Shimokawa and H. Nakano, J. Phys. Soc. Jpn. 81, 084710\n(2012).\n13) T.Tonegawa, K.Okamoto, T.Hikihara, and T.Sakai, J.Phy s.:\nConf. Ser. 828, 012003 (2017).\n14) S. C. Furuya and T. Giamarchi, Phys. Rev. B 89, 205131\n(2014).\n15) S. Sachdev, Quantum Phase Transitions (Cambridge Univer-\nsity Press, Cambridge, New York, Melbourne, Madrid, Cape\nTown, Singapore, S˜ ao Paulo, Delhi, Tokyo, 2011).\n16) G. Sierra, J. Phys. A: Math. Gen. 29, 3299 (1996).\n17) G. Sierra, in Strongly Correlated Magnetic and Superconduct-\ning Systems , ed. G. Sierra and M.A. Martin-Delgado, Lecture\nNotes in Physics (Springer, Berlin, Heidelberg, 1997) Vol. 478.\n18) S. Sugiura and A. Shimizu, Phys. Rev. Lett. 111, 010401\n(2013).\n19) A. Hams and H. De Raedt, Phys. Rev. E 62, 4365 (2000).\n20) J. Jakliˇ c and P. Prelovˇ sek, Phys. Rev. B 49, 5065(R) (1994).\n21) M.Imadaand M.Takahashi,J.Phys.Soc.Jpn. 55,3354 (1986).\n22) D. Shanks, J. Math. Phys. 34, 1 (1955).\n23) M. Takahashi and M. Yamada, J. Phys. Soc. Jpn. 54, 2802\n(1985).\n7"    },    {        "title": "2106.10111v2.Atomistic_spin_model_of_single_pulse_toggle_switching_in_Mn__2_Ru__x_Ga_Heusler_alloys.pdf",        "content": "Atomistic spin model of single pulse toggle switching in Mn 2RuxGa Heusler alloys\nF. Jakobs and U. Atxitia\u0003\nDahlem Center for Complex Quantum Systems and Fachbereich Physik,\nFreie Universität Berlin, 14195 Berlin, Germany\nSingle femtosecond pulse toggle switching of ferrimagnetic alloys is an essential building block\nfor ultrafast spintronics. Very different element-specific demagnetization dynamics is believed to\nbe a hard limit for switching in ferrimagnets. This suggests that ferrimagnets composed of two\nions of different nature, such as rare earth transition metal alloys, are necessary for switching.\nHowever, experimental observation of toggle switching in Mn 2RuxGa Heusler alloys, has contested\nthis limit since Mn ions are of the same nature. To shed some light into this question, we present an\natomistic spin model for the simulation of single pulse toggle switching of Mn 2RuxGa. The magnetic\nparameters entering in our model are extracted from previous experimental observations. We show\nthat our model is able to quantitatively reproduce measured magnetization dynamics of single pulse\ntoggle switching. We demonstrate that differently to previous understanding toggle switching in\nMn2RuxGa is possible even when both Mn sublattices demagnetization at very similar rate.\nSingle pulse femtosecond toggle switching in ferrimag-\nnets has attracted a lot of attention as a promising\nsolution for low energy, faster memory applications. [1–\n4]. It has already been demonstrated in micro and\nnanostructures[5,6], toswitchmagnetictunneljunctions\n[7], as passive component to induce switching in\nferromagnets[8]andevenusingpicosecondelectricpulses\n[9]. Toggle switching has been mostly found in one class\nof material, systems composed of transition metals rare-\nearth, e.g. GdFeCo [1] and TbFeCo [10] alloys, and\nGd/Co[11] and Tb/Co stacks [12]. Integrating all optical\nswitching with spintronics using this class of material\nhas been proposed[13], however, Mn 2RuxGa alloys, the\nsecond class of material showing toggle switching, is\nbettersuited[14]. WhileforGdFeCointenseresearchhas\nprovided a large amount of experimental data permitting\nthe validation of a number of theoretical models, for\nMn2RuxGa, there only exist a few experimental works\nshowing ultrafast magnetization dynamics and switching\n[14–17].\nThe Mn 2RuxGa Heusler alloy crystallises in the cubic\nspace group F 43mwith the magnetic Mn atoms on the\n4aand4csites (Fig. 1a). Spins at 4aand4csites are\ncoupledantiferromagnetically, whereastherespectiveMn\n4a- and 4csublattices are coupled ferromagnetically [18].\nThe Ga atoms appear at the 4bsites and Ru atoms at the\n4dsites of the lattice, and their magnetic contribution\ncan be neglected. Ultrafast magnetization dynamics of\nthe magneto-optically active Mn 4csublattice was first\nmeasured by Bonfiglio et al. [15]. In a subsequent\nwork, they concluded that Mn 4csublattice shows an\nultrafast demagnetization ( \u0018100s fs) followed by either\na secular equilibrium or by a fast remagnetization ( \u0018\n1ps). By using a phenomenological model based on\nthe so-called four temperature – electron, phonon, and\nspin temperatures of Mn 4aand 4c– these dynamics\nwere interpreted as a signature of strong exchange-\ndriven relaxation [16]. Spin temperature models however\nare unable to describe angular momentum transfer and\nFIG. 1. a) Crystal structure of Mn 2RuxGa with the two\nmagnetic Mn-sublattices represented as blue and red arrows.\nb) Schematic of the exchange constants J4a\u00004a,J4a\u00004cand\nJ4a\u00004c. c) Mn-specific atomic magnetic moments, symbols\ncorrespond to experimental data [19] and lines to the values\nused in the atomistic spin model.\nswitching.\nExperiments carried out by Davies et al. demonstrated\nthat switching is possible in Mn 2RuxGa, but only when\nthe initial temperature ( T0) lies below the magnetization\ncompensation temperature ( TM) [20]. Based upon a\nphenomenological model for the magnetization dynamics\nof two-sublattice magnets [21], the authors argued\nthat in order to link static thermodynamic properties\n(equilibrium TM) to highly non-equilibrium dynamics\n(switching), both sublattices should demagnetize at the\nsame rate, conserving the total angular momentum\nduring the whole process. This picture explains theirarXiv:2106.10111v2  [cond-mat.mtrl-sci]  12 Jan 20222\nobservation of the switching onset for a wide range of\nsystem parameters, including switching with picosecond-\nlong pulses. This interpretation collides with the\nphenomenology proposed by Bonfiglio et al. [16], which\nstates that both sublattices only demagnetize at the\nsame rate after \u00181.5 ps. Still, since data on the\nmagnetization dynamics of the switching process was\nmissing, the question remained open. An apparent\nstep forward along this direction was made by Banerjee\net al. [14] by measuring the dynamics of the Mn 4c\nsublattice when switching occurs. They were unable to\nobserve an explicit switching of the sign of the magneto-\noptical signal, however the dynamics of the Mn 4c\nsublattice behaved like those observed by Bonfiglio[16].\nThese observations differ strongly with the well-known\nelement specific signal switching measured in GdFeCo\n[22]. Recently, however, Banerjee and co-workers have\nclearlyobservedthedynamicsofmagnetizationswitching\nof the Mn 4csublattice [17], in clear disagreement to\ntheir own previous observations [14]. This disagreement\ncould be related to the different strength of the resetting\nmagnetic field, stronger in the latter. Similarly to\nthe previous works[14, 16, 20], the understanding of\nthe physics behind the switching process rested on\nphenomenological arguments. An attempt to describe\nswitching in Mn 2RuxGa using a first-principles model\nexists [23]. Although this model provides useful insights\non the potential origin of switching, due to its simplicity\nit is unable to describe the temperature dependence of\nthe switching condition ( T0< TM) observed by Davies\net al. [20] and Banerjee et al. [14]. A quantitative\nmodel able to account for thermodynamic aspects of\nthe magnetization switching dynamics in Mn 2RuxGa is\nnecessary but so far missing.\nIn this work we address this issue by presenting an\natomistic spin model to describe ultrafast magnetization\nand switching in Mn 2RuxGa alloys. Atomistic spin\nmodels have demonstrated themselves to be able to\ndescribe the dynamics of the magnetization switching in\ntransition metal rare-earth alloys[1, 24] and multilayers\n[25, 26]. Atomistic spin models are based on a\nsemiclassical spin Hamiltonian, which is defined by\nthe exchange and anisotropy constants as well as the\natomic magnetic moments (see Supplemental Material\nfor details). In a minimal model for Mn 2RuxGa,\none needs to determine at least three exchange\nconstants, J4a\u00004a,J4c\u00004candJ4a\u00004c(see Fig. 1 b)).\nWithin a Heisenberg spin model, the values of the\nexchange constants determine the critical temperature\nTc, and in ferrimagnets, besides Tc, the magnetization\ncompensationtemperature, TM, ifitexists. Thereported\nTcrange from Tc= 625K [18] (x= 0:68),Tc= 450K\n(x >0:5) [27],Tc= 550K (x= 0:7) [16]. We fix Tcto\nabout630Kwhichisattheupperendofreportedcritical\n200 400 600\nTemperature [K]0123Magnetization [ µB]\nCompensation\nTemperaturea) (b\n|Mn4a|\n|Mn4c|\nMn4a+4c\n0.6 0.8 1.0\nRuthenium Concentration x0100200300400\nCompensation Temperature [K]\nRoom Temperature\nSwitchingExperiment\nSimulationFIG. 2. Magnetic properties of Mn 2RuxGa alloys gained\nfrom atomistic spin model simulations. a) The equilibrium\nmagnetization for a Mn 2Ru0:68Ga alloy as function of\ntemperatureoftheMn 4asublatticeinred,theMn 4csublattice\nand the total magnetization Mn 4a+4c=jMn4aj\u0000jMn4c|. b)\ndisplays the magnetization compensation temperature TMas\nfunction of Ru-concentration, x. Black dots correspond to\nexperimental measurements [14] and red crosses from our\nmodel simulations. The red line is a linear fit of the simulated\nresults and is guidance to the eye.\ntemperatures. Here, we use values of the exchange\nconstants already determined experimentally [18], but\nslightly re-scaled such that our model reproduces the\nexperimental values of both TcandTM(Fig. 2). The\nanisotropy constant in Mn 2RuxGa has been reported to\nbesite-specific, with dz;4c= 1:1664\u000110\u000023Janddz;4a= 0\n[18], which we use in our model. We note that the role of\nthe anisotropy in the ultrafast magnetization dynamics\nis minimal since dz=J\u001c1and it is included to fix the\naverage magnetization towards the z-axis. Any choice of\nrelatively small values of the anisotropy constant would\nyield the same simulation results.\nLastly values for the site-specific Mn atomic magnetic\nmoments are needed. Based upon experimental data, we\nassume that the atomic moment of the Mn 4asublattice\nstays constant at \u0016s;4a= 2:88\u0016Bfor the range of Ru-\nconcentration studied here [19] (Fig. 1c)). Differently,\nthe Mn 4csites have been shown to have a stronger\ndependence on the Ru-concentration [19](Fig. 1c). Here\nweusevaluesof \u0016s;4c= 4:71\u0016B\u0001x(Fig. 1c))thatareboth\nclose to the experimentally measured values and provide\ngood results for TM(see Fig. 2). These values not only\ncompare well to those found in experiments[19, 28] but\nalso agree with the general trend found in first-principles\ncalculations [23, 29]. Simulations of our model are able\nto reproduce well the observation of an increasing TMfor\nan increasing Ru-concentration. For x<0:66, our model\nstarts to deviate from the experiments [14] (Fig. 2 b)).\nThe shared wisdom of single femtosecond pulse toggle\nswitching in ferrimagnets is based on experimental\nobservations in only one class of material, transition\nmetal rare-earth compounds. Despite their structural\nor morphological differences, in those materials the\nconditions for switching are based in the same working3\nprinciples, the rare-earth spin sublattice response to the\nlaser pulse is slower than that of the transition metal.\nThe physical reason behind this difference relies in the\ncore character of the highly localized 4 felectron spins in\nGd, plusthe absence of orbitalmagneticmoment( L= 0)\n[30]. The laser only excites them indirectly, by their\ncoupling to the 5d6s itinerant electrons. The transition\nmetal spins respond quickly to changes in the electronic\nstructure band due to the quick temperature rise.\nBased on this it has been argued for long that toggle\nswitching is only possible for two sublattice compounds\nwith components that demagnetize at different enough\nrates. Since data on site-specific dynamics is unavailable,\nit is unclear whether or not this criterion holds in\nMn2RuxGa. It is tempting to draw similarities to\nGdFeCo to explain switching in Mn 2RuxGa. For\nexample, since Mn 4aspins are localized, they could\nplay the role of the slow rare earth, while Mn 4c\nhave a more delocalized character, and thus play the\nrole of the transition metal. In this work we show\ndifferently, that Mn 2RuxGa switches even though its\ntwo sublattices show similar demagnetization times,\nunlocking different demagnetization times as a necessary\ncondition for switching. Another hard constrain for\nswitching Mn 2RuxGa is that it is only possible when\nthe initial temperature lies below TM[14, 20]. Davies et\nal. [20] found that this condition is robust and that can\nbe explained assuming that the dynamics is exchange-\ndriven, which has as a consequence that dM4a=dt=\n\u0000dM4c=dt. This condition also holds, when the coupling\nto the heat-bath is similar for both sublattices and they\ndemagnetize at similar rates. Here, we demonstrate that\nswitching is possible when the initial demagnetization\ndynamics is dominated by the coupling to the heat-bath\ninstead the exchange relaxation.\nWe use the atomistic stochastic-Landau-Lifshitz-\nGilbert equation (sLLG) [31] to describe the magne-\ntization dynamics of the Mn 2RuxGa alloys and the\ntwo-temperature model (TTM) to describe the electron\ntemperature Teland the phonon temperature Tph(see\nSupplemental Material (SM) for details) [32, 33]. The\nparameters defining the TTM for Mn 2RuxGa are not\nestablished yet. We use parameters similar to GdFeCo\nalloys, which are close to those used by Bonfiglio et\nal. within the 4TM [16]. The complete set of system\nparameter used in our model are summarized in table II\nof the SM. Within atomistic spin dynamics models, the\ndemagnetization time scales with \u001c\u0018\u0016at=(\u000bTc)[21, 34].\nIn two-sublattice magnets, the three parameters, \u000b;Tc\nand\u0016atare sublattice-specific. While Tcand\u0016atare\ndeterminedbyequilibriumpropertiesasdiscussedbefore,\nthe value of the damping parameter is related to the\ncoupling to the heat-bath. One method to estimate this\nvalue from experiments is to photo-excite magnetization\nprecession. These experiments have been conducted\nin Mn 2RuxGa and measured by time resolved Faraday\nFIG. 3. Comparison between the experimentally measured\nMn4cdynamics for a switching event (points) [17] and our\natomistic spin model simulations (lines).\neffect as a function of the applied field and temperature.\nFrom the decay of the precession the intrinsic damping\nparameter has been determined to have values smaller\nthan 0:02far fromTM[15]. We chose a damping value\n(\u000b4c= 0:01and\u000b4c= 0:013) that reproduces the\nultrafast magnetization dynamics of the Mn 4csublattice\nduring a switching event as measured by Banerjee et\nal. [17] and allows switching for x > 0:85similar to\nthe experimental findings of Banerjee et al. [14]. We\nnote, that Davies et al. [20] find switching already for\nx= 0:75which could be reproduced for a different choice\nof damping values (discussion in the SM). With this our\nmodel is able to quantitatively reproduce the recently\nmeasured demagnetization dynamics of the 4 clattice by\nBanerjee et al. (Fig. 3).\nHow does our model compare to the current understand-\ning of switching in Mn 2RuxGa? Previous works [16,\n17, 20] have suggested, that exchange-driven dynamics\ndominate the first step of the demagnetization and\nswitching process. This is assumed since the inter-\nlatticeantiferromagneticexchangeisstrongerthanin(for\nexample) transition metal rare earth alloys. Exchange-\nrelaxation stems from processes driving both sublattices\nto a mutual equilibrium. Thus, it is unlikely that\nexchange processes play a role on the first steps of the\ndynamics. Moreover, since exchange processes describe\ntransfer of angular momentum between sublattices, it\nis more likely that those processes dominate when the\nsublattice magnetic order is small rather than when its\nsaturated. Near equilibrium, relaxation by coupling to\nthe heat-bath dominates, while for situations of non-\nequilibrium and reduced magnetic order, the exchange-\nrelaxation dominates.\nFigure 4 shows the site-specific dynamics for three\ncharacteristic Ru-concentrations of Mn 2RuxGa forx=\n0:76(a),x= 0:86(b) andx= 0:96(c). These three cases4\n-3-2-1012mz[µB]a)\nMn2Ru0.76GaMn4a Mn4c\n-3-2-1012mz[µB]b)\nMn2Ru0.86Ga\n0 1 2 3 4 5\nTime in ps-3-2-1012mz[µB]c)\nMn2Ru0.96Ga-0.5-0.2500.250.5\nmz,total[µB]\nMz,total\n-0.5-0.2500.250.5\nmz,total[µB]\n-0.5-0.2500.250.5\nmz,total[µB]\nFIG. 4. Site-specific magnetization dynamics of the 4aand\n4cMn sites (red, blue; left axis) after a 100 fs laser pulse\nexcitation at t= 0(Gaussian pulse peak) for an increasing\nRuthenium concentration ( \u000b4c= 0:01,\u000b4a= 0:013). The\ntotal magnetization ( mz;total=mz;4a+mz;4c) is shown as a\nblack line (right axis). All laser parameters were the same for\nall three simulations. a) shows the no-switching scenario for a\nMn2Ru0:76Ga alloy. b) shows the non-deterministic scenario\nfor a Mn 2Ru0:86Ga alloy, with a prolonged demagnetization\nstate. c) shows the switching scenario for a Mn 2Ru0:96Ga\nalloy.\nrepresent alloys below/at/above the threshold of x=\n0:9, which defines the experimentally found switching\ncondition [14].\nA closer look to mz;totalin Fig. 4 indicates that\nup to the first picosecond mz;totalis not conserved,\nalthough it only changes slightly due to the similar\ndemagnetization dynamics of the Mn 4aand 4c\nsublattices (note, that the scale of the right y-axis is\nmuch smaller than the scale of the left axis). This\nmeans that relaxation by coupling to the heat-bath\ndominates. This relaxation can in turn be interpreted as\nexcitation of ferromagnetic (optical) magnons (following\nBanerjee and co-workers[17]). However, as the sublattice\nmagnetization reduces to small values ( \u00181 ps),\nthe value of mz;totalstays constant in time. This\nmeans that exchange-relaxation dominates the dynamics\n(conservation of total angular momentum). This can be\ninterpreted as excitation of antiferromagnetic (acoustic)\nmagnons. Total angular momentum is conserved for a\nrelativelylongperiodoftime(fewpicoseconds). Oncethe\nsystem has reached the exchange relaxation dominated\nregime (\u00191 ps) the interpretation of switching of Davies\nand co-workers remains valid. The exchange relaxation\nresults inj\u0001m4aj=j\u0001m4cj, but sincejm4a;0j<jm4c;0j\n240 260 280 300 320 340\nInitial Temperature T0[K]5001000Pulse Duration [fs]SwitchingNo SwitchingTcomp = 375 KFIG. 5. Switching behavior as color of Mn 2Ru0:98Ga\nas function of the starting temperature T0and the pulse\nduration. Red areas indicate switching behavior, blue marks\nareas without switching and grey indicates a prolonged\ntransient ferromagnetic state or a demagnetized state.\nmeans that m4acrosses zero before m4cdoes and\ntherefore giving rise to successful magnetic switching.\nDifferently to the interpretation of Davies et al. [20]\nthis condition,jm4a;0j<jm4c;0j, holds because the first\nsteps of the sublattice demagnetization are driven by\ndirectcouplingtotheheat-bathwithrelaxationatsimilar\nspeeds.\nOur picture can be easily checked. We argue that\nswitching happens if jm4aj<jm4cjwhen the system\nenters the exchange relaxation regime, which dominates\nthe dynamics for small values of the sublattice magnetic\norder. The sign of the total magnetization at that\nmoment can be controlled by the coupling to the heat-\nbath of each sublattice, namely the value of \u000b4a(4c). By\nincreasing\u000b4athe demagnetizaton due to the coupling\nto the heat-bath increases so that jm4aj<jm4cjcan\nbe fullfilled even for smaller values of x. The effects\nof this change of \u000b4aonmtotalwould change it from\na behaviour as shown in Fig. 4 a) towards the one in\nFig. 4 c). The full result of this analysis is shown in\nFig. 6 in the Supplemental Material. Where we observe\nthat by increasing the value of \u000b4a, a wider range of Ru-\nconcentrations switch and thus lowering the switching\nthreshold set by the compensation temperature, ( x=\n0:90in our model).\nSimilar to the experimental work of Davies et al. [20]\nwe investigate switching as function of the starting\ntemperature T0and the pulse duration (Fig. 5). Our\nresultscomparequalitativelywelltotheexperimentalob-\nservations of Ref. 20. Quantitative comparison between\nour model, based on the experimental measurements\nof Banerjee et al. [14], and Davies et al. are difficult\nwithout modifying some of the material parameters\nused in the model. The main difference between\nthe two experiments is the value of the compensation\ntemperature for similar Ru-concentrations ( TM;x=0:75=5\n370K [20] andTM;x=0:7= 245K [14]). In order to mimic\nthe conditions of Davies et al., we use Mn 2Ru0:98Ga,\nwhich features a compensation temperature of TM=\n375K (experimentally TM;x=0:75= 370K). The pulse-\nduration switching threshold decreases as the system\nbase temperature approaches a threshold temperature\nwith a similar slope as experiments. Experimentally\nthis threshold temperature is close to the compensation\ntemperature, whileinourmodelisabitfurther. However\nour findings suggest that the condition T0<TMis not a\nlimiting, sufficient condition for switching in Mn 2RuxGa.\nTo summarize, we have presented an atomistic spin\nmodel able to describe single pulse toggle switching\nin Mn 2RuxGa Heusler alloys. The parameters of the\nspin model are based on experimental measurements\nand can reproduce key material parameters such as Tc,\nM(T)or the Ru-concentration dependence of TM. We\nshow that our model is able to quantitatively reproduce\nmeasured magnetization dynamics of single pulse toggle\nswitchingandwedemonstratethatdifferentlytoprevious\nunderstanding toggle switching in Mn 2RuxGa is possible\neven when both Mn sublattices demagnetization at very\nsimilar rate.\nAcknowledgments\nWe gratefully acknowledge support by the Deutsche\nForschungsgemeinschaft through SFB/TRR 227 \"Ultra-\nfast Spin Dynamics\", Project A08.\nData Availability\nThe data that support the findings of this study are\navailable from the corresponding author upon reasonable\nrequest.\n\u0003unai.atxitia@fu-berlin.de.\n[1] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nU. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui,\nL. Le Guyader, E. Mengotti, L. J. Heyderman,\nF. Nolting, a. Tsukamoto, A. 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Rebei, Ultrafast spin dynamics: The\neffect of colored noise, Physical Review Letters 102,\n057203 (2009).7\nSUPPLEMENTARY MATERIAL\nDetails of the estimation of the system parameters\nWe model Mn 2RuxGa Heusler alloys based on the Heisenberg spin Hamiltonian:\nH=\u0000X\ni6=jJijSi\u0001Sj\u0000X\nidzS2\nz: (1)\nHereSi=\u0016i=\u0016s;irepresents a classical, normalized spin vector at site i. Here,\u0016s;iis the atomic magnetic moment of\nthe sitei.Si, couples to its nearest neighboring spin Sjvia the coupling constant Jij, anddzrepresents the on-site\nanisotropy with easy-axis along the z-axis.\nThe considered Heisenberg exchange parameters are the ferromagnetic exchange couplings J4a\u00004aandJ4c\u00004cas\nwell as the antiferromagnetic J4a\u00004c. The values of the exchange and the uniaxial anisotropy for Mn 2Ru0:68Ga are\nbased on the values of a molecular field model [35] derived in Ref. 18. The exchange parameters are adjusted to yield\na Curie temperature of about 630 K which is at the upper end of reported Curie temperatures, for instance, it is\nclose to the value reported in Ref. 18 of Tc= 625K forx= 0:68, wherexstands for Ru-concentration. Additional\nexperiments find an approximately constant Curie temperature of 450 K for x >0:5[27], whereas Ref. 16 reports\nTc= 550K forx= 0:7. Furthermore the T cmeasured in Ref. 27 features a peak around x= 0:4which could explain\nwhy our parameters only reproduce reproduce Tcompforx>0:68. Thus, a dependence of the exchange parameters on\nRu-concentration may therefore be possible for lower values of x. Since we are mostly interested in Ru-concentration\nfor which switching has been experimentally demonstrated ( x > 0:7), we restrict our study to Ru concentrations\nofx= 0:6\u00001:0. We note that we have increased J4a\u00004aby 10% to reproduce the magnetization compensation\ntemperature as reported in Ref. [14] while keeping T c= 630K. Using exchange constants with the ratios as stated in\nRef. 18 couldn’t reproduce the experimentally measured Mn 4cdynamics of C. Banerjee et al. [17] that are shown in\nFig. 3 (main text). However, the fact that there are reports of largely varying different Curie temperatures [16, 18, 27]\nand compensation temperature [14, 20] indicate that there seems to be a wide spectrum of valid exchange constants\nbetween different experiments, so that an adjustment of 10% of one of the parameters seems reasonable.\nTheuniaxial anisotropyisalso consideredtobesite-specific, inparticular, theanisotropyenergydensity K z;4c= 216\nkJm\u00003and K z;4a= 0kJm\u00003taken from Ref. [18]. Assuming a unit cell size of ( \u00190:6nm)3([36]) we obtain\ndz;4c= 1:1664\u000110\u000023J as on-site anisotropy and dz;4a= 0. The anisotropy is included to yield an alignment\nalong thez-axis after demagnetization or switching. Since it is much smaller the Heisenberg exchange it does not\nhave meaningful impact on the switching itself. Furthermore ref. 15 finds, based on XMCD experiments, different\ng-factors,g4a= 2:05andg4c= 2:00for the 4a- and 4c-sublattice. However since both \rand\u0016sare proportional to\ngithis does not enter the LLG.\nThe spin dynamics of this system are described by the atomistic stochastic-Landau-Lifshitz-Gilbert equation\n(sLLG) [31]\n(1 +\u000b2\ni)\u0016s;i\n\r@Si\n@t=\u0000(Si\u0002Hi)\u0000\u000bi(Si\u0002(Si\u0002Hi)): (2)\nWhere\rrepresents the gyromagnetic ratio and \u000biis a site-specific atomic damping parameter. Here, we also draw\non experimental observations to estimate the values of the damping parameters. Photo-excited spin precession was\nobserved by time resolved Faraday effect as a function of the applied field and temperature. From the decay of\nthe precession the intrinsic damping parameter was also determined to have values smaller than \u000b= 0:02, far from\ncompensation [15]. Here, we decided to use \u000b4c= 0:01and\u000b4c= 0:013since it reproduces experimental observations\nas discussed in the main text. The temperature dynamics are described using the two-temperature model (TTM)\nthat describes the electron temperature Teland the phonon temperature Tphvia a pair of two coupled differential\nequations [32, 33]:\nCel@Tel\n@t=\u0000gep(Tel\u0000Tph) +Pl(t) (3)\nCph@Tph\n@t= +gep(Tel\u0000Tph): (4)\nCelandCphrepresent the specific heat of the electron- and phonon system and Pl(t)describes the absorbed energy of\ntheelectronsystem, comingfromthelaser. Sinceexperimentaldataontheelectronandphonontemperaturedynamics8\nis missing, we use similar parameters to typical GdFeCo values for our TTM and similar ones to the experimental\nwork by Bonfiglio et al. (Ref. [16]). Table I provides an overview of the used TTM-parameters in comparison to\nRef. [16] and to parameters used in GdFeCo from Ref. 37 and Ref. 38.\nTABLE I. Two temperature model parameters comparison between GdFeCo and Mn 2RuxGa between different sources.\nTTM Unit Mn2RuxGaMn2RuxGa[16]GdFeCo[38] GdFeCo[37]\nCphJ/m3K3\u00021062.27\u00021063\u00021063\u0002106\ngphJ/m3Ks6\u000210178\u000210172\u0002101717\u00021017\n\rJ/m3K2350 484 714 700\nThe laser pulse is assumed to be Gaussian shaped with a FWHM of 100 fs. The electron temperature Telyielding\nfrom the TTM is used to scale the temperature effects in the spin system. This is done by including a Langevin\nthermostat, which adds an effective field-like stochastic term \u0010ito the effective field Hi=\u0010i(t)\u0000@H\n@Siwith white noise\nproperties [39]:\nh\u0010i(t)i= 0andh\u0010i(0)\u0010j(t)i= 2\u000bikBTel\u0016s;i\u000eij\u000e(t)=\r: (5)\nThe complete set of system parameter used in our model are summarized in table II.\nTABLE II. Table of the Heisenberg spin Hamiltonian parameters (left) and the two temperature model (TTM) (right).\nH Value Unit TTM Unit\nJ4a\u00004a 1:28\u000210\u000021[J]Cph 3\u0002106[J/Km3]\nJ4c\u00004c 4:0\u000210\u000022[J]Cel\rel\u0001Te[J/Km3]\nJ4a\u00004c\u00004:85\u000210\u000022[J]\rel350 [J/K2m3]\n\r 1:76\u000210\u000021[1\nTs]gep 6\u00021017[J/sKm3]\ndz 1:17\u000210\u000023[J]\n\u0016s;4a 2.88 [\u0016B]\n\u0016s;4c 4:71\u0001x [\u0016B]\n\u000b4a 0:013\n\u000b4c 0:01\nSwitching behaviour in dependence of Gilbert damping parameters\nFigure 6 shows the switching behavior of Mn 2RuxGa alloys as function of the Ru-concentration xand the absorbed\nlaser energy for different dampings \u000b4a. Red areas indicate switching behavior, blue marks non-switched simulations\nand grey areas indicate simulations with a prolonged transient ferromagnetic state or a demagnetized state.\nThe simulation was counted as switched, when the 4a-sublattice crossed mz= 0and reached a threshold of mz;4a<\n\u00000:12after 15 ps (starting at positive mzvalues). Otherwise it was counted as demagnetized if jmz;4aj<0:12, or\nremagnetized if mz;4a>= 0:12(see Fig. 4 for examples).\nThe damping \u000b4c= 0:01was kept constant while \u000b4awas varied from \u000b4a= 0:011(top) to\u000b4a= 0:013(middle)\nto\u000b4a= 0:015(bottom). Figure 6, shows clearly distinguish behaviors for the simulated alloys depending on the Ru-\nconcentration x. For low absorbed laser energies below 5:5\u0001108J/m3no switching occurs for all Ru concentrations\nconsidered here. This is due to the insufficient energy to temporarily demagnetize both sublattices. For all three\ncases, we find that for low Ruthenium concentrations below a damping-dependent threshold value the alloy does not\nswitch, independent of the laser energy. Only above that threshold we find deterministic switching. In the top panel\nwith\u000b4c= 0:01and\u000b4a= 0:011, we find the threshold Ru-concentration for switching to be around x\u00190:9\u00000:95.\nWhen the damping of the 4asublattice is increased to \u000b4a= 0:013the threshold moves to x\u00190:85\u00000:9, which\napproximatelycorrespondstotheswitchingthresholdfoundinRef.14. Finally, for \u000b4a= 0:015(bottom)theswitching\nthreshold decreases to x\u00190:8. Therefore we find, that by increasing the element specific damping discrepancy the\nswitching threshold moves towards lower Ruthenium concentrations. Our results compare best to the experiments\nby Banerjee and co-workers (threshold around x= 0:8\u00000:9) [14] when choosing \u000b4c= 0:01and\u000b4a= 0:013(Fig.\n6 middle). We note that in our model the only parameter that directly depends on xis\u0016s;4c, which impacts the9\n4567Energy [108J/m3]\nNo SwitchingSwitchingα4a= 0.011\n4567Energy [108J/m3]\nNo SwitchingSwitchingα4a= 0.013\n0.6 0.7 0.8 0.9 1.0\nRuthenium Concenentration x4567Energy [108J/m3]\nNo SwitchingSwitchingα4a= 0.015\nFIG. 6. Switching behavior as color of Mn 2RuxGa alloys as function of the Ruthenium concentration xand the absorbed\nlaser energy for different dampings \u000b4aof the 4a-sublattice. Red areas indicate switching behavior, blue marks areas without\nswitching and grey indicates a prolonged transient ferromagnetic state or a demagnetized state. The damping \u000b4c= 0:01of the\n4c-sublattice was kept constant while \u000b4awas varied from \u000b4a= 0:011(top) to\u000b4a= 0:013(middle) to \u000b4a= 0:015(bottom).10\nspeed of the 4c-sublattice. Our model shows that by increasing x,\u0016s;4calso increases and in turn the sublattice\ndemagnetization speed continuously slows down, up to the point where the demagnetization speed difference in the\n4a- and 4c-sublattice is large enough to enable switching behavior. The model also shows that this relatively different\ndemagnetization speed can also be controlled by the intrinsic site-dependent damping parameters, which influences\nthex-dependent threshold between switching and non switching behavior."    },    {        "title": "2204.14010v4.Entangling_mechanical_vibrations_of_two_massive_ferrimagnets_by_fully_exploiting_the_nonlinearity_of_magnetostriction.pdf",        "content": "Entangling mechanical vibrations of two massive ferrimagnets by fully exploiting\nthe nonlinearity of magnetostriction\nHang Qian,1Zhi-Yuan Fan,1and Jie Li1,\u0003\n1Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation,\nand Zhejiang Province Key Laboratory of Quantum Technology and Device,\nDepartment of Physics, Zhejiang University, Hangzhou 310027, China\n(Dated: December 14, 2022)\nQuantum entanglement in the motion of macroscopic objects is of significance to both fundamental studies\nand quantum technologies. Here we show how to entangle the mechanical vibration modes of two massive\nferrimagnets that are placed in the same microwave cavity. Each ferrimagnet supports a magnon mode and a\nlow-frequency vibration mode coupled by the magnetostrictive force. The two magnon modes are, respectively,\ncoupled to the microwave cavity by the magnetic dipole interaction. We first generate a stationary nonlocal\nentangled state between the vibration mode of the ferrimagnet-1 and the magnon mode of the ferrimagnet-2.\nThis is realized by continuously driving the ferrimagnet-1 with a strong red-detuned microwave field and the en-\ntanglement is achieved by exploiting the magnomechanical parametric down-conversion and the cavity-magnon\nstate-swap interaction. We then switch o \u000bthe pump on the ferrimagnet-1 and, simultaneously, turn on a red-\ndetuned pulsed drive on the ferrimagnet-2. The latter drive is used to activate the magnomechanical beamsplitter\ninteraction, which swaps the magnonic and mechanical states of the ferrimagnet-2. Consequently, the previously\ngenerated phonon-magnon entanglement is transferred to the mechanical modes of two ferrimagnets. The work\nprovides a scheme to prepare entangled states of mechanical motion of two massive objects, which may find\napplications in various studies exploiting macroscopic entangled states.\nI. INTRODUCTION\nEntanglement of mechanical motion has been first demon-\nstrated in two microscopic trapped atomic ions [1], then in two\nsingle-phonon excitations in nanodiamonds [2], and recently\nin two macroscopic optomechanical resonators [3–5]. In the\npast two decades, a number of theoretical proposals have been\ngiven in optomechanics [6–35] for preparing the entanglement\nbetween massive mechanical resonators, utilizing the cou-\npling between optical and mechanical degrees of freedom by\nradiation pressure. In particular, the application of reservoir\nengineering ideas [36–40] to optomechanics [15, 16, 20, 22]\nhas led to a significant and robust mechanical entanglement\nand the entanglement has been successfully demonstrated in\nthe experiment [4].\nExploring novel physical platforms that could prepare\nquantum states at a more massive scale is of great signifi-\ncance for the study of macroscopic quantum phenomena [41],\nthe boundary between the quantum and classical worlds [42],\nand gravitational quantum physics [43, 44], etc. Recently, the\nmagnomechanical system of a large-size ferrimagnet, e.g., yt-\ntrium iron garnet (YIG), that has a dispersive magnon-phonon\ncoupling has shown such a potential [45–58]. The magnon\nand mechanical vibration modes are coupled by the nonlin-\near magnetostrictive interaction, which couples ferrimagnetic\nmagnons to the deformation displacement of the ferrimag-\nnet [59–62]. The magnomechanical Hamiltonian takes the\nsame form as the optomechanical one [63] (by exchanging\nthe roles of magnons and photons), which allows us to pre-\ndict many optomechanical analogues in magnomechanics. By\ncoupling the magnomechanical system to a microwave cav-\n\u0003jieli007@zju.edu.cnity, they form the tripartite cavity magnomechanical system.\nThe magnetostriction has been exploited in cavity magnome-\nchanics to generate macroscopic entangled states of magnons\nand vibration phonons of massive ferrimagnets [45, 47, 48,\n51, 52, 56, 58], as well as nonclassical states of microwave\nfields [64, 65]. Therefore, the magnetostrictive nonlinearity\nbecomes a valuable resource for producing various quantum\nstates of microwave photons, magnons, and phonons. These\nnonclassical states may find potential applications in quan-\ntum information processing [66, 67], quantum metrology [65]\nand quantum networks [68]. Despite of the aforementioned\nmany proposals and very limited experiments [60–62] in cav-\nity magnomechanics, there is so far only one protocol [51] for\nentangling two mechanical vibration modes of macroscopic\nferrimagnets. The protocol [51], however, relies on an exter-\nnal entangled resource and transfers the quantum correlation\nfrom microwave drive fields to two vibration modes. There-\nfore, designing more energy-saving protocols without using\nany external quantum resource is highly needed.\nAlong this line, we present here a scheme for generating\na nonlocal entangled state between the mechanical vibrations\nof two ferrimagnets, without the need of any quantum driving\nfield. Two ferrimagnets are placed in a microwave cavity and\neach ferrimagnet supports a magnon mode and a mechanical\nmode. The cavity mode couples to two magnon modes by\nthe magnetic dipole interaction, and the magnon modes cou-\nple to their local vibration modes by magnetostriction, respec-\ntively. We show that the entanglement between the vibration\nmodes of two ferrimagnets can be achieved by fully exploit-\ning the nonlinear magnetostriction interaction (i.e., exploit-\ning both the magnomechanical parametric down-conversion\n(PDC) and state-swap interactions) and by using the common\ncavity field being an intermediary to distribute quantum cor-\nrelations. The mechanical entanglement is established by two\nsteps. We first generate steady-state entanglement between thearXiv:2204.14010v4  [quant-ph]  13 Dec 20222\nmechanical mode of the ferrimagnet-1 and the magnon mode\nof the ferrimagnet-2. We then activate the magnomechani-\ncal state-swap interaction in the ferrimagnet-2, which trans-\nfers the magnonic state to its locally interacting mechanical\nmode. Consequently, the two mechanical modes get nonlo-\ncally entangled.\nThe remainder of the paper is organized as follows. In\nSec. II, we introduce the general model of the protocol that is\nused in the two steps. We then show how to prepare a station-\nary nonlocal magnon-phonon entanglement with a continuous\nmicrowave pump in Sec. III, and how to transfer this entan-\nglement to two mechanical modes with a pulsed microwave\ndrive in Sec. IV. Finally, we discuss and conclude in Sec. V.\nII. THE MODEL\nThe protocol is based on a hybrid five-mode cavity mag-\nnomechanical system, including a microwave cavity mode,\ntwo magnon modes, and two mechanical vibration modes,\nas depicted in Fig. 1. The magnon modes are embodied by\nthe collective motion of a large number of spins (i.e., spin\nwave) in two ferrimagnets, e.g., two YIG spheres [60–62] or\nmicro bridges [69, 70]. They simultaneously couple to the mi-\ncrowave cavity via the magnetic dipole interaction. This cou-\npling can be strong thanks to the high spin density in YIG [71–\n73]. The mechanical modes refer to the deformation vibration\nmodes of two YIG crystals caused by the magnetostrictive\nforce. Due to the much lower mechanical frequency (rang-\ning from 100to 102MHz) than the magnon frequency (GHz)\nin the typical magnomechanical systems [60–62, 69], the vi-\nbration phonons and the magnons are coupled in a dispersive\nmanner [70, 74, 75]. The Hamiltonian of the system reads\nH=~=!aaya+X\nj=1;2\u0012\n!mjmy\njmj+!bj\n2\u0010\np2\nj+q2\nj\u0011\u0013\n+X\nj=1;2\u0012\ngj\u0010\namy\nj+aymj\u0011\n+G0jmy\njmjqj\u0013\n+i\nk\u0010\nmy\nke\u0000i!0kt\u0000mkei!0kt\u0011\n:(1)\nThe first (second) term describes the energy of the cavity\nmode (magnon modes), of which the frequency is !a(!mj)\nand the annihilation operator is a(mj) with the commutation\nrelation [ a;ay]=1\u0010\u0002mj;my\nj\u0003=1\u0011\n. The magnon frequency\n!mjis determined by the external bias magnetic field Hjvia\n!mj=\rHj, where the gyromagnetic ratio \r=2\u0019=28 GHz=T.\nThe third term denotes the energy of two mechanical vibra-\ntion modes with frequencies !bj, and qjandpj([qj;pj]=i)\nare the dimensionless position and momentum of the vibra-\ntion mode j, modeled as a mechanical oscillator. The cou-\npling gjis the linear cavity-magnon coupling rate, and G0j\nis the bare magnomechanical coupling rate. For large-size\nYIG spheres with the diameter in the 100 \u0016m range [60–\n62], G0jis typically in the 10 mHz range [75]. It, however,\ncan be much stronger for micron-sized YIG bridges [69, 70].\nNevertheless, the e \u000bective magnomechanical coupling can be\nsignificantly enhanced by driving the magnon mode with a\nFIG. 1: (a) Sketch of the system. Two YIG crystals are placed near\nthe maximum magnetic fields of a microwave cavity. Each YIG crys-\ntal is in a uniform bias magnetic field, and supports a magnon mode\nand a mechanical vibration mode. The two magnon modes couple to\nthe same cavity field. Two drive fields are applied successively to the\ntwo magnon modes in the two steps of the protocol. Note that though\nYIG spheres are adopted in the sketch, the YIG crystals can be a\nnonspherical structure, e.g., micron-sized YIG bridges [69, 70]. (b)\nInteractions among the subsystems. The magnon mode mj(j=1;2)\ncouples linearly to the cavity mode awith the coupling strength gj,\nand couples dispersively to the mechanical mode bjwith the e \u000bective\nmagnomechanical coupling rate Gj. A nonlocal phonon-magnon ( b1-\nm2) entanglement is created in the first step with a continuous drive\non the magnon mode m1, and the entanglement is transferred to the\ntwo mechanical modes in the second step by using a pulsed drive on\nthe magnon mode m2.\nstrong microwave field [45]. The driving Hamiltonian is de-\nscribed by the last term, and the corresponding Rabi frequency\n\nk=p\n5\n4\rpNkBk(k=1 or 2) [45], with Bk(!0k) being\nthe amplitude (frequency) of the drive magnetic field, and Nk\nbeing the total number of spins in the kth crystal. We re-\nmark that the model di \u000bers from the one used in Ref. [47]\nby including a second mechanical mode, which brings in a\nsignificant amount of additional thermal noise to the system.\nMore importantly, the present work aims to entangle two low-\nfrequency (in MHz) mechanical modes. This is much more\ndi\u000ecult to prepare than the entanglement of two GHz magnon\nmodes studied in Ref. [47].\nIn what follows, we adopt a two-step procedure to prepare\nthe two mechanical modes in an entangled state, and in each\nstep, we apply a single drive field on either magnon mode\nm1orm2. This avoids the complex Floquet dynamics in our\nhighly hybrid system caused by simultaneously applying mul-\ntiple pump tones [15, 16, 20, 22]. We first generate a nonlo-\ncal entangled state between the mechanical mode b1and the3\nFIG. 2: Mode and drive frequencies used in the first step. When\nthe magnon mode m1is (the cavity and magnon mode m2are) reso-\nnant with the blue (red) mechanical sideband of the drive field with\nfrequency!01, the nonlocal phonon-magnon entanglement Eb1m2is\nestablished.\nmagnon mode m2by continuously driving the magnon mode\nm1. After the system enters a stationary state, we then turn\no\u000bthe drive on m1and, simultaneously, turn on a red-detuned\ndrive on the magnon mode m2to activate the magnomechan-\nical state-swap interaction m2$b2. This operation transfers\nthe quantum correlation shared between b1andm2to two me-\nchanical modes, thus establishing a quantum correlation (i.e.,\nentanglement) between the two mechanical modes.\nIII. STATIONARY NONLOCAL MAGNO-MECHANICAL\nENTANGLEMENT\nIn the first step, we aim to entangle the mechanical mode\nb1and the magnon mode m2. This can be realized by driving\nthe magnon mode m1with a strong red-detuned microwave\nfield [45], see Fig. 2. It was shown that a genuine tripar-\ntite magnon-photon-phonon entangled state can be produced\nwithout involving the second YIG crystal [45]. By using the\npartial result that the mechanical mode b1and the cavity a\nare entangled, and by coupling the cavity to the second nearly\nresonant magnon mode m2(which have a beamsplitter inter-\naction realizing the state-swap operation a$m2), the two\nmodes b1andm2thus get entangled. This is confirmed by the\nnumerical results presented in this section.\nIt should be noted that since the strong drive is applied on\nthefirst YIG crystal, the e \u000bective (magnomechanical) cou-\npling to the mechanical mode b2in the second YIG crystal is\nmuch smaller than that in the first YIG crystal, G2\u001cG1, so\nthe presence of the second mechanical mode, or not, will not\nappreciably a \u000bect the entanglement dynamics analysed above.\nBecause in the next step, the coupling to the second mechani-\ncal mode must be turned on ( G02>0), including this coupling\nin the model also in the first step means no additional opera-\ntion (e.g., adjusting the direction of the bias magnetic field\nto activate or inactivate the coupling G02[47, 60]) has to be\nimplemented between the two steps. Another reason is that,\nas will be shown in Sec. IV, the entanglement between b1-m2\nshould be transferred to the two mechanical modes as soon as\npossible because it rapidly decays when the drive in the first\nstep is switched o \u000b.\nIn the frame rotating at the drive frequency !01, the quan-\ntum Langevin equations (QLEs) describing the system dy-namics are given by\n˙a=\u0000(i\u0001a+\u0014a)a\u0000ig1m1\u0000ig2m2+p\n2\u0014aain;\n˙mj=\u0000(i\u0001mj+\u0014mj)mj\u0000iG0jmjqj\u0000igja+ \n j+q\n2\u0014mjmin\nj;\n˙qj=!bjpj;\n˙pj=\u0000!bjqj\u0000\rbjpj\u0000G0jmy\njmj+\u0018j;\n(2)\nwhere \u0001a=!a\u0000!01,\u0001mj=!mj\u0000!01, and\u0014a,\u0014mjand\rbj\nare the dissipation rates of the cavity, magnon and mechanical\nmodes, respectively. The Rabi frequency \nj= \n 1\u000ej1(j=\n1;2) implies only one drive field applied on the magnon mode\nm1.ainandmin\njare the input noise operators a \u000becting the cav-\nity and magnon modes, whose non-zero correlation functions\narehain(t)ainy(t0)i=[Na(!a)+1]\u000e(t\u0000t0),hainy(t)ain(t0)i=\nNa(!a)\u000e(t\u0000t0),hmin\nj(t)miny\nj(t0)i=[Nmj(!mj)+1]\u000e(t\u0000t0)\nandhminy\nj(t)min\nj(t0)i=Nmj(!mj)\u000e(t\u0000t0). The Langevin force\noperator\u0018jis accounting for the mechanical Brownian mo-\ntion, which is autocorrelated as h\u0018j(t)\u0018j(t0)+\u0018j(t0)\u0018j(t)i \u0019\n\rbj[2Nbj(!bj)+1]\u000e(t\u0000t0), where we consider a high qual-\nity factor Qb=!b=\rb\u001d1 for the mechanical oscillators\nto validate the Markov approximation [76]. Here, Nk(!k)=h\nexp(~!k\nkBT)\u00001i\u00001(k=a;mj;bj) are the equilibrium mean ther-\nmal photon, magnon, and phonon numbers, respectively, at\nthe environmental temperature T, with kBbeing the Boltz-\nmann constant.\nThe strong drive field leads to large amplitudes of the\nmagnon modes and cavity mode due to the magnon-cavity\ncoupling,jhmjij;jhaij \u001d 1. This allows us to linearize the\nnonlinear QLEs (2) by writing each mode operator as a classi-\ncal average plus a fluctuation operator with zero mean value,\ni.e., O=hOi+\u000eO(O=a;mj;qj;pj), and by neglect-\ning small second-order fluctuation terms. Substituting the\nabove mode operators into Eq. (2) yields two sets of lin-\nearized Langevin equations, respectively, for classical aver-\nages and fluctuation operators. By solving the former set of\nequations in the time scale where the system evolves into a\nstationary state, we obtain the solution of the steady-state av-\nerages, which are hpji=0,hqji=\u0000G0jjhmjij2=!bj,hai=\n\u0000i\u0000g1hm1i+g2hm2i\u0001=(i\u0001a+\u0014a), and\nhm1i=\n1(i\u0001a+\u0014a)\ng2\n1+(i˜\u0001m1+\u0014m1)(i\u0001a+\u0014a)\u0000g2\n1g2\n2\ng2\n2+(i˜\u0001m2+\u0014m2)(i\u0001a+\u0014a);\nhm2i=\u0000\n1(i\u0001a+\u0014a)g1g2\ng2\n1+(i˜\u0001m1+\u0014m1)(i\u0001a+\u0014a)\ng2\n2+(i˜\u0001m2+\u0014m2)(i\u0001a+\u0014a)\u0000g2\n1g2\n2\ng2\n1+(i˜\u0001m1+\u0014m1)(i\u0001a+\u0014a);\n(3)\nwhere ˜\u0001mj= \u0001 mj+G0jhqjiis the e \u000bective magnon-drive\ndetuning, which includes the magnetostriction induced fre-\nquency shift. Typically, this frequency shift is negligible (be-\ncause of a small G0[60–62]) with respect to the optimal de-\ntuning used in this work, i.e., j˜\u0001mj\u0000\u0001mjj \u001c j \u0001mjj \u0019!bj.\nTherefore, in what follows we can safely assume ˜\u0001mj\u0019\u0001mj.4\nThe set of the linearized QLEs for the system fluctuations\ncan be written in the matrix form\n˙u(t)=Au(t)+n(t); (4)\nwhere u(t)=\u0002\u000eX(t);\u000eY(t);\u000ex1(t);\u000ey1(t);\u000ex2(t);\u000ey2(t);\u000eq1(t);\n\u000ep1(t);\u000eq2(t);\u000ep2(t)\u0003Tis the vector of the quadrature fluc-\ntuations, and n(t)=hp2\u0014aXin(t);p2\u0014aYin(t);p\n2\u0014m1xin\n1(t);p\n2\u0014m1yin\n1(t);p\n2\u0014m2xin\n2(t);p\n2\u0014m2yin\n2(t);0; \u0018 1(t);0; \u0018 2(t)iT\nis the vector of the input noises entering the system,\nwith\u000eX=\u0010\n\u000ea+\u000eay\u0011\n=p\n2,\u000eY=i\u0010\n\u000eay\u0000\u000ea\u0011\n=p\n2,\n\u000exj=\u0010\n\u000emj+\u000emy\nj\u0011\n=p\n2, and\u000eyj=i\u0010\n\u000emy\nj\u0000\u000emj\u0011\n=p\n2,\nand the drift matrix Ais given by\nA=0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@\u0000\u0014a\u0001a 0 g1 0 g2 0 0 0 0\n\u0000\u0001a\u0000\u0014a\u0000g1 0\u0000g2 0 0 0 0 0\n0 g1\u0000km1 \u0001m1 0 0\u0000Re[G1]0 0 0\n\u0000g10\u0000\u0001m1\u0000\u0014m1 0 0\u0000Im[G1]0 0 0\n0 g2 0 0 \u0000\u0014m2 \u0001m2 0 0\u0000Re[G2]0\n\u0000g20 0 0 \u0000\u0001m2\u0000\u0014m2 0 0\u0000Im[G2]0\n0 0 0 0 0 0 0 !b1 0 0\n0 0\u0000Im[G1]Re[G1] 0 0\u0000!b1\u0000\rb1 0 0\n0 0 0 0 0 0 0 0 0 !b2\n0 0 0 0 \u0000Im[G2]Re[G2] 0 0\u0000!b2\u0000\rb21CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (5)\nwith Gj=ip\n2G0jhmjibeing the e \u000bective magnomechanical\ncoupling rate, which is generally complex.\nBecause of the linearized dynamics of the system and the\nGaussian nature of the input noises, the steady state of the\nsystem’s quantum fluctuations is a five-mode Gaussian state.\nThe state can be completely described by a 10 \u000210 covariance\nmatrix (CM)Vwith entries defined as Vi j=hui(t)uj(t0)+\nuj(t0)ui(t)i=2 (i;j=1;2;:::; 10). The CMVcan be obtained\nby directly solving the Lyapunov equation\nAV+VAT=\u0000D; (6)\nwhere the di \u000busion matrix Dis defined as Di j=\nhni(t)nj(t0)+nj(t0)ni(t)i=\u00022\u000e(t\u0000t0)\u0003and given by D =\ndiag[\u0014a(2Na+1);\u0014a(2Na+1);\u0014m1(2Nm1+1);\u0014m1(2Nm1+1);\n\u0014m2(2Nm2+1);\u0014m2(2Nm2+1);0;\rb1(2Nb1+1);0;\rb2(2Nb2+1)].\nWhen the CM of the system fluctuations is obtained, one\ncan then extract the state of the interesting modes by remov-\ning inVthe rows and columns related to the uninteresting\nmodes and study their entanglement properties. We use the\nlogarithmic negativity [77] to quantify the Gaussian bipartite\nentanglement, of which the definition is provided in the Ap-\npendix.\nIn Fig. 3, we show relevant steady-state bipartite entan-\nglements generated in the first step. The blank areas are\nthe parameter regimes in which the system is unstable, re-\nflected by the fact that the drift matrix Ahas at least one pos-\nitive eigenvalue. We have employed the following parame-\nters [60–62, 69, 70]: !a=2\u0019=10 GHz,!b1=2\u0019=17 MHz,\n!b2=2\u0019=12 MHz,\rb=2\u0019=100 Hz,\u0014a=2\u0019=1 MHz,\u0014m=\u0014a,\ng1=2\u0019=5 MHz, g2=2\u0019=1 MHz, G01=2\u0019'G02=2\u0019=10\nHz, and T=10 mK. The amplitude of the drive magnetic\nfield B1=4:8\u000210\u00004T, corresponding to the drive power\nP1'1:1 mW for two YIG micro bridges (approximated as\nFIG. 3: Nonlocal phonon-magnon entanglement Eb1m2versus (a) \u0001a\nand\u0001m1, and (b) \u0001aand\u0001am2\u0011!a\u0000!m2. (c) The phonon-cavity\nentanglement Eb1aand (d) the mechanical entanglement Eb1b2versus\n\u0001aand\u0001m1. In (d) the whole purple region denotes Eb1b2=0. We\ntake\u0001am2=0:9\u0014ain (a), (c) and (d), and \u0001m1=0:95!b1in (b). The\nblank areas denote the unstable region where the stability condition\nis not fulfilled. See the text for other parameters.\ntwo cuboids) with dimensions of 13.7 \u00023\u00021\u0016m3and 16.4\n\u00023\u00021\u0016m3. This milliwatt power might lead to heating in-\nduced temperature rise. To avoid the possible heating e \u000bect,\none should improve the bare coupling G0jto reduce the power.\nSince the mechanical resonance frequency mainly depends on\nthe length and thickness of the beam [69], the sizes of the\nbeams are chosen corresponding to the mechanical frequen-5\ncies we adopted. To determine the power, we have used the\nrelation between the drive magnetic field B1and the power P1,\ni.e.,B1=p\n2\u00160P1=(lwc) [70], with \u00160being the vacuum mag-\nnetic permeability, cthe speed of the electromagnetic wave\npropagating in vacuum, and land wthe length and width\nof the YIG cuboid. The nonlocal phonon-magnon entangle-\nment Eb1m2(Fig. 3(a)-(b)) is the result of the phonon-cavity\nentanglement Eb1a(Fig. 3(c)) and the cavity-magnon ( a-m2)\nbeamsplitter (state-swap) interaction. The phonon-cavity en-\ntanglement Eb1ais obtained by using the results of Ref. [45],\nwhich studies a tripartite magnon-photon-phonon system (i.e.,\nthem1-a-b1subsystem here). As shown in Ref. [45], a red-\ndetuned strong drive field on the magnon mode m1(with\n\u0001m1\u0019!b1) cools the hot mechanical mode b1(a precondi-\ntion for preparing quantum states) and, simultaneously, acti-\nvates the magnomechanical PDC interaction, which produces\nthe magnomechanical entanglement Eb1m1. The latter process\noccurs only when the pump field is su \u000eciently strong, such\nthat the weak coupling condition jG1j\u001c!b1for taking the\nrotating-wave (RW) approximation to obtain the cooling in-\nteraction/my\n1b1+m1by\n1is no longer satisfied, and the counter-\nRW terms/my\n1by\n1+m1b1(responsible for the PDC interac-\ntion) start to play the role. The magnomechanical entangle-\nment is partially transferred to the phonon-cavity subsystem\nwith Eb1a>0 (Fig. 3(c)) when \u0001a\u0019\u0000!b1[45]. The opti-\nmal detunings \u0001m1\u0019 \u0000\u0001a\u0019!b1imply that the cavity and\nthe magnon mode m1are respectively resonant with the two\nmechanical sidebands of the drive field, see Fig. 2.\nSince the cavity-magnon ( m2) state-swap operation works\noptimally when the two modes are resonant. The entangle-\nment Eb1m2is maximized for a nearly zero cavity-magnon de-\ntuning \u0001am2\u00190, as confirmed by Fig. 3(b). The detuning\n\u0001am2up to several cavity linewidth will significantly hinder\nthe transfer of the entanglement. The fact that the entangle-\nment Eb1m2is transferred from the entanglement Eb1acan also\nbe seen from the complementary distribution of the entangle-\nment in Figs. 3(a) and 3(c), indicating the entanglement flow\namong the subsystems switched on by the beamsplitter cou-\npling.\nIt is worth noting that in the first step the two mechani-\ncal modes are not entangled, as confirmed by Eb1b2=0 in\nFig. 3(d) in a wide range of parameters. This is mainly be-\ncause the magnon mode m2is driven by the nearly resonant\ncavity field, which is not the condition for cooling the me-\nchanical mode b2, or realizing the state-swap interaction be-\ntween the two modes m2andb2. Instead, a red-detuned mi-\ncrowave field should be used to drive the magnon mode m2\nto realize the state-swap operation m2$b2, such that the\nphonon-magnon entanglement Eb1m2can be transferred to the\ntwo mechanical modes, as will be discussed in the next sec-\ntion.\nIV . ENTANGLEMENT BETWEEN TWO MECHANICAL\nMODES\nIn this section, we show how to transfer the phonon-\nmagnon entanglement Eb1m2prepared in the first step to the\nFIG. 4: Mode and drive frequencies used in the second step. A red-\ndetuned pulsed drive with frequency !02is used to activate the mag-\nnomechanical state-swap operation m2$b2, which transfers the\nphonon-magnon entanglement Eb1m2generated in the first step to the\ntwo mechanical modes.\ntwo mechanical modes. To this end, we apply a red-detuned\npulsed microwave drive on the magnon mode m2(see Fig. 4)\nto activate the local magnomechanical state-swap interaction\nm2$b2. The pulsed drive should be fast as the entangle-\nment Eb1m2will quickly decay as soon as the continuous drive\nis turned o \u000bin the first step. To simplify the model, we use\na flattop microwave pulse to drive the magnon mode and then\nthe model and the treatment used in Sec. III for a continuous\ndrive are still valid for the case of a flattop pulse drive. Di \u000ber-\nently, we shall solve the dynamical solutions rather than the\nsteady-state solutions.\nThe QLEs remain the same as in Eq. (2), except that all\nthe detunings are redefined with respect to the frequency !02\nof the pulsed drive and the Rabi frequency is associated with\nthe pulsed drive, i.e., \u0001a=!a\u0000!02,\u0001mj=!mj\u0000!02, and\n\nj= \n 2\u000ej2. The dynamics of the quantum fluctuations of the\nsystem can still be described by Eq. (4) but with dynamical\nhmji(t) and thus Gj(t) in the drift matrix, due to a pulsed drive.\nThe dynamical CM V(t) can be obtained by [29]\nV(t)=M(t)V0M(t)T+Zt\n0ds M (s)DM(s)T; (7)\nwhere tis the duration of the pulsed drive, M(t)=eRt\n0A(\u001c)d\u001c,\nandV0is the CM of the initial state of the system when si-\nmultaneously turn on (o \u000b) the drive on the magnon mode m2\n(m1), which is obtained in the first step by solving the Lya-\npunov equation (6). When the dynamical CM is achieved, we\ncan then study the dynamics of the entanglement.\nFigure 5(a) shows the mechanical entanglement Eb1b2is\nmaximized at the detuning \u0001m2\u0019!b2, which is the optimal\ndetuning for realizing the magnomechanical state-swap inter-\naction, by which the previously generated phonon-magnon\nentanglement Eb1m2is transferred to the mechanical modes.\nNote that Fig. 5(a) is plotted at the optimal pulse duration tmax\nfor a given detuning, yielding a maximal Eb1b2. Figure 5(b)\nshows Eb1b2versus the pulse duration tfor the optimal detun-\ning. As is shown, there is a time window for the presence of\nthe entanglement. The mechanical entanglement emerges a\nshort while after the phonon-magnon entanglement dies out\n(c.f. the inset of Fig. 5(b)). When Eb1b2reaching its maxi-\nmum at tmax, we then turn o \u000bthe pulse drive to decouple the\nmechanics from the rest of the system to protect the entan-\nglement. The two mechanical oscillators then evolve almost\nfreely, and their entanglement is a \u000bected by their local thermal\nbaths, which lasts for a much longer time due to a small me-6\nFIG. 5: Mechanical entanglement Eb1b2versus (a) detuning \u0001m2at\nthe optimal pulse duration tmax; (b) pulse duration tat the optimal\ndetuning \u0001m2=!b2. The inset shows that the phonon-magnon en-\ntanglement Eb1m2dies out around t'0:01\u0016s. (c) Eb1b2versus time\nwhen the drive is turned o \u000battmax. Soon after tmax, the two mechani-\ncal oscillators evolve freely with the only coupling to their local ther-\nmal baths. (d) Eb1b2versus bath temperature at the optimal detuning\nand pulse duration. We take \u0001a=\u00000:95!b1,\u0001m1=0:95!b1(as\ndefined in the first step), and \u0001am2=0:9\u0014ain all plots. The other\nparameters are the same as in Fig. 3.\nchanical damping and a low bath temperature. This is clearly\nshown in Fig. 5(c) (c.f. Fig. 5(b)). The mechanical entan-\nglement is robust against bath temperature and the maximal\nentanglement survives up to T'100 mK, as illustrated in\nFig. 5(d). We have used a drive power P2=1:3 mW, giving\nthe amplitude of the drive magnetic field B2=4:8\u000210\u00004T.\nLastly, we discuss how to detect the mechanical entangle-\nment. The entanglement can be verified by measuring the CM\nof the two mechanical modes. The mechanical quadratures\ncan be measured by coupling the deformation displacement to\nan optical cavity that is driven by a weak red-detuned light to\ntransfer the mechanical state to the optical field. By homo-\ndyning two cavity output fields one can then obtain the CM of\nthe mechanical modes [5, 78].V . CONCLUSION AND DISCUSSION\nWe present a protocol to entangle two mechanical vibra-\ntion modes in a cavity magnomechanical system. The pro-\ntocol contains two steps by applying successively two drive\nfields on two magnon modes to activate di \u000berent functions of\nthe nonlinear magnetostrictive interaction, namely, the mag-\nnomechanical PDC and state-swap operations. We show that\nthe entanglement between two mechanical vibration modes of\ntwo YIG crystals can be achieved by fully exploiting the above\nmagnetostrictive functions and the cavity-magnon state-swap\ninteraction. We remark that our protocol is valid for any mag-\nnomechanical system of ferrimagnets or ferromagnets, spher-\nical [60–62] or nonspherical structures [69, 70], as long as\nthey possess the dispersive coupling between magnons and\nphonons. The work may find important applications in many\nstudies that require the preparation of macroscopic entangled\nstates.\nAcknowledgments\nThis work has been supported by National Key Re-\nsearch and Development Program of China (Grant No.\n2022YFA1405200) and National Natural Science Foundation\nof China (Nos. 92265202 and 11874249).\nAppendix\nThe logarithmic negativity is used to quantify the Gaussian\nbipartite entanglement, which is defined as\nEN\u0011max\u00020;\u0000ln 2˜\u0017\u0000\u0003; (8)\nwhere ˜\u0017\u0000=min eigji\n2˜V4j(\n2=\b2\nj=1i\u001byand\u001byis the y-\nPauli matrix) is the minimum symplectic eigenvalue of the\nCM ˜V4=PV 4P, withV4being the 4\u00024 CM of two relevant\nmodes, obtained by removing in Vthe rows and columns of\nthe uninteresting modes, and P=diag(1;\u00001;1;1) being the\nmatrix that implements the partial transposition of the CM.\n[1] J. D. Jost, J. P. Home, J. M. Amini, D. Hanneke, R. Ozeri, C.\nLanger, J. J. Bollinger, D. Leibfried, and D. 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Contrary to previous reports, only\none spin canting angle of Cr ions was observed. The spin canting angles of Mn and Cr ions in\nthe ferrimagnetic spiral obtained at a liquid-He temperature were 43\u000eand 110\u000e, respectively. The\nnuclear spin-spin relaxation was determined by the Suhl-Nakamura interaction at low temperatures\nbut the relaxation rate T\u00001\n2increases rapidly as the temperature approaches Ts. This indicates\nthat the \ructuation of the spiral component becomes faster as the temperature increases but not\nfast enough to leave an averaged hyper\fne \feld to nuclei in the time scale of nuclear spin precession\nin the ferrimagnetic phase, which is on the order of 10\u00008s. The spiral volume fraction measured\nfor various temperatures reveals that the collinear and the spiral ferrimagnetic phases are mixed\nbelow the transition temperature of the spiral order. The temperature hysteresis in the volume\nfraction implies that this transition has \frst-order characteristics.\n1arXiv:1006.1983v1  [cond-mat.str-el]  10 Jun 2010I. INTRODUCTION\nOne of the reasons spinel oxides AB 2O4have been of interest is the various spin structures\nthey show at low temperatures.1{3Cubic spinels MCr2O4(M=Mn, Co) have drawn much\nattention recently in relation with multiferroics since it was found that their spin con\fgu-\nration leads ferroelectric.4The magnetic phase of MCr2O4changes from paramagnetic to\nferrimagnetic and from ferrimagnetic to spiral orders as the temperature decreases, even be-\ncoming glass-like at low temperatures. The complexity in the spin structure of cubic spinels\nresults from geometrical frustration among the spins of B site ions, which form the lattice of\ncorner-sharing tetrahedra. M(Cr) ions occupy the A (B) sites that locate at the center of a\ntetrahedron (octahedron) in the oxygen lattice. Both the Mions and Cr ions are magnetic,\nand their magnetization directions are opposed in the ordered states.\nThe antiferromagnetic exchange interaction JBBamong the spins of the B site ions does\nnot induce ordered states due to the aforementioned frustration.5However, the addition of\nthe exchange interaction JABbetween the A and B site ions can give rise to a magnetic order.\nThe relative strength of JABtoJBBis the key factor determining the state of the magnetic\norder, including the spin canting angle. The N\u0013 eel con\fguration, a collinear ferrimagnet, is\nexpected to be the ground state when JBBis much weaker than JAB. AsJBBincreases,\nboth the Mand Cr spins become canted and the magnetic structure becomes complex.\nA classical theory predicts that the ground state of a cubic spinel is determined by the\nparameteru= 4JBBSB=3JABSA, whereSAandSBare the spin magnitudes at the A and\nB sites, respectively.6The N\u0013 eel con\fguration is the ground state when u < 8=9 and the\nferrimagnetic spiral is locally stable when 8 =9<u< 1:3. It becomes unstable when u>1:3.\nAn early neutron di\u000braction experiment showed that the ferrimagnetic spiral is the ground\nspin state of the cubic spinel MnCr 2O4.7Two di\u000berent spin canting angle values were mea-\nsured for the Cr ions, while only one value was measured for the Mn ions. The cone angles\nof the Mn ions, Cr(I), and Cr(II) in the ferrimagnetic spiral were 24\u000e, 104\u000e, and 152\u000e,\nrespectively. The approximate uvalue estimated by these measurements is 1.6, which pre-\ndicts that the ferrimagnetic spiral is unstable. In contrast, the cone angle of Mn ion spins\nmeasured by NMR was 63\u000eor 42\u000e, while those of Cr(I) and Cr(II) spins were 94\u000eand 97\u000e,\nrespectively.8{10Theory poorly matches these numbers, but it predicts a very unstable spiral\nstate in general. The question as to whether the ferrimagnetic spiral is the stable ground\n2state in the cubic spinels was raised again by a recent neutron di\u000braction experiment in-\nvolving MnCr 2O4.11It was reported that the ferrimagnetic state is long-range ordered for\nall temperatures below TC\u001850 K, and that the spiral component appears in the plane\northogonal to the direction of the ferrimagnetic order below Ts\u001820 K. However, it is only\nshort-range ordered.\nThe present study investigates the characteristics of the ordered spin state of MnCr 2O4\nat a low temperature by nuclear magnetic resonance (NMR). First, the cone angles of Mn\nand Cr ion spins, to which the previous NMR and neutron di\u000braction results gave di\u000berent\nvalues, were measured. Information pertaining to accurate cone angles is important to\nunderstand not only the ground state of the spin order but also the electrical polarization.12\nWe measured the nuclear spin-spin relaxation time T2as a function of the temperature to\nstudy the change in the spin \ructuation with temperature. The result indicates that the\n\ructuation of the spiral component increases rapidly as the temperature approaches the\nphase transition temperature of the spiral order Ts, above which the \ructuation is too fast\nfor the spiral component to be measured by even local probes such as neutron di\u000braction\nor NMR. The temperature-dependence of the NMR signal intensity reveals that the spiral\nphase is mixed with the ferrimagnetic phase at temperatures below Ts.\nII. EXPERIMENT\nA polycrystalline MnCr 2O4sample was synthesized by a solid-state reaction from a molar\nratio mixture of MnO and Cr 2O3powders. The mixture was sintered at 1100\u000eC for 12 hours\nin an Ar environment, for 12 additional hours at 1200\u000eC, and \fnally for 24 hours at 1300\u000eC.\nA single-crystalline sample was grown by the \rux method using a mixture of MnCO 3, Cr 2O3,\nPbF 2and PbO (molar ratio = 1:1:2:2). NMR signals were obtained by the conventional spin\necho method using a custom-made spectrometer in the temperature range of 4 to 20 K. To\nestimate the spin canting angle, the resonance frequency was measured for various magnetic\n\felds up to 4 T. The nuclear spin-spin relaxation time, T2, was obtained by varying the\ntime delay between the 90\u000eand 180\u000epulses. The53Cr NMR spectrum was measured in the\nfrequency range from 60 to 70 MHz, and the55Mn NMR spectrum was assessed from 530\nto 560 MHz. As the spectral width was very broad, the signal intensity was measured as a\nfunction of the frequency after selective excitation.\n3III. RESULTS AND DISCUSSION\nThe magnetization versus the temperature curves show a discrepancy in the transition\ntemperatures of the ferrimagnetic spiral and the collinear ferrimagnet of the polycrystalline\nand single-crystalline samples. In Fig. 1, the thick solid and dashed lines represent the\n\feld cooling (FC) and zero \feld cooling (ZFC) M(T) curves of the polycrystalline sample,\nrespectively, and the thin solid and dotted lines represent those of the single crystal. TCis\nrelatively well de\fned by the abrupt increase in the magnetization in both samples of ap-\nproximately 40 K for the polycrystalline sample and 50 K for the single-crystalline sample.\nTsof the polycrystalline sample is more clearly de\fned by the abrupt decrease in the mag-\nnetization at 20 K compared to that of the single-crystalline sample, whose magnetization\ndecreases smoothly at approximately 12 K only in the ZFC case. The M(T) curves of our\npolycrystalline and single-crystalline samples are in good agreement with those in previous\nreports.11,13One of the reasons for the di\u000berence in the characteristics of the two samples is\nthe site disorder in the single-crystalline sample. X-ray absorption spectroscopy showed that\nthe Mn ions occupy only the A sites and the Cr ions occupy the B sites in our polycrystalline\nsample, whereas both ions are found in both sites in the single-crystalline sample.14All of\nthe experimental data described below were obtained from the polycrystalline sample.\nFigure 2 shows the53Cr NMR spectrum obtained in a zero \feld at 6.5 K for several\ndi\u000berent echo times. The spectrum obtained at the echo time of 20 \u0016sshows a very well-\nde\fned single peak whose width is about 5 MHz. The single peak centered around 67 MHz at\na short echo time of 20 \u0016sappears to split into a double peak as the echo time supasses 90 \u0016s.\nThis is not a splitting but a suppression of the spectral intensity around the center due to\nthe frequency-dependent nuclear spin-spin relaxation rate. In an ordered magnetic insulator\ncontaining a high concentration of identical magnetic nuclear spins, the Suhl-Nakmura (SN)\ninteraction, in which nuclear spins are indirectly coupled by virtual magnons, is expected\nto play a major role in the NMR relaxation at low temperatures. It is known that the SN\ninteraction generates a \feld- and frequency-dependent T2with a minimum occurring in the\ncenter of the spectrum, as the majority of nuclear spins precess at this frequency.15,16The\ndouble peak feature of the spectrum obtained at 90 \u0016sdisappears in the spectrum obtained\nat the same echo time, however, in a magnetic \feld, as denoted by the open circles in the\n\fgure. This is consistent with the fact that the spin-spin relaxation rate due to the SN\n4interaction decreases as the \feld increases. The dependence of T2on the frequency and \feld\ncon\frms that the SN interaction is the main source of Cr nuclear interaction in MnCr 2O4.\nThis most likely explains why a double peak was observed in the previous NMR work, rather\nthan the di\u000berence in the sample quality.10The di\u000berence of only 3 % in the canting angles\nof the two Cr spins associated with the two peaks in the previous NMR report supports this\nclaim, because the experimental error of the canting angle as estimated by NMR is larger\nthan this in general. The55Mn NMR spectrum showed a well-de\fned single peak centered\naround 550 MHz in a zero \feld at the liquid-Helium temperature.\nThe spin canting angles of the Mn2+and Cr3+ions relative to the magnetization direction\nare determined by the shift of the spectrum with an external \feld. The NMR resonance\nfrequencyfis proportional to the magnitude of the total \feld, which is the vector sum of\nthe hyper\fne \feld Hhfand external \feld Hext. This is expressed as follows:\nf=\r=2\u0019\f\f\f\u0000 !Hext+\u0000 !Hhf\f\f\f\n'\r=2\u0019(Hhf\u0000Hextcos\u0012);\nwhere\ris the gyromagnetic ratio, and \u0012is the angle between HhfandHext. The direction\nof the hyper\fne \feld is antiparallel to the local magnetization in most magnetic materials,\nproviding the minus sign in the equation. As the hyper\fne \felds of the Mn2+and Cr3+ions\nin MnCr 2O4are more than one order of magnitude larger than the external magnetic \feld\nused in the experiment, the \frst-order approximation of the total \feld can be taken. The\nslope of the frequency shift with external \feld is then determined as \r=2\u0019cos\u0012.\nIn Fig. 3, the center frequencies of the Cr and Mn NMR spectra obtained at 4.2 K\nare plotted as a function of the external \feld. Both of the frequencies change linearly in\nthe experimental \feld range, as expected. The resonance frequency of the Mn spectrum\ndecreases as the \feld increases, whereas that of the Cr spectrum increases. This visually\nshows that the spin directions of the Mn and Cr ions are opposite to each other, because\nthe signs of the hyper\fne \felds of the ions are identical. From the slope of the linear \ft\nto the data, the spin canting angles of the Mn and Cr ions were determined to be 43 \u00065\u000e\nand 110\u00065\u000e, respectively. This is contrary to the previous neutron di\u000braction or NMR\nexperiments that reported two di\u000berent values of spin canting angles for Cr ions. The\ncanting angle of the Mn spin is identical to that in one of the previous NMR reports9\nbut twice as large as the value of the neutron result7. The canting angle of the Cr spin\n5is consistent with the previous NMR measurement10and one of the values given by the\nneutron di\u000braction7. Considering that no single value of ucan result in these cone angles,\nthe classical theory fails to explain the result. However, both of the values corresponding\nto the Mn and Cr spin canting angles indicate that the ferrimagnetic spiral con\fguration is\nunstable in MnCr 2O4. This reminds us of the fact that a long range order of the ferrimagnetic\ncomponent accompanies a short range order of the spiral component below Tsin MnCr 2O4.11\nFigure 4 shows the nuclear spin-spin relaxation rate T\u00001\n2of Cr ions at temperatures\nranging from 6.5 K to 14.5 K. The relaxation rate increases relatively slowly with the tem-\nperature below 11 K, above which the slope becomes steep. The main interaction a Cr\nion nucleus experiences is the interaction with the electron spins of magnetic ions and the\nprominent relaxation source is SN interaction that is mediated by spin wave as mentioned\nabove. The relaxation due to SN interaction is normally temperature independent. There-\nfore, the weakly temperature dependent relaxation with the rate of 2 \u0002104sec\u00001near 7 K\nshould be mainly due to SN interaction. The additional relaxation increasing with tempera-\nture indicates that the spin \ructuation becomes too large to be described in the framework\nof spin waves as the temperature approaches Ts. The previous observation that the spi-\nral component of the spin order is unstable and short ordered implies that it is the spiral\ncomponent of the Cr ion spins that \ructuates. This interpretation is also consistent with\nthe saturation magnetization plotted together with the nuclear relaxation rate in Fig. 4.\nThe value of the saturation magnetization stays at 1 :1\u0016Bindependent of temperature even\nwhen the temperature crosses Ts, where the spiral component is generated or vanishes. This\nmeans that the component along the easy axis remains the same while the spiral component\nperpendicular to it is averaged out by \ructuation crossing Ts, leaving only the ferrimagnetic\norder.\nThe NMR signal intensity is in general a function of the temperature, T2, and the number\nof nuclei. The data points in Fig. 5, where the Cr NMR signal intensity vs. the temper-\nature is plotted, were obtained from the raw experimental data after temperature and T2\ncorrection. Thus, they represent the number of the nuclei producing the signal. The sig-\nnal intensity obtained while warming the sample stays constant while that obtained while\ncooling it changes. It is worthwhile to note that the NMR signal is observed not in the\nferrimagnetic phase but only in the spiral phase of MnCr 2O4. Therefore, the corrected sig-\nnal intensity in the \fgure is proportional to the volume of the ferrimagnetic spiral phase.\n6Upon warming, the entire volume of the MnCr 2O4sample maintains its ferrimagnetic spiral\nphase until the change to the ferrimagnetic phase at Ts. Upon cooling, however, MnCr 2O4\nremains in the ferrimagnetic phase well below Ts. In the temperature region where the signal\nintensity changes, the two phases coexist. Depending on the temperature change history,\nthe ferrimagnetic spiral phase is embedded in the matrix of the collinear ferrimagnet phase.\nThis mixed phase might have caused some error in the measurement of various quantities\nin the previous neutron di\u000braction and NMR experiments. The temperature hysteresis in\nthe volume of the ferrimagnetic spiral phase can be ascribed to a \frst-order transition at Ts.\nAn ESR work on the similar cubic spinel CoCr 2O4showed an abrupt shift of the frequency\natTs, also indicating a \frst-order transition.18The experimental evidence is in con\rict to\nthe second-order transition on which the classical theory is based. The Mn NMR signal\nintensity also showed a similar temperature hysteresis.\nIV. CONCLUSION\nThe spin canting angles of Mn and Cr ions and the nuclear spin-spin relaxation rates were\nmeasured in this study. Only one canting angle of Cr spins was observed, contrary to that\nobserved in previous neutron and NMR experiments. The measured canting angles predict\nan unstable ferrimagnetic spiral state at a low temperature. This instability is consistent\nwith the measurement of the spin-spin relaxation rate. The relaxation rate increases more\nrapidly as the temperature increases until it appears to diverge at Ts. The rapid increase\nin the relaxation rate near Tscan be explained by the \ructuation of the spiral component.\nThe NMR signal is not observed due to this strong relaxation near Ts. The NMR signal\nis unobservable above Tsas well, where the magnetic phase is collinear ferrimagnetic. The\nmagnetization remains the same, crossing Ts, indicating that the canted spins in the ferri-\nmagnetic spiral phase do not line up along one direction, entering the ferrimagnetic phase;\ninstead, the \ructuation of the spiral component averages out to leave only the magnetic com-\nponent along the easy axis. The \ructuation accelerates as the temperature approaches Ts,\nand pastTs, it becomes fast enough to make the spiral component unobservable in neutron\ndi\u000braction experiments but not fast enough to leave an averaged hyper\fne \feld to nuclei in\nthe time scale of nuclear spin precession, which is on the order of 10\u00008s. 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Heeger, J. Phys. Chem. Solids 29, 1085 (1968).\n9T. Tsuda, A. Hirai, and T. Tsushima, Solid State Commun. 9, 2207 (1971).\n10H. Nagasawa and T. Tsushima, Phys. Lett. 15, 205 (1965).\n11K. Tomiyasu, J. Fukunaga, and H. Suzuki, Phys. Rev. B 70, 214434 (2004).\n12Hosho Katsura, Naoto Nagaosa, and Alexander V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).\n13E. Winkler, S. Blanco Canosa, F. Rivadulla, M. A. L\u0013 opez-Quintela, J. Rivas, A. Caneiro, M.\nT. Causa, and M. Tovar, Phys. Rev. B 80, 104418 (2009).\n14J.H. Park (private communication).\n15R. R. Arons, H. G. Bohn and H. L utgemeier, Physica 80B, 12 (1975).\n16J. Barak and N. Kaplan, Phys. Rev. Lett. 23, 925 (1969).\n17M. Shaham, J. Barak, and U. El-Hanany, W. W. Warren, Jr., Phys. Rev. B 22, 5400 (1980).\n18T. A. Kaplan and N. Menyuk, Philosophical Magazine 87, 3711 (2007).\n801 02 03 04 05 06 00200400600800100012001400 \nFC  (poly) \nZFC (poly) \nFC  (single) \nZFC (single) \n M (arb. units)T\nemperature (K)FIG. 1: (Color online) Magnetization vs. temperature curves obtained at 50 Oe: the thick solid\nand dashed lines (black) represent the FC and ZFC M(T) curves of the polycrystalline sample,\nrespectively, and the thin lines (blue) represent those of the single crystal.\n9606 26 46 66 87 07 27 40.00.51.01.52.02.53.03.5z\nero field     20 msec \n                   90 msec \n                 160 msec \n   5 kG        90 msec \n Signal Intensity (arb. units)N\nMR frequency (MHz)FIG. 2: (Color online) The \flled circles represent the zero-\feld Cr NMR spectrum obtained at the\necho time of 20, 90, and 160 \u0016s, and the open circles represent the Cr NMR spectrum obtained in\nthe external \feld of 5 kG and at the echo time of 90 \u0016s.\n100.00 .51 .01 .52 .02 .53 .03 .567686970 \n Frequency (MHz)E\nxternal field (T)0.00 .51 .01 .52 .02 .53 .0530540550560(\na)(\nb)   \nFrequency (MHz)FIG. 3: (Color online) (a) The \flled circles represent the central frequency of the Mn NMR\nspectrum obtained in the external \feld at 4.2 K. (b) The open circles represent the central frequency\nof the Cr NMR spectrum obtained in the external \feld at 4.2 K. The red lines denote the linear\n\ft, of which the slope is \r=2\u0019cos\u0012.\n1146 8 1 01 21 41 61 82 02 22 40246810 T\nemperature (K)Relaxation rate /s40104 sec-1/s41T\nS0\n.00.20.40.60.81.01.21.4s\naturation magnetization /s40 mB /s41FIG. 4: The \flled circles show the relaxation rate T\u00001\n2at the central frequency of the Cr NMR\nspectrum with the temperature. The open circles denote the saturation magnetization.\n1268 1 01 21 41 60.00.20.40.60.81.01.21.41.6 \ncooling \nwarming \n Volume fraction (arb. units)T\nemperature (K)FIG. 5: The volume fraction of the ferrimagnetic spiral phase vs. the temperature. The open\ncircles represent the volume fraction of the ferrimagnetic spiral obtained while cooling, and the\n\flled circles represent that obtained while warming.\n13"    },    {        "title": "0911.1216v2.Frustrated_spin_ladder_with_alternating_spin_1_and_spin_1_2_rungs.pdf",        "content": "arXiv:0911.1216v2  [cond-mat.str-el]  30 Jan 2010Frustrated spin ladder with alternating spin-1 and spin-1/ 2 rungs\nV. Ravi Chandra,1,2N. B. Ivanov,3,4and J. Richter5\n1Max-Planck-Institute for Physics of Complex Systems, N¨ ot hnitzer Str-38, D-01187, Dresden, Germany\n2Physics Department, The Technion, Haifa, 32000, Israel\n3Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, D-33501 Bi elefeld, Germany\n4Institute of Solid State Physics, Bulgarian Academy of Scie nces, Tzarigradsko chaussee 72, 1784 Sofia, Bulgaria\n5Institut f¨ ur Theoretische Physik, Universit¨ at Magdebur g, PF 4120, D-39016 Magdeburg, Germany\n(Dated: November 1, 2018)\nWe studytheimpact ofthe diagonal frustratingcouplings on the quantumphase diagram of atwo-\nleg ladder composed of alternating spin-1 and spin-1/2 rung s. As the coupling strength is increased\nthe system successively exhibits two gapped paramagnetic p hases (a rung-singlet and a Haldane-\nlike non-degenerate states) and two ferrimagnetic phases w ith different ferromagnetic moments per\nrung. The first two states are similar to the phases studied in the frustrated spin-1/2 ladder, whereas\nthe magnetic phases appear as a result of the mixed-spin stru cture of the model. A detailed char-\nacterization of these phases is presented using density-ma trix renormalization-group calculations,\nexact diagonalizations of periodic clusters, and an effecti ve Hamiltonian approach inspired by the\nanalysis of numerical data. The present theoretical study w as motivated by the recent synthesis\nof the quasi-one-dimensional ferrimagnetic material FeIIFeIII(trans-1,4-cyclohexanedicarboxylate)\nexhibiting a similar ladder structure.\nPACS numbers: 75.10.Jm, 75.50.Gg, 64.70.Tg\nI. INTRODUCTION\nOver the past two decades there has been an increas-\ning interest in quantum spin systems with competing\nexchange interactions.1,2Quantum spin chains and lad-\nders with frustration, both for half-integer and integer\nspins, set up an important part of this research since\nthey provide a unique testing ground based on the avail-\nable powerful analytical and numerical techniques for\none-dimensional (1D) systems. In particular, the frus-\ntrated ladder models haveallowedcontrolled calculations\nto examine topological order,3dimer order,4as well as\nthe appearance of fractional excitations in spin models.5\nMost of previously studied frustrated chain and ladder\nmodels havebeen related to uniform-spin structures with\nall the spins same. In comparison, till now much less\nexperimental as well as theoretical work concerning the\nimpact of competing interactions in quasi-1D mixed-spin\nsystems has been accomplished.6Often these systems ex-\nhibit quasi-1D ferrimagnetic ground states with a net\nferromagnetic moment, so that apart from rich quan-\ntum phase diagrams they might be expected to provide\ngeneric examples of 1D magnetic-paramagnetic quantum\nphase transitions.7\nOn the experimental side, during the past two decades\nit has become possible to synthesize a large variety of\nquasi-1D materials with ferrimagnetic properties. Most\nof these materials are heterometallic molecular magnets\ncontainingdifferenttransitionmetalionsintheunitcell.8\nA generic spin model describing these materials is the\nquantum Heisenberg spin chain with antiferromagnetic\nnearest-neighbor exchange interactions and two types of\nalternating quantum spins with magnitudes S1andS2\n(S1> S2).9–11In the extreme quantum case of spins\n(1,1/2), the latter model was shown to provide an ex-cellent description of the thermodynamic parameters of\nthe recently synthesized quasi-1D bimetallic compound\nNiCu(pba)(D 2O)3·2D2O(pba=1,3-propylenebis).12An-\nother important class of quasi-1D ferrimagnets – the so-\ncalled topological ferrimagnets – is related to some ho-\nmometallic materials exhibiting composite chain struc-\ntures with different magnetic sublattices.13The ho-\nmometallic material A 3Cu3(PO4)4(A=Ca,Sr,Pb) is an\nexample of such quasi-1D ferrimagnets: In this com-\npound, the Cu2+ions form diamond chains with strongly\ncoupled trimersbridged by oxygenions.14Since quasi-1D\nhomometallic materials usually have rich exchange path-\nway structures, they may be expected to provide some\nreal examples of quasi-1D ferrimagnets with magnetic\nfrustration. To the best of our knowledge, the recently\nsynthesized mixed-valent magnetic material FeIIFeIII\n(trans-1,4-cyclohexanedicarboxylate)15provides the first\nreal example of a quasi-1D Heisenberg ferrimagnet with\nmagnetic frustration.15The experimentally established\nmagnetic structure for temperatures larger than 36 K\ncorresponds to the mixed-spin ladder with diagonal ex-\nchange bonds shown in Fig. 1, where the site spins\nS1= 5/2 andS2= 2 are respectively related to the\nmagnetic ions FeIIIand FeII.15\nThe mentioned experimental achievements motivated\na series of theoretical studies on quantum mixed-spin\nchains and ladders with geometric frustration. The\nsymmetric diamond chain with antiferromagnetic ver-\ntical bonds was probably the first studied model of a\n1D quantum ferrimagnet with competing interactions.16\nA variant of this model, the distorted spin-1/2 dia-\nmond chain, has received special theoretical17as well\nas experimental18interest due to its rich quantum\nphase diagram19and the relevance for the real mate-\nrial Cu 3(CO3)2(OH)2. The diamond Heisenberg chain2\nJ\nJ J\nJ1 23\n4σ\nσS1,2n\nS2,2n1,2n−1\n2,2n−1φ\nθ\nφ\nθz\nx\nFIG. 1: The mixed-spin ladder considered in the paper. The\narrows show the classical canted state described by the an-\ngles 0< φ < π/ 2 and 0< θ < π/ 2 for the classical spins with\nmagnitudes S1andS2, respectively. The other two classical\nphases correspond to spin configurations with ( φ,θ) = (0,0)\n(antiferromagnetic state) and ( φ,θ) = (π/2,π/2) (ferrimag-\nnetic state).\nis also one of the simplest quantum spin models admit-\nting four-spin cyclic exchange interactions.20A generic\nquantum spin model of a frustrated 1D ferrimagnet is\nthe mixed-spin Heisenberg chain composed of two types\nof alternating spins interacting via competing nearest-\nneighbor and next-nearest-neighbor antiferromagnetic\nexchange bonds.21This model may also be considered\nas a mixed-spin zigzag ladder and is a ferrimagnetic\nanalogue of the frustrated Heisenberg chain with ferro-\nmagnetic nearest-neighbor and antiferromagnetic next-\nnearest-neighbor exchange bonds. The spin-1/2 frus-\ntratedJ1−J2ferromagnetic chain has recently at-\ntracted much attention,22as it is supposed to describe\na number of quasi-1D edge-sharing cuprates, such as\nRb2Cu2Mo3O1223, Li2ZrCuO 4,24and LiCuVO 4.25The\nlatter material exhibits multiferroic properties26as well\nas an interesting specific phase transition in a magnetic\nfield from an ordered spiral to an ordered modulated-\ncollinear magnetic phases.27There are other two generic\ntypes of frustrated mixed-spin ladder models describing\ntwo interacting mixed-spin alternating chains. The first\none is the checkerboard mixed-spin Heisenberg ladder\nwith frustrating diagonal exchange couplings,28and the\nsecond one is the two-leg ladder model with two types of\nalternating rungs presented in Fig. 1. Finally, there has\nbeen a lot of recent work reporting interesting quantum\nphase diagrams in different composite Heisenberg chains\nwith ferrimagnetic ground states29\nIn this study we focus on the effects of frustration on\nthe ground state phase diagram of the mixed-spin ladder\nshown in Fig. 1. In addition to the theoretically inter-\nesting question of the effects of frustration in this sys-\ntem, an experimental realization of a closely related sys-\ntem in a mixed-valence iron polymer further motivates\nus.15In the next section we introduce the model and\nstudy some relevant properties of its Hamiltonian. In\nSection III, we give a detailed description of the quan-\ntum phases by using an effective Hamiltonian approach\ninspired by the analysis of data obtained using density-\nmatrix renormalization-group (DMRG) and exact diag-onalization (ED) techniques. We conclude in Section IV\nwith a brief summary of the results.\nII. THE MODEL\nThe system under consideration (see Fig. 1) consists of\ntwo equivalent mixed-spin Heisenberg chains (character-\nized by the nearest-neighbor exchange constant J3>0)\ncoupled via rung ( J1,J2>0) as well as diagonal( J4≥0)\nexchange bonds. The Hamiltonian of the system reads\nas\nH=H12+H3+H4, (1)\nwhere\nH12=L/2/summationdisplay\nn=1(J1s1,2n·s2,2n+J2σ1,2n−1·σ2,2n−1),\nH3=J3L/2/summationdisplay\nn=12/summationdisplay\nm=1[sm,2n·(σm,2n−1+σm,2n+1)],\nH4=J4L/2/summationdisplay\nn=1[s1,2n·(σ2,2n−1+σ2,2n+1)\n+s2,2n·(σ1,2n−1+σ1,2n+1)].\nHeresk,2nandσk,2n−1(k= 1,2) are, respectively, spin-\nS1and spin- S2operators ( S1> S2), andLis the number\nof rungs.\nIt is instructive to present the Hamiltonian in the fol-\nlowing form\nH=H12+L/2/summationdisplay\nn=1[Jss2n·(σ2n−1+σ2n+1)]+JaV ,(2)\nwhereJs,a= (J3±J4)/2, ands2n=s1,2n+s2,2nand\nσ2n+1=σ1,2n+1+σ2,2n+1are rung spin operators. The\noperator Vreads as\nV=L/2/summationdisplay\nn=1=L2n·(l2n−1+l2n+1), (3)\nwhereL2n=s1,2n−s2,2nandl2n±1=σ1,2n±1−σ2,2n±1\nare rung vector operators. The following analysis of the\nzero-temperature quantum phase diagram addresses the\nextreme quantum case of spins S1= 1 andS2= 1/2, and\nis mainly restricted to the parameter subspacedefined by\nJ1=J2=J3>0 andJ4≥0. To some extent, such a\nchoice of the parameters is motivated by the experimen-\ntally established strengths of the exchange couplings in\nthe ferrimagnetic ladder material FeIIFeIII(trans-1,4-\ncyclohexanedicarboxylate).15\nA. Symmetries of the model\nThe mixed-spin system inherits some important sym-\nmetries of the parent uniform-spin Heisenberg ladder3\nwith diagonal interactions.30First, if the parameters\nJ3andJ4inHare exchanged, one can recover the\noriginal Hamiltonian by exchanging either the spins on\ntheS1rungs (s1,2n←→s2,2n), or the spins on the\nS2rungs (σ1,2n−1←→σ2,2n−1). This means that\nH(J1,J2,J3,J4) =H(J1,J2,J4,J3). Therefore, the\nstudyofthemodelcanberestrictedintheregion J4/J3≤\n1 since the model with J4/J3>1 maps onto the one with\nJ4/J3<1. Because of the same symmetry, the Hamilto-\nnian (2) does not contain mixed products of rung spins\nand rung vector operators.\nThe second property of Hconcerns the subspace J3=\nJ4(Ja= 0), when the last term in Eq. (2) disappears.\nAs is the uniform-spin case,31in this parameter subspace\nthe Hamiltonian Hcommutes with the local operators\ns2\n2nandσ2\n2n−1(n= 1,2,...,L/2), which means that the\nrung spins s2nandσ2n−1[defined as s2\n2n=s2n(s2n+1)\nandσ2\n2n−1=σ2n−1(σ2n−1+1)] are good local quantum\nnumbers. Thus in every sector of the Hilbert space, de-\nfined by the sequence [ σ1,s2,...,σ L−1,sL], the first two\nterms in Eq. (2) reduce to the constant\nE0=−L\n2[J1S1(S1+1)+J2S2(S2+1)]\n+1\n2L/2/summationdisplay\nn=1[J1s2n(s2n+1)+J2σ2n−1(σ2n−1+1)].\nThus Eq. (2) takes the simple form of a Heisenberg spin\nchain\nH0=E0+L/2/summationdisplay\nn=1Jss2n·(σ2n−1+σ2n+1).(4)\nThe above expression for E0implies that for strong\nenough rung interactions ( J1/J3,J2/J3≫1) the sin-\nglet eigenstate of Eq. (4), defined as a product of local\nrung-singlet states, becomes an exact ground state of the\nmodel. This state belongs to the sector [0 ,0,...,0,0] and\ncanbeconsideredasaprototypeoftherung-singletphase\nof Eq. (2) discussed below. The following analysis of the\nquantum phase diagram of Eq. (2) implies that in the\nextreme quantum limit ( S1,S2) = (1,1/2) the sectors\n[1,1,...,1,1], [1,2,...,1,2], and [1 ,1,1,2,...,1,1,1,2]\nalso play an important role: In the first sector, the model\ndefined by Eq. (4) is equivalent to the spin-1 Haldane\nchain, whereas in the last two sectors Eq. (4) represents\nspin-alternating ferrimagnetic chains. The ground states\nrelated to these models appear in the quantum phase di-\nagram of the discussed system.\nB. Classical phase diagram\nThe classical phases of Eq. (1) can be described by the\nanglesφandθ(see Fig. 1) which determine the orienta-\ntions of the classical spins in the xzplane. We consider\nthe parameter subspace defined by J1=J2=J3= 1 andJ4≥0. The expression for the ground-state energy per\ncell containing two rungs is seen to be\nEc\nS1S2=−S1\nS2cos(2φ)−S2\nS1cos(2θ)\n−4cos(φ−θ)+4J4cos(φ+θ).(5)\nA minimization using the independent angle variables φ\nandθgives the following equations:\ncos(φ+θ) =c1\nκJ4−c2\ncos(φ−θ) =c2J4−c1κ, (6)\nwherec1=σ−σ−1,c2=σ+σ−1, andσ=S1/S2>1.\nThe parameter κ=κ(J4) readsκ= (4J2\n4/3−1/3)1/2.\nThe lower ( J(d)\n4) and the upper ( J(u)\n4) phase bound-\naries of the classical canted phase shown in Fig. 1 are\nrelated to the inequalities |cos(φ+θ)|,|cos(φ−θ)|≤1\nimplying\nJ(d)\n4=c2+1/radicalbig\n4(c2+1)2−3c2\n1,\nJ(u)\n4=c2−1/radicalbig\n4(c2−1)2−3c2\n1. (7)\n4φ\nJ(a)\nθ [deg]θ,φ 40 80\n 0\n 0.4  0.6  0.8  1\n4JM1\n21(b)Classical magnetizations2\n−MM = M + M\n 0.5 1 1.5\n 0.4  0.6  0.8  1 2\n 0\nFIG. 2: (a) The classical phase diagram described by the\nanglesφandθvs.J4, as obtained from Eq. (6) for the system\nwithS1= 1 and S2= 1/2. (b) z components of the classical\nmagnetizations in the S1(M1) andS2(M2) sites of the same\nsystem. The filled circles on the J4axis correspond to the\nclassical transition points J(d)\n4= 7/13 andJ(u)\n4= 1.\nForJ4< J(d)\n4, we get states of zero magnetization\nin which the two spins on any rung and spins along a\nleg are antiferromagnetically aligned. The canted state\nrealized for J(d)\n4< J4< J(u)\n4has a net magnetization\nthat takes a maximal value at some intermediate J4be-\ntween both boundaries [see Fig. 2(b)]. For J4> J(u)\n4\nthis classical canted phase gives way to a ferrimagnetic\nstate where all the spins of the same magnitude are fer-\nromagnetically aligned but the relative alignment of S1\nandS2is antiferromagnetic. Notice that the magnetic\nmeasurements in Ref. 15 indicate the discussed ferrimag-\nneticconfiguration–eventuallywith asmallcantingofthe\nclassical spins– as the most probable spin configuration\nrealized in the real material FeIIFeIII. ForS1= 1 and\nS2= 1/2, the above equations give J(d)\n4= 7/13≈0.538\nandJ(u)\n4= 1. For the real material studied in Ref. 15\n(S1= 5/2,S2= 2), one has J(d)\n4= 61/121≈0.504 and\nJ(u)\n4= 21/39≈0.553.4\n0 0.5 1 1.5−2−1.5−1−0.500.51\nJJJ J\n444 4Unit−cell correlations\n00.511.5012\n(1,2)(3,4) (1,4)\n(1,3)\n3\n41\n2c2c1 c3\nFIG. 3: Unit-cell isotropic spin-spin correlations as a fun c-\ntion of the frustration parameter J4, as obtained from the\nDMRG method for open boundary conditions ( L= 100).\nJc1\n4= 0.710,Jc2\n4= 0.875, and Jc3\n4= 0.975 are the special\npoints identified as phase-transition points between differ ent\nground states. The inset shows the difference in the spin-1\nrung correlations in two neighboring cells. Note that the pr e-\nsented spin-spin correlations belong to unit cells far from the\nends.\nInterestingly,thediscussedclassicalferrimagneticstate\nappears only for relatively small values of σ. For larger\nσ, the lowest energy collinear configuration for large J4\nis a non-magnetic state with ferromagnetically arranged\nlegs pointing in opposite directions (i.e., antiferromag-\nnetically aligned rungs). Comparing the energies of both\nconfigurations ( E(1)\nc=S2\n1+S2\n2−4S1S2−4S1S2J4,\nE(2)\nc=−S2\n1−S2\n2+ 4S1S2−4S1S2J4, respectively), we\nsee that the ferrimagnetic configuration is realized only\nin the interval 1 < σ≤2 +√\n3≈3.73. In the large σ\ncase, the cantedphaseis alsomodified: Onincreasingthe\nparameter J4fromJ(d)\n4up toJ(u)\n4, theS2spins smoothly\nchangetheirorientationby π, whereasthenetorientation\nof the larger S1spins coincides at the phase boundaries.\nIn both variants of the classical phase diagram the phase\nboundaries are defined by Eq. (7).\nFinally, the discussed classical phase diagramswere in-\ndependently confirmed by our classical Monte-Carlo sim-\nulations. Below we argue that the classical ferrimagnetic\nphase survives quantum fluctuations, whereas both the\nantiferromagnetic as well as the canted classical phases\nare completely destroyed.\nIII. QUANTUM PHASE DIAGRAM\nWe consider the parameter subspace defined by J1=\nJ2=J3≡1 and 0≤J4≤1.5, and use the DMRG\nmethod32foropen boundaryconditionssupplemented by\nED data for periodic clusters containing up to L= 140 0.5 1 1.5−2.2−2.0−1.8−1.6\nJJJJJ\n4444c1c2c34mE/La\nb\nRS HL2 1FF\nFIG. 4: Ground-state energy per rung as a function of the\nfrustration parameter J4(DMRG, L= 90). Jm\n4= 0.723\ndenotes the location of the maximum. The positions of the\nspecial points identified in Fig. 3 separate different ground\nstates: RS (rung-singlet), HL (Haldane-like ), and two dif-\nferent ferrimagnetic states ( F1andF2). The straight line ab\nrepresents the energy of the Haldane state ( |ΨH/angbracketright) defined by\nEH=/angbracketleftΨH|H|ΨH/angbracketright.\nrungs. DMRG is carried out for this system for a range\nof lattice sizes up to L= 100 rungs with the spin values\nS1= 1andS2= 1/2,respectively. Upto320densityma-\ntrix eigenvectors were retained. Depending on the value\nofJ4, the truncation errors are between 10−7and 10−12.\nThe DMRG results presented in Fig. 3 reveal three\nspecial points on the J4axis separating regions with\ndifferent characteristics of the short-range correlations:\nJc1\n4= 0.710,Jc2\n4= 0.875, and Jc3\n4= 0.975. The same\npoints are also presented in Fig. 4 which shows DMRG\nresults(L= 90)fortheground-stateenergyofthemixed-\nspinmodel (1). Adetailed numericalanalysis, usingboth\nthe DMRG and ED methods, predicts singlet ground\nstates in the entire region 0 ≤J4< Jc2\n4. ForJ4> Jc2\n4,\nthesameanalysissuggestsgroundstatescharacterizedby\nnet ferromagnetic moments. Below we argue that these\nspecial points are related to quantum phase transitions\nbetween different ground states.\nA. Mapping onto the frustrated spin-1/2 ladder\nAn inspection ofthe short-rangecorrelationspresented\nin Fig. 3 implies that the weight of the local rung quintet\n(i.e.,s2n= 2) states on the spin-1 rungs is negligible\nalmost in the whole interval 0 ≤J4< Jc2\n4. Indeed, by\nusing the identity ∝angbracketlefts1,2n·s2,2n∝angbracketright=/parenleftbig\n∝angbracketlefts2\n2n∝angbracketright−3/2/parenrightbig\n/2−5/4,\none finds that the following relation between the average\nrung correlations should be satisfied for any state with a\nzero weight of the rung quintet states:\n∝angbracketlefts1,2n·s2,2n∝angbracketright=∝angbracketleftσ1,2n−1·σ2,2n−1∝angbracketright−5\n4.(8)5\nAs seen from the numerical results, the above relation is\nalmost perfectly fulfilled in the entire region 0 ≤J4<\nJc2\n4, excluding some narrow vicinity of the point Jc2\n4\nwhere the correlations ∝angbracketlefts1,2n·s2,2n∝angbracketrightabruptly change to\n≈1. The extremely small contribution of the quintet\nrung states in the region 0 ≤J4< Jc2\n4can be explained\nby the peculiarities of the energy spectrum of the mixed-\nspin plaquette, where the lowest quintet state happens to\nbe well separated from the low-lying triplet and singlet\nstates. Note that the excitation of local quintet states is\ncontrolled by the last term ( V) in the Hamiltonian (2).\nThus, starting from an eigenstate belonging to the sector\ns2n,σ2n−1= 0,1 (n= 1,...,L/2), the first-order correc-\ntions to the wave function of this eigenstate will contain\nrelativelysmallamountofconfigurationsbelongingtothe\nsectors with local quintet states due to the larger energy\ndenominator in the perturbation expression.\nThese observations suggest, in particular, that in\nthe discussed region the ground-state properties of the\nmixed-spin system may be approximately interpreted by\nprojecting out the local quintet states in the mixed-spin\nHamiltonian (2). Up to first order in Ja, the projected\nHamiltonian reads as (see the Appendix)\nHeff=−5\n8JL+L/summationdisplay\nn=1/bracketleftBig\nJ′\n⊥σ1,n·σ2,n\n+J′\nsσn·σn+1+J′\naln·ln+1/bracketrightBig\n,(9)\nwhereσ1,nandσ2,nare spin-1/2 operators, σn=σ1,n+\nσ2,n,ln=σ1,n−σ2,n,J′\n⊥=J⊥,J′\ns=Js, andJ′\na=\n−2/radicalbig\n2/3Ja. For simplicity, we have restricted ourselves\nto the case of equal rung couplings ( J1=J2≡J⊥). The\neffective Hamiltonian (9) describes a frustrated spin-1/2\nHeisenbergladdercharacterizedbythreeparameters,i.e.,\nthe strength of the rung ( J′\n⊥), leg (J′\n3=J′\ns+J′\na), and\ndiagonal ( J′\n4=J′\ns−J′\na) exchange bonds. Using the same\nreasoning, it may be safely suggested that the next-order\ncorrections in Jado not change substantially the singlet\nground states, so that the effective Hamiltonian (9) may\nbe used (i) to identify the singlet ground states of the\noriginal Hamiltonian (2) in the region 0 ≤J4< Jc2\n4and\n(ii) to analyze the related quantum phase transitions.\nAs is well-known, as a function of the frustration pa-\nrameter J′\n4the model (9) exhibits the so-called rung-\nsinglet (RS) and Haldane-like(HL) phases.4,30,33–37Both\nground states are non-degenerate and exhibit finite\nsinglet-triplet gaps. The character of the quantum RS-\nHL transition in the weak-coupling limit is still under de-\nbate: Some of the cited works30,33,35,36suggest a direct\nfirst-order transition between these phases, but the oth-\ners predict an intermediate columnar dimer phase.4,34,37\nThus the mapping of Eq. (2) implies that the special\npointJ4=Jc1\n4can presumably be identified as a quan-\ntum phase transition point separating similar phases. Of\ncourse, such an analysis does not exclude the presence\nof some intermediate singlet phases in a tiny interval be-\ntween the RS and HL states. Some hints in this directioninspired by the DMRG results for the ground-state en-\nergy (Fig. 4) will be discussed below in more detail.\nThe established connection with the frustrated spin-\n1/2 ladder model is additionally supported by the fact\nthat the special point Jc1\n4perfectly maps on the RS-HL\nphase boundary in the phase diagram of the frustrated\nspin-1/2 ladder model.30Indeed, taking the parameters\ny1=J′\n⊥/J′\n3andy2=J′\n4/J′\n3used in Ref. 30, the estab-\nlished relations J′\ns=JsandJ′\na=−2/radicalbig\n2/3Jabetween\nthe parameters of the original and the projected Hamil-\ntonians take the form\ny1=J⊥/J3\nb2J4/J3−b1, y 2=b2−b1J4/J3\nb2J4/J3−b1,(10)\nwhereb1=/radicalbig\n2/3−1/2 andb2=/radicalbig\n2/3+1/2. Note that\nthe change of J4(at fixed J⊥=J3= 1) corresponds to a\nruninthe( y1,y2)planeonthe abline(seeFig.5)defined\nbyy2= (b1/b2+ 1)y1−b1/b2. Following Ref. 30, we\nmayidentifythepositionofthequantumphasetransition\nwith the point J4=Jc1\n4≡0.710 where the spin-1/2 rung\ncorrelations change their sign (see Fig. 3). We find that\nthe (y1,y2) image Aof the transition point Jc1\n4maps\nperfectly on the phase boundary in the ( y1,y2) plane. In\nFigure 5, we also show the symmetric point A′obtained\nby the coordinate transformations y1→y1/y2andy2→\n1/y2,whicharerelatedtotheexchangesymmetry J3←→\nJ4ofthe Hamiltonian. As expected, the symmetric point\nA′also lies on the phase boundary.\nA\nA'a\nbHL\nRS\ny1y\n2\nFIG. 5: Phase diagram of the effective spin-1/2 ladder\nmodel with diagonal bonds.30The point Awith coordinates\n(y1,y2) = (1.618,1.766) is the image of the special point\nJc1\n4= 0.710 obtained by using Eq. (10). The point A′is\nan image of Acorresponding to the symmetry transformation\nJ′\n3←→J′\n4.abis the path in the ( y1,y2) plane corresponding\nto the change of J4at fixedJ⊥=J3= 1.6\n0 0.2 0.4 0.6 0.8 1−2−1.5−1−0.500.5\nJ3Unit−cell correlations0 0.5 10.40.81.2(1,4)\n(1,3)\n(3,4)\n(1,2)−2.5−2−1,5\nSpin gap\n00.5 1E/L−1\nFIG. 6: Unit-cell isotropic spin-spin correlations of the m odel\n(1) as a function of J3(J4= 0,L= 100). The notations are\ndefined in Fig. 3. The inset on the left shows the ground-state\nenergy per rung vs.J3and the inset on the right shows the\nvariation of the singlet-triplet excitation gap with J3.\nB. Rung-singlet and Haldane-like phases\n1. Rung-singlet phase\nThe RS phase, originally studied in the two-leg\nspin-1/2 ladder without diagonal bonds,38,39is a non-\ndegenerate singlet state with a finite singlet-triplet gap.\nThe existence of a spin gap in this model can be eas-\nily anticipated by using a strong-coupling analysis:39For\nJ′\n3/J′\n⊥≪1, the ground state is a simple product of rung\nsinglet bonds. The lowest rung excited states are local\ntriplets with a characteristic gap ∝J′\n⊥which survives\nthe perturbation in J′\n3/J′\n⊥. On the other hand, the per-\nturbation produces an energy band (with a bandwidth\n∝J′\n3) of triplet excitations.\nThe same physics can be easily extracted from a\nstrong-couplinganalysisofthe mixed-spin ladder (2). In-\nsteadofdoingthis, wepresentinFig.6DMRGresultsfor\nthe short-range correlations as a function of J3(J4= 0).\nThe state at ( J3,J4) = (1,0) is known to be gapped.40\nThe essential information in Fig. 6 is that the curves are\ndevoid of any features that might suggest a change of the\nphase. Thus we can assert that the phase at J3= 1 is\nsmoothly connected to the phase at J3= 0, which is a\nRS phase. The variation of the gap with J4is shown\nin Fig. 7. We see that the gap goes to zero around the\npointJ4= 0.710 identified above as a phase transition\npoint to another singlet phase. Below we discuss in more\ndetail the structure of the low-lying excitations close to\nJ4=Jc1\n4.2. Haldane-like phase\nThe discussed mapping of Eq. (2) on the frustrated\nspin-1/2ladder model suggests that the HL phase should\noccupy some region in the phase diagram for J4> Jc1\n4.\nTo reveal the peculiarities of the suggested HL phase –\nas compared to the well-known Haldane phase of the pe-\nriodic spin-1 Heisenberg chain – notice that in the sector\n[1,1,...,1] the Haldane state |ΨH∝angbracketrightis the exact ground\nstate ofthe mixed-spin Hamiltonian(2) at the symmetric\npointJ3=J4. In the general case ( J3∝negationslash=J4), the energy\nof this state EH=∝angbracketleftΨH|H|ΨH∝angbracketrightreads as\nEH\nL=−J1\n2+J2\n8+1\n2(J3+J4)εH,(11)\nwhereεH=−1.40148403897(4) is the the ground-\nstate energy per bond of the periodic spin-1 Heisenberg\nchain.41Here, we have used the fact that the operator V\n[Eq. (3)] does not have non-zero matrix elements in the\nsector [1,1,...,1]: In particular, we have ∝angbracketleftΨH|V|ΨH∝angbracketright=\n0. The energy of the Haldane state EHas a function of\nJ4(J1=J2=J3= 1) is shown in Fig. 4 (the abline).\nInterestingly, at the special point J4=Jc2\n4≡0.875– also\nrelated to an abrupt change of the spin-1 rung correla-\ntions – the DMRG estimate for the ground-state energy\nof the Hamiltonian (2) E/L=−1.6899 almost coincides\nwith the energy of the Haldane state ( EH/L=−1.6889)\nobtained from Eq. (11). As already mentioned above,\nthe numerical analysis implies that the special point Jc2\n4\nis a quantum phase-transition point from a singlet non-\ndegenerate state to a state exhibiting a net magnetic mo-\nment. The above remarks suggest that the HL phase ap-\npears as a good candidate for the phase diagram of the\nmixed-spin model.\nFurther qualitative information about the characteris-\ntics of this phase can be extracted from a perturbative\nanalysis starting from the symmetric point J3=J4and\nbased on the Haldane state in a periodic spin-1 chain.\nNote that in some interval ( J4< Jc2\n4) the parameter Ja,\nwhich controls the Vterm in Eq. (2), may be used as\na small parameter (e.g., Ja= 0.0625 for J4= 0.875).\nThus, up to second order in Ja, the ground-state energy\ntakes the form E=EH−const(1−J4)2L, whereconst\nis some positive number of order one. Qualitatively, this\nresultreproducesthe behavioroftheground-stateenergy\nin the interval Jc1\n4< J4< Jc2\n4extracted from the DMRG\nanalysis (see Fig. 4). To some extent, this result also val-\nidates the choice of |ΨH∝angbracketrightas a starting unperturbed state.\nAs compared to the Haldane state, some peculiarities\nof the HL phase can be revealed by looking at the first-\norder correction in Jato the wave function |ΨH∝angbracketright,\n|Ψ∝angbracketright=|ΨH∝angbracketright+Ja/summationdisplay\nn/negationslash=0∝angbracketleftΨn|V|ΨH∝angbracketright\nE0−En|Ψn∝angbracketright+O/parenleftbig\nJ2\na/parenrightbig\n.(12)\nHere the sum runs over the excited eigenstates |Ψn∝angbracketrightof\nthe Hamiltonian (2) at J3=J4, andE0≡EH. The\nmatrix elements of V(see the Appendix) admit only two7\ntypes of excited states ( |Ψ1,2∝angbracketright) defined, respectively, in\nthe sectors [1 ,...,1,0,0,1,...,1] (two neighboring rungs\nin singlet states) an [1 ,1,...,1,2,0,1,...,1] (one rung\nin a a quintet state an a neighboring rung in a singlet\nstate). The weightsof both types of defect configurations\nin the HL state change in the interval Jc1\n4< J4< Jc2\n4:\nWhile the weight of the |Ψ1∝angbracketrightconfigurations grows in a\nregion around the transition point Jc1\n4, the|Ψ2∝angbracketrightcon-\nfigurations (containing spin-2 defects) become visible in\nthe DMRG result for the spin-1 rung correlations only\nin a short interval preceding the transition to a mag-\nnetic state (see Fig. 3). Note that the observed increase\nof the weight of the |Ψ2∝angbracketrightconfigurations formally con-\ntradicts the perturbation result in Eq. (12), which pre-\ndicts the opposite behavior. A reasonable resolution for\nthis is provided by the guess that close to the transi-\ntion point Jc2\n4some of the eigenenergies Enrelated to\nthe sector [1 ,1,...,1,2,0,1...,1] soften. As of now we\ndo not have firm numerical results in favor of such a sug-\ngestion, although some preliminary DMRG results, using\nopen boundary conditions, seem to predict strong reduc-\ntions of the singlet-quintet and triplet-quintet gaps close\ntoJc2\n4.\n3. The RS-HL transition\nTurning to the region around the transition point Jc1\n4,\nit is instructive to comment on our numerical results\nfor the excitation gaps (Fig. 7) in the light of the dis-\ncussed mapping to the spin-1/2 ladder model. For the\nlatter model, it has been numerically established30that\n(i) the lowest state above the singlet ground states close\nto the phase boundary is a singlet excitation and (ii)\nthe low-lying triplet excitations are gapped in the whole\nregion of the phase diagram in Fig. 5, including the\nphase-transition boundary. Such a structure of the low-\nlying excitations is consistent with the established first-\norder quantum phase transition, which is described as\na level crossing of two singlet ground states. As al-\nready mentioned, the character of the RS-HL transition\nin the weak-coupling limit ( J′\n⊥,J′\n4≪J′\n3) is still under\ndebate.4,36,37As a matter of fact, there are some indica-\ntions for a second-order RS-HL transition4and an inter-\nmediate dimer phase34,37, but the debate concerns only\nthe weak-coupling part of the phase boundary. Looking\nat the coordinates of the AandA′images of the tran-\nsition point Jc1\n4(Fig. 5), it is clearly seen that the dis-\ncussed RS-HL transition at J4=Jc1\n4does not belong to\nthe weak-coupling region. Hence, one may expect a first-\norder RS-HL transition at Jc1\n4related to a level crossing\nof singlet ground states.\nFigure 7 presents our numerical (DMRG and ED) re-\nsults for the singlet (∆ s) and triplet (∆ t) gaps of the\nlowest excited modes above both singlet ground states.\nLet us first discuss the ED data for the gaps. As clearly\nseen, both minima, related to the ∆ sand ∆ tdata points,\nare located close to the expected transition point at 0 0.2 0.4 0.6 0.8 1\n 0 0.2  0.4  0.6 0.8 Gaps \n J4∆\n∆tsED\nED(L = 14)(L = 12)DMRG\nED(L = 12)\nFIG. 7: DMRG and ED numerical results for the singlet-\nsinglet (∆ s) and singlet-triplet (∆ s) excitation gaps in the\nmixed-spin model (1) vs.J4. The DMRG data points corre-\nspond to extrapolated values of ∆ sobtained by a polynomial\nfit (up to L= 90) for open boundary conditions. The ED\ndata concerns periodic clusters with L= 10,12,14.\nJ4= 0.710. More importantly, an extrapolation of the\nED data for J4= 0.710 implies that the ∆ spoints scale\nto smaller values than ∆ t. This observation is consistent\nwith the expected low-energy structure close the first-\norder transition point between the RS and HL phases.\nTurning to the DMRG results for ∆ t(J4), one observes\nthat the triplet gap of the RS phase takes very small val-\nues close to the suggested transition point ( J4= 0.710).\nWe could not conclusively exclude the possibility of a\ngapless triplet excitation at the transition point. In any\ncase, such a behavior indicates some peculiarities of the\nRS-HL transition in the mixed-spin system, as compared\nto the uniform-spin case. Another issue to be noticed\nis the steep (but definitely finite) slope of the function\n∆t(J4) at the transition point. This suggests a relatively\nlarge correlation length of this triplet excitation close to\nJc1\n4.\nC. Ferrimagnetic phases\nLooking at the DMRG results for the short-range cor-\nrelations(Fig. 3), it is easy to realize that a ferrimagnetic\nphase, closely related to the ferrimagnetic ground state\nof an antiferromagnetic Heisenberg chain with alternat-\ning (2,1) spins, is stabilized around the symmetric point\nJ4= 1. Exactly at J4= 1, the ground state of the\nHamiltonian (2) belongs to the sector [1 ,2,...,1,2], so\nthat both models are equivalent in the low-energy sector\nof the spectrum. The discussed ferrimagnetic phase ( F1)\nexhibits the magnetic moment per rung M0= 1/2 and8\n020406080100−0.400.40.8\nSite indexLocal magnetisation(a)\n020406080100−0.8−0.400.40.8\nSite indexSpin−1 rung correlations(b)\nFIG. 8: (a) The local magnetizations /angbracketleftsz\n1,2n/angbracketrightand/angbracketleftσz\n1,2n+1/angbracketright\n(n= 1,...,50) along the first leg as a function of the site\nindex. The data shown is for J4= 1.55. (b) The spin-1\nrung correlations along the length of the ladder ( L= 100) at\nJ4= 0.90. The values show a clear alternation between ≈1\nand≈−1 which indicates a two sublattice structure and a\ndoubled unit cell containing four rungs.\nsurvives almost in the entire region after Jc2\n4, excluding\nsome narrow interval in the vicinity of the latter point.\nThis is also seen in Fig. 8(a) which shows a typical be-\nhavior of the local magnetizations ∝angbracketleftsz\n1,2n∝angbracketrightand∝angbracketleftσz\n1,2n+1∝angbracketright\n(n= 1,...,L/2) along the first leg at J4= 1.55. The\nvalues of the spin-1 and spin-1/2 magnetic moments are\n0.866950 and−0.366950, respectively. We see that the\nsum of the local magnetic moments is 1 /2, as expected in\na Lieb-Mattis type ferrimagnetic state with a quantized\nmagnetic moment per rung M0= 1/2. The deviations at\nthe end are essentially because of open boundary condi-\ntions. We have verified numerically that these values do\nnot change much after J4= 1.\nFor the region close to Jc2\n4, the DMRG results pre-\nsented in Fig. 8(b) demonstrate the appearance of an-\nother ferrimagnetic phase ( F2) in a narrow range of J4\nstarting from the transition point Jc2\n4= 0.875 and ter-\nminating at Jc3\n4= 0.975. The F2phase is characterized\nby the magnetic moment per rung M0= 1/4. As clearly\nseen in Fig. 8(b), in the F2phase the space variation of\nthe spin-1 rung correlationsfollow strictly the periodicity\nof the spin structure in the sector [2 ,1,1,1,...,2,1,1,1].\nSuchabreakingofthetranslationalsymmetryisalsoseen\nin the inset of Fig. 3, where on the vertical axis we have\nplotted the magnitude of the difference ofthe spin-1 rung\ncorrelations in two neighboring unit cells for all values of\nJ4. Clearly, the F2phase represents a two-fold degen-\nerate ground state, which is invariant under the transla-\ntion by two lattice periods. Our numerical analysis does\nnot support the appearance of ferrimagnetic phases with\nlarger periods.\nIV. CONCLUSION\nIn conclusion, we have analyzed the combined effect\nof the quantum fluctuations and the competing inter-\nactions in a mixed-spin ladder composed of spin-1 and\nspin-1/2 rungs which is closely related to a recently syn-\nthesized quasi-1D ferrimagnetic material. A comparisonof the classical and quantum phase diagrams reveals the\nfollowing changes in the related quantum system. As ex-\npected, the classical ferrimagnetic phase also presents in\nthe quantum phase diagram, but there appears another\ntwo-fold degenerate ferrimagnetic state which breaks the\ntranslational symmetry. As may be expected, the clas-\nsical N´ eel state does not survive quantum fluctuations.\nMore interestingly, the classical canted state also com-\npletely disappears. This is in contrast to some other 1D\nspin systems exhibiting classical canted states,6where\nthis type of classical magnetic order partially survives\nquantum fluctuations. In the present case, both the clas-\nsicallong-rangeorderedstatesarereplacedbytwosinglet\nnon-degenerate gapped states (RS and HL).\nTurning to the weakly frustrated region, it has been\nestablished that the behavior of the system strongly re-\nsemblesthat ofatwo-legspin-1/2Heisenbergladderwith\nfrustrating diagonal interactions. However, concerning\nthe quantum phase transition between the RS and HL\nphases, we have found a few indications demonstrating\nsome peculiarities (such as the extremely small triplet\ngap at the transition point) of the mixed-spin system.\nThese issues deserve further investigations.\nFinally, although the available experimental results on\nthe ferrimagnetic ladder material FeIIFeIII(trans-1,4-\ncyclohexanedicarboxylate) seam to point toward the re-\nalization of the F1ferrimagnetic state,15a detailed com-\nparison with the experiment requires a more extensive\nanalysis of the quantum phase diagram including, e.g.,\ndifferent rung couplings J1∝negationslash=J2, different pairs of rung\nspin magnitudes, and some anisotropies. Concerning the\ncondition J3= 1, as shown in Fig. 5 it simply restricts\nthe path in the more general parameter space ( J3∝negationslash= 1) to\na straight line crossing one and the same phase bound-\nary. Therefore, there should be a relatively large region\nwithJ3∝negationslash= 1 showing the same structure of the phase\ndiagram. As to the second restriction ( J1=J2), its re-\nmoval may be generally expected to bring new quantum\nspin phases. However, in both cases we have numerically\nchecked that relatively small deviations from the condi-\ntionsJ1=J2=J3do not bring qualitative changes on\nthe established quantum phase diagram.\nAcknowledgments\nThis work has been supported by the Bulgarian Sci-\nence Foundation (Grant DO02-264/18.12.08). J. R. is\nalso indebted to the DFG for financial support (project\nRI615/16-1). V. R. C. thanks the MPIPKS in Dresden\n(Germany) for financial support and computational re-\nsources for most of the duration of the project and ac-\nknowledges being supported in part at the Technion by\na Fine Tust when the manuscript was being finalized.\nHe thanks Andreas L¨ auchli and Masaaki Nakamura for\nuseful discussions.9\nAppendix: Projection onto the spin-1/2 ladder\nWe have to project the spin-1 rung states onto the\nstates of the spin-1/2 rungs. To this end, we use the\nprojection operator P=P1P2,...,P L, where the rung\nprojection operator Pnreads as\nPn=/summationdisplay\nα|Tα\n2n∝angbracketright∝angbracketleftTα\n2n|, α= 0,x,y,z. (A.1)\nHere|T0\n2n∝angbracketrightdenotes the singlet state of the 2 nth spin-1\nrung and|Tk\n2n∝angbracketright= (i/√\n2)ǫklm|l∝angbracketright|m∝angbracketrightare the triplet states\nof the same rung in a vector basis which is a tensor prod-\nuct of the vector bases of the spin-1 objects (i.e., |x∝angbracketright,|y∝angbracketright,\nand|z∝angbracketright). In the following, the Greek indices take the\nvalues 0,x,y, andz, whereas the Latin ones – x,y, and\nz.\nUptofirstorderin Ja, theprojectedHamiltonianreads\nas\nHeff=PHP. (A.2)By using the expressions for the matrix elements\n∝angbracketleftTm\n2n|s2\n2n|Tn\n2n∝angbracketright= 2δmn,∝angbracketleftT0\n2n|Lk\n2n|T0\n2n∝angbracketright=∝angbracketleftTm\n2n|Lk\n2n|Tl\n2n∝angbracketright=\n0, and∝angbracketleftTm\n2n|Lk\n2n|T0\n2n∝angbracketright=−2/radicalbig\n2/3δmk, one obtains\nPns2\n2nPn= 2/summationdisplay\nk|Tk\n2n∝angbracketright∝angbracketleftTk\n2n|=σ2\n2n,(A.3)\nwhereσ2nis an effective rung-1/2 spin operator, and\nPnVnPn=\n−2/radicalbigg\n2\n3/summationdisplay\nk/bracketleftbig\n|T0\n2n∝angbracketright∝angbracketleftTk\n2n|+|Tk\n2n∝angbracketright∝angbracketleftT0\n2n|/bracketrightbig/parenleftbig\nlk\n2n−1+lk\n2n+1/parenrightbig\n.\nNote that the operator in the square brackets is an ef-\nfectivel2nrung vector operator for spin-1/2 rungs. Sum-\nming the above results, we obtain the effective spin-1/2\nladder model presented in Eq. (9).\n1Frustrated Spin Systems , edited by H. T. 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B 48, 3844 (1993)."    },    {        "title": "1410.4313v2.Vigorous_thermal_excitations_in_a_double_tetrahedral_chain_of_localized_Ising_spins_and_mobile_electrons_mimic_a_temperature_driven_first_order_phase_transition.pdf",        "content": "arXiv:1410.4313v2  [cond-mat.stat-mech]  6 Feb 2015APS/123-QED\nVigorous thermal excitations in a double-tetrahedral chai n of localized Ising spins\nand mobile electrons mimic a temperature-driven first-orde r phase transition\nLucia G´ alisov´ a1,∗and Jozef Streˇ cka2,†\n1Department of Applied Mathematics and Informatics, Facult y of Mechanical Engineering,\nTechnical University, Letn´ a 9, 042 00 Koˇ sice, Slovak Repu blic\n2Department of Theoretical Physics and Astrophysics, Facul ty of Science,\nP. J.ˇSaf´ arik University, Park Angelinum 9, 040 01, Koˇ sice, Slo vak Republic\n(Dated: May 28, 2022)\nA hybrid spin-electron system defined on one-dimensional do uble-tetrahedral chain, in which the\nlocalized Ising spin regularly alternates with two mobile e lectrons delocalized over a triangular pla-\nquette, is exactly solved with the help of generalized decor ation-iteration transformation. It is shown\nthat a macroscopic degeneracy of ferromagnetic and ferrima gnetic ground states arising from chiral\ndegrees of freedom of the mobile electrons cannot be lifted b y a magnetic field in contrast to a\nmacroscopic degeneracy of the frustrated ground state, whi ch appears owing to a kinetically-driven\nfrustration of the localized Ising spins. An anomalous beha vior of all basic thermodynamic quanti-\nties can be observed on account of massive thermal excitatio ns, which mimic a temperature-driven\nfirst-order phase transition from the non-degenerate frust rated state to the highly degenerate ferri-\nmagnetic state at non-zero magnetic fields. A substantial di fference in the respective degeneracies\nis responsible for an immense low-temperature peak of the sp ecific heat and very abrupt (almost\ndiscontinuous) thermal variations of the entropy and subla ttice magnetizations.\nPACS numbers: 05.50.+q, 75.10.Jm, 75.10.Pq, 75.30.Kz, 75. 40.Cx, 75.30.Sg\nKeywords: spin-electron chain, spin frustration, first-or der phase transition, magnetization plateau, chirality\nI. INTRODUCTION\nExactlysolvable models areof greatimportance in sta-\ntistical physics because they offer a valuable insight into\ndiverse aspects of quantum, cooperative and critical phe-\nnomena [1–3]. It is worthwhile to remark that an exact\nsolvability of the most famous lattice-statistical models\nis usually restricted to one dimension only, while the list\nof two- and three-dimensional rigorously solved models\nis much more limited [2]. This fact closely relates to a\nrather intricate nature of mathematical treatment, which\nmust be employed in seeking an exact solution of even\nrelatively simple interacting many-body systems [4]. A\nparticularly fruitful idea for suggesting novel exactly sol-\nuble models with peculiar quantum manifestations con-\nsists in linking relatively small quantum systems through\nclassical Ising spins. To get a closed-form exact solu-\ntion for these hybrid classical-quantum models one may\ntake advantage of generalized algebraic transformations,\nwhich establish a rigorous mapping correspondence with\na simpler (fully classical) lattice-statistical model with\nthe known exact solution [5–8].\nUntil recently, the concept of algebraic mapping trans-\nformations has been widely applied mainly to the Ising-\nHeisenberg spin systems, which are composed of small\nclusters of quantum Heisenberg spins coupled together\nthrough classical Ising spins only (see, e.g., Refs. [8–12]\nandreferencestherein). However,it hasbeen shownlater\n∗Electronic address: galisova.lucia@gmail.com\n†Electronic address: jozef.strecka@upjs.skon that this conceptually simple approach is also appli-\ncable for spinless fermion models when ignoring the hop-\nping term on particular lattice sites [13, 14], or for hy-\nbrid spin-electron systems, where finite clusters includ-\ning a few mobile electrons are mutually inter-connected\nthrough the localized Ising spins in order to form either\none- [15–21] or two-dimensional [22–25] lattice.\nIn the present work, we will propose and exactly solve\nthe hybrid spin-electron system on a double-tetrahedral\nchain in a magnetic field. To achieve an exact solv-\nability of this model, we will suppose that the local-\nized Ising spins placed at nodal lattice sites regularly\nalternate with triangular plaquettes available to mobile\nelectrons. It is worth mentioning that the geometry of\ndouble-tetrahedral chain was theoretically introduced by\nMambrini et al.[26]whenexaminingtheresidualentropy\nand spin gap in the respective Heisenberg model. Since\nthat time, several other models with this lattice geome-\ntry havebeen discussed in literature, namely, the spinless\nfermion model [14], the Heisenberg and Hubbard models\n[27–30] and the Ising-Heisenberg model [31, 32]. A pos-\nsible experimental realization of the double-tetrahedral\nchain is realized in the copper-based polymeric chain\nCu3Mo2O9[33–35].\nThe outline of this paper is as follows. In Sec. II\nwe will describe in detail the investigated spin-electron\ndouble-tetrahedral chain and then, the most important\nsteps of an exact mapping method will be clarified. In\nSec.III wewill discussthe most interestingresultsforthe\ngroundstate, themagnetizationprocessandtemperature\ndependences of basic thermodynamic quantities (magne-\ntization, entropy, specific heat). The paper ends up with\na brief summary of our findings in Sec. IV.2\nFIG. 1: (Color online) A part of the spin-electron system on\na double-tetrahedral chain. Full circles denote nodal latt ice\nsites occupied by the localized Ising spins, while the empty\ncircles forming triangular plaquettes are available to mob ile\nelectrons.\nII. SPIN-ELECTRON\nDOUBLE-TETRAHEDRAL CHAIN\nLetusconsidertheone-dimensionaldouble-tetrahedral\nchain, in which one localized Ising spin placed at nodal\nlattice site regularly alternates with a triangular plaque-\ntte consisting of three equivalent lattice sites available to\ntwo mobile electrons (see Fig. 1). This one-dimensional\nspin-electron system may alternatively be viewed as the\nspin-1/2 Ising linear chain, the bonds of which are dec-\norated by triangular plaquettes available to two mobile\nelectrons. From this perspective, the total Hamiltonian\ncan be defined as a sum over cluster Hamiltonians Hk:\nH=N/summationdisplay\nk=1Hk, (1)\nwhereas each cluster Hamiltonian Hkinvolves all the in-\nteraction terms connected to the mobile electrons from\nthekth triangular plaquette:\nHk=−t/summationdisplay\nα=↑,↓(c†\nk1,αck2,α+c†\nk2,αck3,α+c†\nk3,αck1,α+h.c.)\n+J\n2(σz\nk+σz\nk+1)3/summationdisplay\nj=1(nkj,↑−nkj,↓)+U3/summationdisplay\nj=1nkj,↑nkj,↓\n−HI\n2(σz\nk+σz\nk+1)−He\n23/summationdisplay\nj=1(nkj,↑−nkj,↓).(2)\nAbove,c†\nkj,αandckj,αrepresent usual fermionic creation\nand annihilation operators for mobile electrons from the\nkth triangular plaquette with spin α=↑or↓,nkj,α=\nc†\nkj,αckj,αis the respective number operator, σz\nk=±1/2\nlabels the Ising spin placed at the kth nodal lattice site\nandNdenotes the total number of nodal lattice sites.\nThe hopping parameter t>0 takes into account the ki-\nnetic energy of mobile electrons delocalized over trian-\ngular plaquettes, U≥0 represents the on-site Coulomb\nrepulsion between two electrons of opposite spins occu-\npying the same lattice site and Jstands for the Ising\ncoupling between the mobile electrons and their nearest\nIsing neighbors. Finally, HIandHeare the Zeeman’s\nterms accounting for the magnetostatic energy of the lo-\ncalized Ising spins and mobile electrons in a presence of\nthe external magnetic field.A crucial step of our calculations lies in the evaluation\nofthepartitionfunctionfortheinvestigatedspin-electron\ndouble-tetrahedralchain. Withregardtoavalidityofthe\ncommutation relation between different cluster Hamilto-\nnians [Hk,Hl] = 0 (k/ne}ationslash=l), the partition function Zcan\nbe partially factorized into a product of cluster partition\nfunctions Zk:\nZ=/summationdisplay\n{σk}N/productdisplay\nk=1Trke−βHk=/summationdisplay\n{σk}N/productdisplay\nk=1Zk,(3)\nwhereβ= 1/Tis the inverse temperature (we set\nkB= 1), the symbol/summationtext\n{σk}denotes a summation overall\npossible states of the localized Ising spins and the symbol\nTrklabels a trace over degrees of freedom of two mobile\nelectrons from the kth triangular plaquette. The clus-\nter partition function Zkcan be subsequently acquired\nby a diagonalization of the cluster Hamiltonian (2). The\nrelevant calculation is easy to accomplish in a matrix\nrepresentation of the Hilbert subspace corresponding to\nthe cluster Hamiltonian (2), which is spanned over the\nfollowing orthonormal basis of the electron states:\n|ψk/an}bracketri}ht={c†\nk1,↑c†\nk2,↑|0/an}bracketri}ht,c†\nk2,↑c†\nk3,↑|0/an}bracketri}ht,c†\nk3,↑c†\nk1,↑|0/an}bracketri}ht,\nc†\nk1,↓c†\nk2,↓|0/an}bracketri}ht,c†\nk2,↓c†\nk3,↓|0/an}bracketri}ht,c†\nk3,↓c†\nk1,↓|0/an}bracketri}ht,\nc†\nk1,↑c†\nk1,↓|0/an}bracketri}ht,c†\nk2,↑c†\nk2,↓|0/an}bracketri}ht,c†\nk3,↑c†\nk3,↓|0/an}bracketri}ht,\nc†\nk1,↑c†\nk2,↓|0/an}bracketri}ht,c†\nk2,↑c†\nk3,↓|0/an}bracketri}ht,c†\nk3,↑c†\nk1,↓|0/an}bracketri}ht,\nc†\nk1,↓c†\nk2,↑|0/an}bracketri}ht,c†\nk2,↓c†\nk3,↑|0/an}bracketri}ht,c†\nk3,↓c†\nk1,↑|0/an}bracketri}ht}.(4)\n(|0/an}bracketri}htlabels the vacuum state). A straightforward diago-\nnalization of the cluster Hamiltonian (2) in the relevant\nHilbert subspace gives fifteen eigenenergies:\nE1,2=−hI+he−t,\nE3,4=−hI−he−t,\nE5,6=−hI±he+2t,\nE7,8=−hI−t,\nE9=−hI+2t,\nE10,11=−hI+1\n2/bracketleftBig\nt+U+/radicalbig\n(U−t)2+8t2/bracketrightBig\n,\nE12,13=−hI+1\n2/bracketleftBig\nt+U−/radicalbig\n(U−t)2+8t2/bracketrightBig\n,\nE14,15=−hI−1\n2/bracketleftBig\n2t−U±/radicalbig\n(U+2t)2+32t2/bracketrightBig\n.(5)\nHere, we have introduced the following notation hI=\nHI(σz\nk+σz\nk+1)/2 andhe=J(σz\nk+σz\nk+1)−Hein order\nto write the eigenvalues (5) in a more abbreviated form.\nThe complete set ofthe eigenvalues(5) allowus to obtain\nthe resultingexpressionfor the clusterpartition function:\nZk=15/summationdisplay\nj=1e−βEj= eβhI/braceleftBig/parenleftbig\n2eβt+e−2βt/parenrightbig\n[1+2cosh(βhe)]\n+4e−βt/2−βU/2cosh/bracketleftbiggβ\n2/radicalbig\n(U−t)2+8t2/bracketrightbigg3\n+2eβt−βU/2cosh/bracketleftbiggβ\n2/radicalbig\n(U+2t)2+32t2/bracketrightbigg/bracerightBig\n.(6)\nIt should be pointed out that the cluster partition func-\ntion (6) still depends through the newly defined param-\netershIandheon the Ising spins σkandσk+1attached\nto the mobile electrons from the kth triangular plaque-\ntte. Next, one can perform the generalized decoration-\niteration mapping transformation [5–8]:\nZk=15/summationdisplay\nj=1e−βEj= eβhI/braceleftBig/parenleftbig\n2eβt+e−2βt/parenrightbig\n[1+2cosh(βhe)]\n+4e−βt/2−βU/2cosh/bracketleftbiggβ\n2/radicalbig\n(U−t)2+8t2/bracketrightbigg\n+2eβt−βU/2cosh/bracketleftbiggβ\n2/radicalbig\n(U+2t)2+32t2/bracketrightbigg/bracerightBig\n=Aexp/bracketleftbig\nβJeffσz\nkσz\nk+1+βHeff(σz\nk+σz\nk+1)/2/bracketrightbig\n,(7)\nwhich provides an exact mapping relation between\nthe partition function Zof the spin-electron double-\ntetrahedral chain and, respectively, the partition func-\ntionZICof the spin-1 /2 Ising chain with the effective\nnearest-neighbor coupling Jeffand the effective magnetic\nfieldHeffafter substituting Eq. (7) into Eq. (3):\nZ(β,J,t,U,H I,He) =ANZIC(β,Jeff,Heff).(8)\nThe mapping parameters A,JeffandHeffemerging in\nEq. (8) can be obtained from the ’self-consistency’ con-\ndition ofthe applied decoration-iterationtransformation:\nA=4/radicalBig\n(W−+W)(W++W)(W0+W)2,\nJeff=Tln/bracketleftbigg(W−+W)(W++W)\n(W0+W)2/bracketrightbigg\n,\nHeff=HI+Tln/parenleftbiggW−+W\nW++W/parenrightbigg\n, (9)\nand the functions W∓,W0andWare defined as:\nW∓=/parenleftbig\n2eβt+e−2βt/parenrightbig\n[1+2cosh( βJ∓βHe)],\nW0=/parenleftbig\n2eβt+e−2βt/parenrightbig\n[1+2cosh( βHe)],\nW= 4e−βt/2−βU/2cosh/bracketleftbiggβ\n2/radicalbig\n(U−t)2+8t2/bracketrightbigg\n+2eβt−βU/2cosh/bracketleftbiggβ\n2/radicalbig\n(U+2t)2+32t2/bracketrightbigg\n.(10)\nNote that the partition function of the spin-1 /2 Ising\nchain in a magnetic field has exactly been calculated us-\ning the transfer-matrix method [1, 36]. From this point\nof view, an exact calculation of the partition function\nof the spin-electron double-tetrahedral chain is also for-\nmally completed.\nExact results for other thermodynamic quantities fol-\nlow directly from the mapping relation (8). Actually,the Gibbs free energy Gof the spin-electron double-\ntetrahedral chain takes the form:\nG=−TlnZIC−NTlnA, (11)\nwhich can be further used for the calculation of the en-\ntropySand the specific heat C:\nS=−/parenleftbigg∂G\n∂T/parenrightbigg\nH, C=−T/parenleftbigg∂2G\n∂T2/parenrightbigg\nH,(12)\nas well as the sublattice magnetizations mIandmenor-\nmalized per one localized Ising spin and mobile electron,\nrespectively:\nmI=−1\nN/parenleftbigg∂G\n∂HI/parenrightbigg\n, me=−1\n2N/parenleftbigg∂G\n∂He/parenrightbigg\n.(13)\nIn view of this notation, the total magnetization nor-\nmalized per one magnetic particle of the spin-electron\ndouble-tetrahedral chain can be expressed as:\nm=1\n3(mI+2me). (14)\nIII. RESULTS AND DISCUSSION\nIn this section, we will proceed to a discussion of the\nmost interesting results for the investigated spin-electron\ndouble-tetrahedral chain by considering the particular\ncase with the antiferromagnetic interaction J >0 be-\ntween the localized Ising spins and mobile electrons. To\nreduce the total number of free interaction parameters,\nwe will assume equal magnetic fields acting on the Ising\nspins and mobile electrons, i.e. HI=He≡H≥0.\nA. Ground state\nTo get the ground state of the investigated spin-\nelectron model, it is sufficient to find the lowest-energy\neigenstate of the cluster Hamiltonian (2) that can be\nsimply extended to the whole double-tetrahedral chain\ndue to the commuting character of the cluster Hamilto-\nnians. The lowest-energy eigenstate can be obtained by\ninspection from the full spectrum of eigenvalues (5) of\nthe cluster Hamiltonian (2) after taking into account all\nfour available states of two nodal Ising spins σkandσk+1\ninvolved therein. In this way, one finds three different\nmacroscopically degenerate ground states: the ferromag-\nnetic (FM) state, the ferrimagnetic (FRI) state and the\nfrustrated (FRU) state, which are unambiguously char-\nacterized by the following eigenvectors and energies:\n|FM/an}bracketri}ht=N/productdisplay\nk=1|↑/an}bracketri}htσk⊗/braceleftbigg|φ+\n↑/an}bracketri}htk\n|φ−\n↑/an}bracketri}htk,\nEFM=N\n2(2J−3H−2t); (15)4\n|FRI/an}bracketri}ht=N/productdisplay\nk=1|↓/an}bracketri}htσk⊗/braceleftbigg|φ+\n↑/an}bracketri}htk\n|φ−\n↑/an}bracketri}htk,\nEFRI=−N\n2(2J+H+2t); (16)\n|FRU/an}bracketri}ht=/braceleftBiggN/productdisplay\nk=1|↑/an}bracketri}htσk\n|↓/an}bracketri}htσk/bracerightBig\n⊗ |φ0/an}bracketri}htk,H= 0\nN/productdisplay\nk=1|↑/an}bracketri}htσk⊗ |φ0/an}bracketri}htk,H >0\nEFRU=N\n2/parenleftBig\nU−H−2t−/radicalbig\n(U+2t)2+32t2/parenrightBig\n.(17)\nIn above, the product runs over all primitive unit cells,\nthe state vector |↑/an}bracketri}htσk(|↓/an}bracketri}htσk) determines up (down) state\nof thekth localized Ising spin σz\nk= 1/2 (σz\nk=−1/2).\nThe state vectors |φ+\n↑/an}bracketri}htkand|φ−\n↑/an}bracketri}htkemerging in Eqs. (15)\nand (16) label two eigenstates of the mobile electrons\nwith a positive and negative chirality:\n|φ+\n↑/an}bracketri}htk=1√\n3(c†\nk1,↑c†\nk2,↑+ωc†\nk2,↑c†\nk3,↑+ω2c†\nk3,↑c†\nk1,↑)|0/an}bracketri}ht,\n|φ−\n↑/an}bracketri}htk=1√\n3(c†\nk1,↑c†\nk2,↑+ω2c†\nk2,↑c†\nk3,↑+ωc†\nk3,↑c†\nk1,↑)|0/an}bracketri}ht,\nω= e2πi/3(i =√\n−1), (18)\nwhile the last eigenstate of the mobile electrons |φ0/an}bracketri}htk\nappearing in Eq. (17) refers to a non-chiral state with a\nzero current:\n|φ0/an}bracketri}htk=1√\n6/bracketleftbig\nsinϕ(c†\nk1,↑c†\nk2,↓+c†\nk2,↑c†\nk3,↓+c†\nk3,↑c†\nk1,↓\n−c†\nk1,↓c†\nk2,↑−c†\nk2,↓c†\nk3,↑−c†\nk3,↓c†\nk1,↑)\n+√\n2cosϕ3/summationdisplay\nj=1c†\nkj,↑c†\nkj,↓/bracketrightbig\n|0/an}bracketri}ht. (19)\nThe mixing angle ϕdetermining a quantum entangle-\nment of the relevant electron states within the eigen-\nstate (19) depends on a mutual competition between the\nhopping term and Coulomb term through the relation\ntanϕ=√\n2\n8t(U+2t+/radicalbig\n(U+2t)2+32t2).\nIt can be easily understood from Eqs. (15) and (16)\nthat the common feature of FM and FRI ground states\nis a quantum entanglement of three ferromagnetic states\nc†\nk1,↑c†\nk2,↑|0/an}bracketri}ht,c†\nk2,↑c†\nk3,↑|0/an}bracketri}ht,c†\nk3,↑c†\nk1,↑|0/an}bracketri}htof the mobile elec-\ntrons and hence, both ground states differ from each\nother just by the respective spin arrangement of the lo-\ncalized Ising spins. While the Ising spins are aligned\ninto a direction of the external magnetic field within\nthe FM ground state appearing at high enough mag-\nnetic fields, they point in opposite direction in the FRI\nground state with regard to the antiferromagnetic cou-\npling with the mobile electrons. It should be stressed,\nmoreover, that the lowest-energy eigenstate (18) of the\nmobile electrons is two-fold degenerate at each triangu-\nlar plaquette due to two possible values of the scalar chi-/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s49/s50/s51/s52/s56\n/s70/s82/s73/s70/s82/s85/s70/s77\n/s32/s32/s72 /s32 /s47/s32/s74\n/s116/s32 /s47/s32/s74/s85/s32/s47/s32/s74/s32/s61/s32/s48\n/s53/s50/s48\nFIG. 2: The ground-state phase diagram in the t/J−H/J\nplane for the spin-electron double-tetrahedral chain with the\nantiferromagnetic coupling J >0 upon varying a relative\nstrength of the Coulomb term U/J= 0, 5, 20, ∞.\nrality, which consequently leads to a substantial resid-\nual entropy S/3N= ln21/3of the FM and FRI ground\nstates.\nHowever, the most spectacular ground state is real-\nized in the FRU phase, in which the mobile electrons\nunderlie a quantum superposition of six intrinsic antifer-\nromagneticstates c†\nk1,↑c†\nk2,↓|0/an}bracketri}ht,c†\nk2,↑c†\nk3,↓|0/an}bracketri}ht,c†\nk3,↑c†\nk1,↓|0/an}bracketri}ht,\nc†\nk1,↓c†\nk2,↑|0/an}bracketri}ht,c†\nk2,↓c†\nk3,↑|0/an}bracketri}ht,c†\nk3,↓c†\nk1,↑|0/an}bracketri}htand three non-\nmagnetic ionic states c†\nkj,↑c†\nkj,↓|0/an}bracketri}ht(j= 1,2,3). The na-\nture of the FRU ground state consists in a kinetically-\ndrivenspinfrustration,theoriginofwhichisquitesimilar\nto that reported previously by Pereira et al. for the anal-\nogous spin-electron diamond chain [15, 16]. As a matter\nof fact, the localized Ising spins are at a zero magnetic\nfieldcompletelyfreetoflip inarbitrarydirectionowingto\nthe kinetically-driven spin frustration caused by the anti-\nferromagnetic alignment of the mobile electrons, whereas\narbitrarybut non-zeromagnetic field tends to align them\ninto the external-field direction. The magnetic field thus\nlifts the macroscopicdegeneracyofthe FRUgroundstate\n(and the associated residual entropy S/3N= ln21/3)\nin contrast with two previously discussed FM and FRI\nground states, where the macroscopic degeneracy relates\ntochiraldegreesoffreedomofthe mobile electronsrather\nthan to a kinetically-driven spin frustration of the local-\nized Ising spins.\nThe ground-state phase diagram in t/J−H/Jplane\ninvolving all three possible ground states is depicted on\nFig. 2 for several values of the Coulomb term U/J. Ev-\nidently, the FRI phase becomes a ground state at low\nenough magnetic fields whenever the influence of the an-\ntiferromagnetic Ising interaction Joverwhelms the effect\nof the hopping term t, i.e. whenever the hopping term is\nsmallert<tbthan the boundary value\ntb=−U\n18+1\n18/radicalbig\n(U+6J)2+24UJ. (20)5\nIf the reverse condition t > tbholds, then, the FRU\nphase is preferred as a ground state at sufficiently low\nmagnetic fields due to a predominant influence of the\nkinetically-driven spin frustration. Of course, the inves-\ntigated spin-electron double-tetrahedral chain undergoes\nathighenoughmagneticfieldsafield-inducedphasetran-\nsition towards the FM ground state with all nodal Ising\nspins andmobile electronsfully polarizedtothe external-\nfield direction. Analytic expressions of the relevant first-\norder phase transitions read:\nFRI−FM:H= 2J, (21)\nFRU−FM:H=J−U\n2+1\n2/radicalbig\n(U+2t)2+32t2.(22)\nIt is worthwhile to remark that the macroscopic degen-\neracyS/3N= ln41/3≈0.4621 at the FRI–FM phase\nboundary is greater than the macroscopic degeneracy\nS/3N= ln31/3≈0.3662 at the FRU–FM phase bound-\nary.\nTo complete our ground-state analysis, let us make\na few comments on a special limit of infinitely strong\nCoulomb repulsion U/t→ ∞when a mutual effect of\nthe hopping and Coulomb term should be equivalent\nto the antiferromagnetic Heisenberg coupling as it can\nbe proved within the second-order perturbation theory\n[37]. The ground-state phase diagram of the investigated\nspin-electron double-tetrahedralchain with two electrons\nper triangular plaquette indeed becomes identical in the\nU/t→ ∞limit with the ground-state phase diagram of\nthe spin-1/2 Ising-Heisenberg diamond chain (compare\nFig. 2 with Fig. 2b in Ref. [38]). However, there is\na fundamental difference in a character of the relevant\ngroundstates: allthreegroundstatesofthe spin-electron\ndouble-tetrahedral chain exhibit a remarkable quantum\nentanglement in contrast to a classical nature of spin ar-\nrangements of two ground states (FRI and SPP) of the\nspin-1/2 Ising-Heisenberg diamond chain [38]. Besides,\nit is quite obvious that essential magnetic features of the\nmodel under investigation do not qualitatively change\nwith the Coulomb term and hence, our further analy-\nsis will be restricted to the particular case with the fixed\nvalue of the on-site Coulomb repulsion U/J= 5.\nB. Magnetization process\nThe spin-electron double-tetrahedral chain bears a\nclose relation to the spin-1/2 Ising-Heisenberg diamond\nchain [38] as far as the behavior of magnetic quantities is\nconcerned. To illustrate this point, the total magnetiza-\ntion is plotted in Fig. 3 against the magnetic field for the\nfixed value of the Coulomb term, two values of the hop-\nping term and a few temperatures. As one can see, the\ndisplayed magnetization curves are quite similar to that\nof the spin-1/2 Ising-Heisenberg diamond chain (cf. with\nFig. 3 of Ref. [38]). In fact, the intermediate one-third\nplateau can always be detected in low-temperature mag-\nnetization curvesirrespective ofwhether the FRI or FRU/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s70/s82/s73/s45/s70/s77/s109 /s32 /s47/s32/s109\n/s115/s97/s116\n/s32/s48/s46/s51\n/s48/s46/s53\n/s84/s32 /s47/s32/s74 /s32/s61/s32/s48\n/s32/s32\n/s32/s32 /s116/s32 /s47/s32/s74 /s32/s61/s32/s48/s46/s53\n/s85/s32 /s47/s32/s74 /s32/s61/s32/s53\n/s48/s46/s49\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s70/s82/s85/s45/s70/s77/s48/s46/s51\n/s48/s46/s53\n/s84/s32 /s47/s32/s74 /s32/s61/s32/s48\n/s32/s32/s109 /s32 /s47/s32/s109\n/s115/s97/s116\n/s72/s32 /s47/s32/s74/s32/s32 /s116/s32 /s47/s32/s74 /s32/s61/s32/s48/s46/s54\n/s85/s32 /s47/s32/s74 /s32/s61/s32/s53\n/s48/s46/s49\nFIG. 3: (Color online) The total magnetization normalized\nwith respect to its saturation value as a function of the\nmagnetic field at a few temperatures, the fixed value of the\nCoulomb term U/J= 5 and two different values of the hop-\nping term t/J= 0.5 (upper panel), t/J= 0.6 (lower panel).\nphase is realized as the ground state before the magne-\ntization reaches saturation at sufficiently high magnetic\nfields. In addition, the zero-temperature magnetization\ncurve starts from non-zero value in the asymptotic limit\nof vanishing external field if the ground state is formed\nby the FRI phase (the upper panel in Fig. 3), while it\nstarts from zero asymptotic limit on assumption that the\nFRU phase constitutes the ground state (the lower panel\nin Fig. 3). Both aforementioned features have been al-\nready found and discussed in depth in our previous work\nconcerned with the spin-1/2 Ising-Heisenberg diamond\nchain [38]. In agreement with common expectations, the\none-third plateau as well as a steep increase in the mag-\nnetization observable near zero and saturation fields are\ngradually smoothened upon increasing temperature.\nC. Low-temperature thermodynamics\nLet us examine in detail temperature variations of ba-\nsic thermodynamic quantities such as the specific heat\nand entropy. Fig. 4 shows temperature dependences of\nthe zero-fieldspecific heat for a few different values of the\nkinetic term t/J. According to Eq. (20), a relativesize of\nthe hopping term as compared with the boundary value6\n/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s116/s32 /s47/s32/s74 /s32/s61/s32/s48/s46/s52/s32/s70/s82/s73/s67 /s32 /s47/s32/s51 /s78\n/s32/s40/s50/s52/s49/s49/s47/s50\n/s45/s53/s41/s47/s49/s56/s48/s46/s53/s53 /s32/s48/s46/s53\n/s32/s32\n/s72/s32 /s47/s32/s74 /s32/s61/s32/s48/s32\n/s85/s32 /s47/s32/s74 /s32/s61/s32/s53\n/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s40/s50/s52/s49/s49/s47/s50\n/s45/s53/s41/s47/s49/s56/s72/s32 /s47/s32/s74 /s32/s61/s32/s48/s32\n/s85/s32 /s47/s32/s74 /s32/s61/s32/s53/s32/s70/s82/s85\n/s32/s32/s67 /s32 /s47/s32/s51 /s78\n/s84/s32 /s47/s32/s74/s116/s32 /s47/s32/s74 /s32/s61/s32/s48/s46/s56\n/s48/s46/s54 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s46/s55\nFIG. 4: (Color online) Temperature variations of the zero-\nfield specific heat for the special value of the Coulomb term\nU/J= 5 and a few different values of the hopping term.\nThe upper(lower) panel shows thermal dependenceswhen the\nhopping term is smaller (greater) than the boundary value\ntb/J= (√\n241−5)/18. The lowest dotted curve illustrates\nthermaldependenceexactlyattheFRI–FRUphaseboundary.\ntb/J= (√\n241−5)/18≈0.5847is conclusive whether the\nFRI or FRU phase represents the actual ground state\nwhen considering the special value of Coulomb repul-\nsionU/J= 5. The upper (lower) panel thus demon-\nstrates temperature variations of the zero-field specific\nheat, which are quite typical for the particular case with\nthe FRI (FRU) ground state. It is quite apparent from\nFig. 4 that temperature dependences of the zero-field\nspecific heat generally exhibit one broad maximum in a\nhigh-temperature region regardless of whether the FRI\nor FRU phase constitutes the ground state. Moreover,\ntherealsomayappearoneadditionalSchottky-typemax-\nimum at a lower temperature if the hopping term t/Jis\nselected sufficiently close to the FRI–FRU phase bound-\nary. A position of the low-temperature peak moves to-\nwards zero temperature as the kinetic term approaches\nthe boundary value tb/Jgiven by Eq. (20) at which it\ncompletely disappears (see dotted curve in Fig. 4). It\ncan be also seen from Fig. 4 that the low-temperature\nSchottky-type peak gradually merges with a round high-\ntemperature maximum when the size of hopping term\nvaries further apart from the FRI–FRU phase boundary.\nFurthermore, let us turn our attention to the effect of/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s85/s32 /s47/s32/s74 /s32/s61/s32/s53/s46/s48\n/s32/s116 /s32/s47/s32 /s74 /s32/s61/s32/s48/s46/s54\n/s32/s48/s46/s53\n/s32/s32/s67/s32 /s47/s32/s51 /s78/s32\n/s72/s32 /s47/s32/s74 /s32/s61/s32/s48/s46/s50\n/s49/s69/s45/s51 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s50/s46/s48/s51/s52/s85/s32 /s47/s32/s74 /s32/s61/s32/s53/s46/s48\n/s32/s116 /s32/s47/s32 /s74 /s32/s61/s32/s48/s46/s54\n/s72/s32 /s47/s32/s74 /s32/s61/s32/s49/s46/s57\n/s32/s32/s84/s32 /s47/s32 /s74/s50/s46/s49/s32/s67/s32 /s47/s32/s51 /s78/s32/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s48/s46/s48/s48/s46/s48/s48/s53\n/s32/s48/s46/s48/s50/s72/s32 /s47/s32/s74 /s32/s61/s32/s48/s46/s49\n/s32/s32/s67/s32 /s47/s32/s51 /s78/s32/s85/s32 /s47/s32/s74 /s32/s61/s32/s53/s46/s48\n/s32/s116 /s32/s47/s32 /s74 /s32/s61/s32/s48/s46/s54\nFIG. 5: (Color online) Temperature dependences of the spe-\ncific heat for the special value of the Coulomb term U/J= 5,\nthe hopping term t/J= 0.6 and a few different values of the\nexternal magnetic field H/J. The lowest dotted curve in the\nupper panel shows the relevant zero-field dependence, while\nthe lowest short-dashed curve in the lower panel shows the\nrelevant dependence at the saturation field.\nexternal magnetic field on temperature variations of the\nspecific heat. A few typical dependences are depicted\nin Fig. 5 for the particular case with the fixed values\nof Coulomb and hopping terms, which drive the model\nsystem to the FRU ground state in a vicinity of the FRI–\nFRU phase boundary. Under this condition, a relatively\nsmall applied magnetic field is responsible for the ap-\npearance of a remarkable triple-peak dependence of the\nspecific heat, whereas the maximum found at the lowest\ntemperature can be ascribed to the Zeeman’s splitting of\nenergy levels of the frustrated Ising spins (see the curve7\nH/J= 0.005intheupperpanelofFig. 5). Inaccordance\nwith this statement, the maximum gradually shifts to-\nwardshighertemperatureswith increasingmagneticfield\nuntil it merges with the intermediate maximum, which is\nalso present in the relevant zero-field dependence (dot-\nted curve in the upper panel). The intermediate maxi-\nmum appears due to a thermal excitation from the FRU\ngroundstatetowardsthelow-lyingFRIexcitedstateonce\nthe hopping term is selected slightly above the boundary\nvalue (20) of the FRI–FRU phase boundary. However,\nthe most surprising finding is a rather abrupt rise of this\nlow-temperature peak at moderate values of the mag-\nnetic field (see the central panel of Fig. 5), the origin of\nwhich will be examined in the following. Although the\nheightofthelow-temperaturepeakisgreatestaroundthe\nmoderate magnetic fields H/J≈1, the low-temperature\npeak persists up the magnetic fields slightly above the\nsaturation field and it does not disappear neither at the\nsaturation field (see the short-dashed curve in the lower\npanel of Fig. 5).\nLet us provide a comprehensive understanding of the\norigin of the sizable low-temperature peak, which occurs\nin a thermal dependence of the specific heat at moderate\nvalues of the magnetic field. To clarify this issue, we de-\npict in Fig. 6 temperature variations of the specific heat,\nentropy and sublattice magnetizations for the set of pa-\nrametersU/J= 5,t/J= 0.6 andH/J= 1 for which\nthis peculiar phenomenon is especially pronounced. It\nis quite clear from the inset of Fig. 6 that the spe-\ncific heat shows a very sharp peak in a relatively narrow\ntemperature range, which could easily be confused either\nwith theλ-type divergence accompanying a second-order\nphase transition or the anomalous peak accompanying a\nfirst-order phase transition. Although the temperature\ndependence of the total magnetization does not exhibit\nany striking feature, both sublattice magnetizations per-\ntinent to the localized Ising spins and mobile electrons\nexhibit very abrupt thermal variations (almost discontin-\nuous jumps) from the values typical for the FRU phase\nto the values typical for the FRI phase (see the lower\npanel in Fig. 6). This result affords a convincing evi-\ndence that the anomalous peak in the specific heat can\nbe attributed to vigorous thermal excitations from the\nFRU ground state to the FRI excited state. A rather\nsteep increase observed in the relevant temperature de-\npendenceofentropy(the centralpanelofFig. 6)isalsoin\naccordance with this statement, because almost discon-\ntinuous jump between zero and ln21/3is consistent with\na zero-point entropy of the FRU phase and the residual\nentropy of the FRI phase at non-zero magnetic fields.\nIt could be concluded that a substantial difference be-\ntween degeneracies of the FRU and FRI states is re-\nsponsible for vigorous thermal excitations manifested as\nthe anomalous peak of the specific heat, which mimic a\ntemperature-inducedfirst-orderphasetransitionbetween\nthe FRU and FRI states. In the consequence of that, one\nmay estimate a pseudo-critical temperature correspond-\ning to the massive thermal excitations from the FRU/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32\n/s32/s32/s83/s32 /s47/s32/s51 /s78/s32\n/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s32/s109\n/s73\n/s32/s109\n/s101\n/s32/s109 /s47/s109\n/s115/s97/s116\n/s32/s32/s84/s32 /s47/s32 /s74/s32/s109\n/s73/s32/s44/s32/s109\n/s101/s32/s44/s32/s109/s47/s109\n/s115/s97/s116/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32\n/s32/s32/s67/s32 /s47/s32/s51 /s78/s32\n/s48/s46/s48/s49 /s48/s46/s49 /s49/s49/s69/s45/s54/s49/s69/s45/s52/s48/s46/s48/s49/s49/s49/s48/s48/s32\n/s100/s101/s116/s97/s105/s108\nFIG. 6: (Color online) Temperature dependences of the spe-\ncific heat (upper panel), entropy (central panel) and sublat -\ntice magnetizations (lower panel) for the fixed values of the\nCoulomb term U/J= 5, the hopping term t/J= 0.6 and the\nmagnetic field H/J= 1. The inset shows a detailed plot of\nthe specific heat in a low-temperature region within a log-lo g\nscale.\nphase to the FRI phase from the equality of Gibbs free\nenergies:\nTpc=/radicalbig\n(U+2t)2+32t2−U−2J\nln4.(23)\nEven if a temperature change of the enthalpy and en-\ntropy has been completely neglected by a derivation of\nthe pseudo-critical temperature, the formula (23) pro-\nvides a very rather accurate estimate of the peak posi-\ntion provided that the system is sufficiently close to the\nFRU–FRI phase boundary, i.e. vigorous thermal excita-\ntions between both states occur near zero temperature.8\n/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52\n/s48\n/s53\n/s85/s32 /s47/s32/s74 /s32/s61/s32/s50/s48/s48/s85/s32 /s47/s32/s74 /s32/s61/s32/s50/s48\n/s32/s32/s67/s32/s112/s101/s97/s107\n/s47/s32/s51 /s78\n/s116/s32 /s47/s32/s74 /s32/s61/s32 /s116\n/s98/s32 /s47/s32/s74 /s32/s43/s32/s48/s46/s48/s50/s53\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56\n/s32/s32/s84/s32/s112/s101/s97/s107\n/s47/s32/s74\n/s72/s32 /s47/s32/s74/s116/s32 /s47/s32/s74 /s32/s61/s32 /s116\n/s98/s32 /s47/s32/s74 /s32/s43/s32/s48/s46/s48/s50\nFIG. 7: (Color online) The height (upper panel) and the\nposition (lower panel) of the low-temperature peak of the\nspecific heat, which occurs due to massive thermal excita-\ntions from the non-degenerate FRU ground state towards\nthe highly degenerate FRI ground state when the hopping\nterm is selected slightly above the FRI–FRU phase boundary\nt/J=tb/J+0.02 and the Coulomb term varies U/J= 0, 5,\n20.\nFig. 7 illustrates changes in the height and position of\nthe low-temperature maximum arising when the hopping\ntermt/J=tb/J+ 0.02 is selected slightly above the\nFRI–FRU phase boundary for a few specific values of\nthe Coulomb term U/J= 0, 5 and 20. In agreement\nwith previous argumentation, Fig. 7 confirms our state-\nment that the low-temperature peak shows the great-\nest height around the moderate field H/J≈1. More-\nover, it can be also seen from the upper panel in Fig. 7\nthat the peak height generally increases with increasing\nthe Coulomb term. This result would suggests that the\nCoulomb term supports temperature-induced excitations\nbetween the FRU and FRI states. On the other hand,\nthe peak position remains unchanged around H/J≈1\nwith an accuracy up to three decimal places, whereas it\nshifts towards lower temperatures as the Coulomb term\nincreases (see the lower panel in Fig. 7).\n/s32/s32\n/s85/s32 /s47/s32/s74 /s32/s61/s32/s53/s46/s48\n/s32/s32 /s116/s32 /s47/s32/s74 /s32/s61/s32/s48/s46/s53\n/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48/s48/s46/s51/s53/s48/s46/s52/s48/s48/s46/s52/s53/s48/s46/s53/s48/s48/s46/s53/s53/s48/s46/s54/s48\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s32/s32/s84 /s32 /s47/s32/s74\n/s72/s32 /s47/s32/s74/s85/s32 /s47/s32/s74 /s32/s61/s32/s53/s46/s48\n/s32/s32/s116/s32 /s47/s32/s74 /s32/s61/s32/s48/s46/s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s32/s32\n/s32/s84 /s32 /s47/s32/s74/s83/s32 /s47/s32/s51 /s78\nFIG. 8: (Color online) A density plot of the entropy as a\nfunction of the magnetic field and temperature by assuming\nthe constant value of the Coulomb term U/J= 5 and two\ndifferent values of the hopping term t/J= 0.5 (upper panel)\nandt/J= 0.6 (lower panel). The displayed curves correspond\nto isentropy lines, namely, S/3N= 0.25,0.3,...,0.55 (black\nsolid curves), S/3N= ln41/3(white solid curve in the upper\npanel),S/3N= 0.001 (black broken curve), and S/3N=\nln21/3,ln31/3(white solid curves in the lower panel).\nD. Enhanced magnetocaloric effect\nLast but not least, we turn to a discussion of the mag-\nnetocaloric effect in its classical interpretation as an adi-\nabatic change of temperature achieved upon varying the\nexternal magnetic field. For this purpose, the density\nplot of the entropy is depicted in Fig. 8 as a function\nof the magnetic field and temperature for two different\nmagnetization scenarios discussed previously. Isentropic\nchanges of temperature upon varying the magnetic field\ncanbe identifiedin Fig. 8ascontoursofconstantentropy\ndisplayed by solid lines. The adiabatic demagnetization\nrelated to the field-induced phase transition between the\nFRI and FM phase can be analyzed from dependences\nshown in the upper panel of Fig. 8. It is quite evident\nthat the model under investigation exhibits an enhanced\nmagnetocaloric effect in a vicinity of the relevant field-\ninducedtransitionwhenevertheentropyissetsufficiently\nclose to the value S/3N= ln41/3≈0.4621, but temper-\nature finally tends towards a finite value as the magnetic\nfield vanishes. On the other hand, the enhanced mag-9\nnetocaloric effect during the adiabatic demagnetization\ncan be found at zero field as well as the relevant critical\nfield for another magnetization scenario pertinent to the\nfield-induced phase transition between the FRU and FM\nphase (see the lower panel of Fig. 7). Under this circum-\nstance, the most abrupt drop in temperature is achieved\nunder the adiabatic condition if the entropy is set suffi-\nciently close to the values S/3N= ln21/3≈0.231 and\nS/3N= ln31/3≈0.3662, respectively. The adiabatic\nchange of temperature in response to the variation in a\nmagnetic field is unusually striking for the isentropy line\nS/3N= ln21/3≈0.231 (the lower white curve in the\nlower panel of Fig. 8), for which temperature remains\nnearly constant in a relatively wide range of the mag-\nnetic fields H/J∈(1/2,3/2). This intriguing feature\ncan be attributed to the anomalous behavior of specific\nheat discussed previously in Sec. IIIC, because the adi-\nabatic change of temperature with the magnetic field is\ninversely proportional to the specific heat and it is there-\nfore negligible due to a sizable value of the specific heat.\nIV. CONCLUSIONS\nThe present work deals with magnetic properties\nof a one-dimensional double-tetrahedral chain of lo-\ncalized Ising spins and mobile electrons, which can\nbe exactly treated through the generalized decoration-\niteration transformation establishing a rigorous mapping\ncorrespondence with a simple spin-1/2 Ising chain with\nthe effective nearest-neighbor interaction and effective\nmagnetic field. Our exact calculation have allowed us to\nexamine in detail the ground-state phase diagram, mag-\nnetization process, magnetocaloric effect, entropy and\nspecific heat. Although the investigated spin-electron\nmodel on a double-tetrahedral chain resembles to a cer-\ntain extent some magnetic features of the spin-1/2 Ising-Heisenberg diamond chain (e.g., an intermediate one-\nthird plateau in a low-temperature magnetization curve,\nenhanced magnetocaloric effect during the adiabatic de-\nmagnetization and temperature variations of the specific\nheat with one, two or three separate peaks) [38], it also\ndisplays a lot of other remarkable features not reported\nin the literature hitherto.\nIn particular, we have found three different ground\nstates with an interesting quantum entanglement be-\ntween states of the mobile electrons and a high macro-\nscopic degeneracy. The ferromagnetic and ferrimagnetic\nground states are macroscopically degenerate due to chi-\nral degrees of freedom of the mobile electrons, while the\nfrustrated state displays a macroscopicdegeneracyowing\nto a kinetically-driven frustration of the localized Ising\nspins. It has been evidenced that the residual entropy\ndue to the kinetically-drivenspin frustration can be thor-\noughly lifted by the magnetic field unlike the residual en-\ntropyconnectedtochiraldegreesoffreedomofthemobile\nelectrons. However, the most spectacular finding con-\ncernswith the anomalousbehaviorof all basic thermody-\nnamic quantities if the hopping and Coulomb terms drive\nthe system to the frustrated ground state in a close vicin-\nityofthe phaseboundarywith theferrimagneticstate. A\nsubstantial difference in the respective ground-state de-\ngeneraciesisresponsibleforan immense low-temperature\npeak of the specific heat and very abrupt (almost dis-\ncontinuous) thermal variations of the entropy and sub-\nlattice magnetizations. The caution before interpreting\nall aforementioned features as typical manifestations of\na phase transition should be accordingly made [39, 40],\nbecause our exactly solved spin-electron chain serves in\nevidence that all those outstanding features can origi-\nnate from vigorous thermal excitations between a non-\ndegenerate ground state and a highly degenerate (low-\nlying) excited state due to a high entropy gain.\n[1] R.J. Baxter, Exactly Solved Models in Statistical Me-\nchanics (Academic Press, New York, 1982).\n[2] D.C. Mattis, The Many-Body Problem: An Encyclope-\ndia of Exactly Solved Models in One Dimension (World\nScientific, Singapore, 1993).\n[3] B. Sutherland, Beautiful Models: 70 Years of Exactly\nSolvedQuantumMany-BodyProblems (WorldScientific,\nSingapore, 2004).\n[4] A.J. Guttmann, Pramana 64, 829 (2005).\n[5] M.E. Fisher, Phys. Rev. 113, 969 (1959).\n[6] I. Syozi, Phase Transition and Critical Phenomena, Vol.\n1, editedbyC. Domb, andM. 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Jpn. 83, 074001 (2014).\n[21] R.C.P. Carvalho, M.S.S. Pereira, M.L. Lyra, O. Rojas, J .\nStreˇ cka, Acta Phys. Polonica A 126, 12 (2014).\n[22] J. Streˇ cka, A. Tanaka, L. ˇCanov´ a, T. Verkholyak, Phys.\nRev. B80, 174410 (2009).\n[23] J. Streˇ cka, A. Tanaka, M. Jaˇ sˇ cur, J. Phys.: Conf. Ser .10\n200, 022059 (2010).\n[24] L. G´ alisov´ a, J. Streˇ cka, A. Tanaka, T. Verkholyak, J .\nPhys.: Condens. Matter 23, 175602 (2011).\n[25] F.F. Doria, M.S.S. Perreira, M.L. Lyra, J. Magn. Magn.\nMater.368, 98 (2014).\n[26] M. Mambrini, J. Tr´ ebosc, F. Mila, Phys. Rev. B 59,\n13806 (1999).\n[27] O. Rojas, F.C. Alcaraz, Phys. Rev. B 67, 174401 (2003).\n[28] C.D. Batista, B.S. Shastry, Phys. Rev. Lett. 91, 116401\n(2003).\n[29] M. Maksymenko, O. Derzhko, J. Richter, Eur. Phys. J.\nB84, 397 (2011).\n[30] M. Maksymenko, O. Derzhko, J. Richter, Acta Phys.\nPolonica A 119, 860 (2011).\n[31] D. Antonosyan, S. Bellucci, V. Ohanyan, Phys. Rev. B\n79, 014432 (2009).\n[32] V. Ohanyan, Phys. Atom. Nucl. 73, 494 (2010).\n[33] M. Hase, H. Kitazawa, K. Ozawa, T. Hamasaki, H.\nKuroe, T. Sekine, J. Phys. Soc. Jpn. 77, 034706 (2008).\n[34] H. Kuroe, T. Hosaka, S. Hachiuma, T. Sekine, M. Hase,\nK. Oka, T. Ito, H. Eisaki, M. Fujisawa, S. Okubo, H.\nOhta, J. Phys. Soc. Jpn. 80, 083705 (2010).\n[35] M. Matsumoto, H. Kuroe, T. Sekine, M. Hase, J. Phys.\nSoc. Jpn. 81, 024711 (2012).[36] H.A. Kramers, G.H. Wannier, Phys. Rev. 60, 252 (1944).\n[37] P. Fazekas, Lecture Notes on Electron Correlation and\nMagnetism (Singapore: World Scientific, 1999).\n[38] L.ˇCanov´ a, J. Streˇ cka, M. Jaˇ sˇ cur, J. Phys.: Condens.\nMatter18, 4967 (2006).\n[39] F. Mancini, F.P. Mancini, Phys. Rev. E 77, 061120\n(2008).\n[40] F. Mancini, E. Plekhanov, G. Sica, Eur. Phys. J. B 86,\n224 (2013).\nAcknowledgments\nThisworkwasfinanciallysupportedbythegrantofthe\nSlovakResearchandDevelopmentAgencyunderthecon-\ntract No. APVV-0097-12 and by the ERDF EU (Euro-\npean Union European regional development fond) grant\nprovided under the contract No. ITMS26220120005 (ac-\ntivity3.2). J.S.wouldliketothankDr. TarasVerkholyak\nfor his helpful comments and encouraging discussions\nconcerning with the topic of this work."    },    {        "title": "2208.04539v2.Hybrid_spin_Hall_nano_oscillators_based_on_ferromagnetic_metal_ferrimagnetic_insulator_heterostructures.pdf",        "content": " 1 Hybrid spin Hall nano -oscillators based on ferromagnetic metal/ferrimagnetic insulator heterostructures  \nHaowen Ren1*, Xin Yu Zheng2, Sanyum Channa3, Guanzhong Wu1, Daisy A. O’Mahoney4, Yuri Suzuki2, and  \nAndrew D. Kent1+ \n1Center for Quantum Phenomena, Department of Physics, New York University, New York, NY 10003, \nUSA \n2Department of Applied Physics  and Geballe Laboratory for Advanced Materials , Stanford University, \nStanford, CA 94305, USA  \n3Department of Physics  and Geballe Laboratory for Advanced Materials , Stanford University, \nStanford, CA 94305, USA  \n4Department of Materials Science and Engineering  and Geballe Laboratory for Advanced Materials , \nStanford University, Stanford, CA 94305, USA  \nCorresponding author s: *haowren@gmail.com , +andy.kent@nyu.ed u \nAbstract  \nSpin -Hall nano -oscillators (SHNOs) are promising spintronic devices to realize current controlled GHz \nfrequency signals in nanoscale devices for neuromorphic computing and creating Ising  systems. However, \ntraditional SHNOs --- devices based on transition metals --- have high auto -oscillation threshold currents as \nwell as low quality factors and output powers. Here we demonstrate a new type of  hybrid SHNO  based on a  \npermalloy (Py) ferromag netic -metal nanowire and low-damping ferrimagnetic insulator, in the form of \nepitaxial lithium aluminum ferrite (LAFO) thin films . The superior  characteristics of such SHNOs  are \nassociated with the excitation of larger spin -precession angles and volumes . We further find that the \npresence of the ferrimagnetic insulator enhances the auto -oscillation amplitude of spin -wave edge modes, \nconsistent with our micromagnetic modeling. This hybrid  SHNO expands spintronic applications, including \nproviding new means of coupling multiple SHNOs for neuromorphic computing and advancing magnonics.  \n   2 Introduction . \nHigh  efficiency oscillators  are essential to accelerate the application of spintronics for neuromorphic \ncomputing1–4, Ising systems5 and magnonic device s6–8 among other applications. Spin -Hall nano -oscillator s \n(SHNO s) are one of the important  approaches  to achieve these application s due to their two-dimensional \ngeometry, which permits coupling multiple SHNOs in a plane9–11, as well as their ease of fabrication.  Several \ngeometries of SHNOs have been proposed in previous studies, such as nanodisk12–14, nanowire15–17, and \nnanoconstriction types18–22. However, these SHNOs generally have high threshold currents, low emission \npower s and poor  quality  factor s because of the nature of their constituent materials, specifically t he large \nmagnetic damping in transition metal ferromagnets. In recent years, attention has focused on ferrimagnetic \ninsulators23–26 due to their extremely low damping and consequently high magnon conductivity27, which is \nvery favorable for spintronic applications. At the same time, this low damping characteristic facilitates the \nformation of spin -Hall effect induced auto -oscillations; inde ed, ferrimagnetic insulator -based nano -\noscillators have been demonstrated with yttrium iron garnet/Pt bilayers25,26. Nevertheless, they suffer from \nlow power emission due to their small inverse spin Hall effect signals28. Joule heating also limits their \napplication at room temperatu re28. One way to overcome these drawbacks is by creating a new type of \nhybrid SHNOs based on ferromagnetic metal -ferrimagnetic insulator  heterostructures . Interesting physics \nemerges when cou pling thin layers of these two types of materials29–32. When the two layers are weakly \ncoupled, there are two distinct spin resonances, associated with acoustic and optical modes. However, when \nthey are strongly coupled, the two layers act collectively, leading to magnetic properties inherited from both \nlayers33, specifically a lower effective damping. Thus, SHNOs fabricated from such hetero structures can take \nthe advantage of the low damping from the ferrimagnetic insulator layer and yet maintain a strong electrical \nsignal from the ferromagnetic metal layer.  \nTheoretical studies have shown that a uniform spin current applied to an extended magnetic thin film does \nnot support the formation of auto -oscillations due to the emergence of nonlinear damping from magnon -\nmagnon interactions34. However, by concentrating spin curren t in a small region, linear spin -wave mode \nauto -oscillation states can be stabilized35. Later, it was shown that nonlinear localized modes36 can be also \nachieved due to t he suppression of magnon -magnon interactions, which has been experimentally \ndemonstrated in point -contact type and disk-type SHNOs12–14,37. Meanwhile, if the device geometry is \nconfined (e.g. a nanowire or nanoconstriction), auto -oscillations can still be excited in a localized region that \nleads to a potential well that limits spin -wave propagation16,20. These self -localized modes can have much \nsmaller threshold current s than linear modes due to lower radiative loss in an au to-oscillation state.  \nIn this article, we demonstrate a new type of SHNO that combi nes a  ferromagnetic transition metal Py with \nan epitaxial thin film ferrimagnetic insulator , lithium aluminum ferrite (LAFO) . This hybrid  SHNO expand s \nspintronic applications, including providing new means of coupling multiple SHNOs for neuromorphic \ncomputing and can advance designs for magnonics.  Furthermore, c ompared to conventional Py/Pt  SHNOs,  \nthis hybrid SHNO is superior in all important characteri stics  having  a reduced threshold current, stronger \nemission power and higher quality factor.  \n \nResults  and Discussion . \nOur heterostructures are composed of two different lithium aluminum ferrite compositions  (Li0.5Al1.0Fe1.5O4 \n(LAFO) or Li 0.5Al0.5Fe2O4 (LFO)) ( x nm)/Py(5nm)/Pt(5nm) layers with varied LAFO or LFO thickness x (including \nx=0, i.e. , just Pt/Py layers).  The Py/Pt layers are patterned into 400 nm wide nanowires with a 400 nm gap \nbetween two Au contact pads as shown in Fig. 1a. Detailed deposition and fabrication conditions are in \nMethods. We fabricated devices with LAFO : LAFO4/Py5/Pt5, LAFO10/Py5/Pt5, and LAFO20/Py5/Pt5, and LFO:  3 LFO15/Py5/Pt5 with the numbers being the layer thicknesses in nm . Lastly,  a Py5/Pt5 reference device  was \ndepo sited on a sapphire substrate.  \n \n \nFigure 1 . (a) Schematic of the hybrid SHNO  device and power spectral density (PSD) measurement setup. (b) \nFMR frequency versus resonance field for various unpatterned thin films and heterostructures, including for \nreferenc e, Py5/Pt5 bilayers and LAFO and LFO thin films. (c) FMR linewidth as a function of frequency for the \nsame samples.  \n \nTo determine the magnetic properties in different Py/LAFO samples , ferromagnetic resonance spectroscopy \n(FMR) measurements were carried out  on unpatterned thin films and heterostructures via a vector network \nanalyzer (VNA) technique38. Effective magnetization 𝑀eff and anisotropy field 𝐻𝑎 are obtained by fitting \nresonan ce peaks to the Kittel model 𝑓=𝜇0𝛾/2𝜋√(𝐻+𝐻𝑎)(𝐻+𝐻𝑎+𝑀eff), where 𝐻 is the external \nmagnetic field,  𝑀eff is the effective magnetization, 𝛾 is the gyromagnetic ratio , and 𝜇0 is the vacuum \npermeability. Both LAFO and LFO have magnetocrystalline an isotropy with an easy axis along <110> and \nhard axis along <100> directions that is characterized by an in -plane anisotropy field 𝐻𝑎. Gilbert damping \nconstants 𝛼 are obtained by measuring the FMR linewidth as a function of the frequency.  The data and fi ts \nare shown in Fig. 1b&c and the fitting parameters are listed in Supplementary Table S1 . There was always \nonly one FMR absorption peak observable, indicating that the two magnetic layers in our Py/LAFO \nheterostructures are strongly coupled. Further, the 𝑀eff of Pt/Py/LAFO falls between the 𝑀eff of bare LAFO \nor LFO and Py layers, as expected for  two ferromagnetically coupled magnetic layers. To analyze the change \n 4 of 𝑀eff and 𝛼 in the heterostructures, a macrospin model based on Landau -Lifshitz -Gilbert (LLG) equation \nconsidering two strongly coupled magnetic layers is used (see Supplementary Note 7). When two magnetic \nlayers are strongly ferromagnetic coupled  the acoustic mode resonance condition will be set by the \nweighted mean of the magnetic propertie s of the two individual layers, 𝑀eff=(𝑡Py𝑀𝑠,Py𝑀eff,𝑃𝑦+\n𝑡LAFO𝑀𝑠,LAFO𝑀eff,LAFO)/(𝑡Py𝑀𝑠,Py+𝑡LAFO𝑀𝑠,LAFO)  and 𝛼=(𝑡Py𝑀𝑠,Py𝛼𝑃y+𝑡LAFO𝑀𝑠,LAFO𝛼LAFO)/\n(𝑡Py𝑀𝑠,Py+𝑡LAFO𝑀𝑠,LAFO), where 𝑀eff and 𝛼 are the weighted effective magnetization and damping \nconstant of the  bilayers, 𝑡Py(𝑡LAFO ) and 𝑀𝑠,Py(𝑀𝑠,LAFO ) are the thickness and saturation magnetization of  \nthe Py(LAFO) layer, respectively, consistent with previous models of coupled layers33. The measured values \nare listed in Supplementary Table S1  and are compared to the model’s 𝑀eff and 𝛼.  𝑀eff obtained from the \nsimple model is always smaller than the actual measured value, while 𝛼 is always larger than the measured \nvalue, which indicates that the exchange coupling is smaller than that of the harmonic mean of  the two \nlayers. As the damping of t he coupled magnetic layers decreases, a spin current injected into the Py layer in \na LAFO/Py/Pt heterostructures can excite a larger magnetic volume , which , as we show , greatly improves \ndevice performance.  \nTo compare the magnetic excitations of thin films with patterned structures we conducted spin -torque FMR \n(ST-FMR) on both 2 µm wide stripe devices and 400nm  wide nanowire devices. Figures 2a and b show the ST -\nFMR spectra of 400nm wide nanowire devices.  In contrast to the FMR spectra, ST -FMR shows two domin ant \nresonances, whose linewidth and peak amplitude are sensitive to bias current. In contrast, for 2µm width \nstripe devices only one ST -FMR peak is seen (Supplementary Fig. S2a).  \nFigure 2c shows the ST -FMR frequency -resonance field spectra of a 400nm nanow ire and a 2µm stripe \nLAFO20/Py5/Pt5 device. We find that the dispersion of the higher frequency mode of the two devices \noverlap, with a fit to the Kittel model giving 𝜇0𝑀eff=0.86 T, close to that found from the FMR spectra of the \nassociated unpatterned f ilm. We thus attribute this feature to a bulk mode (BM), a spin excitation that is \nmost uniform across the width of the device. The lower frequency mode only appears in the 400nm wide \ndevice and is associated with a much lower 𝜇0𝑀eff=0.65 T. We attribute  this lower frequency mode to an \nedge mode (EM), as indicated in Figs. 2a&b, and this conclusion is supported by micromagnetic simulations \nas discussed below. Two modes of this type have been reported in previous studies15–17,20. \nHybrid SHNO devices  show the onset of auto -oscillations at a threshold current. Figures 2d -g shows the \npower spectral density (PSD) as a function of bias current at fixed field 𝐻=0.0817 T for 𝜙=70o. In all the \ndevices, the auto -oscillation frequency redshifts with increasi ng bias current, a characteristic of localized \nmodes in nanowire SHNOs16,17. This is not a pure heating effect  (see Supplementar y Fig. S4).  Interestingly, \nthe threshold current 𝐼𝑡ℎ drops dramatically between the reference sample, Py5/Pt5, and the sample with \nLAFO, LAFO4/Py5/Pt5, and then the 𝐼𝑡ℎ slowly increases with the thickness of LAFO. The slow increase of 𝐼𝑡ℎ \nwith the th ickness of LAFO layer agrees well with the expectations of the macrospin model that predicts, \n𝐼𝑡ℎ∝(𝛼Py𝑡Py𝑀𝑠,Py+𝛼LAFO𝑡LAFO𝑀𝑠,LAFO)𝑀eff, consistent with ST-FMR results obtained from the 2µm \nwidth stripe  and 400nm width nanowire  samples  (Supplementary Fig. S2 b,c). However, the drop of 𝐼𝑡ℎ from \nPy5/Pt5 to LAFO4/Py5/Pt5 cannot be explained by this model. We note this decrease in 𝐼𝑡ℎ is observed in ST -\nFMR studie s conducted on both  400nm nanowire and 2µm width strip samples. So it does not depend \nsensitively on sample geometry. It is thus possible that  the spin current generated from the Py  layer itself \nacts on the LAFO to increase the spin torques and reduce 𝐼𝑡ℎ.39–42 Previous stud ies have experimentally \nshown spin-orbit torque can be generated from a single magnetic layer19,43. In addition,  the edge mode  can \nbe dominant in  hybrid devices  and caus e a mode -related change of nonlinear damping44,45, which  would  \nreduc e radiative loss  and thus 𝐼𝑡ℎ. In addition, in hybrid devices the dominant EM  will cause mode -related \nnonlinear damping change, which  would reduce radiative loss and thus 𝐼𝑡ℎ. Nevertheless, further study is \nrequired to explain this drop of 𝐼𝑡ℎ.   5 Notice that compared to the Py5/Pt5 sample, the slopes of the redshift increase in all LAFO samples. This is \nlikely due to larger spin prece ssion angles and the emergence of a nonlinear self -localized mode18. \nInterestingly, the relative magnitude of EM and BM measured from ST -FMRs and PSDs both follow the same \ntrend: the dominant mode transitions from a BM in Py5/Pt5 to an EM in LAFO20/Py5/Pt5, which \nsimultaneously increases the performance of the oscillators. This transition will be discussed in detail in the \nnext section.  Threshold current s and auto -oscillation current s for different devices  are listed in \nSupplementary Table S2.  \n  \nFigure 2.  (a) ST -FMR measurements of 400nm nanowire device for (a) Py5/Pt5 and (b) LAFO20/Py5/Pt5 at 7 \nGHz with the applied field at an angle 𝜙=70o to the wire. (c) Kittel model fitting curves of LAFO20/Py5/Pt5 \nfor 2µm stripe (blue circles) device and 400nm nanowire device . Green diamond shows fit for the peaks from \nbulk mode and red square for the peaks from edge mode. Maps of PSDs as a function of frequency and dc \nbias at a fixed field 𝐻=0.0817 T for 𝜙=70o for nanowire devices consisting of (d) Py5/Pt5, (e) LAFO4/Py5/Pt5, \n(f) LAFO10/Py5/Pt5, and (g) LAFO20/Py5/Pt5.  The output power increases significantly for the thickest LAFO \nsample studied  as indicated by the colorscales above each PSD map.  \n \nTo investigat e the spin -wave modes of Py and Py/LAFO SHNO heterostructures, micromagnetic simulations \nwere carried out using MuMax3 (see Methods)46. Spin currents were applied solely to the 400 nm Py \nnanowire’s ce nter region to mimic the device’s current distribution. The simulation is run until a steady state \nresponse is observed . The time evolution of magnetiza tion was then converted to the frequency domain by \nFast Fourier transform (FFT). Figure 3a shows the spatial-average d FFT amplitude in the center region of Py \nfor different samples , confirming the experimental observation of two dominant auto -oscillation modes . \nFrom these simulations, we can find that the auto -oscillation frequencies and their trends are in excellent \nagreement with our experimental results: (i) the resonance frequency  is almost no t changed from  the \nPy5/Pt5 device to the LAFO4/Py5/Pt5 device, then increase s with the thickness of  LAFO . This is mainly due to \n 6 variations in  the net 𝑀eff, as the resonance frequency depends strongly on 𝑀eff; and we find that the  \nresonance frequency changes closely follow the trends in  𝑀eff determined by  FMR and ST -FMR , shown in  \nSupplementary  Table S1 ; (ii) the high -frequency mode has a higher amplitude in Py5/Pt5, while the low -\nfrequency mode gradually becomes dominant with increasing LAFO thickness ; (iii) the peak amplitude and \nquality factor for LAFO20/Py5/Pt5 is significantly higher than that of Py5/Pt5. To identify the reason behind \nthis transition, pixel -wise  spatial FFTs were conducted on Py5/Pt5 and LAFO20/Py5/Pt5 simulations.  Spatial \nFFT image s of the Py layer from Py5/Pt5 at the auto -oscillation frequencies, obtained from Fig. 3a, are shown \nin Fig. 3b. We find that the low -frequency peak at 𝑓=6.21 GHz is concentrated on the edge of the Py stripe, \nwhile the peak at 𝑓=7.17 GHz is dominant in the middle. This observation is consistent with our assignment \nof the modes in the previous section and earlier st udies15,16. \nMore interesting magnetic behavior occurs in magnetic heterostructures . Spatial FFT images of Py  and LAFO \nlayers from the LAFO20/Py5/Pt5 simulation are shown in Fig. 3c. Compared to Py5/Pt5, the EM and BM of \nthe Py layer excited in LAFO20/Py5/Pt5 shows a much larger oscillation amplitude. The magnetization of the \nLAFO layer oscillates coherently with  that of the Py layer. The increased auto -oscillation amplitude is mainly \ndue to a larger precession cone angle in the Py layer, caused by the lower effective damping constant. \nMeanwhile, the area of the EM expands in the bilayer system, which leads to a l arger excited volume of \nmoments, consistent with a higher quality factor. The out-of-plane expansion of both BM and EM is caused \nby the strong ferromagnetic coupling between  the two layers, while the in-plane expansion of EM can be \nunderstood in this way: the exchange field generated by the LAFO layer tends to align the moments at the \nedge of Py nanowires against the demagnetization field, which decreases the effective field inhomogeneity, \nthus increasing the EM coherent oscillation volume. This is confirme d by plotting the transverse \nmagnetization profile of the devices at the equilibrium ( Supplementary Fig. S3a), where one can observe a \nsmoother transition near the edge of Py  layer in the LAFO20/Py5/Pt5 device. This expands the area of  the \nlocalized mode, especially the EM, greatly enhancing the coherence  of each mode and thus increasing the \nmaximum power and quality factor of signals emitted from the oscillators.  \n   \n  \nFigure 3.  (a) FFT amplitude  spectrum  as a function of frequency from micromagnetic simulations of different \ndevices. The spectrum is acquired by doing FFT on the time revolution of spa tial-average d magnetization  \n 7 𝑚̅𝑧(𝑡)  in the center  region of nanowire excited by a spin current. (b) Top views of spatial FFT images on the \nPy layer of Py5/Pt5 obtained at EM and BM resonance frequencies. (c) Top view of spatial FFT images of the \nPy layer (top) and LAFO layer (bottom) of a LAFO20/Py5/Pt5 device.  The image size for the Py5 layer  is \n1500×400 nm2 and for the LAFO20 layer is 1500×1500 nm2. The logarithmic colorscale is on the right  where \nthe color represents the FFT amplitude  of 𝑚𝑧(𝑥,𝑦). \nTo better understand the properties of the auto -oscillation mod es, maps of the PSD as a function of \nmagnetic field at fixed bias current (1.15  times 𝐼𝑡ℎ) are plotted in Figs. 4a-d. Similar to the current dependent \nPSD map (Figs. 2d to g), BM and EM are observed. By fitting the two PSD peaks to Lorentzian functions, we \ncan obtain the auto -oscillation frequencies and dispersion curves for the BM and EM as shown in Figs. 4e-h. \nThis is show n in comparison to the FMR and ST -FMR results. In the Py5/Pt5 sample (Fig. 4a), only the BM is \ndetectable , and its dispersion curve is slightly redshifted compared to that of FMR and ST -FMR data . \nHowever, contrary to the commonly seen self-localize d mode47,48 in magnetic thin film s, in a magnetic wire  \nthe propagation of spin  waves is restricted in the transverse direction but allowed along the wire direction, \npreventing  mode  local ization . This BM has a similar 𝐼𝑡ℎ compared to  the uniform mode35, which is confirmed \nby the ST -FMR on 2µm stripe s (Supplementary Fig. S2(b)). Instead, the EM is localized due to a self -induced \npotential well, leading to a localized quasi -linear auto -oscillation mode.  \nIn LAFO/Py bilayers  (Fig.  4b-d), due to the strong coupling between two magnetic layers, the center region of \nLAFO will precess coherently with Py. The exchange fields generated from the LAFO layer change the auto -\noscillations frequencies in LAFO/Py/Pt samples. The increased difference between the dispersion curves of \nauto -oscillation and those obtained from FMR and ST -FMR supports the idea that the spin -wave modes are \nmore localized in the LAFO containing devices . As shown in Figs. 4e-h, compared to the Py5/Pt5 sample, the \ndispersion curve of the BM of the LAFO4/Py5/Pt5 sample from PSD measurements is mu ch lower than the \nFMR mode. This is one of the key characteristics of a mode that is more strongly localized . \n \nFigure 4.  PSD maps as a function of external magnetic field and frequency for (a) Py5/Pt5, (b) LAFO4/Py5/Pt5, \n(c) LAFO10/Py5/Pt5, and (d) LAFO20 /Py5/Pt5  SHNOs . Resonance frequency as a function of external \nmagnetic field for the device (e ) Py5/Pt5, (f) LAFO4/Py5/Pt5, (g) LAFO10/Py5/Pt5, and (h) LAFO20/Py5/Pt5 \nobtained by FMR (brown dash dot), ST -FMR (green dash), and PSD (blue and red solid) measurements. \n 8 Dominant resonance modes are highlighted by a wider line. Notice the resonance frequen cy obtained from \nPSD has two distinctive peaks, which are associated with bulk mode and edge modes.  \n \nAccording to  the discussion  in the  previous section s, we determined a few critical properties of the LAFO \nlayer  which can guide us to design  a better hybr id SHNOs : (i) low 𝛼 ferrimagnetic insulator to have lower 𝛼, \n(ii) higher 𝑀eff to make the auto -oscillation EM more localized and (iii) appropriate thickness to not increase \nthe threshold current. To meet th ese criteria , LFO (15nm), which possess these properties (properties listed \nin Supplementary  Table S1 ), was used in place of LAFO as the ferrimagnetic insulator in our device. As shown \nin Figs . 5a&b, strong auto -oscillation signals up to 30 dB over the noise floor (NF) are detected only \nassociat ed with EMs. Compared to the LAFO samples, the auto -oscillations occur at higher frequency due to \nthe larger 𝑀eff of the LFO layer. Figure 3c is the dispersion curves obtained from FMR, ST -FMR, and PSD \nmeasurements for LFO15/Py5/Pt5 sample. Since FMR spec tra are  measured at 𝜙 = 0o (magnetic hard axis) \nand the ST -FMR spectra are measured at 𝜙= 70o (closer to the magnetic easy axis  direction ), a significant  \ndifference between these results occur s due to the large in-plane crystalline anisotropy of the LFO layer.  The \nstrong anisotropy also cause s a crossing  between  the maximum PSD signal  and ST -FMR curve s at low field s \nin Fig. 5c. At this low field region,  the sample is not  magnetically  saturated  as we are not measuring the PSD \nwith the field along the easy axis of LFO . This leads to multidomain states at low field and a resonance \nfrequency that does not dependent monotonically on applied field.  However, a t higher field,  the PSD \ndispersion curve is closer to what we obtained from ST -FMR. The auto -oscilla tion dispersion in this field \nrange is redshifted relative to ST-FMR dispersion, which  again  indicates the formation of localized auto -\noscillation modes. To systematically compare the  performance of different samples, PSDs at a fixed field \n𝐻=0.045 T obtained from the field dependent PSD maps are plotted in Fig. 5 d. Clearly, with the optimization  \nof the LAFO layer, the maximum power in LAFO/Py/Pt samples can be at least a 1000 times larger than that \nof the Py5/Pt5. We note that the anisotrop ic magnetoresistance (AMR) does not vary significantly in the \ndifferent samples ( see Supplementary Fig. S1 ) and is th us not an important factor in the change in device  \noutput power. By fitting the dominant peak of the PSDs with a Lorentzian function, we obtained both the \nmaximum power and maximum quality factor from each device as shown in Fig. 5e. From all thes e results, \ncompared to the conventional Py/Pt SHNOs, we can obtain orders of magnitude higher emission power and \nquality factor in hybrid low damping ferrimagnetic insulator (LAFO) ferromagnetic metal (Py)  \nheterostructures , which provides a new platform fo r SHNOs.   9   \nFigure 5.  PSD maps for LFO15/Py5/Pt5 as a function of (a) bias current and (b) magnetic field. (c) Dispersion \ncurves of LFO15/Py5/Pt5 obtained from FMR, ST -FMR, and PSDs measurements. (d) PSD spectrums of \ndifferent samples at fixed 𝐻=0.045 T. Lines are shifted upward 1 5 dB for each spectrum (e) Max signal (dB \nover NF) and max Q factor for different devices obtained from the PSDs in (a) and Fig. 4a -d. \n \nIn summary, our work presents a new hybrid  type of SHNOs, which shows superior perfor mance compared \nto conventional Py/Pt  spin oscillators. In hybrid SHNOs much  higher power emission and quality factor  can \nbe obtained  relative to  conventional Py/Pt SHNOs . To understand the mechanism behind the improved  \nperformance , ferromagnetic resonance measurements  and micromagnetic simulations were carried out  on \nboth conventional Py/Pt SHNO and hybrid  SHNOs.  Results show that the two layers precess coherently in \nbulk mode and edge modes.  Meanwhile,  the localization of auto -oscill ation s reduce s the threshold current \nand makes the edge mode the dominant power emission source rather than the bulk mode. Further, by \ndesigning the composition and thickness of the ferrimagnetic insulator layer, we successfully fabricated \nhybrid SHNOs wit h better performance by replacing LAFO with LFO. Our work expands the possibility of \nSHNOs for many types of spintronic applications, such as synchronizing electrically isolated SHNOs, for \nneuromorphic computing and for magnonic logic circuits.  \n  \n 10 Methods  \nSample deposition and fabrication. Epitaxial L i0.5Al1.5Fe1.5O4 and Li 0.5Al0.5Fe2O4 films  are grown on (001) \nMgAl 2O4 (MAO) substrates at 400° C in 15mTorr O 2 at a laser fluence of 1.9J/cm2 by pulsed laser deposition.  \nThe deposition of epitaxial LAFO with different compositions follows the previous study49,50. After the growth \nof the ferrite thin film s of varying  thickness and composition, Py(5nm)/Pt(5nm) bilayers  are deposited via a \nKurt Lester magnetron sputtering system at room temperature. The reference sample Py5/Pt5 is deposited \non a c -sapphire (0001) substrate. The as -deposited samples are then spin -coated with PMMA 495 4A and \nexposed by an Elionix 50 keV E -beam lithography system for the nanowires patterning. After  developing, the \nsamples are transferred to the Kurt Lester system for Ar plasma dry etching. After cleaning the residual \nresists, waveguides with 400  nm gaps  between two contact pads are patterned again by E -beam lithography. \nFinally, Cr( 5nm )/Au(50nm) contacts are deposited.  \nExperimental techniques. VNA -FMR is used for detecting the thin film samples ferromagnetic resonance. For \npure LAFO samples, a f ield-modulated technique is used to achieve detection of the low linewidth resonance \npeaks. The samples are always mounted with the dc magnetic field along the in -plane [100] hard axis (𝝋=0o) \nof the LAFO thin film. ST -FMR measurements are carried out in a probe station with the external field always \napplied at 𝝋=70o with respect to the current direction. The field is modulated with a coil and signal is \ndetected by a lock -in. The DC is applied via a K eithley 2400. PSDs are measured via Keysight N9030B \nspectrum analyzer with a noise floor extension option. Input signals are amplified by an internal 29 dB low -\nnoise amplifier. During the measurement, the resolution bandwidth is always kept at 1MHz.  A Keithley 2400 \nis used for applying a DC into the SHNOs. The noise floor in this setup is -125 dBm.  To exclude  the sample -to-\nsample variation s of resistance s, ST-FMR s, and PSD maps, additional samples with the same geometry and \ncomposition are measured  and show n in Supplementary Note  6. \nMicromagnetic simulations. Micromagnetic simulations are run by Mumax3 micromagnetic simulator46. The \nmesh size is set to 300 × 300 × 5, and each cell size is 5 × 5 × 5 nm3. This length is smaller than the \nexchange length of Py and LAFO. The top Py layer is designed as a stripe in the center with dimension 400 × \n1500 × 5 nm3, while the bottom LAFO layer is extended to the boundaries and varied in thickness. Periodic \nboundary condition along the long axis of the Py nanowire are used to eliminate the demagnetization field \nfrom the end of Py stripe. The exchange con stant between t he Py and LAFO layers  is taken to be half of the \nharmonic mean of two layers. To reduce the spin -wave reflection at the boundary, we set an exponentially \nincreased damping region near the boundaries of the simulated region.  The spin current applied is restricted \nto the center region of the Py nanowire with dimension 500 × 400 × 5 nm3, since most of the spin current is \nconcentrated between two Au contacts. Threshold currents are found by running simulation s over 200 ns \nand slowly increasing th e applied current until 𝑚𝑧 starts to converge to a stable auto -oscillation state. In \norder to determine  the auto -oscillation spectrum in  the frequency domain, we set the current to be 1.2 \ntimes the threshold current found above and run the simulation for  500 ns. And then we use FFT algorithms \nto convert the magnetization evolution in the time -domain to  the frequency -domain. This method can be \nused to generate the auto -oscillation spectrum of the full device using the spatial averaged 𝑚̅𝑧(𝑡) or to \ngenera te the spatial profile of each auto -oscillation modes from  𝑚𝑧(𝒓,𝑡). 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(2022).  \n \nAcknowledgements.  \nThis research was supported by the Quantum Materials for Energy Efficient Neuromorphic Computing (Q -\nMEEN -C), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of \nScience, Basic Energy Sciences (BES) , under Award DE -SC0019273. Work at Stanford is supported by the U.S. \nDepartment of Energy, Director, Office of Science, Office of Basic Energy Sciences, Division of Materials \nSciences and Engineering under Contract No. DESC0008505  (X.Y.Z.).  S.C. was supported by the Air Force \nOffice of Scientific  Research under Grant No. FA 9550 -20-1-0293. D.A.O. was supported by the National \nScience Foundation under award DMR -2037652.  \nAuthor Contributions Statement.  \nH.R., Y.S. and A.D.K. conceived the experiment, X. Y.Z., S.C., and D.A.O synthesized the LAFO and LFO thin \nfilms  and performed part of the FMR characterization , while H.R. deposited the Py and Pt thin films. H.R. \nfabricated the SHNOs, performed the transport experiments and analyzed the data, including FMR , ST-FMR \nand PSD data. G.W. and H.R. performed the micromagnetic simulations. The manuscript was prepared by \nH.R. and A.D.K. in consultation with all other authors. All authors read and commented on the manuscript.  \nCompeting Inter ests Statement . \n \nThe autho rs declare no competing interests.  \n  14 Figures’ captions.  \n \nFigure 1 . (a) Schematic of the hybrid SHNO device and power spectral density (PSD) measurement setup. (b) \nFMR frequency versus resonance field for various unpatterned thin films and heterostructures, including for \nreference, Py5/Pt5 bilayers and LAFO and LFO thin films. (c) FMR linewidth as a function of frequency for the \nsame samples.  \nFigure 2.  (a) ST -FMR measurements of 400nm nanowire device for (a) Py5/Pt5 and (b) LAFO20/Py5/Pt5 at 7 \nGHz with the ap plied field at an angle 𝜙=70o to the wire. (c) Kittel model fitting curves of LAFO20/Py5/Pt5 \nfor 2µm stripe (blue circles) device and 400nm nanowire device. Green diamond shows fit for the peaks from \nbulk mode and red square for the peaks from edge mode. Maps of PSDs as a function of frequency and dc \nbias at a fixed field 𝐻=0.0817 T for 𝜙=70o for nanowire devices consisting of (d) Py5/Pt5, (e) LAFO4/Py5/Pt5, \n(f) LAFO10/Py5/Pt5, and (g) LAFO20/Py5/Pt5. The output power increases significantly for the thickest LAFO \nsample studied as in dicated by the colorscales above each PSD map.  \n \nFigure 3.  (a) FFT amplitude spectrum as a function of frequency from micromagnetic simulations of different \ndevices. The spectrum is acquired by doing FFT on the time revolution of spatial -averaged magnetizat ion \n𝑚̅𝑧(𝑡)  in the center region of nanowire excited by a spin current. (b) Top views of spatial FFT images on the \nPy layer of Py5/Pt5 obtained at EM and BM resonance frequencies. (c) Top view of spatial FFT images of the \nPy layer (top) and LAFO layer (b ottom) of a LAFO20/Py5/Pt5 device. The image size for the Py5 layer is \n1500×400 nm2 and for the LAFO20 layer is 1500 ×1500 nm2. The logarithmic colorscale is on the right where \nthe color represents the FFT amplitude of 𝑚𝑧(𝑥,𝑦). \n \nFigure 4.  PSD maps as a function of external magnetic field and frequency for (a) Py5/Pt5, (b) LAFO4/Py5/Pt5, \n(c) LAFO10/Py5/Pt5, and (d) LAFO20/Py5/Pt5 SHNOs. Resonance frequency as a function of external \nmagnetic field for the device (e ) Py5/Pt5, (f) LAFO4/Py5/Pt5, (g) LAFO10/P y5/Pt5, and (h) LAFO20/Py5/Pt5 \nobtained by FMR (brown dash dot), ST -FMR (green dash), and PSD (blue and red solid) measurements. \nDominant resonance modes are highlighted by a wider line. Notice the resonance frequency obtained from \nPSD has two distinctive peaks, which are associated with bulk mode and edge modes.  \n \nFigure 5.  PSD maps for LFO15/Py5/Pt5 as a function of (a) bias current and (b) magnetic field. (c) Dispersion \ncurves of LFO15/Py5/Pt5 obtained from FMR, ST -FMR, and PSDs measurements. (d) PSD spe ctrums of \ndifferent samples at fixed 𝐻=0.045 T. Lines are shifted upward 15 dB for each spectrum (e) Max signal (dB \nover NF) and max Q factor for different devices obtained from the PSDs in (a) and Fig. 4a -d. \n "    },    {        "title": "2201.03883v4.Effect_of_a_Gaussian_random_external_magnetic_field_with_spatio_temporal_variation_on_compensation_in_Ising_spin_1_2_trilayered_square_ferrimagnets.pdf",        "content": "arXiv:2201.03883v4  [cond-mat.stat-mech]  29 Jun 2023Effect of a Gaussian random external magnetic field with\nspatiotemporal variation on compensation in Ising spin-1/ 2\ntrilayered square ferrimagnets\nSoham Chandra∗1\n1Department of Physics, Presidency University, 86/1 Colleg e Street, Kolkata -700 073, India\nAbstract\nIn this work, an extensive Metropolis Monte Carlo simulatio n is performed to investigate the steady-state mag-\nnetic and thermodynamic behaviour of a trilayered spin-1/2 Ising ferrimagnet with square monolayers, driven by\nexternal Gaussian random magnetic field with certain spatio -temporal variations. Such thinferrimagnetic systems\nexhibit compensation phenomenon and thus are potentially i nteresting candidates for several technological applica-\ntions. Here, two distinct theoretical atoms, A and B make up t heABAandAABtypes of configurations in which\nthe like atoms (A-A and B-B) ferromagnetically interact and the unlike atoms (A-B) interact antiferromagnetically.\nDepending upon the strength of the spatio-temporally varyi ng Gaussian random field, the compensation and criti-\ncal points shift and steady-state magnetic behaviours chan ge between the different distinct types of ferrimagnetic\nbehaviours. The compensation phenomenon even vanishes aft er crossing a finite threshold of the standard deviation\nof the magnetic field for particular choices of the other cont rolling parameters. Consequently, in the Hamiltonian\nparameter space of both configurations, islands of ferrimag netic phase without compensation appear within the\nphase area with compensation of field-free case. The areas of such islands grow with an increasing standard devi-\nation of the external field, σ, obeying the scaling relation: f(σ,A(σ)) =σ−bA(σ) withbABA= 1.913±0.137 and\nbAAB= 1.625±0.066 . These values of exponents match within the statistical interval with those obtained with\nthe uniform random magnetic field.\nKeywords: Spin-1/2 Ising square trilayer; Gaussian random external magnet ic field; Spatio-temporal variation in\nfield; Metropolis Monte Carlo simulation; Compensation temperature ; No-compensation islands\n∗E-mail addresses: soham.rs@presiuniv.ac.in ; sohamc07@g mail.com\n11 Introduction\nIn the last few decades, research on layered magnetic su-\nperlatticeshasshownthattheycanhavelowdensity,trans-\nparency and mechanical strength, which find potential ap-\nplications in magnetic recording, information storage and\nmagneto-resistive sensors [1]. Amongst them, few-layered\nferrimagnetic materials are often found to have physical\nproperties very different from the bulk. Though ferrimag-\nnetism was discovered in 1948 [2], the experimental inter-\nest in ferrimagnetism has grown up rapidly with the dis-\ncovery of thin film growth techniques, like, metalorganic\nchemicalvapourdeposition(MOCVD)[3],molecular-beam\nepitaxy (MBE) [4], pulsed laser deposition (PLD) [5], and\natomic layer deposition (ALD) [6]. Such experimental ad-\nvancements have made the growth of bilayered [7], tri-\nlayered [8], and multilayered [9–11] systems with desired\ncharacteristics a reality. Expectedly, theoretical and com-\nputational studies of layered magnets have also gained\nmomentum. For a multilayered ferrimagnet, magnetiza-\ntions of each of the monolayers may evolve differently\nwithtemperature. Combinationofsuchdifferentmagnetic\nbehaviours in specific cases, exhibit compensation . The\nCompensation point for layered magnets is that specific\ntemperature, lower than the critical temperature, where\nthe total magnetization of the system vanishes but indi-\nvidual layers remain magnetically ordered [2].\nThe temperature dependence of total magnetization\nof layered magnets with antiferromagnetic interlayer cou-\npling, exhibiting a ferrimagnetic ground state, may show\nmagnetic compensation. Compensation is not related to\nthe criticality of the system but the magnetic coerciv-\nity shows singularity at the compensation point [12,13]\nfor some ferrimagnetic materials. Strong temperature de-\npendence of the coercive field around the compensation\npoint and compensation point about the room tempera-\nture, make such ferrimagnets useful for thermomagnetic\nrecording [12]. At the compensation point, a small driv-\ning field can reverse the sign of magnetization. That is\nwhy, the Magnetocaloric Effect in the vicinity of com-\npensation temperature is studied in [14]. So the control\nand manipulation of the compensation phenomenon be-\ncomes an important topic of research from the point of\nview of theoreticians and experimentalists. A few related\nexamples, in this direction, follow. In [15], it has been\nshown that, polycrystalline molecular magnets, for ex-\nample,N(n−CmH2m+1)4FeIIFeIII(C2O4)3[m= 3−5]\nhave compensation temperatures near 30 K depending on\nthe type of cation A+. This particular kind of system\nwas simulated by Monte Carlo Simulation with a mixed\nspin model of spin-2 and spin-5 /2 on a layered honeycomb\nstructurewithnearestneighbourinteractionstoclarifythe\neffects ofinterlayerinteractionsandsingle-ionanisotropies\noncompensation[16]. Inthelastdecade,theferrimagnetic\ntrilayered structure,\nFe3O4(25nm)/Mn3O4(50nm)/Fe3O4(25nm) , prepared\nbyoxygen-plasma-assisted MBE, was shown to have mag-\nnetic compensation due to the formation of domain-wall-\nlike configurations, mainly in Fe3O4[17].\nTo examine compensation, numerical studies on the\nequilibrium (field-free and in the presence of static fields)properties of layered Ising ferrimagnets on various lattice\ngeometries have been performed [18–25]. The trilayered\nferrimagnetic spin-1/2 Ising superlattices on square sub-\nlayers of ABA and AAB type [Figure 1] in the current\nstudy have an advantage as, in field-free cases they show\ncompensation effect [21–25], even without site-dilution or\nmixed-spin structures. So, they are among the simplest\nsystems to display compensation. The magnetic descrip-\ntion provided by traditional Monte Carlo Simulation is in\ngood agreement with the description provided by Inverse\nabsolute of reduced residual magnetisation (IARRM) and\nTemperature interval between Critical and Compensation\ntemperatures (TICCT) [23–25] for both types of sand-\nwiched configurations with square and triangular mono-\nlayers. The equilibrium studies are now well established.\nHowever,realsystemscannotpreservethepristinechar-\nacter, and disorder is almost unavoidable in the descrip-\ntion of any spin model. The effect of spin-0 impurities (a\nkind of static disorder) on compensation in trilayered fer-\nrimagnets (with triangular monolayers) is numerically in-\nvestigatedinarecentarticle[26]. Thedisorderforsystems\nstudied in this work can as well be time-dependent. A\nfew sources of time-dependent disorders [27] are: (a) time\nvarying interaction strength between pairs of spins; (b)\nthenumberofinteractingspinsforaparticularsitemaybe\nchangingwith time; (c) Eventhe natureofthe spins(mag-\nnitude of spin, different magnetic atoms etc.) in the or-\nderedstructuremayvarywithtime. Asaresult,modelling\nsuch temporallyvariablecompositional and morphological\ndisorders is extremely difficult. In an attempt in this di-\nrection[27], thetrilayeredspin −1/2superlatticeshasbeen\nsubjected to a uniform random external magnetic field\nwith spatio-temporal variations. So the dynamic Hamilto-\nnian for these systems then is a Random Field Ising Model\n(RFIM). RFIM was developed by Larkin in 1970 [28] and\nis historically used to model many remarkable static and\ndynamicbehavioursindisorderedsystems[29]. Prominent\nexamples of experimentally observed Random field type\nphenomenologyindisorderedsystemsrangefromdisorder-\ninduced frustration and electronic transport in disordered\ninsulators to melting of intercalates in layered compounds\ne.g.TiS2, [30–37] to name a few. The simulational results\nand subsequent analyses in [38] and references therein ex-\nplain how RFIM may describe various types of noises in\nmagnets.\nFor a large variety of random field distributions, the\ncriticalexponentsofthe powerlawsareindependent ofthe\nparticular choice [38]. But it is yet to be established how\na change in the nature of the continuous random external\nfield, from uniform random [27] to Gaussian random, af-\nfects the compensation phenomenon in the superlattices\nof Figure 1. Uniform random field has a lower and upper\ncut-off whereas the Gaussian random field admits all the\npossible real values of the field. Evidently, only one physi-\ncalmeasurei.e. thestandarddeviationofthesecontinuous\nrandom field distributions is common to both of these and\nprovides us with a description of randomness, irrespective\nof the nature of the distribution. Thus the objective of\nthe current study is to examine the influence of the Gaus-\nsian random external magnetic field (or, more specifically,\nthe standard deviation of the Gaussian distribution) with\n2certainspatiotemporalvariationonthecompensationphe-\nnomenon associated with a trilayered spin-1 /2 Ising ferri-\nmagnet with square monolayers. The plan of the paper\nfollows. The layered magnetic model and the dynamic\nHamiltonian are described, in detail, in Section 2. The\nsimulational details are described in Section 3. Section 4\ncontains the numerical results and associated discussions.\nIn Section 5, the summary of the work is presented.\n2 Outline of the Model\nThe ferrimagneticIsingsuperlatticein thisstudy issimilar\nto the one used in [27]. Each site has spin value, s= 1/2,\nand contains three magnetic sub-layers on square lattice.\nEach alternate layer is exhaustively composed of by either\nA or B type of atoms. The magnetic atoms on the top\nand bottom layers do not interact. [Fig.-1]. The mag-\nnetic interaction between the like atoms (A-A and B-B) is\nferromagnetic and between dislike atoms (A-B) is antifer-\nromagnetic. Additionally to the cooperative interactions,\nthez-component of spins, Sz\niat each site couples with\na longitudinal Gaussian random external magnetic field,\nhi(t). At a particular site, this external field varies in time\nand at any time instant, the values of this local field are\ndifferent over the lattice sites.\nThe spins interact Ising-like, limited to the nearest\nneighbours only, in-plane as well as inter-plane. Recent\ndiscoveries of CrGeTe 3[39],CrI3[40,41] and FeX2(X=\nCl,Br,I) [42] show the nearest-neighbour Ising interac-\ntions in few-layer limits of a magnetic material to be a\nreality. The time dependent Hamiltonian for such a tri-\nlayered ferrimagnetic system is:\nH(t) =−J11/summationdisplay\n<t,t′>Sz\ntSz\nt′−J22/summationdisplay\n<m,m′>Sz\nmSz\nm′\n−J33/summationdisplay\n<b,b′>Sz\nbSz\nb′−J12/summationdisplay\n<t,m>Sz\ntSz\nm−J23/summationdisplay\n<m,b>Sz\nmSz\nb\n−/summationdisplay\nihi(t)Sz\ni (1)\n/an}bracketle{tt,t′/an}bracketri}ht,/an}bracketle{tm,m′/an}bracketri}ht,/an}bracketle{tb,b′/an}bracketri}htarenearest-neighborpairsin the top,\nmid and bottom layers respectively and /an}bracketle{tt,m/an}bracketri}ht,/an}bracketle{tm,b/an}bracketri}htare,\nrespectively, pairs of nearest-neighbor sites in, top & mid\nand mid & bottom layers. The first three terms are for\nthe intra-planar ferromagnetic interactions. The fourth\nand fifth terms are for the inter-planar nearest neighbour\ninteractions, between top and mid layers and mid and\nbottom layers, respectively. The sixth term denotes the\nspin-field interaction term of all the spins to the exter-\nnal Gaussian random magnetic field, at time instant t.\nBecause of the interactions: JAA>0 ,JBB>0, and\nJAB<0. For an ABA type system :J11=J33=JAA;\nJ22=JBBandJ12=J23=JAB. For anAAB type sys-\ntem:J11=J22=J12=JAA;J33=JBBandJ23=JAB.\nThe local, Gaussian random external magnetic field\nvalueshi(t) at any site, iat time instant t, is drawn from\nthe following probability distribution:\nPGaussian(hi(t)) =1√\n2πσ2exp/parenleftbigg−h2\ni(t)\n2σ2/parenrightbigg\n(2)\nBox-Muller algorithm [43] is used to get a Gaussian\ndistribution of zero mean and standard deviation, σ. Thesimulational details of implementation and associated im-\nportant characteristicsof such a time-varying field are dis-\ncussed, in detail, in Appendix A.\n3 Simulation Protocol\nThe Metropolis single spin-flip algorithm [44,45] is em-\nployed for simulation of the system. The three square\nmonolayers has L2sites with L= 100. The z-components\nof spin projections of nearest neighbours, Sz\ni(Sz\ni=±1)\ncontribute to the cooperative and spin-field interactions.\nAt each site i, a local, time-varying, Gaussian random\nfield,hicouples with each spin. In [22], Compensation\ntemperature has been found to be practically constant for\nL/greaterorequalslant60, for the systems of this study in field-free study.\nFrom Appendix B, we see the compensation point is still\nindependent of the system size in the vicinity of L= 100,\nin presence of the external Gaussian field of this study.\nSo the chosen system size is statistically reliable for sim-\nulational investigation. For both the configurations, the\nsystems are initiated at a high temperature paramagnetic\nphase, with randomly selected half of the total spin pro-\njections being “UP” (with Sz\ni= +1) and the rest being\n“DOWN” (with Sz\ni=−1) (Using 1 for 1 /2 fixes up the\nenergy scale). At a fixed temperature T, the spin flipping\nis governedby the Metropolis rate [46,48], of Equation [3]:\nP(Sz\ni→ −Sz\ni) = min{1,exp(−∆E/kBT)}(3)\nwhere the associated change in internal energy in flipping\nthei-th spin projection from Sz\nito−Sz\ni, is ∆E. Simi-\nlar 3L2individual, random single-spin updates constitute\none Monte Carlo sweep (MCS) of the entire system (unit\nof time in this study). Periodic boundary conditions in-\nplane and Open boundary conditions along the vertical\nare employed.\nThe systems are kept for 105MCS at every temper-\nature step. The last equilibrium configuration at the\nprevious highertemperature acts as the starting configu-\nration at a new lowertemperature. For the first cumu-\nlative 5×104MCS: the system is allowed to equilibrate\nin the field-free environment first and then the system at-\ntains steady-state in the presence of the external field (the\nprovided time, for attaining eqilibrium and steady state,\nis sufficient [Refer to (a) Figures 9 & 10 and discussions\ntherein and (b) Appendix C]). After that the externalfield\nis kept switched on for the next 5 ×104MCS. So for the\nsystems, theexposuretimeintervalin thefield, δis5×104.\nThe temperatures are measured in units of JBB/kB. For\neach of the fixed standard deviation of the Gaussian ran-\ndom field, the system is observed for seven equidistant\nvalues of JAA/JBB, from 0.04 to 1.0 with an interval of\n0.16. For each fixed value of JAA/JBB,JAB/JBBis de-\ncreased from −0.04 to−1.0 with a step of −0.16.\nFor any combination of JAA/JBBandJAB/JBB, and\na fixed standard deviation of the external field σ, the\ntime averages of the following quantities are calculated\nafter equilibration at any temperature, ( T) in the follow-\ning manner [27,49]:\n(1) Sublattice magnetisations are calculated at time\ninstant say, t, by:\nMq(T,t) =1\nL2L/summationdisplay\nx,y=1/parenleftbig\nSz\nq(T,t)/parenrightbig\nxy(4)\n3Figure 1: (Colour Online) Miniaturised versions (3 ×4×4) of (a) ABA and (b) AAB square trilayered ferrimagnet\nwith two types of theoretical atoms, AandB. Each of the sublattices of the ferrimagnetic systems are formed on\nsquare lattice. The actual simulation is carried out on a system with Nsites= 3×100×100 . Courtesy: [27]\nThe time averaged sublattice magnetizations is calculated\nby:\n/an}bracketle{tMq(T)/an}bracketri}ht=1\nδ/integraldisplayt0+δ\nt0Mq(T,t)dt (5)\nwhereqis to be replaced by t,morbfor top, mid and\nbottom layers.\n(2) The order parameter ,O(T), forthe trilayerattem-\nperature, Tis defined as:\nO(T) =1\n3(/an}bracketle{tMt(T)/an}bracketri}ht+/an}bracketle{tMm(T)/an}bracketri}ht+/an}bracketle{tMb(T)/an}bracketri}ht) (6)\n(3) Fluctuation of the order parameter, ∆O(T) at\ntemperature, Tas follows:\n∆O(T) =/radicalBigg\n1\nδ/integraldisplayt0+δ\nt0[M(T,t)−O(T)]2dt(7)\nwhereM(T,t) is the total magnetisation of the whole sys-\ntem, attemperature, T,calculatedatthe t-thtimeinstant.\n(4) The time averaged value of cooperative energy\nper site ,/an}bracketle{tE(T)/an}bracketri}ht, at temperature, T, is determined by:\n/an}bracketle{tE(T)/an}bracketri}htABA=−1\n3L2δ/integraldisplayt0+δ\nt0dt[JAA(/summationdisplay\n<t,t′>Sz\ntSz\nt′\n+/summationdisplay\n<b,b′>Sz\nbSz\nb′)+JBB/summationdisplay\n<m,m′>Sz\nmSz\nm′\n+ JAB(/summationdisplay\n<t,m>Sz\ntSz\nm+/summationdisplay\n<m,b>Sz\nmSz\nb)] (8)\nand\n/an}bracketle{tE(T)/an}bracketri}htAAB=−1\n3L2δ/integraldisplayt0+δ\nt0dt[JAA(/summationdisplay\n<t,t′>Sz\ntSz\nt′\n+/summationdisplay\n<m,m′>Sz\nmSz\nm′/summationdisplay\n<t,m>Sz\ntSz\nm)\n+ JBB/summationdisplay\n<b,b′>Sz\nbSz\nb′+JAB/summationdisplay\n<m,b>Sz\nmSz\nb](9)(5) The fluctuation of the cooperative energy per\nsiteat temperature, T, by:\n∆E(T) =/radicalBigg\n1\nδ/integraldisplayt0+δ\nt0[E(T,t)−/an}bracketle{tE(T)/an}bracketri}ht]2dt(10)\nwhereE(T,t) is the instantaneous cooperative energy per\nsite, for the system at time tand temperature, T, within\nthe exposure interval, δ.\nAt the pseudo-critical temperatures, the fluctuations\npeak. Around this temperature close range simulations\nwere performed to narrow down the position of the re-\nportedcriticaltemperatureswith anaccuracyof, ∆ Tcrit=\n0.04 . Compensation temperature ( < Tcrit), where the\ntotal magnetisation again becomes zero, is determined by\nlinearinterpolationfromthetwoneighbouringpointsacross\nthe zero of magnetization in the plots of order parameter\nversus temperature [e.g. Figure 2(a)]. The upper bounds\nof linear interpolation provide us with an estimate of the\nerrors with the values of compensation points [47]. The\nJackknife method [48] is used to provide an estimate of\nthe errors with the magnetizations and fluctuations.\n4 Results and discussions\n4.1 Thermodynamic Response\n4.1.1 Magnetization versus temperature:\nIn the few cases in Figure 2, for a fixed standard deviation\nof the external field with characteristics of Section A, we\nsee the compensation and critical temperatures shift as we\nincrease the magnitude of any of the coupling strengths.\nAs we increase the magnitude of either of the coupling\nstrengths, we can identify the nature of the magnetization\ncurves by the N` eel classification scheme. A detailed dis-\ncussion on the classification schemes (e.g. P-type, N-type,\nR-type etc.) can be found out [50–52]. For the ABA\nconfiguration :in Figure 2 (a) withJAA/JBB= 0.20\nandσ= 0.60: forJAB/JBB=−0.04 we see a P-type\n4magnetization; all the intermediate curves are of N-type\nand for JAB/JBB=−1.00 we see a R-type magneti-\nzation; and in Figure 2 (b) withJAB/JBB=−0.20\nandσ= 0.60: for JAA/JBB= 0.04 we see a P-type\nmagnetization; the intermediate curves up to are of N-\ntype and for JAB/JBB= 1.00 we see a Q-type magne-\ntization. The L-type, within braces and not explicitly\nshown, would be encountered while moving from the for-\nmer type to the latter. For the weakest combination of\ncoupling strengths, we witness the field-driven vanishing\nof compensation. For the AAB configuration :in Fig-\nure 2 (c) withJAA/JBB= 0.20 andσ= 0.76: for\nJAB/JBB=−0.04 we see a P-type magnetization and for\nJAB/JBB=−0.20 we see an L-type magnetization and\nall other curves are of N-type and in Figure 2 (d) with\nJAB/JBB=−0.20 andσ= 0.76: forJAA/JBB= 0.04\nwe see a P-type magnetization; for JAA/JBB= 0.20 we\nsee an L-type; the JAA/JBB= 0.36,0.52 curves are of N-\ntype; the JAA/JBB= 0.68,0.84 curves are of Q-type and\nforJAB/JBB= 1.00 we see a P-type magnetization again.\nFor the two weakest combinations of coupling strengths,\nwe again witness the field-driven vanishing of compensa-\ntion inthe AAB configuration. The magneticresponseun-\nder the influence of Gaussian random magnetic field with\nspatio-temporal variation is quite similar to what we see\nfor the spatio-temporally varying uniform random mag-\nnetic field in [27].\nNow we will focus on the effects of the randomness\nof the external Gaussian random field has on the mag-\nnetic response. For any fixed combination of the coupling\nstrengths, an increase in the value of the standard de-\nviation of the external Gaussian random field decreases\nthe compensation and critical temperatures for both the\nABA and AAB type of sandwiched structures [Figures 3\nand 4]. Similar to the uniform random field [27], as we\nincrease the randomness of the external Gaussian random\nfield the decrement for the compensation temperatures is\nmuch more visible than the decrement of critical temper-\nature, with or without compensation. The field driven ab-\nsence of compensation phenomenon is also present in Fig-\nure 3(a) for ABA and Figures 4(a)&(b) for AAB configra-\ntion. Like the uniform random external field, we can see\nand identify the nature of ferrimagnetic curves and field-\ndriventransitionsamongtheminFigures3and4. Accord-\ning to the classification schemes of references [50–52]: (A)\nFor ABA , in Figure 3(a), the magnetic response changes\nfrom type- N(σ= 0,0.20) to type- L(σ= 0.40) to type- P\n(σ= 0.60,0.76,1.00); In Figure 3(b), for all the fields only\ntype-Nresponse is witnessed; and in Figures 3(c)&(d),\nwe see only type- Qresponse for all the fields. (B) For\nAAB, in Figure 4(a), the transition happens from type- N\n(σ= 0.00,0.20) to type- P(σ= 0.40,0.60,0.76,1.00) via\ntype-L; Similar transitions are witnessed in Figure 4(b);\nin Figure 4(c) & (d), all the magnetic responses are of\ntype-Q.\n4.1.2 Fluctuations versus temperature:\nTo better understand (a) the shifts of compensation and\ncritical temperatures and (b) the reason behind the field\ndriven vanishing of compensation, we will now observe\nboth the fluctuations: fluctuation of the order parame-ter and fluctuation of the cooperative energy per site, as\nfunctions of temperature while standard deviation of the\nexternal Gaussian random field acts as the parameter. We\nwitness a plateau with a smeared peak in the vicinity of\ncompensation point for both the fluctuations of order pa-\nrameter and energy in Figures 5 and 6 for the ABA and\nAAB configurations respectively. We clearly see the com-\npensation and critical temperatures moving towards lower\ntemperature values, with the increase in the standard de-\nviation of the Gaussian field. Even the smeared peaks at\nthe low temperature segments flatten out as the standard\ndeviation is increased in steps, which signifies the vanish-\ning of compensation. So we again witness a field-driven\nvanishing of compensation. At the lower parts of the tem-\nperature axis, the increase in both the fluctuations imply\nthat magnetic ordering is gradually decreasing with the\nincrease of the standard deviation of the external field.\nTo understand these arguments in the context of van-\nishing of compensation, let’s study the lattice morphol-\nogy (or, spin-density plots) at the zero-field compensation\npointsfor both ABA and AAB configurations with σ=\n{0.00,0.20,0.76}(σ= 0.00 means absense of the external\nfield). For the ABA configuration, as the external field is\nswept with σ= 0.20, the surface A-layers lose significant\nmagnetisation(i.e. decreaseinmagneticordering/increase\nin randomisation) at the Tcomp(σ= 0) = 0 .502 [Refer to\nFigure 7]. Consequently we need to lower the temper-\nature further to achieve the required magnetic ordering\n(i.e. magnetisation) in the surface A-layers, so that they\ncumulatively cancel out the magnetisation of the B-layer\nto produce a compensation point at a lower temperature\nvalue [Refer to Figure 3]. The surface A-layers almost be-\nhave identically. When the standard deviation of the ex-\nternal field is increased to σ= 0.76, we can see (from the\nvalues of magnetisation beneath the spin-density plots),\nthe magnetisation is very much reduced for the surface A-\nlayers and conseqently, lowering of the temperature even\nfurther isn’t able to generate enough magnetic ordering in\nthe A-layersto cancelout the magnetizationofthe B-layer\nto create a compensation point leading to the field-driven\nvanishing of the compensation point in the steady state.\nThereductionofmagneticorderingleadstoincreaseinthe\nfluctuations of both, order parameter (equivalently, mag-\nnetisation) and cooperative energy per site. Following the\nsimilar argument we can readily understand the shift and\nvanishing ofcompensation in the AAB configurationwhen\nthe standard deviation of the external field is increased for\na fixed combination of coupling strengths [Refer to Figure\n8].\n4.1.3 Magnetization versus time:\nThe section is devoted to the behaviour of sublattice mag-\nnetisations with time asthe field is switched ON, at a suit-\nablyverylowtemperature T= 0.01. Wehavechosenthree\ndistinctcombinationsofthecouplingratios, JAA/JBBand\nJAB/JBB, as (0.04,-0.20), (0.20,-0.20), (1.00,-0.52) for the\ntwo values of the standard deviation, σ={0.20,0.60 }of\nthe external Gaussian random field for both the ABA and\nAAB configurations [Figures 9 and 10]. At T= 0.01, till\nt= 5×104MCS, the Hamiltonian only has the coopera-\ntive part and the sublattice and total magnetisation of the\nsystem remains in equilibrium. Just after t=t0= 5×104\n5-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)JAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00ABA (a)\nJAA/JBB=0.20\nσ=0.60\nP,(L),N,R-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)JAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00ABA (b)\nJAB/JBB=-0.20\nσ=0.60\nP,(L),N,Q\n-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)JAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00AAB (c)\nJAA/JBB=0.20\nσ=0.76\nP,L,N-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)JAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00AAB (d)\nJAB/JBB=-0.20\nσ=0.76\nP,L,N,Q,P\nFigure 2: (Colour Online) Plots of Order parameter versus reduced temperature for a 3 ×100×100 system of\ntype: (a) ABA with JAA/JBB= 0.20 and variable JAB/JBBforσ= 0.60; (b) ABA with JAB/JBB=−0.20 and\nvariable JAA/JBBforσ= 0.60; (c) AAB with JAA/JBB= 0.20 and variable JAB/JBBforσ= 0.76; (d) AAB\nwithJAB/JBB=−0.20 and variable JAA/JBBforσ= 0.76 . Shift of Compensation and Critical temperatures\ntowards higher temperature end are witnessed with increase in any of the coupling ratios. The field-driven vanishing\nof compensation is witnessed for the weakest combination of couplin g strengths.\nMCS, the external field is switched ON and the sublayer\nand total magnetisations start to react. As in the case\nwith uniform random field [27], both the systems, ABA\nand AAB, reach the steady state very quickly. Conclusive\nfeatures are unraveled in these cases.\nFor the ABA configuration , we see in the top\npanel: Figure9(A) ,withJAA/JBB= 0.04andJAB/JBB=\n−0.20, both the surface A-layers react magnetically for\nboth the standard deviations of the external Gaussian\nfield. The reason is, the per site cooperative energy of the\nA-layers is comparable to the spin-field energy. But the\nmid B-layer with dominant in-plane coupling, remains in\nits equilibrium magnetic state. So it is evident that the re-\nduction (or, destruction) of magnetic order in the sublay-\nersistheresultofthecompetitionbetweenthecooperative\nand spin-field energies and spin-field energies dominating\nthe cooperative part in the relevant cases. The similar in-\nference is valid for the middle panel: Figure 9 (B) ,\nwith the σ= 0.60, where per site spin-field energies are\ncomparable to the cooperative part of the Hamiltonian for\nthe surface A-layers. In the bottom panel: Figure 9\n(C), with the σ= 0.20 andσ= 0.60, all the three sublay-\ners don’t react. Here, the cooperative part of the Hamil-\ntonian for the surface A-layers dominates the steady state\nper site spin-field energies. The magnetization curves forthe top and bottom layers overlap for most of the times as\nthey have identical interacting magnetic neighborhood. In\nFigure 9(A) withJAA/JBB= 0.04,JAB/JBB=−0.20\nandσ= 0.60, weseethereductioninmagnetizationofthe\nsurface layers in presence of the external field leads to the\nvanishingofcompensationeveninthe lowestpossibletem-\nperature in simulation. Even in the lowest simulational\ntemperature, the cumulative steady state value of mag-\nnetization of the surface A-layers becomes smaller than\neven the steady state value of magnetization of the mid-\ndle B-layer (that is why the total magnetisation remains\npositive, same signature as the B-layer, in the presence of\nthe field). Even in the lowest temperature,\n|/an}bracketle{tMt(T)+Mb(T)/an}bracketri}ht|<|/an}bracketle{tMm(T)/an}bracketri}ht|\nwhich causes compensation to disappear in the presence\nof the field. This is the reason behind all the instances\nwith field-driven vanishing of compensation.\nFor the AAB configuration , the influence of the\nbottom B-layer is limited to the middle A-layer (because\nof the nearest neighbour Ising interactions). So the top\nA-layer gets much more affected by the external field than\nthe middle A-layer. In Figure 10, we can understand it by\nsimply noticing the orange line for the magnetisation of\nthe top A-layer. In the top panel: Figure 10(A) , with\n6-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (a)\nJAA/JBB=0.04\nJAB/JBB=-0.20 N,L,P-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (b)\nJAA/JBB=0.04\nJAB/JBB=-0.84 N\n-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (c)\nJAA/JBB=0.84\nJAB/JBB=-0.20Q -0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (d)\nJAA/JBB=0.84\nJAB/JBB=-0.84Q\nFigure 3: (Colour Online) Magnetic response of the ABA trilayered (3 ×100×100) system. The shift of both,\nthe compensation (where it is present) and critical temperatures towards the low temperature ends and shift of the\nmagnetic behaviours between N,L,P,Q etc. type of ferrimagnetism, w ith increase in the standard deviation of the\nuniform random external magnetic field, are clearly visible in all these plots. The type L within brackets is explicitly\nnot seen in the plots but encountered in-transition. Where, the er rorbars are not visible, they are smaller than the\narea of the point-markers.\nJAA/JBB= 0.04,JAB/JBB=−0.20 andσ= 0.60,1.00,\nthe spin-field energies per site dominates the cooperative\nenergy per site of the A-layers. Consequuently, the A lay-\ners lose much of the magnetic ordering at even the lowest\ntemperature, the extent of randomisation is much more\nprominent for the top A-layer and the top A-layer is al-\nmost completely randomised when σ= 0.60. So the com-\nbined magnetisation of A-layers is not enough to nullify\nthe magnetisation of the bottom B-layer , leading to van-\nishing of compensation. In Figures 10(B) and (C) ,\nwe can explain the behaviour in light of the discussions\nabove. A few more instances of both the configurations\nforσ= 1.00 are presented in Appendix D to supplement\nthis discussionforabetter understanding. Suchbehaviour\nis qualitatively similar to what we have seen with uniform\nrandom field [27].\nThis is another example of dynamic field-driven\nvanishing of compensation in the Ising spin-1/2 tri-\nlayers, driven by Gaussian random external field with spa-\ntiotemporalvariation. Thebottompanelwith JAA/JBB=\n1.00 andJAB/JBB=−0.52 for both the configurations\nsupports that vanishing of compensation is a result of the\ncompetition between the cooperative and spin-field ener-\ngies.4.2 Phase Diagram and Scaling\nThe phase diagrams in Figure 11 depict the effects of a\nGaussian random external field on the Hamiltonian pa-\nrameter space for both the trilayered ferrimagnetic sys-\ntems. Compensation temperature merges with the critical\ntemperatureforhighervaluesof JAA/JBBwhen|JAB/JBB|\nis fixed or vice-versa, just like in the zero-field case. In\nFigure 11, the phase diagrams are drawn following the\nzero-field cases [21,22,25] where, compensation is present\n(marked by P) within the orange areas and absence of\ncompensation is marked by white areas (marked by A).\nThe presence of the external Gaussian field [From σ=0.2,\nonwards]createstheenclaveorislandswithno-compensation\nwithin the phase area where parameters support compen-\nsation. These closed areas or No-Compensation Islands\n(NCI) grow as the randomness (equivalently, standard de-\nviation) of the external field increases, similar to the case\nwith uniform random external field [27]. In Figure 12, we\npresent the plots of absolute area and rate of increase of\nabsolute area versus the standard deviation of the applied\nfield. Linear interpolation/extrapolation is employed to\nobtain the closed curve for the boundary of the NCI, and\nthe fractional area is obtained by Monte Carlo Integra-\ntion [53]. Central difference formula is employed to find\nout the rate of increase of the area of NCIs to determine\n7-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (a)\nJAA/JBB=0.04\nJAB/JBB=-0.20 N,(L),P-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (b)\nJAA/JBB=0.04\nJAB/JBB=-0.84 N,(L),P\n-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (c)\nJAA/JBB=0.84\nJAB/JBB=-0.20Q -0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (d)\nJAA/JBB=0.84\nJAB/JBB=-0.84Q\nFigure 4: (Colour Online) Magnetic response of the AAB trilayered (3 ×100×100) system. The shift of both,\nthe compensation (where it is present) and critical temperatures towards the low temperature ends and shift of the\nmagnetic behaviours between N,L,P,Q etc. type of ferrimagnetism, w ith increase in the standard deviation of the\nuniform random external magnetic field, are clearly visible in all these plots. The type L within brackets is explicitly\nnot seen in the plots but encountered in-transition. Where, the er rorbars are not visible, they are smaller than the\narea of the point-markers.\nthe nature of curve of the area of NCI versus standard\ndeviation of the external field.\nTo analyse the nature of the curves of absolute area\nversusthestandarddeviation,wenote,thecurve(inRED),\ncomes out to be a mixture of superlinear and sublinear in\nnature,for the ABA configuration . From the BLUE\ncurve ofslope versusstandard deviation ofthe field in Fig-\nure 12(a), in the vicinity of σ= 0.60 we find the curve is\nsublinear (almost linear). Again at and after σ= 0.88,\nthe area changes in a prominent sublinear manner. At\nthe low random fields around σ= 0.20 and around mod-\nerately high randomness around σ= 0.70, the nature is\nsuperlinear. For the AAB configuration , the area of\nNCIs increase superlinearlyon averagetill σ= 0.64. After\nthat the behaviour is almost linear. Now the scaling be-\ntween the magnitude of the area of NCIs and the standard\ndeviation of the field can be performed by the following\nscaling function [27]:\nf(A(σ),σ) =σ−bA(σ) (11)\nThe scaling exponents come out to be: for ABA: bABA=\n1.913±0.137 and for AAB: bAAB= 1.625±0.066. A\nfaithful estimate of error in the values of the exponents is\nobtained by the standard deviation among all the sets of\ndata.5 Summary and Conclusion\nIn the Ising model, the coupling constants are tradition-\nallytakentobe translationallyinvariant. Alongwith that,\ncompeting ferromagnetic and antiferromagnetic interac-\ntions in the systems of Figure 1 throw up exciting bulk\nbehaviour e.g. Compensation. The equilibrium studies on\nthese systems [21–24] have shown us the prevalent com-\nplexity in deriving conditions for the existence of compen-\nsation. Now, in the current article, a Metropolis Monte\nCarlostudy hasbeen performed onthe magneticand ther-\nmodynamic responses ofthe systems ofFigure 1 under the\ninfluence ofa GaussianRandom external field with spatio-\ntemporal variations.\nIt is time to discuss the implications of the current\nwork. From [27], we have a fair idea about how such sys-\ntems react under the influence of uniform random mag-\nnetic field. From Section 4.1.1 we find that the mag-\nnetic response is qualitatively similar to the reponses un-\nder the uniform random field. The compensation and\ncritical temperatures shift towards the low temperature\nends and even results in the vanishing of compensation\nin proper cases as we increase the standard deviation of\nthe external Gaussian random magnetic field. Similar in-\nference can be drawn from the thermodynamic behaviour\n8 0 0.005 0.01 0.015 0.02 0.025\n 0  1  2  3  4∆O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (a)\nJAA/JBB=0.04\nJAB/JBB=-0.20\n 0 0.003 0.006 0.009 0.012 0.015\n 0  1  2  3  4∆E(T) (JBB)\nT (JBB/kB)σ=0.00\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (b)\nJAA/JBB=0.04\nJAB/JBB=-0.20\n 0 0.005 0.01 0.015 0.02 0.025\n 0  1  2  3  4∆O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (c)\nJAA/JBB=0.04\nJAB/JBB=-0.84\n 0 0.003 0.006 0.009 0.012 0.015\n 0  1  2  3  4∆E(T) (JBB)\nT (JBB/kB)σ=0.00\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (d)\nJAA/JBB=0.04\nJAB/JBB=-0.84\nFigure 5: (Colour Online) Temperature dependence of Fluctuation o f order parameter, ∆ O(T) and Fluctuation of\ncooperative energy per site, ∆ E(T), for ABA type (3 ×100×100) configuration in (a)-(d) with JAA/JBB= 0.04 and\nJAB/JBB=−0.20 and with JAA/JBB= 0.04 andJAB/JBB=−0.84. Where, the errorbars are not visible, they\nare smaller than the dimension of the point-markers. The nature of the curves prominently shows the shift of critical\ntemperatures and even reason for absence of compensation can be understood from the low temperature segment of\nthe curves.\nof the suitably defined fluctuations of magnetization and\ncooperative energy [Refer to Section 4.1.2]. The effect of\nthe time-dependent part of the Hamiltonian is established\nin Section 4.1.3. We observe that the systems react very\nquickly after switching ON the external field and the dy-\nnamics is governed by the competition between spin-field\nenergies and cooperativeenergies. Thus the Gaussian ran-\ndom field-driven vanishing of compensation, observed in\nthis work, is also a dynamic phenomenon like it was in [27]\nwith auniform randomexternalfield. The phasediagrams\nin Figure 11, for both the ABA and AAB configurations,\nhavesimilarkindofappearancewithNo-CompensationIs-\nlands engraved within the phase area with compensation.\nAs we investigate the plot of the magnitude of the area\nof NO-Compensation Islands versus the standard devia-\ntion of the external field and find out the scaling exponent\naccording to the Equation 11, we find the responses are\nqualitatively similar for both the continuous field distribu-\ntions: UniformandGaussian. Aquickcomparisonfollows:\nbUniform\nABA= 1.958±0.122 and bGaussian\nABA = 1.913±0.137\n(For ABA ) andbUniform\nAAB= 1.783±0.118andbGaussian\nAAB =\n1.625±0.066(For AAB ), where datafor the uniform ran-\ndom field are taken from [27]. So, there exists a very good\nagreementforthe scalingexponents asthey overlapwithin\nthe statistical interval of one-another. So the dynamic re-sponse of the trilayered Ising spin-1/2 square ferrimagn-\nnetic systems, in this article, are quite similar under the\ninfluence of Gaussian random external magnetic field with\nspatiotemporal variations listed in Section A to the Uni-\nform random external field [27]. The exact nature of the\nexternal continuous field distribution, for these two types\nof distributions, does not show a distinguished effect on\nthe qualitative and quantitative features of such systems.\nBut the results also pose a difficulty for technological ap-\nplications. It is difficult to create a source of purely static\nmagnetic field, as some kind of ripple may exist. That rip-\nple or time dependent part, following uniform or Gaussian\ndistribution with characteristics described here or in [27],\nmayshiftthecompensationandcriticaltemperaturesfrom\ndesignated values.\nStill we have unanswered questions. For example, how\nwould the system behave under the influence of an exter-\nnal magnetic field following a Simpson distribution [54,\n55], with spatio-temporal variation of similar kind of the\npresent work? The results would definitely help us com-\nment strongly on the behaviour of the scaling exponent, b,\nunder a wide variety of continuous field distributions for\nboth the ABA and AAB type of trilayered stackings. The\nresponses under the discrete distributions are also yet to\nbe reported. these areplanned for the future. In realmag-\n9 0 0.005 0.01 0.015 0.02 0.025\n 0  1  2  3  4∆O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (a)\nJAA/JBB=0.04\nJAB/JBB=-0.20\n 0 0.003 0.006 0.009 0.012 0.015\n 0  1  2  3  4∆E(T) (JBB)\nT (JBB/kB)σ=0.00\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (b)\nJAA/JBB=0.04\nJAB/JBB=-0.20\n 0 0.005 0.01 0.015 0.02 0.025\n 0  1  2  3  4∆O(T) (µB)\nT (JBB/kB)σ=0.00\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (c)\nJAA/JBB=0.04\nJAB/JBB=-0.84\n 0 0.003 0.006 0.009 0.012 0.015\n 0  1  2  3  4∆E(T) (JBB)\nT (JBB/kB)σ=0.00\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (d)\nJAA/JBB=0.04\nJAB/JBB=-0.84\nFigure 6: (Colour Online) Temperature dependence of Fluctuation o f order parameter, ∆ O(T) and Fluctuation of\ncooperative energy per site, ∆ E(T), for AAB type (3 ×100×100) configuration in (a)-(d) with JAA/JBB= 0.04 and\nJAB/JBB=−0.20 and with JAA/JBB= 0.04 andJAB/JBB=−0.84. Where, the errorbars are not visible, they\nare smaller than the dimension of the point-markers. The nature of the curves prominently shows the shift of critical\ntemperatures and even reason for absence of compensation can be understood from the low temperature segment of\nthe curves.\nnetic systems, impurities, compositional disorder, lattice\ndislocations etc. modify the Hamiltonian to a transla-\ntionally non-invariant kind. Such a complexity, with Ising\nmechanics, may be described by a dynamic Hamiltonian\nsuch as Equation 1 with a time dependent part, where the\nexternalfieldischaracterizedbyaprobabilitydistribution.\nConflicts of interest\nThere are no conflicts of interest to declare.\nData availability statement\nThe data that support the findings of this study are avail-\nable from the author upon reasonable request.\nAcknowledgements\nThe author acknowledgesthe financial assistance from the\nUniversity Grants Commission, India in the form of Re-\nsearch Fellowship and extends his thanks to Dr. Tam-\naghna Maitra for feedback and technical assistance.Appendix\nA Characteristics of the External\nMagnetic Field\nThe local, Gaussian random external magnetic field\nvalues [of Equation 1] at any site, at a time instant follow\na Gaussian probability distribution. Box-Muller\nalgorithm [43] is used to get such a distribution G0of\nstandard deviation, σand zero mean:\nG0=σ/radicalbig\n−2ln(U1)cos(2πU2) (12)\nHereU1andU2are two uniform random distributions\nbetween [0 ,1] .\nA few additional characteristics are also added to the\nexternal field distribution [27]:\n(a) At different lattice sites, the values of the external\nfield are uncorrelated at any time instant. Again at\na lattice site, the values of the external field are\nuncorrelated for different time instants. So :\nhm(t)hn(t′) =a(t)δmnδ(t−t′), where m,nare\ntwo different lattice sites and t,t′are two different\ntime instants.\n10ABA:JAA/JBB=0.04;JAB/JBB=−0.20andT=Tcomp(σ=0) =0.502\nTop layer Mid layer Bottom layer\n(a)σ=0.00\n 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection\nMt(T,σ) =−0.484 Mm(T,σ) = +1.000 Mb(T,σ) =−0.502\n(b)σ=0.20\n 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection\nMt(T,σ) =−0.394 Mm(T,σ) = +1.000 Mb(T,σ) =−0.398\n(c)σ=0.76\n 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection\nMt(T,σ) =−0.201 Mm(T,σ) = +1.000 Mb(T,σ) =−0.192\nFigure 7: For ABA configuration : Lattice morphologies of top layer (at Left) ;mid layer (at Middle) and\nbottom layer (at Right) att=tmorph= 105MCSforJAA/JBB= 0.04 andJAB/JBB=−0.20 and a few standard\ndeviations of the external field. The magnetisations are rounded- off to three decimal places. The shift and vanishing\nof compensation in the respective cases (b) and (c) is due to the sig nificant reduction of magnetic ordering in the top\nand bottom layers i.e. surface A-layers .\n(b) The following conditions are also satisfied:\n(i) After the field is switched ON, the bulk\naverage of the Gaussian field at a time instant\nt, is zero:/summationdisplay\nmhm(t) = 0\n.\nSo,/summationdisplay\nm,nhm(t)hn(t)δmn= 3L2σ2\n.\n(ii) At the m-th site, the temporal mean of thelocal field over the exposure interval, δ, is\nzero:/an}bracketle{thm(t)/an}bracketri}ht=1\nδ/integraltextt0+δ\nt0hm(t)dt= 0 .\nAt a few randomly chosen time instants within the\nexposure interval, the implementation of the desired field\ndistribution is checked by the Cumulative Distribution\nFunction (CDF) [56], the Kernel Density Estimate\n(KDE) [57] and the Histogram.\n11AAB:JAA/JBB=0.04;JAB/JBB=−0.20andT=Tcomp(σ=0) =0.296\nTop layer Mid layer Bottom layer\n(a)σ=0.00\n 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection\nMt(T,σ) =−0.189 Mm(T,σ) =−0.814 Mb(T,σ) = +1.000\n(b)σ=0.20\n 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection\nMt(T,σ) =−0.091 Mm(T,σ) =−0.615 Mb(T,σ) = +1.000\n(c)σ=0.76\n 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection 25 50 75 100\n 25 50 75 100y\nx\n-1 1\nspin-projection\nMt(T,σ) =−0.022 Mm(T,σ) =−0.240 Mb(T,σ) =−1.000\nFigure 8: For AAB configuration : Lattice morphologies of top layer (at Left) ;mid layer (at Middle) and\nbottom layer (at Right) att=tmorph= 105MCSforJAA/JBB= 0.04 andJAB/JBB=−0.20 and a few standard\ndeviations of the external field. The magnetisations are rounded- off to three decimal places. The shift and vanishing\nof compensation in the respective cases (b) and (c) is due to the sig nificant reduction of magnetic ordering in the top\nand middle A-layers.\nB On Compensation point and\nsize of the system\nWe have mentioned that the system size in this study is\nfixed atL= 100 in Section 3. The size of the system is a\nkey factor in influencing the thermodynamic response in\nthe context of a trilayered Ising system’s critical\nproperties in a field-free environment, according to [22].\nDue to the opposing magnetic moments of the sublayers,\ncompensation happens when the net magnetization of\nthe system is zero. In [22], for L≥60, we have observed\nthat the compensation point is immune to the linear\nsystem size in a field-free environment. But under theinfluence of a Gaussian random external field with\nspatio-temporal variation, we still need to find out\nwhether the compensation point of the steady state\nmagnetisations depends on the linear system size. To\naddress this issue, in this section, a few representative\ncases are discussed in Figure 13. For both the\nconfigurations, ABA and AAB, the combination of\nmoderate ferromagnetic-moderate antiferromagnetic\ncoupling ratio is shown with JAA/JBB= 0.36 and\nJAA/JBB=−0.36 and system sizes varied from L= 30\ntoL= 100. From these cases, we see the fluctuations in\nthe values of compensation points have been confined to\nwithin 1% across system sizes. For very few other\n12-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.60\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.60\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n(A) Top panel: JAA/JBB= 0.04andJAB/JBB=−0.20\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.60\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.60\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n(B) Middle panel: JAA/JBB= 0.20andJAB/JBB=−0.20\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.60\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.60\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01Part: A Part: B\n(C) Bottom panel: JAA/JBB= 1.00andJAB/JBB=−0.52\nFigure 9: (Colour Online) Plots of Magnetisations for square monolay ers (sublattices) and total magnetisation of\nthe bulk versus time in MCS, for the ABAconfiguration , whereMt(t): Magnetization of the top layer; Mm(t):\nMagnetization of the mid layer; Mb(t): Magnetization of the bottom layer are all functions of time, t, in units of MCS.\nIn these figures, Part: A describes the equilibrium (Zero-field) and transient (Field: ON) beha viour whereas, Part:\nBdescribes the steady state behaviour (Field: ON). The magnetisat ion curves for the surface A-layers (orange and\ngreen) of ABA configuration overlap for the most of the times.\nrandomly checked combinations of coupling strengths,\nthe same feature is present. So, within the scope of\navailable limited computational resources, using only the\nvalues of compensation temperatures for linear system\nsizesL= 100 doesn’t compromise much on the accuracy\nfor all the combinations of coupling strengths and\nstandard deviation of the external Gaussian randommagnetic field. That is a strong compendium in support\nof choosing L= 100, in this study.\nC On the transient behaviour\nIn this study, we have magnetic systems which are\nresponding to time-dependent external fields and\n13-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.60\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.60\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n(A) Top panel: JAA/JBB= 0.04andJAB/JBB=−0.20\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.60\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.60\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n(B) Middle panel: JAA/JBB= 0.20andJAB/JBB=−0.20\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.60\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.60\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01Part: A Part: B\n(C) Bottom panel: JAA/JBB= 1.00andJAB/JBB=−0.52\nFigure 10: (Colour Online) Plots of Magnetisations for square monola yers (sublattices) and total magnetisation of\nthe bulk versus time in MCS, for the AABconfiguration , whereMt(t): Magnetization of the top layer; Mm(t):\nMagnetization of the mid layer; Mb(t): Magnetization of the bottom layer are all functions of time, t, in units of MCS.\nIn these figures, Part: A describes the equilibrium (Zero-field) and transient (Field: ON) beha viour whereas, Part:\nBdescribes the steady state behaviour (Field: ON).\nachieving steady-state. So we need to figure out the time\ninterval the systems usually consume to die out the\ntransient behaviours. A few selected examples are\nprovided here which shows the reason behind the choice\nof transient time interval (∆ T) in Section 3. We can\nroughly estimate that ∆ T∼50 MCS, and that’s why 250\nMCS is consumed to reach the steady state (and record\ndata) for surity.D Magnetic behaviour for σ= 1.00\nIn this section, a small add-on is provided for better\nunderstanding of how magnetic order diminishes when\nthe standard deviation is increased to σ= 1.00 and\ndimensionless temperature, T= 0.01.In the field-free\nenvironment, all the sublayers are perfectly\nordered in such a nearly athermal condition . A\nfew examples are provided in Figure 15 where the\nexternal field affects the magnetic behaviour even in such\nlow temperature. We understand now that when in-plane\ncoupling strengths and corresponding cooperative\nenergies are comparable to the steady state spin-field\n14(a)\n (b)\n(c)\n (d)\nFigure 11: (Colour Online) Phase diagram for the: ABA trilayered fer rimagnetic system when: (a) σ= 0.40; (b)\nσ= 1.00 and AAB trilayered ferrimagnetic system when: (c) σ= 0.40; (d)σ= 1.00, in presence of the uniform\nrandom external magnetic field. A: Compensation is absent; P: Com pensation is present. With an increase in the\nstandard deviation of the external field, the magnitude of the are a of the no-compensation island have grown. The\nblue segment of the phase separation curves are obtained via linear extrapolation. All these plots are obtained for a\nsystem of 3 ×100×100 sites. Where the errorbars are not visible, they are smaller tha n the point markers.\nenergies, the corresponding sublayer magnetisations\ndon’t deviate much from equilibrium values. But when\nthe spin-field term dominates, the steady state sublayer\nmagnetisation diminishes gradually as we increase the\nrandomness of the external Gaussian random field. If the\nin-plane coupling strength is very weak, e.g.\nJAA/JBB= 0.04 for the AAB configuration in Figure\n15(B), magnetic ordering in the steady state almost\nvanishes (steady-state sublayered magnetisation ≈0).\nReferences\n1. Barbic M., Schultz S., Wong J., Scherer A., IEEETransactions on Magnetics 37, 1657 (2001).\n2. Cullity B.D. and Graham C.D., Introduction to\nMagnetic Materials, second ed. (John Wiley &\nSons, New Jersey, USA, 2008).\n3. Stringfellow G.B., Organometallic Vapor-Phase\nEpitaxy: Theory and Practice (Academic Press,\n1999).\n4. Herman M.A. and Sitter H., Molecular Beam\nEpitaxy: Fundamentals and Current Status, Vol. 7\n(Springer Science & Business Media, 2012).\n5. Singh R.K. and Narayan J., Phys. Rev. B 41,\n8843 (1990).\n6. George S.M., Chem. Rev. 110, 111 (2010).\n15(a)\n0.000.100.200.300.40\n0.000.250.500.751.00|Aσ| , |dAσ|/dσ \nσ (sd)|Aσ|\n|dAσ|/dσ\nABA(b)\n0.000.100.200.300.40\n0.000.250.500.751.00|Aσ| , |dAσ|/dσ \nσ (sd)|Aσ|\n|dAσ|/dσ\nABA\nFigure 12: (Colour Online) Plots of: Magnitude of the area of the no- compensation islands versus standard deviation\nof the field (in RED) and the rate of increase in the magnitude of the a rea of the no-compensation islands versus\nstandard deviation of the field (in BLUE) for (a) ABA and (b) AAB con figurations for a system of 3 ×100×100sites.\n 1.6 1.7 1.8 1.9\n 30 40 50 60 70 80 90 100Tcomp(L)\nLTcomp(L=100)\nTcompABA\nJAA/JBB = 0.36\nJAB/JBB = -0.36\nσ = 0.20 (a) 1.1 1.2 1.3 1.4\n 30 40 50 60 70 80 90 100Tcomp(L)\nLTcomp(L=100)\nTcomp ABA\nJAA/JBB = 0.36\nJAB/JBB = -0.36\nσ = 0.80 (b)\n 1.4 1.5 1.6 1.7 1.8\n 30 40 50 60 70 80 90 100Tcomp(L)\nLTcomp(L=100)\nTcomp AAB\nJAA/JBB = 0.36\nJAB/JBB = -0.36\nσ = 0.20 (c)  1 1.1 1.2 1.3\n 30 40 50 60 70 80 90 100Tcomp(L)\nLTcomp(L=100)\nTcomp AAB\nJAA/JBB = 0.36\nJAB/JBB = -0.36\nσ = 0.80 (d)\nFigure 13: (Colour Online) Compensation temperatures versus linea r system sizes of Ising trilayered square stacking\nof ABA type (a & b) and AAB type (c & d). The reported value of comp ensation temperature (L = 100) is confined\nwithin 1% across all the sizes.\n7. Stier M., and Nolting W., Phys. Rev. B 84,\n094417 (2011).\n8. Leiner J., Lee H., Yoo T., Lee S., Kirby B. J.,\nTivakornsasithorn K., Liu X., Furdyna J. K., and\nDobrowolska M., Phys. Rev. B 82, 195205 (2010).\n9. Sankowski P., and Kacmann P., Phys. Rev. B 71,\n201303(R) (2005).10. Pradhan A., Maitra T., Mukherjee S., Mukherjee\nS., Nayak A., et al., Mater Lett. 210, 77 (2018).\n11. Maitra T., Pradhan A., Mukherjee S., Mukherjee\nS., Nayak A., and Bhunia S., Phys. E 106, 357\n(2019).\n12. Connell G., Allen R., and Mansuripur M., J. Appl.\nPhys.53, 7759 (1982).\n16-1.2-0.6 0 0.6 1.2 1.8\n 49500  49750  50000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:\n(a) σ=0.20JAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 49500  49750  50000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:\n(a) σ=0.20JAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 49500  49750  50000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:\n(b) σ=1.00JAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 49500  49750  50000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:\n(b) σ=1.00JAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 49500  49750  50000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:\n(c) σ=1.00JAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 49500  49750  50000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:\n(c) σ=1.00JAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 49500  49750  50000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:\n(d) σ=0.60JAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 49500  49750  50000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:\n(d) σ=0.60JAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01Field: OFF Field: ON\nFigure 14: (Colour Online) Transient behaviour of the trilayered sys tems: ABA (a & b) and AAB (c & d) for a few\ncases.\n13. 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B 26, 1495 (1982).\n17ABA:σ= 1.00\nJAA/JBB=0.04 J AA/JBB=0.20 J AA/JBB=1.00\nJAB/JBB=−0.20 J AB/JBB=−0.20 J AB/JBB=−0.52\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01Part: A Part: B\nAAB:σ= 1.00\nJAA/JBB=0.04 J AA/JBB=0.20 J AA/JBB=1.00\nJAB/JBB=−0.20 J AB/JBB=−0.20 J AB/JBB=−0.52\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=0.04\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=0.20\nJAB/JBB=-0.20\nT=0.01Part: A Part: B\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0  25000  50000  75000  100000Magnetisations ( µB)\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=1.00\nJAB/JBB=-0.52\nT=0.01Part: A Part: B\nFigure 15: (Colour Online) Plots of Magnetisations for square monola yers (sublattices) and total magnetisation of\nthe bulk versus time in MCS, for theABAandAABconfigurations withσ= 1.00 . Here, Mt(t): Magnetization\nof the top layer; Mm(t): Magnetization of the mid layer; Mb(t): Magnetization of the bottom layer are all functions\nof time,t, in units of MCS. In these figures, Part: A describes the equilibrium (Zero-field) and transient (Field: ON)\nbehaviour whereas, Part: B describes the steady state behaviour (Field: ON).\n38. Sethna J. P., Dahmen K. A., and Perkovi´ c O., in\nThe Science of Hysteresis, Vol. II, pp. 107-179\n(2006).\n39. Gong C., Li L., Li Z., Ji H., Stern A., et al., Nature\n546 (7657) , 265 (2017).\n40. Huang B., Clark G., Navarro-Moratalla E., Klein\nD. R., Cheng R., et al., Nature 546(7657) , 270\n(2017).\n41. Song T., Fei Z., Yankowitz M., Lin Z., Jiang Q., et\nal., Nat. Mater. 18, 1298 (2019).\n42. McGuire M. A., Crystals 7(5), 121 (2017).\n43. Box G. E. P. and Muller M. E., Ann. Math.\nStatist. 29(2), 610 (1958).\n44. Landau D. P. and Binder K., A Guide to Monte\nCarlo Simulations in Statistical Physics\n(Cambridge University Press, New York, 2000).\n45. Binder K. and Heermann D. W., Monte Carlo\nSimulation in Statistical Physics (Springer, New\nYork, 1997).46. Metropolis N., Rosenbluth A. W., Rosenbluth M.\nN., Teller A. H., and Teller E., J. Chem. Phys.\n21, 1087 (1953).\n47. Scarborough J. B., Numerical Mathematical\nAnalysis (Oxford & Ibh, London, 2005).\n48. Newman M. E. J. and Barkema G. T., Monte Carlo\nMethods in Statistical Physics (Oxford University\nPress, New York, 1999).\n49. Robb D. T., Rikvold P. A., Berger A., and Novotny\nM. A., Phys. Rev. E 76, 021124 (2007).\n50. N´ eel M. L., Ann. de Phys. 12, 137 (1948).\n51. Chikazumi S., Physics of Ferromagnetism (Oxford\nUniversity Press, Oxford, 1997).\n52. Streˇ cka J., Physica A 360, 379 (2006).\n53. See, e.g., Krauth W., Statistical Mechanics:\nAlgorithms and Computations (Oxford University\nPress, New York, 2006).\n54. Wentzel E. S., Probability Theory (first steps) (Mir\nPublishers, Moscow, 1986)\n1855. Alder H. L., and Roessler E. B., Introduction to\nProbability and Statistics (W. H. Freeman and Co.,\nSan Francisco, 1975)\n56. Deisenroth M. P., Aldo Faisal A., and Ong C. S.,\nMathematics for Machine Learning (Cambridge\nUniversity Press, New York, 2020).\n57. See, e.g., Rosenblatt M., Ann. Math. Statist. 27,\n832 (1956); Parzen E., Ann. Math. Statist. 33,\n1065 (1962).\n19"    },    {        "title": "2309.08990v1.Antiferromagnetic_to_Ferrimagnetic_Phase_Transition_and_Possible_Phase_Coexistence_in_Polar_Magnets__Fe___1_x__Mn__x____2_Mo__3_O__8_.pdf",        "content": " \nAntiferromagnetic to Ferrimagnetic Phase Transition \nand Possible Phase Coexistence in Polar Magnets (Fe\n1-xMn x)2Mo 3O8 (0≤x≤1)  \n \nYuting Chang1#, Lei Gao1#, Yunlong Xie2, Bin You1, Yong Liu3, Rui Xiong3, Junfeng Wang1, \nChengliang Lu1*, and Jun-Ming Liu2,4 \n \n1 School of Physics & Wuhan National High Magneti c Field Center, Huazhong University of Science \nand Technology, Wuhan 430074, China \n2 Institute for Advanced Materials, Hube i Normal University, Huangshi 435001, China  \n3 School of Physics and Technology, and the Key Labor atory of Artificial Micr o/Nano structures of \nMinistry of Education, Wuhan University, Wuhan 430072, China \n4 Laboratory of Solid State Micros tructures and Innovation Center of  Advanced Microstructures, \nNanjing University, Nanjing 210093, China    \nKeywords: Magnetoelectric effect, phase transition, antiferromagnetic, ferrimagnetic, polar magnet \n \n   \nAbstract:  \nIn the present work, magnetic pr operties of single crystal (Fe 1-xMn x)2Mo 3O8 (0≤x≤1) have been \nstudied by performing extensive measurements. A detailed magnetic phase diagram is built up, in which antiferromagnetic state dominates for  x≤0.25 and ferrimagnetic phase arises for x≥0.3. \nMeanwhile, sizeable electric polarization of spin origin is commonly observe d in all samples, no \nmatter what the magnetic state is. For the samples hosting a ferrimagnetic state, square-like magnetic hysteresis loops are reve aled, while the remnant magnetizat ion and coercive field can be \ntuned drastically by simply vary ing the Mn-content or temperatur e. Possible coexistence of the \nantiferromagnetic and ferrimagnetic phases is propos ed to be responsible for the remarkable \nmodulation of magnetic prope rties in the samples.  \n   \n \n  I. Introduction \nMagnetoelectrics has been providing a fer tile ground for studying cross-coupling among \nvarious ferroic-orders, which is fundamentally in teresting and practically important for developing \nconceptually new devices 1-4. Recently, a group of newly discovered magnetoelectric (ME) \nmaterials, i.e. the polar magnets A2Mo 3O8 (A: transition metal), have been drawing considerable \nattention, because of their concurrent spontaneous electric polarization ( P) due to the \ncrystallographic polarity, spin-driven electric P, and linear ME effect 5-11. This is rarely seen in the \nexisting materials such as other ferrites 12-13. Note that the linear magnetoelectrics usually have zero \nspontaneous P and thus are regarded as non-ferroelectric (FE) 14.  \nThe family of A2Mo 3O8 belongs to a pyroelectric space group P63mc 15-17. As illustrated in Fig. \n1(a), there are two ineq uivalent sites of the A ions, i.e. the octahedra and the tetrahedra, and the \npolar arrangement of the AO4 tetrahedra induces crystallographic polarity along the c-axis. \nMagnetism of  A2Mo 3O8 is dominated by the A-ions, while the Mo ions are trimerized. The two \nmagnetic sublattices com posed of tetragonal- A and octahedral- A are coupled in an \nantiferromagnetic (AFM) manner. In each of the sublattices, magnetic moments can be either \nparallel or antiparallel, depending on the A ions. For instance, Fe 2Mo 3O8 and Co 2Mo 3O8 have two \ninteracted AFM sublattices, and Mn 2Mo 3O8 hosts a ferrimagnetic (FRM ) ground state where two \ntypes of ferromagnetic (FM) orders  are coupled antiferromagnetically 9, 18. In Ni 2Mo 3O8, a specific \nnoncollinear magnetic configuration is stabilized due to the coupli ng of magnetic sublattices of Ni2+ \n11. The different magnetic configurations give rise  to strikingly distinct ME effect and thereby \nunderlying physics in these materials.  \nBeyond the conventional cross-control between ma gnetic and electric variables, some more fascinating properties such as the optical ME effect which is an extension of the classic ME to \ndynamic region, orbitally selective Mott feature, and giant thermal Hall effect have been revealed \ntypically in Fe 2Mo 3O8. Importantly, all these properties are found to closely relate to the magnetic \nstructure. The band gap of Fe 2Mo 3O8 exhibits notable narrow down at the AFM transition TN~60 K, \nimplying intimate coupling between the ch arge and spin degrees of freedom 19. The optical ME \ndepends on the relative arra ngement between magnetization ( M) and electric polarization P, e.g. the \ndiagonal term MꞏP leads to the gyrotropic birefri ngence and the off-diagonal term  M×P determines \nthe nonreciprocal di rectional dichroism 20-21. For the thermal Hall effect, the transverse thermal \nconductivity jumps to a unprecedentedly large value driven by the AFM to FRM transition 7. \nMotivated by these observations, it is of highly interesting and practical to study the modulation of \nmagnetic order of Fe 2Mo 3O8, which may extend the ME scenar io to a much broader scope.  \nFe2Mo 3O8 hosts a rather robust AFM ground state,  and the critical magnetic field ( Hcri) of the \nAFM-FRM transition is quite high indeed 5, 8. In this sense, chemical substitution may be a more \nfavorable route to tailor the magnetism of Fe 2Mo 3O8, considering the very different magnetic states \namong the members of A2Mo 3O8 as mentioned above. In particular , it was found that the Fe ions at \nthe octahedral (Fe O) and the tetrahedral (Fe T) sites could be selectiv ely substituted by properly \nchoosing the doping species 22. This is fundamentally important, because of the distinct electronic \nand magnetic properties between Fe O and Fe T. In comparison with non-magnetic ions, doping with \nmagnetic ions could better preserve  the fascinating properties th at essentially associate with \nmagnetic order. Especially, the effect of Mn-substitution at Fe-site on the magnetic properties deserves special attention, based on three motivations. First, Fe\n2Mo 3O8 hosts an AFM ground state, \nwhile Mn 2Mo 3O8 has a FRM ground state, wh ich facilitates the modula tion of magnetic structure efficiently. Second, although both materials have the same c-axis magnetic anisotropy, Fe2+ has \nmuch stronger spin anisotropy than Mn2+. For instance, the c-axis magnetic anisotropy remains \nnotable up to room temperature ( T) in Fe 2Mo 3O8, but disappears right ab ove the FRM temperature \nTC~41 K in Mn 2Mo 3O8 6, 8. Third, in Fe 2Mo 3O8, Fe T (~4.6 μB/Fe) has much larger magnetic moment \nthan Fe O (~4.2 μB/Fe) 22. Differently, Mn O and Mn T have the same magnetic moments of ~5 μB/Mn \nin Mn 2Mo 3O8, which are expected to fully compensate at T=0 K. Moreover, it was found that \nMn-doping in Fe 2Mo 3O8 indeed causes interesting sign-reversal  of the linear ME coefficient from \nnegative to positive, while the detailed magnetic properties of (Fe 1-xMn x)2Mo 3O8 remains largely \nunexplored 6.  \n In the present work, a series of single crystal (Fe\n1-xMn x)2Mo 3O8 (0≤x≤1) were synthesized, and \nextensive characterizations including struct ure, magnetization, a nd pyroelectric current \nmeasurements were carried out. Doping induced AFM to FRM transition is identified at x=0.25, i.e. \nthe bicritical point. It is found that the FRM pha se can be modulated remarkably, and the remnant \nmagnetization ( Mr) and coercive field ( Hc) exhibit strong dependence on the Mn-content x and T. \nInterestingly, possible coexistence of AFM and FR M phases may widely exist in the materials with \ndifferent x.      \n \nII. Experimental details \nThe (Fe 1-xMn x)2Mo 3O8 single crystals were s ynthesized using chemical vapor transport method \n23-24. High purity MnO 2, Mn, Fe 2O3, CoO, and MoO 2 powders were mixed and sealed in evacuated \nsilica tubes, which then were pl aced in a two-zone furnace for th e crystal growth. Chemical vapor transport method is more favorab le for the synthesis of (Fe 1-xMn x)2Mo 3O8 polar magnets than other \nroutes 25-26. The obtained crystals are hexagonal plate-like with typical size of 2×2×0.5 mm3. In \norder to determine the crystalline structure, the crystals were crashed thoroughly, and then powder \nX-ray diffraction (XRD) measurements we re performed at room temperature ( T). Energy Dispersive \nX-ray Spectroscopy (EDS) and X-ray Photoelect ron Spectroscopy (XPS) measurements were \nperformed to check the stoichio metry and valance state of the single crystals. The magnetization \nmeasurements as a function of T and magnetic field H were carried out using a Quantum Design \nphysical property measurement system (PPMS). For both zero field cooling (ZFC) and field cooling \n(FC) measurements, the measuring field was fixed at H=0.1 T. The M(H) measurements were \ncarried out at various temperatures for the sa mples. All the magnetization measurements were \nperformed with H//c-axis. Gold electrodes were prepared for pyroelectric current measurements. \nElectric polarization along the c-axis as a function of T were obtained by integrating the \npyroelectric current which were measured usin g Keithely 6517 electrometer during the warming \nprocess. Pulsed high magnetic field measurements  for magnetization were performed by means of a \nstandard inductive method employing a couple of coaxial pickup coils. The pulsed magnetic field \nup to ~58 T with a duration time of ~10 ms was generated by using a nondestructive long-pulse \nmagnet energized by a 1 MJ capacitor bank.   \nIII. Results and Discussions \nThe collected powder XRD spectra confirm that  all samples with different Mn-content x are \npure phase. To acquire more details of the crystalline structure, Rietveld-profile refinements of the \nXRD data were performed for all samples. The refined result for the sample with x=0.9 is shown as an example in Fig. 1(b). The difference between the measured and refined spectra is small with \nreliability parameter Rwp = 3.86%. For all other samples, the refinements are all high quality, and \nthe Rwp values are at similar levels. According to the refinements, derived lattice constants are \na=5.773 Å and c=10.054 Å for Fe 2Mo 3O8 (i.e. x=0), and a=5.799 Å and c=10.267 Å for Mn 2Mo 3O8 \n(i.e. x=1), which are in good agreem ent with previous studies 18. The obtained a, c, and cell volume \nV are plotted as a function of x in Fig. 1(c)-(e), respectiv ely. It is seen that a ll lattice parameters vary \nwith x continuously, without showing any apparent a nomaly. Since stoichiometry and valance state \nare crucial ingredients in determining the properties of oxides 27-30, further EDS and XPS \nmeasurements have been performed in the crysta ls (shown in the Supporting Information). It is \nconfirmed that no apparent deviatio n of the concentration of the orig inal cations from a given value, \nand Fe2+ and Mn2+ are quite stable in the crystals.   \n \nBecause of the ine quivalent sites, Fe 2Mo 3O8 has two AFM sublatti ces consisting of Fe O and \nFeT, respectively, which are coupled antiferromagnetically. Fe 2Mo 3O8 undergoes a paramagnetic \n(PM) to AFM transition at TN~60 K, evidenced by sharp peaks in Mc(T) curves measured under \nboth ZFC and FC sequences, as show n in Fig. 2(a). By introducing Mn2+ into the Fe2+ sites, TN is \ngradually shifted toward low- T region. For the sample with x=0.2 shown in Fig. 2(c), the peak of \nMc(T) gets evidently higher than the samples with  lower doping content, indicating the occurrence \nof ferrimagnetism in the sample. At x=0.25, the Mc(T) curves exhibit drastic enhancement ensuing \nat TC~53 K, and the subsequent peak is about ten times higher than that of the samples with x<0.2, \nevidencing development of the FRM phase. The FRM transition temperature TC is determined by \nthe minima of T derivative of Mc. Further cooling the sample causes steep decrease in Mc at TN~39 K, owing to the FRM to AFM transition.  \nAt x=0.3, the FRM phase is well stabilized below TC~53 K, and no further AFM transition can \nbe seen in the low- T region according to the FC curve. In ferrimagnets , there are generally two \nways to induce net M 31. One is due to the different magnetic moments of the interacted sublattices, \nand the uncompensated M emerges below TC and then persists down to low T. The other way is \nrelated to different T-dependence of the sublattices hosting  the same magnetic moment. In this \nsense, the M(T) curve usually first increases fast at TC, and then gradually decreases with T upon \ncooling the system. Eventually, M goes to zero as T→0, because of cancellation of the two \nsublattices. For a mixed case where the sublattices have different T-dependence and magnetic \nmoments, M is expected to be reduced to a finite value at low- T, such as the FC curve at x=0.3 \nshown in Fig. 2(e). Another phenomenon is that the ZFC curve shows tremendous difference from \nthe FC curve and becomes negative below TB~20 K. This has been wi dely observed in other \nferrimagnets such as spinels, and can be underst ood based on the classic molecular field theory 32. \nThe two FM sublattices are aligned in opposite directions, and the sign of net M depends on the \ncompetition between Zeeman energy and magnetic  anisotropy. As a consequence, negative M can \nappear. Once the Zeeman energy is beyond the anisotropy energy, the net M is thus reversed, \nleading to the sh arp variation of Mc from negative to positive around TB~20 K in the ZFC curve.   \nFurther increasing x causes continuous decrease in TC. Meanwhile, the large bifurcation \nbetween ZFC and FC curves can be seen in all samples with x>0.3, and TB is changed modestly \nwith x. Obviously, the FC curves of th e samples commonly exhibit non-zero Mc as T→0 K, \nresulting from the different magnetic  moments of the sublattices. In Mn 2Mo 3O8 (x=1), the PM-FRM \ntransition is identified at TC~41 K, consistent with previous study. Although Mn O and Mn T have the same magnetic moment of 5 μB/Mn, the different T-dependence of two FM s ublattices leads to the \nstriking Mc-enhancement at TC, a typical example of ferrima gnetism as mentioned above. \nNevertheless, a fully compensated stat e would be eventually achieved at T=0 K, revealed by the fast \napproaching between the FC and ZFC curves below TB~12 K shown in Fig. 2(m).   \n To better understand the ma gnetic properties of (Fe\n1-xMn x)2Mo 3O8, extensive M(H) \nmeasurements were carried out, and typical resu lts are shown in Fig. 3. For the non-doped sample \nFe2Mo 3O8, sharp AFM to FRM transition is trigge red by relatively large critical field Hcri, e.g. \nHcri~12 T at T=35 K. The jump at Hcri in M(H) is caused by the different magnetic moments \nbetween Fe T and Fe O, which has similar value of about ~0.5 μB/f.u. at various temperatures. \nIncreasing x causes significant reduction in Hcri. For instance, Hcri is decreased from ~12 T to ~3 T \nat T= 35 K as x is increased from 0 to 0.2, indicating stro ng suppression of the AFM phase of the \nsystem. At x=0.2, additional kink anomaly arises in M(H) above Hcri. This is getting more clear at \nx=0.25 where multiple step-like transitions accom panied by evident magnetic hysteresis can be \nidentified, shown in Fig. 3(d). In  fact, weak but unambiguous magnetic  hysteresis also exists at the \nvery low- H region. These phenomena imply coexisten ce of the AFM and FRM phases in the \nmaterial, and the steps are due to successive tran sitions from AFM to FRM domains, consistent with \nthe observation of both TC and TN from the M(T) shown in Fig. 2(d).  \nFor the sample with x=0.3 shown in Fig. 3(e), square-like M(H) hysteresis loops are obtained, \nconfirming the emergence of the FRM phase. Interestingly, the initial branch of M(H) at T=15 K \nshows step-like transition at  a reduced critical field H~0.5 T, possibly due to the H-driven \nAFM-FRM transition akin to the cases at x≤0.25. Similar phenomenon is generally seen in the samples with higher doping content. Meanwhile, the doped samples with 0.3 ≤x≤0.9 exhibit very \nlarge coercive field ( Hc) shown in Fig. 3(e)-(k), which is about  one order of magnitude larger than \nthat of Mn 2Mo 3O8 hosting a pure FRM state  (Fig. 3(l)). Therefore, strong competition between \nAFM and FRM phases may be a common feature in the (Fe 1-xMn x)2Mo 3O8 samples with \nintermediate doping levels.  \nIn addition, the M(H) curve at T=15 K of Mn 2Mo 3O8 displays another clear upturn above H~5 \nT, shown in Fig. 3(l). To understand this, high fi eld magnetization measurements were carried out, \nwhich are able to capture much more de tails of the sudden upturn in a broad H-range. As shown in \nFig. 4(a), the critical field of the upturn is moved to higher H side with increasing T. This is \nbasically different from conventio nal FRM to FM transition which is usually promoted by thermal \nfluctuation, e.g. the critical fiel d would be lowered by increasing T of the sample. Moreover, M(H) \nshows continuous linear increase after the upturn, without any trace of satu ration. This resembles \nthe typical antiferromagnetic beha vior. Therefore, the upturn-a nomaly should be assigned to a \nspin-flop transition from the c-axis FRM to the ab-plane FRM. The critical field of the upturn at \nvarious temperatures is summarized in Fig. 4(b).  \n Remnant magnetization M\nr and coercive field Hc are two key parameters of magnetic materials. \nIn fact, enhancing the macroscopic magnetization has been one of the primary goals of the ME \nresearch, which is crucial to combine the advant ages of both magnetic and ferroelectric memories. \nTypical data of Mr and Hc derived from M(H) at various temperatures are plotted as a function of \nMn-content x in Fig. 5(a) and (b), respectivel y. Similar behavior is observed for Mr(x) and also Hc(x) \nat different T. As an example, at T=15 K, Mr jumps from zero to ~0.2 μB/f.u. at x=0.3 where the doping-induced FRM phase is stabilized, and then decreases gradually for 0.4 ≤x≤1. Hc(x) displays \ndifferent evolving way that it increases quickly as the FRM phase is induced, and then reaches a \nplateau for 0.5 ≤x≤0.9. After that, Hc drops to a small value of ~0.2 T in Mn 2Mo 3O8, i.e. x=1. The \nfull data of both Mr and Hc at various T and x are summarized in Fig. 5(c) and (d), respectively. Mr \nincreases with T monotonously for all samples, and it can be as large as ~0.5 μB/f.u. for the samples \nwith 0.3≤x≤0.5. Regarding Hc(T), successive decrease can be seen for all samples hosting the FRM \nphase.  \n The above magnetization characterizations ha ve demonstrated highly tunable magnetic \nproperties in (Fe\n1-xMn x)2Mo 3O8. Further, it is confirmed that the magnetically driven electric \npolarization is maintained in all samples, no ma tter what kind of magnetic  state is hosted. Each \nsample was first cooled down far above TN or TC, and then the pyroelectric current was collected \nduring the warming process. For both the cooling and warming processes, no magnetic and electric \nfields were applied. Because of the polar crys talline structure, all samples display notable \npyroelectric current even far above TN. Importantly, additional shar p valley emerges in each \npyroelectric current curve, which matches well w ith the AFM or FRM transition of the sample, \nshown in Fig. 6(a). By integrating th e pyroelectric current versus time, T-dependence of electric \npolarization is obtained, shown in Fig. 6(b). The marked drop of ΔP(T) at TC or TN suggests a spin \norigin of the induced P, probably due to an exchange stri ction mechanism proposed by previous \nfirst principles calculations 8. Moreover, magnitude of the P-drop is about ΔP ~1000 μC/m2 for the \nsamples, which is larger than most of the multiferroics hosting a spin-origin 2, 33-37.  \n Based on the comprehensive characterizations of magnetization and pyr oelectric current, a T-x \nphase diagram is built up for (Fe 1-xMn x)2Mo 3O8 shown in Fig. 7(a). Three regions can be seen in the \ndiagram, which are dominated by PM, FRM, and AFM phases, respectively. At the phase boundary \nx~0.25, coexistence of AFM and FRM phases is identified. Interestingly, such  phase coexisting \nfeature may widely happen in the (Fe 1-xMn x)2Mo 3O8 samples, especially for the ones with middle \ndoping levels. Moreover, for the FRM phase, very large remnant magnetization Mr and coercive \nfield Hc are identified, and both parameters can be tuned remarkably by varying the Mn-content x \nand T. To understand these phenomena,  the occupation ratios of Mn2+ at the two inequivalent sites \ncould be a pivotal factor, because of the very differ ent properties of ions at the two sites in the polar \nmagnets.   \nIn the FRM phase, Mr is in principle determined by the difference of magnetic moments \nbetween Fe T (Mn T) and Fe O (Mn O), and thus can be given as:  \nMr = 2x ꞏ PO ꞏ MMn-O + (1 - 2 x ꞏ PO) ꞏ MFe-O - [2x ꞏ PT ꞏ MMn-T + (1 –2 x ꞏ PT) ꞏ MFe-T]    (1)    \nwhere P O and P T (=1- P O) are the occupation ratios of Mn2+ locating at the octahedral site and \ntetragonal site, respectively. MFe-O and MMn-O (MFe-T and MMn-T) are the magnetic moment of Fe2+ \nand Mn2+ on the octahedral (tetragonal ) site, respectively. Equation (1 ) can also be written as:   \nMr = 2x ꞏ PO ꞏ (MMn-O – MFe-O) - [2 x ꞏ PT ꞏ (MMn-T – MFe-T)] + (MFe-O – MFe-T)           ( 2 )  \nAs revealed by previous Möessbauer spectroscopy on Fe 2Mo 3O8, MFe-O and MFe-T are ~4.8 µ B and \n~4.2 µ B, respectively 22, resulting in ( MFe-O – MFe-T)~0.6 µ B. This is close to our experimental result \nof ~0.5 µ B, determined by the jump in M(H) due to the AFM-FRM transition. For Mn 2Mo 3O8, \nMMn-O and MMn-T have the same magnitude of M~5 µ B. With these parameters, it is now able to \nsimulate the Mr(x). The Mr(x) data typically measured at T=40 K close to TC (Fig. 5(a)) are used for the simulation, in order to make sure that the majo rity of the domains have  been converted to be \nFRM and uniformly aligned. The ob tained occupation ratios of Mn2+ are roughly distributed around \nPT ~ 0.6 and P O ~ 0.4 in all samples, as shown in Fig. 7(b). This is consistent with previous \nMöessbauer spectroscopy characterizations giving that Mn2+ has slight preference to occupy the \ntetragonal sites in (FeMn) 2Mo 3O8 6, 18. For Mn 2Mo 3O8 (x=1), the occupation rati os slightly deviate \nfrom 0.5, which is caused by the different T-dependence of MMn at different sites.  \nAt low- T, the difference between MMn-O and MMn-T becomes negligible, and thus Mr is \ndominated by the term ( MFe-O – MFe-T). For instance, if all doma ins are FRM, the ideal low- T Mr is \n~0.35 μB/f.u in the sample with x=0.3, by taking ( MFe-O – MFe-T)~0.5 µ B and the derived occupation \nratios of Mn2+. This is about two times larg er than the measured value  Mr~0.18 μB/f.u at T=10 K. \nThe discrepancy can be even remarkable at lower T, because the measured Mr decreases with T \nlinearly, shown in Fig. 5(c). Therefore, it is natu rally expected that AFM orders coexist with the \nFRM phase in the sample with x=0.3, responsible for the significantly reduced Mr. Difference \nbetween the ideal and measured Mr can also be found in the samples with higher doping contents, \nwhile the discrepancy is gradually descended. Taking this scenario that AFM and FRM phases \ncoexist, the giant coercive field Hc can be well understood by the strong phase competition in the \nsystem. In addition, the polar magnets (Fe 1-xMn x)2Mo 3O8 exhibit large remnant magnetization and \nelectric polarization simultaneously, which is usually accessible in composite ME systems rather than the single-phase materials \n38-39, and therefore may have interesting potentials in developing \nmicrowave materials 40-41.   \n \nIV . Conclusion A series of single crystal (Fe 1-xMn x)2Mo 3O8 (0≤x≤1) have been synthesized, and characterized \nby performing structure, magnetization, and electric  polarization measurements. It is found that \nMn-substitution of Fe can induce antiferromagne tic to ferrimagnetic phase transition at x=0.25, and \ncoexistence of AFM and FRM phases may be a comm on feature in the materials. For the samples \nwith x≥0.3 where ferrimagnetic state is stabilized, square-like magnetic  hysteresis loops with highly \ntunable remnant magnetization Mr and coercive field Hc are identified. In particular, remarkable \nelectric polarization driven by ma gnetic ordering is found in a ll samples hosting various magnetic \nstates. These findings indicate that (Fe 1-xMn x)2Mo 3O8 possessing highly tunabl e properties could be \nunique candidate for exploring em ergent physics and functions.   \n   ASSOCIATED CONTENT \nSupporting Information \nThe Supporting Information is available free of charge at *** \nAdditional characterizations such  as energy dispersive X-ray sp ectroscopy and X-ray photoelectron \nspectroscopy of the materials s upplied as Supporting Information. \n AUTHOR INFORMATION \nCorresponding Author \n* Authors to whom correspondence should be addressed, email: cllu@hust.edu.cn;  \nAuthor Contributions \n# Yuting Chang and Lei Gao contri buted equally to this work.  \nC.L.L. designed and supervised the experiments. Y.T.C. performed the measurements. L.G., B.Y., \nY.L., R.X., and J.F.W. contributed to the measurem ents and data analysis. Y.L.X. contributed to \ndata analysis. C.L.L. and J.M.L. wrote the manuscript.  \nNotes \nThe authors declare no comp eting financial interest. \n ACKNOWLEDGMENT \nThis work is supported by the National Nature Science Foundation of China (Grant Nos. 12174128, \n12074291, 92163210, and 11834002), Hubei Province Natural Science Foundation of China (Grant \nNo. 2020CFA083), and the Fundamental Research Funds  for the Central Universities (Grant No. \n2019kfyRCPY081, 2019kfyXKJC010). \n  Reference:  \n1. Lu, C.; Wu, M.; Lin, L.; Liu, J.-M., Single-Phas e Multiferroics: New Materials, Phenomena, \nand Physics. National Science Review 2019,  6 (4), 653-668. \n2. Dong, S.; Liu, J.-M.; Cheong, S. -W.; Ren, Z., Multiferroic Ma terials and Magnetoelectric \nPhysics: Symmetry, Entanglement, Excitation, and Topology. Advances in Physics 2015,  64 (5-6), \n519-626. \n3. 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Scientific Reports 2021,  11 (1), 18342. \n  Figures:  \n \n \nFigure 1. (a) Sketch of cr ystalline structure of Fe 2Mo 3O8 (top) or Mn 2Mo 3O8 (bottom). Magnetic \nmoments of Fe2+ and Mn2+ have also been labeled correspondingl y. One of the magnetic sublattices \nis indicated by dashed lines. (b) Rietveld refinement of the XRD spectra of sample \n(Fe 0.1Mn 0.9)2Mo 3O8. (c)-(e) present evalua ted lattice parameters a, c, and cell volume V as a \nfunction of Mn-content x, respectively.  \n \n   \nFigure 2. Temperature dependence of magnetization with H along the c-axis for (Fe 1-xMn x)2Mo 3O8 \n(0≤x≤1) single crystals: (a) x=0, (b) x=0.1, (c) x=0.2, (d) x=0.25, (e) x=0.3, (f) x=0.4, (g) x=0.5, (h) \nx=0.6, (i) x=0.7, (j) x=0.8, (k) x=0.9, and (l) x=1. TC and TN represent the ferrimagnetic and \nantiferromagnetic transitions, respectively. TB indicates the reversal of net magnetization.  \n \n   \nFigure 3. Magnetizati on as a function of H // c-axis measured at various temperatures for \n(Fe 1-xMn x)2Mo 3O8 (0≤x≤1) single crystals: (a) x=0, (b) x=0.1, (c) x=0.2, (d) x=0.25, (e) x=0.3, (f) \nx=0.4, (g) x=0.5, (h) x=0.6, (i) x=0.7, (j) x=0.8, (k) x=0.9, and (l) x=1.  \n  \nFigure 4. (a) High field magneti zation measurements of Mn 2Mo 3O8, in which the critical field Hcri \nof spin flop transition is indicated. (b) Summari zed magnetic phase diagram, based on the pulsed \n(black squares) and static (red do ts) magnetic fields measurements.  \n \n \nFigure 5. (a) and (b) present obtained remnant magnetization Mr and coercive field Hc at a function \nof x at various temperatures, re spectively. (c) and (d) show T-dependence of Mr and Hc, \nrespectively.    \nFigure 6. (a) Measured pyroelectric current as a function of T. (b) Integrated electric polarization as \na function of T.  \n \n  \n \nFigure 7. (a) T-x phase diagram is summarized for (Fe 1-xMn x)2Mo 3O8 (0≤x≤1), based on the data of \nmagnetization (solid squares and triangles) and pyroelectric current (stars and crosses). (b) \nCalculated Mn2+ occupation ratios P O and P T as a function of x.  \n \n  \nTable of Contents Graphic \n \n "    },    {        "title": "1102.4414v1.Frustration_Induced_Ferrimagnetism_in_S_1_2_Heisenberg_Spin_Chain.pdf",        "content": "arXiv:1102.4414v1  [cond-mat.str-el]  22 Feb 2011Typeset with jpsj3.cls <ver.1.1> Letter\nFrustration-Induced Ferrimagnetism in S= 1/2 Heisenberg Spin Chain\nTokuro Shimokawa∗and Hiroki Nakano†\nGraduate School of Material Science, University of Hyogo, K amigori, Hyogo 678-1297, Japan\n(Received November 17, 2018)\nThe ground-stateproperties ofthe S= 1/2 frustrated Heisenbergspinchain withinteractions\nuptofourthnearest neighbors areinvestigated bytheexact -diagonalization methodanddensity\nmatrix renormalization group method. Ournumerical calcul ations clarify that theferrimagnetic\nstate is realized in the ground state in spite of the fact that a multi-sublattice structure in the\nshape of the system is absent. We find that there are two types o f ferrimagnetic phases: one\nis the well-known ferrimagnetic phase of the Lieb-Mattis ty pe and the other is the nontrivial\nferrimagnetic phase that is different from that of the Lieb-M attis type. Our results suggest\nthat a multi-sublattice structure of the shape is not necess arily required for the occurrence of\nferrimagnetism.\nKEYWORDS: quantum spin chain, frustration, ferrimagnetis m, DMRG, exact diagonalization\nFerrimagnetism is one of fundamental phenomena in\nthe field of magnetism. A typical case showing ferrimag-\nnetism is that when a system includes spins of two types\nthat antiferromagnetically interact between two spins\nof different types in each neighboring pair. The sim-\nplest example is an ( S,s)=(1, 1/2) antiferromagnetic\nmixed spin chain, in which two different spins are ar-\nranged alternately in a line and coupled by the nearest-\nneighborantiferromagneticinteraction.1)Theoccurrence\nof ferrimagnetism in this case is understood within the\nMarshall-Lieb-Mattis theorem concerning quantum spin\nsystems.2,3)Even though a system includes spins of one\ntype, this theorem also derives the presence of ferrimag-\nnetism when the system includes more than one sub-\nlattice of spin sites, for example, the spin system in a\ndiamond chain.4–8)From these two mechanisms, the ex-\nistence of a multi-sublattice structure is very important\nfor the occurrence of ferrimagnetism.\nAt this stage, one asks a fundamental question: Is a\nmulti-sublattice structure in the shape of a Hamiltonian\nessential and necessary for the occurrence of ferrimag-\nnetism? The purpose of the present study is to answer\nthis question. Our following demonstration will clarify\nthat the answer is no. In this study, we find that ferri-\nmagnetism can appear due to the effect of magnetic frus-\ntrationevenintheabsenceofamulti-sublatticestructure\nin the shape of a system.\nIn this study, we examine the model whose Hamilto-\nnian is given by\nH=J/summationdisplay\ni[Si·Si+1+1\n2Si·Si+2] (1)\n−J′/summationdisplay\ni[Si·Si+3+1\n2(Si·Si+2+Si·Si+4)],\nwhereSiis theS= 1/2 spin operator at the site i. The\nsystemsizeisdenoted by N.We emphasizeherethat this\nmodel has only one spin in a unit cell, namely, it has no\nsublattice structure. Energies are measured in units of\n∗E-mail address: rk09s002@stkt.u-hyogo.ac.jp\n†E-mail address: hnakano@sci.u-hyogo.ac.jpJ; therefore, we set J= 1 hereafter. We have a control-\nlable parameter, J′, in the Hamiltonian (1). This model\nwas originally introduced in ref. 11 detailing the study\nof constructing a model Hamiltonian as a generalization\nfrom the Majumdar-Ghosh model.12)The Hamiltonian\n(1) includes two cases in which the ground state of the\nsystem is exactly obtained. For J′= 0, the system is re-\nduced to the Majumdar-Ghosh model,12)whose ground\nstateis describedbydirectproductsofspin-singletstates\nin nearest-neighbor pairs of S= 1/2 spins. The ground\nstate is called the dimer (DM) state. Note that even if\nJ′takes a nonzero value, this DM state is still an eigen-\nstate of the system. The DM state becomes an excited\nstate when J′increases. In the limit of a large J′, on\nthe other hand, the ferromagnetic (FM) state becomes\nthe ground state. Although the wavefunctions of these\nlimits are well known, the ground state in the interme-\ndiate region is not sufficiently understood. In ref. 11, it\nwas reported that the spontaneous magnetization in the\nintermediate region appears and that the magnetization\nchanges gradually. In the present study, we investigate\nthe magnetic structure of the ground state in this inter-\nmediate region by some numerical calculations. We show\nthat our results lead to the conclusion that the ferrimag-\nneticstatecanappearinthegroundstate,evenofmodels\nconsisting of only a spin in each unit cell.\nWe employ two reliable numerical methods, the ex-\nact diagonalization (ED) method and density matrix\nrenormalization group (DMRG) method.13,14)The ED\nmethod can be used to obtain precise physical quanti-\nties for finite-size clusters. This method does not suffer\nfrom the limitation of the shape of the clusters. It is\napplicable even to systems with frustration, in contrast\nto the quantum Monte Carlo (QMC) method coming\nacross the so-called negative-sign problem for a system\nwith frustration. The disadvantage of the ED method\nis the limitation that the available sizes are only small.\nThus, we should pay careful attention to finite-size ef-\nfects in quantities obtained from this method. On the\nother hand, the DMRG method is very powerful when a\nsystem is one-dimensionalunder the open-boundarycon-\n12 J. Phys. Soc. Jpn. Letter Online-Journal Subcommittee\n\tB\n\tC\n 0 10 20\nSztot–40–30E nergy N=96 DMR G \nMS=1500\nSW=50\nJ’ =1.5 J’ =0.87J’ =0.685 \n0 2 4\nJ’ 00.51M/M sN=24(E D,periodic) N=24(E D,open)\n0.5 1 1.5 2 J’ 00.20.4M/M s\nN=24(E D,open)\nN=96(DMR G) \nFig. 1. (Color) (a) Lowest energy in each subspace divided by\nStot\nz. Results of the DMRG calculations are presented when the\nsize system is N= 96 for J′= 0.685,0.87, and 1.5. Arrows in-\ndicate the spontaneous magnetization Mfor a given J′;Mis\ndetermined to be the highest Stot\nzamong the values taking the\nlowestcommon energy.(b) J′dependence ofthe normalizedmag-\nnetization M/Msin the ground state. Red circles (black squares)\ndenote the results obtained by ED calculations for a size sys tem\nofN= 24 under the open (periodic)-boundary condition. In the\ninset of (b), blue diamonds show the results obtained by DMRG\ncalculations fora sizesystemof N= 96underthe open-boundary\ncondition accompanied by red circles denoting the results o b-\ntained by ED calculations for a size system of N= 24 under the\nopen-boundary condition.\ndition. The method can treat much larger systems than\ntheEDmethodandisapplicableeventoafrustratedsys-\ntem. In the present research, we use the ”finite-system”\nDMRG method.\nIn the present study, two quantities are calculated by\nthe two methods mentioned above. One is the lowest en-\nergy in each subspace divided by Stot\nzto determine the\nspontaneous magnetization M, whereStot\nzis thezcom-\nponent ofthe total spin. We can obtain the lowestenergy\nE(N,Stot\nz,J′) for a system size Nand a given J′. For ex-\nample, the energies of each Stot\nzin the three cases of J′\nare presented in Fig. 1(a). This figure is obtained by our\nDMRG calculations of the system of N= 96 with the\nmaximum number of retained states ( MS) 1500, and a\nnumber of sweeps ( SW) 50. The spontaneous magneti-\nzationMfor a given J′is determined as the highest Stot\nz\namongthoseatthelowestcommonenergy.(Seearrowsin\nFig. 1(a).) The other quantity is the local magnetization\nin the ground state for investigating the spin structure\nof the state. The local magnetization is obtained by cal-\nculating /angbracketleftSz\ni/angbracketright, where/angbracketleftA/angbracketrightdenotes the expectation value\nof the physical quantity AandSz\niis thez-component of\nthe spin at the site i.\nFirst, let us examine the J′dependence of M/Msto\nconfirm the existence of the intermediate phase between\nthe FM phase and the nonmagnetic DM phase irrespec-\ntive of the boundary conditions, where Msis the satu-\nration value of the magnetization. Results are presented0 0 .05 0.1 \n1/N0.511.522.5J’ \nJ’ 1J’ 2J’ 2J’ 5J’ 3J’ 4\nJ’ 1\nFig. 2. (Color) Size dependences of the phase boundaries. Th e\nresults presented are those of N= 12,18,24, and 30 from the ED\ncalculations and those of N= 48,72, and 96 from the DMRG\ncalculations. Squares (circles) denote results in the case s under\nthe periodic (open)-boundary condition. Dotted lines are d rawn\nas guides for the eyes between the data from the ED and DMRG\ncalculations. In the limit N→ ∞, the phase boundary between\nthe 0< M/M s<1/3 phase and the M/Ms= 1/3 phase seems\nto converge to approximately J′= 1.30.\nforN= 24 from our ED calculations under the open\nand periodic boundary conditions in Fig. 1(b). We suc-\ncessfully observe the intermediate-magnetization phase\nirrespective of the boundary conditions. We also include\nin Fig. 1(b) some DMRG results of N= 96, which sug-\ngests a weak size dependence of M/Msas a function of\nJ′. Careful observation of the region of 0 < M/M s≤1/3\nenables us to find that the intermediate-magnetization\nphase consists of two phases. One is the phase where\nM/Msis fixed at 1 /3; this feature is that of the ferri-\nmagnetism of the so-called Lieb-Mattis (LM) type, in\nwhich the spontaneous magnetization is fixed to be a\nsimple fraction of the saturated magnetization.2,3)The\nother is the phase where M/Mschanges continuously\nwith respect to the strength of J′. This feature is cer-\ntainly different from that of the LM ferrimagnetism; the\ncontinuous change in M/Msis observed as the ferrimag-\nnetism of the non-Lieb-Mattis (NLM) type in several\nmodels.15–22)We will determine later whether or not the\nphase of 0 < M/M s<1/3 in the present model is of the\nNLM type. Note here that these two phases are observed\nunder both boundary conditions. On the other hand, the\nregion of 1 /3< M/M s<1 is observed near M/Ms= 1\nonly under the open-boundary condition. At present, it\nis unclear whether or not this phase survives in the limit\nN→ ∞.\nNext, we study the size dependences of the bound-\naries between the phases observed above. We investigate\nfive boundaries: J′=J′\n1between the DM phase and the\nphase of 0 < M/M s<1/3,J′=J′\n2between the phase\nof 0< M/M s<1/3 and the phase of M/Ms= 1/3,\nJ′=J′\n3between the phase of M/Ms= 1/3 and the\nphase of 1 /3< M/M s<1,J′=J′\n4between the phase\nof 1/3< M/M s<1 and the FM phase, and J′=J′\n5\nbetween the phase of M/Ms= 1/3 and the FM phase\nwithout the phase of 1 /3< M/M s<1. Note that J′\n3and\nJ′\n4appear under the open-boundary condition, whereas\nJ′\n5appears under the periodic-boundary condition. Fig-\nure 2 shows the results of N= 12,18,24, and 30 from\nthe ED calculations and those of N= 48,72, and 96J. Phys. Soc. Jpn. Letter Online-Journal Subcommittee 3\n\tB\n\tC\n\tD\n00.20.4<S iz>\nJ’ =1.5, M =16\n00.20.4<S iz>J’ =0.87, M =12\n20 40 60 80 \ni–0.1 00.10.2<S iz>\nJ’ =0.685, M=8 \nFig. 3. (Color) Local magnetization /angbracketleftSz\ni/angbracketrightunder the open-\nboundary condition: (a) for J′= 1.5, (b) for J′= 0.87, and\n(c) forJ′= 0.685 from the DMRG calculation for N= 96. The\nsite number is denoted by i, which is classified into i= 3n−2,\n3n−1, and 3 n, where nis an integer. Squares, circles, and tri-\nangles mean i= 3n−2, 3n−1, and 3 n, respectively.\nfrom the DMRG calculations. One finds that J′\n1from the\nED calculations under the periodic-boundary condition\nand that from the DMRG calculations under the open-\nboundary condition are consistent with each other; we\nhaveJ′\n1∼0.59 as an extrapolated value. Concerning the\nboundary J′\n2, there exists a not so small difference be-\ntween the result under the open-boundary condition and\nthat under the periodic-boundary condition for a given\nN; however, J′\n2seems to converge to 1.30 irrespective of\nthe boundary condition. On the other hand, the situa-\ntions of the boundaries of the phase of M/Ms= 1/3 and\nthe FM phase are slightly complicated in our results. It\nseems that J′\n3andJ′\n4become farther away from each\nother with increasing Nand that J′\n3andJ′\n5converge to\nthe same value of 1.77. We also have J′\n4converging to\n2.06. From these results of the extrapolation, it is evi-\ndent that the phase of M/Ms= 1/3 and the phase of\n0< M/M s<1/3 exist in the thermodynamic limit. On\nthe other hand, it is difficult to determine whether or\nnot the phase of 1 /3< M/M s<1 is present. There is a\npossibility that this phase merges with the FM phase in\nthe thermodynamic limit for two reasons:one is that this\nphase appears only near M/Ms= 1 and the other is that\nit is observed only under the open-boundary condition.\nThe issue of whether or not this phase survives should\nbe clarified in future studies; hereafter, we do not pay\nfurther attention to this phase.\nNext, we examine the local magnetization /angbracketleftSz\ni/angbracketrightin the\ntwo phases of 0 < M/M s<1/3 andM/Ms= 1/3 to\ndetermine the magnetic properties in each phase. We\npresent our DMRG results of /angbracketleftSz\ni/angbracketrightof the system ofǰ\n\u0015P̂\u0014 \n\u0015P̂\u0013 \u0015P \tB\n\tC\n0 1 2 3 \nJ’ –1 –0.5 0E nergy EFM Eθ=0 \nEθ=π/3 \nFig. 4. (a) Spin configuration from the point of view of classi cal\nvectors. The site number iin the Hamiltonian (1) is classified\ninto 3n, 3n−1, and 3 n−2, where nis a positive integer. The\nangleθforJ′is determined by minimizing the classical energy.\n(b)J′dependences of classical energies of eq. (2) for θ= 0 (LM,\ndotted line), π/3 (NM, dotted chain line), and FM (solid line)\nenergy of eq. (3).\nN= 96. Note here that we calculate /angbracketleftSz\ni/angbracketrightwithin the\nsubspace of the highest Stot\nzcorresponding to the spon-\ntaneous magnetization Mobtained for a given J′. The\nresultsof /angbracketleftSz\ni/angbracketrightareshowninFigs.3(a)-3(c)for J′= 0.685,\n0.87, and 1.5, respectively. In each case, one can observe\na three-sublattice structure of the spin state clearly. In\nFig. 3(a), the idependence of /angbracketleftSz\ni/angbracketrightin each of the sub-\nlattices of the spin structure is weak around the center\nof the system, although the edge effect spreads into a\nwide range from the edges. This behavior suggests that\nthe spin state forms the LM ferrimagnetic state of up-\nup-down, which is consistent with M/Ms=1/3 in the pa-\nrameterregionnearapproximately J′= 1.5.InFigs.3(b)\nand 3(c), on the other hand, we find that the local mag-\nnetizationshowsalonger-distanceperiodicityinaddition\nto the three-sublattice structure. The longer-distancepe-\nriodicity changes when J′is changed within the phase of\n0< M/M s<1/3, the periodicity suggests an incom-\nmensurate modulation. A similar feature of this local\nstructure was reported in some one-dimensional quan-\ntum frustrated spin systems.18,19)Therefore, the phase\nof 0< M/M s<1/3 is considered as the NLM-type ferri-\nmagnetic phase. This incommensurate feature originates\nfrom the effects of quantum fluctuation and frustration.\nWe also calculate /angbracketleftSz\ni/angbracketrightfor different system sizes, N= 48\nand 72. At least from these data (not shown in this pa-\npar), the periodicity and amplitude of the modulation\nseem to show only weak dependences on the system size.\nNote that the behavior of long-distance periodicity ac-\ncompanied by the three-sublattice structure at the same\ntime is different from the wave functions with a long pe-\nriodicity reported in ref. 23.\nHere, let us discuss the behavior of the intermediate\nphase between the FM phase and the nonmagnetic phase\nfrom the viewpoint that spins in the Hamiltonian (1) are\nassumedtobeclassicalvectors.Weconsiderthespincon-\nfiguration of the classical vectors depicted in Fig. 4(a),4 J. Phys. Soc. Jpn. Letter Online-Journal Subcommittee\nwhere the characteristic angle θis defined. This classical\nspin arrangement has been determined from our obser-\nvation in Fig. 3 that the three-sublattice spin structure\nis realized in the intermediate region. The case of θ= 0\nmeans that this classical state is the LM-type ferrimag-\nnetic state with the ratio of the spontaneous magnetiza-\ntion to the saturated magnetization to be 1 /3. On the\nother hand, θ=π/3 means that the state is in a non-\nmagnetic (NM) state. The classical energy per spin site\nunder the periodic-boundary condition is given by\nE(J′,θ) =1\n24[(6−4J′)cos2θ−(6−4J′)cosθ+(−3−4J′)],\n(2)\nand the energy of the ferromagnetic state is given by\nEFM= (3−4J′)/8. (3)\nThe dependences of the energies shown in eqs. (2) and\n(3) are shown in Fig. 4(b). The FM (NM) phase appears\natJ′>1.5 (J′<1.5). One finds that J′=1.5 is the\nboundary of the FM and NM phases. At exactly J′=1.5,\nmany states degenerate, including not only the FM and\nNM states but also the ferrimagnetic state with an ar-\nbitrary angle θ. There is no intermediate phase between\nthe two phases. It is worth emphasizing here that even\nthe LM ferrimagneticphase does not appear.This argue-\nment suggests that the occurrence of the intermediate-\nmagnetization state observed in the Hamiltonian (1) of\nthe quantum system is a consequence of the quantum\neffect induced by frustration.\nFinally, we mention another case when the\nintermediate-magnetization phase appears in the\nfrustrated spin system in one dimension with anisotropic\ninteractions.24–28)Note here that this phase disappears\nin the isotropic case of interactions, which suggests that\nthe origin of this phase is the anisotropy. However, it\nhas not been examined yet whether or not this model\nshows a similar incommensurate modulation. Such\nexamination would clarify the relationship between the\nintermediate magnetization of this model and the NLM\nferrimagnetism studied in the present case.\nIn summary, we study the ground-state properties\nof anS= 1/2 frustrated Heisenberg spin chain with\nisotropic interactions up to the fourth nearest neighbor\nby the ED and DMRG methods. In spite of the fact that\nthis system consists of only a single spin site in each unit\ncell determined from the shape of the Hamiltonian, the\nferrimagnetic ground state is surprisingly realized in a fi-\nnite region between the ferromagnetic and nonmagnetic\nstates. This result is in contrast to that of other systems\nof translationally invariant chains.29,30)We find that the\nintermediate region consists of phases of two ferrimag-\nnetic types, the Lieb-Mattis type and non-Lieb-Mattis\ntype. In the latter phase, we confirm that the local mag-\nnetization shows characteristic incommensurate modula-\ntion. The presence of the ferrimagnetic state without a\nsublattice structure of the shape of the system is a con-\nsequence of the strong quantum effect induced by frus-\ntration. Our findings shed light on a new aspect of the\neffect of frustration in quantum systems.\nAcknowledgmentsWewishtothankProf.K.HidaandProf.T.Tonegawa\nfor fruitful discussions. This work was partly supported\nby a Grant-in-Aid (No.20340096) from the Ministry of\nEducation, Culture, Sports, Science and Technology of\nJapan. This work was partly supported by a Grant-in-\nAid (No. 22014012) for Scientific Research and Priority\nAreas “Novel States of Matter Induced by Frustration”\nfrom the Ministry of Education, Culture, Sports, Science\nand TechnologyofJapan.Diagonalizationcalculationsin\nthe present work were carried out based on TITPACK\nVersion 2 coded by H. Nishimori. 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Venkatesan1,3,4,5,6, and Hyunsoo  \nYang1,3* \n \n1Department  of Electrical  and Computer  Engineering,  National  University  of Singapore,  117576,  \nSingapore  \n2Shanghai Key Laboratory of Special Artificial Microstructure Materials & School of Physics \nScience and Engineering,  Tongji University, Shanghai 200092, China  \n3NUSNNI -NanoCore, National University of Singapore, 117411, Singapore  \n4Department of Physics, National University of Singapore, Singapore 117542, Singapore  \n5Department of Materials Science and Engineering, National University of Singapore, \nSingapore 117542, Singapore  \n6Integrated Science and Engineering Department, National University of Singapore, \nSingapore 117542, Singapore  \n \nWhile  current -induced  spin-orbit  torques  (SOTs)  have  been  extensively  studied  in ferromagnets  \nand antiferromagnets,  ferrimagnets  have  been  less studied . Here  we report  the presence  of \nenhanced  spin-orbit  torque s resulting  from  negative  exchange  interaction  in ferrimagnet s. The \neffective  field and switching  efficiency  increase  substantially  as CoGd  approaches  its \ncompensation  point , giving  rise to 9 times  larger  spin-orbit  torques  compared  to that of non-\ncompensated  one. The macrospin  modelling  results  also support  efficient  spin-orbit  torques  in a \nferrimagnet . Our results  suggest  that ferrimagnet s near compensation  can be a new route  for spin-\norbit  torque  application s due to their high thermal  stability  and easy current -induced  switching  \nassisted  by negative  exchange  interaction.   \n \n*eleyang@nus.edu.sg  \n \n 2 \n SOTs have emerged as proficient means of manipulating the magnetization  [1–9], in which \nthe spin current generated from a heavy metal transfers its angular momentum to the adjacent \nmagnet. A w ide range of SOT experiments have demonstrated  that the  spin current  can be \nmodulated  by materials  [10–12], structural [13–15] and interface  [16–18] engineering . On the \nother hand , a magnet itself can also play an important  role in modulati ng the spin  torques  as \nsuggested by recent  experiments and theory  [19–23]. A majority  of these report s use magnetic \nlayers  with an antiferromagnetic ordering . \nA key characteristic of the materials with antiferromagnetic ordering is the strong negative \nexchange interaction. Due to th e negative exchange, antiferromagnets and ferrimagnets show  \nfeatures distinct to their ferromagnetic counterparts. For example , the time scale of magnetization \ndynamics and reversal for antiferromagnets and ferrimagnets  (ps) is much lower compared to that \nof ferromagnets (ns)  [19,24,25] . Similarly, a faster demagnetization is observed in the case of \nantiparallel coupling as compared to parallel coupling in Co/Pt multilayers separated by a Ru \nspacer  [26]. Negative exchang e coupling is also found to assist in efficient domain wall motio n in \nsynthetic antiferro magnets  [20]. In light of the above observations , an active role of \nantiferromagnetic exchange is expect ed in modulating  SOTs in magnetic switching devices.  \nFerrimagnets provide an ideal platform to explore this possibility due to their tun able exchange \ninteraction  [27]. Recent SOT studies on collinear ferrimagnets were limited to  one or two fixed \ncomposition s [28,29] . However, in order to evaluate the  effect of exchange interaction on SOTs , \na comparative study across different ferrimagnetic compositions  is required .   \nIn this Letter, we demonstrate that the negative exch ange interaction torque can enhance \nthe SOTs in a  thermally stable , thick (6 nm) collinear ferrimagnet , CoGd . Unlike a ferromagnetic \nsystem, the SOT effective fields in Pt/CoGd are found not to scale inversely with Ms. As the 3 \n ferrimagnet approaches compensation, the longitudinal SOT effective field ( HL) and switching \nefficiency ( ) increase  ~ 9 and 6  times , respectively , while the decrease in MS is only two fold. \nThe anomalous increase of the SOT efficiency ( HL and ) near the compensation is attributed to  \nthe presence of an additional torque in ferrimagnets  which increases as the ferrimagnet approaches \ncompensation. This additional torque which we refer to as exchange interaction torque is due to \nthe negative exchange interaction between the ferrimagnetic sub -lattices .  \nCoGd is a rare earth -transition metal ( RE-TM) ferrimagnet in which the Co and Gd are \ncoupled antiferromagnetically  [27,30]. Therefore, the magnetization of CoGd can be tuned to be \ndomin ated either by Co or Gd depending on their relative composition. At a magnetic \ncompensation point, the total Co and Gd moments are equal, so the net magnetic moment of the \nCoGd system is zero  [27]. CoGd also has a bulk perpendicular magnetic anisotropy ( PMA) for \nvarious compositions depending on the deposition parameters  [27,30,31] , and consequently a thick \nmagnetic layer can be grown. In spite of almost zero saturation magnetization ( Ms) near \ncompensation, CoGd has a finite spin polarization which results in a non -zero tunnel \nmagnetoresistance ( TMR ) in magnetic tunnel junctions  [32], which facilitates the reading \noperation of ferrimagnet based memory devices.  \nThe film stacks of Si substrate/Pt (10 nm)/Co 1-xGdx (6 nm)/TaO x (1 nm) were deposited on \nthermally oxidized Si substrates using magnetron sputtering with a base pressure less than 5 10-\n9 Torr. The CoGd layer was deposited by co -sputtering from Co and Gd targets. The sputtering  \npower of Co target was fixed at 120 W w hile varying the sputtering power of Gd target  from 60 to \n120 W. The Co and Gd composition in the films was determined to be in the range from Co83Gd17 \nto Co 64Gd36 by Rutherford backscattering spectrometry. All the films showed PMA, confirmed by \npolar magneto -optic Kerr effect (MOKE) measurements. Subsequently the films were patterned 4 \n into Hall bar devices with a width of 10 m, using photolithography and ion -millin g. A thick Pt \nchannel (10 nm) was used to ensure  that the resistance of the patterned devices does not  vary \nwidely  and the current distribution profile remains  similar  for different samples.  \nFigure 1(a) shows the schematic of the transport measurement geom etry. Anomalous Hall \nmeasurements were performed by sweeping the magnetic field out of plane and measuring the Hall \nvoltage. The samples with x (Gd % in CoGd) ranging from 20 to 24.2 were identified to be Co \ndominated  using the anomalous Hall resistance, RAHE (Fig. 1(b)) and MOKE (inset of Fig. 1(c))  \nmeasurements , while samples  with x ranging from 25.5 to 34 were Gd  dominated  [33,34] . A \nnegative RAHE polarity for Gd rich sample s can be attributed to the fact that  the RAHE is dominated \nby Co in the CoGd  ferrimagnetic system  [35,36]. Figure 1(c) shows the value of coercivity for \ndifferent x, obtained from the MOKE and RAHE measurements.  As it is known that the coercivity \ndiverges and reaches its peak near compensation for RE -TM alloys  [30,3 7], there is a peak at x ~ \n25. Near compensation, the Ms is minimum, measured by a vibrating sample magnetometer \n(VSM), as shown in Figure 1(d). Throughout the above material characterization, we identify x ~ \n25 as a crossover composition between  a Co to Gd rich state.  \nWe have then performed harmonic (Fig. 2(a,b)) and SOT switching measurements (Fig. \n2(c,d)) on patterned Hall bar devices to determine current -induced effective fields and SOT \nswitching efficiency ( ), respectively. The data from two  representative samples for Co and Gd \nrich regime are shown in Fig. 2. The s econd harmonic ac measurements were performed to \nevaluate the longitudinal ( HL) and transverse ( HT) effective fields [5,6,15,38–40]. Low frequency \n(13.7 Hz) ac current s with an amplitude  of 10 mA (6.251010 A/m2) was passed through the Hall \nbar. The 1st (V) and 2nd (V2) harmonic Hall voltages were measured simultaneously using two \nlock-in amplifiers. Two sets of second harmonic measurements were performed by sweeping a \n5 \n small in-plane magnetic field in the longitudinal  (||) and transverse  () direction to the current \nflow. Figure 2( a,b) show the data  for sample s with x = 21.5   (Co rich) and 28  (Gd rich) . The net \neffect of spin -orbit torques on the ferrimagnet is determined by the dominating sub -lattice  \n(direction of net m) [28,4 1]. The slopes of second harmonic straight line s for x = 21.5 is opposite \nto that of sample x = 28 as a result of opposite RAHE polarity .  \nFor the switching measurements, an in -plane magnetic field of 1000 Oe was applied in the \ndirection of current flow and switching loops were obtained by probing the RAHE while sweeping \nthe current pulses. Figure s 2(c,d) show that the switching loop s obtaine d for samples with x = 21.5  \nis opposite to that of x = 28  due to the opposite sign of RAHE , similar to the harmonic measurements . \nParabolic background due to Joule heating has been removed to obtain  a clear switching \npicture  [42]. The switching was found to be gradual rather than abrupt. This type of gradual \nswitching behaviour was also found in other Pt /ferromagnet (F M) systems [23,4 5,46]. For our \nPt/CoGd devices , the switching happens through domain wall nucleation followed by e xpansion , \ngiving rise to a gradual switching slope. MOKE imaging of current induced switching was carried \nout to confirm this behaviour  [42]. \nMagnetization switching in the presence of SOT s is dominated by the damping -like torque \nor its equivalent effectiv e field, HL. The r ight axis of Fig. 3(a) shows  HL with various compositions  \nfor a current density of 1012 A/m2. From a value of  0.7 kOe for  x = 20, HL reaches a peak value of \n6.1 kOe for x = 24.2 near compensation and then decreases with further increasing x, reaching a \nvalue of 0.8 kOe for x = 34. The e ffect of the planar Hall effect ( PHE ) has been taken into account \nwhile extracting the SOT effective fields  [42]. A longitudinal temperature gradient in the device \ncan affect the second harmonic Hall voltage due to anomalous Nernst effect (ANE) ; however , the \neffect of ANE is found to be minimal  [42]. The value of HL near compensation is ~ 3 to 12 times 6 \n larger compared to  that of  Pt (3)/Co  (0.9)/Ta  (4 nm) [13] and Ta  (3)/CoFeB  (0.9)/MgO  (2 nm) [6] \nsystem s. The peak value of HL and HT correspond s to the spin efficiency (akin to the spin Hall \nangle) of 0.52 and 0.44 , respectively . These values are at least three tim es higher  compared to \nother Pt /FM systems  [42,47,48].  \nThe S OT efficiency of the system can also be evaluated  by measuring the switching \nefficiency parameter , . For a SOT switching through domain wall nucleation and propagation , \nthe depinning field is of essential  importance in SOT driven magnetic  reversal  [18,49]. Therefore,  \n is defined as HP/JS, where HP is the depinning field and JS is the switching current density [42]. \nRight axis of Fig . 3(b) shows the value of  for different  compositions.  follows  the same trend \nas HL with changing the composition . The peak value of  near compensation is found to be  \n96 10\n Oe/Am-2. When normalized by the thickness of magnetic layer , this value is 1-2 order (at \nleast 40 times) of magnitude larger compared to traditional FM  systems  [13,50]. \nBoth the SOT effective field (\n/2L sh e sH J eM d\n ) [51] and switching efficiency ( ) are \ninversely proportional to  Ms. Since Ms has a  minimum value at a compensation point, the observed \nenhancement of HL and  could be attributed to the change of Ms. However, we find that the amount \nof increase of HL and  is notably  higher than the decrease of Ms. For example , Ms decreases by \n2.1 times from 161 to 75 emu/cc  when the composition changes from x = 20 to 24.2 , whereas the \ncorresponding increase of HL  is ~9 times from 0.7 to 6.1 kOe  for a current density of 1012 A/m2 \nand  increase s ~6 times from \n90.6 10  to \n93.6 10  Oe/Am-2. Even after considering the effect \nof Joule heating on Ms and Hp during switching, a similar disproportiona l scaling of   is \nobserved  [42]. HL,  and 1/Ms values have been  plotted after normali zing with the corresponding \nvalues for Co 80Gd20 in Fig. 3 using the left y-axis. It is evident that the increase in HL and  is 7 \n significantly  higher  compared to the decrease of Ms as we approach compensation. A similar \ndisproportionate scaling trend i s observed using   =\n222/K ext sH H J , which is a simplified \nparameter based on a macrospin model to evaluate the SOT efficiency (inset of Fig. 3(b))  [52].  \nAnother  series of sample s also showed a qualitatively similar scaling trend of HL,  and  \nwith respect to Ms [42]. To further verify the efficient SOT scaling near compensation , a \ncomplementary  approach  based on chiral domain wall motion was used  to measure HL [48]. The \nenhancement in HL evaluated using this method  is ~ 6 times compared to 1.6 times decrease in Ms \nas sample approaches compensation [42]. This unusual and disproportion ate (to 1/Ms) scaling trend \nof  and HL observed in our ferrimagnetic Co 1-xGdx system cannot be understood in the framework \npreviously discussed in ferromagnetic SOT systems.   \nWe attribute  the observed  anomalous SOT scaling behaviour in the ferrimagnet to the \nnegative exchange interaction between the Co and Gd sub -lattice s. This antiferromagnetic \nexchange interaction field adds up with the existing longitudinal effective SOT field (\nSOT\nLH ), \nthereby enhancing the overall effective field  experience d by the dominant magnetization . In order \nto explain the augmented effect of SOT in ferrimagnet s, we consider two possible coupling cases \nbetween two sub -lattices A and B as shown in Fig. 4 (a). For the case (i) the sub -lattices are coupled \nantiferromagnetically  (A being dominating sub -lattice) , while they are coupled ferromagnetically  \nin the case of (ii). For fair comparison, n et Ms of the system is considered equal for bo th cases  \n(equal  in both cases ). In Fig. 4 (a), the yellow arrow represents the applied external field , \nHext (excluding exchange and SOT field) . In the present illustration, the anisotropy field ( Hk) is \nignored  for simplicity  leading to an initial magnetization direction along the x direction  (including \nanisotropy also gives a similar result  [42]). When the current is applied along the x direction, the \nSOT\nLH8 \n longitudinal SOT effective field ( ) acts on the two sub -lattices in a direction given by  m, \nwhere   is the direction (+y direction) of spins incoming from Pt  and m is the direction of \nindividual magnetization. Therefore , \nSOT\nLH  acts in opposite direction for the two sub -lattices in the \ncase (i),  while it acts in the same direction for the case (ii).   \nThe red, green and purple arrow s in Fig. 4(a) indicate the SOT, exchange and external \nfields  respectively,  acting along \nmy direction on individual moments at equilibrium. \nex\naH  and \nex\nbH\n are the exchange field acting on sub -lattice A and B, respectively. It is evident that for the \ndominating sub -lattice in case (i), \nsin( )ex\na b aH  adds up with \nSOT\nLH  thereby giving rise to a \nlarger effective HL. However , for case (ii) only \nSOT\nLH acts on the system. Even for the non -\ndominating sub -lattice in case (i) , the net effective HL is combination of \nSOT\nLH and \nsinext bH . The \nstrength of net current -induced effective field can be gauged by the value of tilt angle ( ). For this \npurpose, force balance is applied along \nmy  to quantify  [5,40]. In small angle approximation, \nan analytical solution of force balance equation s reveals that \n,a b fm  as shown  by\n()\n()ex ex SOT\na b ext L\na ex ex\next b a extH H H H\nH H H H\n, \n()\n()ex ex SOT\na b ext L\nb ex ex\next b a extHHH H\nH H H H , and \n/SOT\nfm L extHH . Using \nabove equations  it can be deduced that the net current -induced longitudinal effective field \nexperienced by a ferrimagnet is \n()\n()ex ex\nSOT a b ext\nLL ex ex\nb a extH H HHHHHH . Likewise , for the case when both \nanisotropy and external field s (Hext << Hk) are considered  [42], HL can be expressed \nas \n()\n()ex ex\nSOT b k a\nLL ex ex\nb k aH H HHHH H H . \nSOT\nLH9 \n The magnitude of \nex\naH (\nex\nbH ) is proportional to the individual  saturation magnetization of \nB(A) [53]. Therefore , for the case when the magnetization is dominated by A, \nex ex\nbaHH . As a result,  \nHL keep s on increasing as the ferrimagnet approaches compensation due to an increase of \nex\naH (see \nabove  equations of HL) and \nSOT\nLH  (inversely  proportional to Ms). Due to a larger net HL as \nexplained above, switching in a ferrimagnet is more efficient  compared to the FM case . It should \nbe noted that even though the negative exchange torque is present for all the composition s of a \nferrimagnet, its effect becomes more pronounced as the ferrimagnet approaches compensation \nbecause of increase in the value of the negative exchange. Too far away from the compensation , \nthe system is dominated by one of the sub -lattices and the effects of the negative exchange are \nnegligible due to its small value . In such a scenario, the SOT behaviour  will be closer to that of a \nferromagnetic system.   \n To validate the above model , macrospin simulation s were  performed  by solving  two \ncoupled Landau -Lifshitz -Gilbert (LLG) equation s [54,55]. Each LLG equation simulate s the \ndynamics of sub -lattices A and B individual ly. The two equations are coupled by  the exchange \nfield,  \n,ex\nabH , which was included in the net effective field  [42]. Figure 4(b) shows a plot of HL and \n1/Ms obtained from simulation  results . The trend  is qualitatively similar to what we observe  in \nexperiments as shown in Fig. 3(a). From simulations , it is clear that a system of two sub -lattice s \nwith negative exchange interaction has a higher net current -induced longitudinal effective field \ncompared to a ferromagnetic system with an equivalent  Ms.  \nIt is interesting to note  that in experiments the scaling of HL for Gd rich samples (x = 32.5 \nand 3 4) is found to be less than that  of 1/ Ms. One possible reason is low exchange interaction at \nroom temperature for Gd rich CoGd alloys  [27]. It is also possible  that incoming spin s do not 10 \n completely transfer their angular momentum to the Gd sub -lattice because the electron s carrying \nmagnetic moment s in Gd reside in inner 4 f shell , which  can result in a lower scaling of  HL. For B \ncomposition more than ~ 30% in A1-xBx, the s imulated value of HL is found to be  less than that of \nequivalent ferromagnet , when SOT act ing on sub-lattice B is considered zero. The scaling of HL \nfor A rich composition still remains far larger compared to 1/ Ms [42]. Another notable point  is that \nthe analyse s and results present ed in this work  holds true for collinear ferrimagnets like CoGd.  A \ndrastic enhancement of the SOT efficiency is not observed for the case of CoTb [ 56] which is a \nnon-collinear ferrimagnet (sperimagnet) [5 7]. In CoTb  or other non -collinear RE -TM \nferrimagnets , where  RE mom ents are distributed in opposite half sphere relative to TM due to non -\nzero orbital moment of RE, the effect of negative exchange may not be straight forward to analyse \nwith the macrospin model  and may possibly result in a different SOT behaviour .  \nIn conc lusion, we have evaluated the  role of negative exchange in enhancing the efficiency \nof ferrimagnetic SOT devices. It is found that the SOT efficiency increase s anomalously  near \ncompensation compared to the scaling of Ms due to the inc rease in the negative exchange . The \nlongitudinal ( HL) and transverse ( HT) effective fields as well as the switching efficiency ( ) are \nsignificantly higher compared to  conventional  ferromagnetic SOT systems. 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Oezelt, A. Kovacs, F. Reichel, J. Fischbacher, S. Bance, M. Gusenbauer, C. Schubert, M. Albrecht, and \nT. Schrefl, J. Magn. Magn. Mater. 381, 28 (2015).  \n[55] I. Firastrau, L. D. Buda -Prejbeanu, B. Dieny, and U. Ebels, J. Appl. Phys. 113, 113908 (2013).  \n[56] J. Finley and L. Liu, Phys. Rev. Appl . 6, 054001  (2016).   \n[57] M. Schlenker, J. Pelissier, B. Barbara, J. P. Guigay, G. Fillion, R. H. Geiss, A. Liénard, and B. Blanchard, J. \nPhys. 51, 483 (1990).  \n   \n  14 \n Figure captions  \n \nFig. 1. (a) Schematic of film stack and transport measurement geometry. (b) RAHE plotted as a \nfunction of Gd concentration. The i nsets show  opposite RAHE hysteresis loop s for the Co and Gd \ndomi nated  samples . (c) Coercivity of deposited films using MOKE and that of the  fabricated \ndevices using AHE measurement s. The left (right) inset shows a hysteresis loop for Co (Gd) rich \nsample using MOKE. Note the reversal of hysteresis loop s across compensation. (d) Saturation \nmagnetization versus Gd concentration measured by VSM.  \nFig. 2. (a,b) 1st and 2nd harmonic Hall voltage measurements with an in -plane field applied parallel \n(||) and perpendicular  () to the current direction. Colored line s represent  the quadratic and linear \nfitting for extracting the HL and HT. Note that  the net magnetization  (M) is in upwards direction  \nduring measurements for both the cases.  (c,d) Current -induced switching loops for the Co and Gd \nrich samples . An o ffset due to Hall bar misalignment has been removed.   \nFig. 3. (a) Normalized longitudinal effective field ( HL) and switching efficiency  () with \nnormalized value of 1/ Ms. HL, , and 1/ Ms for various samples are normalized to their respective \nvalue s of Co80Gd20 sample . The i nset in (b) shows  the normalized macrospin switching efficiency \n(\n) for various compositions . \nFig. 4. (a) Schematic representation of various fields acting on the  two sub -lattice s with (i) negative \nand (ii) positive exchange interaction . (b) Macrospin simulation  results . Simulation s were  carried \nout in the presence of  anisotropy and external fields. Plot shows normalized HL of ferrimagnet  \ncompared with normalized 1/ Ms for different B concentration s. The scaling trend of normalized \nHL of an equivalent ferromagnet ( similar  Ms) is also shown. Values are normalized with respect to \nthe values for A 99B1.  15 \n  \nFigure 1  \n \n \n \n \n \n \n \n \n \n \n15 20 25 30 35-0.050.000.05\nx (Gd %)RAHE()\nCo rich\nGd rich-1 0 1\nHext (kOe)-1 0 1\nHext (kOe)\n15 20 25 30 3504008001200\nx (Gd %) MOKE\n VAHECoercivity (Oe)-1 0 1\nHext (kOe)-1 0 1\nHext (kOe)\nVI\nPt (10 nm)TaOx(1 nm)\nCoxGd1-x(6 nm)(a)\n(c)\n15 20 25 30 350100200Ms (emu/cc)\nx (Gd %)(b)\n(d)Si/SiO2substrate16 \n  \n \nFigure 2  \n \n \n \n \n \n \n \n \n-0.050.000.05RH ()\n-60 -30 30 60-0.050.000.05RH ()\nI (mA)MMM\nMCo78.5Gd21.5\nCo72Gd28(c)\n(d)\n-1 0 10400404408412416V (V)\n-0.30-0.150.000.150.30\n H || I\n H  I\nV2 (V)MCo78.5Gd21.5\n-2 -1 0 1 2-316-312-308-3040V (V)\nHext (KOe)-0.30-0.150.000.150.30\nV2 (V)\nCo72Gd28(a)\n(b)\nM17 \n  \nFigure 3 \n \n \n \n \n20 25 30 3502468Normalized and 1/Ms\nx (Gd %) \n 1/Ms\n0246\n ( 10-9 Oe/Am-2)\n2025303502040\n \n 1/MsNormalized \nx (Gd %)\n0246810Normalized HL & 1/Ms\n0246\n HL\n 1/Ms\nHL (kOe)(a)\n(b)18 \n  \nFigure 4 \n \n(a)\nTaOx\na\nb\nsin( )ex\nb b aH\nsin( )ex\na b aH\nSOT\nLH\nSOT\nLH\nsinext aH\nsinext bH\nextH\nxz(i)\nPtCoGd\nSi/SiO2currentxz\n0 20 400255075100Normalized HL and 1/Ms\nx (B % in A1-xBx)  Ferrimagnet\n Equivalent ferromagnet\n 1/Ms\n(b)\nSOT\nLH\nsinext aHxz (ii)\nextH\nfm\n"    },    {        "title": "1209.0003v1.Magnetic_symmetry_of_the_plain_domain_walls_in_ferro__and_ferrimagnets.pdf",        "content": " 1 \nMagnetic symmetry of the plain domain walls \nin ferro- and ferrimagnets \n \nB. M. Tanygin 1 and O. V. Tychko 2* \n1Kyiv Taras Shevchenko National University, Radiophy sics Faculty, Glushkov av.2, build.5, \nKyiv, Ukraine, 01022 \n1E-mail: b.m.tanygin@gmail.com  \n2E-mail: pasat@univ.kiev.ua \n \nAbstract.  Magnetic symmetry of all possible plane domain wal ls in ferro- and ferrimagnets is \nconsidered.   Magnetic symmetry classes of non 180 degree (inclu ding 0 degree) domain walls are \nobtained.  The domain walls degeneracy is investigated.  The symmetry classification is applied for \nresearch of  all possible plane domain walls in crystals of the hexoctahedral crystallographic class.  \nPACS: 61.50 Ah, 75.60 Ch \nKeywords: domain wall type, symmetry transformation , magnetic symmetry class, degeneracy \n \n1.  Introduction \nThe investigation of static and dynamic properties [1,2] of domain walls (DWs) in magnetically \nordered media is of considerable interest for the p hysical understanding of medium behavior  and  it is also \nimportant for applications. For sequential examinat ion of these properties it is necessary to take int o \naccount the magnetic symmetry [3,4] of the media. D etermination of the DW magnetic symmetry allows  \n \n*Corresponding author. O.V. Tychko. Address:  64 Vladimirskaya str., Taras Shevchenko Kyiv Natio nal \nUniversity, Radiophysics Faculty. 01033 Kyiv, Ukrai ne. Tel/fax : +38-044-526-03-49  E-mail : \npasat@univ.kiev.ua , a.tychko@mail.ru   2 \nto characterize qualitatively some elements of the DW structure  and  their  change.  The  complete  \nsymmetry classification of plane 180 degree DWs (18 0 0-DWs) in magnetically ordered crystals [5] and \nsimilar classification of these DWs with Bloch line s in ferromagnets and ferrites [6] were carried out  \nearlier. The plane DWs with width δ[1,7] exceeding the characteristic size a of a unit magnetic cell were \nconsidered. Properties of these DWs in ferro- and f errimagnets are described by the density of magneti c \nmoment M [8] . Their symmetry can be characterized by the magnetic  symmetry classes (MSCs) [9] of a \ncrystal containing a DW [5]. The building of a tota lity of the MSCs of all possible [1] plane (i.e. DW  with \n0r>> δ, where 0r is the curvature radius of the DW [5]) DWs in ferr o- and ferrimagnets is the purpose of \nthis work. \n \n2. Domain wall symmetry in the magnetically ordered  media \nLet m be the unit time-odd axial vector [9] along the ma gnetization vector M: M/Mm=, where \nM is the saturation magnetization. Then 1m and 2m are unit time-odd axial vectors along magnetization  \nvectors 1M and 2M in neighboring domains: M/1 1Mm= , M/2 2Mm= . The vectors 1m and 2m \ncoincide with different easy magnetization axes (EM A) of the medium. The angle α2 between these \nvectors determines the DW type ( α2-DW): ()2 1 arccos 2 mm =α . A unit polar time-even vector Wn \nindicates the DW plane normal. It is directed from domain with 1m to domain with 2m. In order to define \nthe unified co-ordinate system we introduce the vec tors 1a and 2a as well as the parameters \n[]Σ Σ×=mnWb  and []mnΔ ×=Δ Wb . The unit vectors of the co-ordicate system z y x O~~~ are chosen as \n[][ ]W zyx naaeee ,,,,12~~~ −= . Here the unit vector 1a coincides with the direction of the vector \n()mnnm Δ−ΔWW  (at 0≠Δb and 0=Σb) or []Wna×2 (at 0=Δb or 0≠Σb). The unit vector 2a coincides \nwith the direction of vector ()Σ Σ−mnnmWW  (at 0≠Σb) or []1an×W (at 0≠Δb and 0=Σb) or else with \nan arbitrary direction in the DW plane ( Wna⊥2 at 0==ΔΣbb ). The time-odd axial vectors mΔ and Σm \nare determined by equalities 12mmm−=Δ  and 21mmm+=Σ  respectively.   3 \n  The MSC kG (here k is a MSC number) of a α2-DW is the magnetic symmetry group including \nall symmetry transformations  (here and hereinafter all translations are consider ed as unit operations)  that \ndo not change the spatial distribution of magnetic moments in the crystal with DW [5]. The above-\nmentioned group is a subgroup of the magnetic (Shub nikov’s) symmetry group of the crystal \nparamagnetic phase [10]. These transformations do n ot change DW boundary conditions and can be \nclassified by two types [5]. The first type transfo rmations ()1g do not change the directions of the vectors \n1m, 2m and Wn: ()\nWgn1=Wn, ()\n111mm=g , ()\n2 21mm=g . The second type transformations ()2g change \nthese directions: ()\nWgn2=Wn−, ()\n2 12mm=g , ()\n1 22mm=g . In conformity with the terminology of [6] the \nMSC BG of DW boundary conditions is the totality of all t ransformations of the magnetic symmetry \ngroup of the crystal paramagnetic phase that satisf y the mentioned six conditions. It is the MSC of th e \nmaximum possible symmetry of a α2 -DW in the given crystal for a particular mutual o rientation of the \nvectors 1m, 2m and Wn. The other possible MSCs kG of a α2-DW with fixed directions of the vectors \n1m, 2m and Wn result by enumeration of the subgroups of BG : P B k GGG ⊂⊆ , where PG is the MSC \nof the crystal paramagnetic phase. The mutual orien tation of the vectors 1m, 2m and Wn is determined by \nthe set of parameters ()Σ Σ=mnWa , ()mnΔ=Δ Wa , ()CW Camn= , Σb and Δb, where time-even axial \nvector Cm is determined by equality []21mmm ×=C . \nThe possible MSCs kG (1 ≤k≤42) of 180 0-DWs were found earlier [5]. All possible MSCs of \nα2-DWs with α2≠180 0 are presented in table 1. \nFor a certain α2-DW the different MSCs are different groups of magn etic point symmetry \ntransformations. Their representations [11,15] are written in the co-ordinate system z y x O~~~. All \nrepresented MSCs are not interrelated by a rotation  over an arbitrary angle around Wn. Also the above-\nmentioned MSCs are not reduced with each other by u nit vectors transformation 2 1aa↔. \nThe possible transformations ()1g or ()2g (column “Symmetry elements” of table 1) of α2-DWs \nwith α2≠180 0  are rotations around two-fold symmetry axes n2 , n2′ or 12 , 12′ or else 22 , 22′ that are  4 \ncollinear with the unit vectors Wn or 1a or else 2a, respectively, reflections in planes n2, n2′ or 12′ or else \n22, 22′ that are normal to the above mentioned vectors, re spectively, rotations around three-, four-, six-\nfold symmetry axes n3, n4, n6 that are collinear with the vector Wn, rotations around three-, four-, six-\nfold inversion axes n3 , n4 , n6  that are collinear with the vector Wn, inversion in the symmetry center 1 \nand identity (symmetry element 1). Here an accent a t symmetry elements means a simultaneous use of the  \ntime reversal operation R [9]. For MSCs with 24 ≤k≤39 and 52 ≤k≤64 only generative symmetry \nelements [11] are represented in table 1. \nThere is a correspondence between MSCs of 180°-DWs ( i.e. at 1m=2m−[1]), 0°-DWs (i.e. at \n1m=2m[13]) and α2-DWs with non-collinear orientation of vectors 1m and 2m[1] (hereinafter the last \nDWs will be marked as α′2-DWs). The above mentioned determinations of criteri ons for transformations \n()1g and ()2g can be represented in another identical form: ()\nWgn1=Wn, ()\nΣ Σ=mm1g , ()mmΔ =Δ1g  and \n()\nWgn2=Wn−, ()\nΣ Σ=mm2g , ()mmΔ − =Δ2g . These criterions restrict an ensemble of MSCs symm etry \ntransformations for an arbitrary α′2-DW. We have 0=Δm and 0=Σm for 0°- and 180°-DWs, \nrespectively. A pair from the above mentioned crite rions does not restrict the MSCs  symmetry \ntransformations of 0°- or 180°-DWs. Therefore the ma gnetic symmetry of α′2-DWs does not exceed the \nmagnetic symmetry of 0°- and 180°-DWs generically. The MSCs of 180°-DWs are the MSCs of α′2-\nDWs if their transformations do not break the symme try of the vector Σm of the α′2-DW (i.e. these \nMSCs must be subgroup of the group mm/m ′′∞, where the infinite-fold symmetry axis is collinea r with \nthe vector Σm).   \nThere is an analogy between MSCs of 180°- and 0°-DWs : their transformations ()1g are the same \nsince they belong to a subgroup of axial time-odd v ector symmetry group (MSC  mm/m ′′∞), where the \ninfinite-fold symmetry axis is collinear with mΔ or Σm for 180°- or 0°-DWs, respectively . Therefore if \nMSCs consist of the transformations ()1g only then these MSCs are common for 180°- and 0°-D Ws. They \nare marked with sign “-” in column “DW center” of t able 1. A conversion of MSC of 180°-DW into MSC  5 \nof 0°-DW is simply a change of the criterion ()mmΔ − =Δ2g  by the criterion ()\nΣ Σ=mm2g . The \ntransformations of corresponding MSCs of these α2-DWs are different by the substitution ()()Rgg ⋅→2 2 \nonly. Therefore, if a pair of MSCs of 180°-DWs and a pair of MSCs of 0°-DWs is connected by the \nabove-mentioned substitution, then these MSCs are c ommon for 180°- and 0°-DWs.  \nAs a result the lists of MSCs of  0°-, 180°- and α′2-DWs  are intersected in general. Total number \nof MSCs of a α2-DW with arbitrary α2 value (including α2=180 0) in ferro- and ferrimagnets is equal \nto 64. General enumeration of MSCs of 180°-DWs cont ains 42 MSCs: 42 1≤≤k [5]. This enumeration \nholds also for MSCs of α2-DW with °≠180 2α(MSC numbers are bold type in column “MSC number \nk” of table 1) . There are 10 MSCs of α′2 -DWs: 13 7≤≤k and 18 16 ≤≤k. The general list of MSCs of \n0°-DWs includes all 42 MSCs of table 1: k=2, 13 6≤≤k, 19 16 ≤≤k, k=22, 24, 26, 30, 32, 37, 39 and \n64 43 ≤≤k. \n \n3. Domain wall structure \nThe α2-DWs with δ>> a in ferro- and ferrimagnets are described by the ma croscopic density of \nmagnetic moment M(z~) [5]. The transformations ()1g and ()2g (()\nkGg∈1;()\nkGg∈2) impose restrictions \non the kind of coordinate dependence of ()z~m components ( ()()()()zzzzz y x~~~~~ ~ ~ mmmm ++= ) in the DW \nvolume and allow to find this dependence [5]. For t he determination of the kind of coordinate dependen ce \nof  ()z~m component of 0°- and α′2-DWs for each MSC (column “Coordinate dependences o f ()z~m \ncomponents” in table 1) the next rules are used: a)  if an axial time-odd vector along unit vectors re \n(r≡x~,y~ or  z~) is not an invariant of the transformation ()1g then there is no component ()zr~m (figure (-) \nin column “Coordinate dependences of ()z~m components” of table 1); b) if the axial time-odd vector \nalong re is inverted by the transformation ()2g then the component ()zr~m is an odd (A) function of \ncoordinate z~; c) if the axial time-odd vector along re is an invariant of the transformation ()2g then \n()zr~m is an even (S) function of coordinate z~; d) if the axial time-odd vector along re is an invariant of \nthe transformation ()1g then transformation ()1g does not restrict the kind of function ()zr~m (A,S).   6 \nIf the MCS of a α2-DW includes transformations that transpose adjacen t magnetic domains then \nthis DW has a center of symmetry [5]. These MSCs en close the symmetry transformations ()2g. They  are \nmarked by coordinate 0~=z in column “DW center” of table 1. \nAs in the case of 180 0-DWs [5], the 00- DWs can be pulsating (i.e. DW with collinear dire ctions of \nvectors M and M≠const  in its volume [5]) DWs. The MSCs with k=2, 6, 19-45, 49-64 describe \nsymmetry of pulsating DWs only. In contrast with  180°- and 0°-DWs there are no pulsating DWs among \nthe α′2-DWs, since α′2-DWs require the presence of two “nonzero” ()z~m  components. The α′2-DWs \nare rotary (i.e. DW with M=const  in its volume) or semi-rotary [5] DWs only. Among r otary or semi-\nrotary DWs there are DWs with only Bloch (i.e DWs w ith MWn=const) [1,14] ( k=7, 8 or 46) and only \nNeel (i.e. DWs with m rotation in the plane containing Wn) [1,15] ( k=9, 12, 17 or 47) laws of m rotation \nin their volume. \nCrystal magnetic ordering is accompanied by phase t ransition and change of crystal magnetic \nsymmetry [3]. In a  magnetically ordered crystal kq-multiply degenerate α2-DWs with fixed α2 can be \nobtained [6], where ()()k P k GGq /ord ord = . Functions ()PGord  and ()kGord  give the order [11] of the \nmagnetic point group of the  crystal  paramagnetic phase [9,10] and of a α2 -DW in this crystal, \nrespectively. These α2 -DWs have the same energy but different structures  (magnetization distribution, \nplane orientation, etc.). The minimum value of kq is 2 in accordance with the invariance of energy f or \ntime reversal operation R. \nAt representation of the PG as the totality of kG (with fixed value k and different symmetry \nelements orientations) the lost transformations (me mbers of adjacent classes) lg [6,12] interrelate the \nabove mentioned kq-multiply degenerate α2-DWs  (i.e. lg operation converts an one of such α2-DWs \ninto another). \nThe degeneracy kq of a α2-DW can be written in the form k B kqqq′= (kq′≤kq), where \n()()k B k GGq /ord ord =′  is the number of equal-energy α2 -DWs with fixed boundary conditions,  7 \n()()B P B GGq /ord ord =  is the number of possible boundary conditions. Her e ()BGord  is the order of the \npoint group of the maximum magnetic symmetry of the  α2-DW in the given crystal.  \nThe α2-DWs of MSC 16 G (MSC 1) have the maximum degeneracy kq. For 180°- and α′2-DWs \nit is equal to 16 (crystallographic class mmm), 48 (crystallographic class 6/mmm) and 96 \n(crystallographic class m3m) in crystals of lower, medium and higher symmetry singonies (in conformity  \nwith terminology of [11]), respectively. The 0°-DWs  are formed in spatially inhomogeneous media [13]. \nConditions of occurrence and existence of such DWs demand to take into account medium peculiarities. \n \n4. Magnetic symmetry classes of domain walls in hex octahedral crystals \nAs an example let's consider MSCs of all possible D Ws in magnetically ordered crystals of \nhexoctahedral class (crystallographic point symmetr y group m 3 m in the paramagnetic phase [3]) . This \nclass is assumed to exhibit the largest variety of possible DWs. Furthermore it encompasses widely \ninvestigated and used magnetic media (all cubic sym metry metals, specifically iron and nickel [6], \nmagnetic oxides, specifically ferrites with structu res of spinel [4] and garnet [16], perovskite, magn etite \nand others). \nThe magnetic anisotropy (MA) energy Ke is the invariant of the initial paramagnetic phase  of \ncrystal. For the m 3 m crystal this energy is given by ()321,,αααKe =1Ks+2Kp+3K2s+4Ksp +..., \nwhere 1K, 2K, 3K and 4K are first, second, third and fourth MA constants, s=2\n22\n1αα+2\n32\n2αα+2\n32\n1αα, \np=2\n32\n22\n1ααα, 1α, 2α and 3α are the direction cosines of  m [16].  The absolute minimum of this energy \ncorresponds to EMAs. Signs of MA constants and rela tion between their values determine EMAs \ndirections. In the framework of the (1K, 2K, 3K) approximation the EMAs directions can coincide wi th \nboth high-symmetric and low-symmetric crystallograp hic directions [17]. In the framework of the two-\nconstant ( 1K,2K) approximation the EMA directions can coincide onl y with high-symmetric <111> or \n<110> or else <100> like crystallographic direction s at 1K≤-32K or 0 ≥1K≥-22K or else 1K≥0 \nrespectively [1,18]. At that 71 0-, 109 0- and 180 0-DWs or 60 0-, 90 0-, 120 0- and 180 0-DWs or else 90 0- and  8 \n180 0-DWs are realized in a m 3 m crystal, respectively [1]. The MSCs and degeneracy  Bq of a α2-DW \nboundary conditions with °>90 2α and °≤90 2α are presented in tables 2 and 3 respectively.  \nThe earlier obtained MSCs of merely 180°-DWs (bold type numbers in table 2) include elements \n[5]: k=1 - ()()1 , 12 , 2 , 2 , 121′×n; k=4 - ()112 , 2 , 1 , 1′ ′; k=5 -()nn2 , 2 , 1 , 1′ ′; k=14 - ()222 , 2 , 1 , 1′′; k=15 - ()1 , 1′; \nk=23 - ( )()1 , 12 , 2 , 2 , 121′×n; k=29 - ()12 , 3′′n; k=34 - ()n n2 , 2 , 41′′.  \nOnly generative symmetry elements are presented for  k=29 and 34. Other MSCs of tables 2 and 3 \nare presented in table 1. In these tables the DW pl ane orientation is assigned by different Miller ind exes \nh,k,l> 1. A simultaneous change on negative and/or cyclic permutation of all indexes doesn’t change \nMSCs. \nThere are no common MSCs of maximum symmetrical 180 °- and α′2-DWs in the m3m crystal. It \nis connected with the presence of the 1′ transformation ( α′2-DW vector Σm is changed by this \ntransformation) in the MSCs of such 180°-DW. \n \n5. Conclusions \nThe full magnetic symmetry classification of all po ssible domain walls in ferro- and ferrimagnet \ncrystals includes 64 magnetic symmetry classes: 42 classes of 0 0- DWs, 10 classes of α2-DWs with \n00<α2 <180 0 and 42 classes of 180 0-DWs. Lists of magnetic symmetry classes of all abo ve mentioned \ntypes of DWs are intersected in general case.  \n00- DWs can be pulsating,  rotary or semi-rotary DWs.  The α2-DWs with 0 0<α2<180 0 are rotary \nor semi-rotary DWs only. Among rotary or semi-rotar y DWs there are DWs with Bloch or Neel laws of \nmagnetization rotation in their volume.  Pulsating, rotary or semi-rotary DWs can have a cen ter of \nsymmetry in their volume.  \nAll possible 180 0- and α2 -DWs with 0 0<α2 <180 0 have even degeneracy (its value is between 2 \nand 96 in general case).  \nMagnetic symmetry classes of maximum symmetrical 18 0°-DWs do not meet with such classes of \nα2-DWs with 0 0<α2<180 0 in a m3m crystal.  9 \nReferences \n[1] A. Hubert, Theorie der Domanenwande in Geordnet en Medielen  (Theory of Domain Walls in Ordered \nMedia), Springer, Berlin, Heidelberg, New York, 1974 \n      A  Hubert and R. Shafer, Magnetic Domains. Th e Analysis of Magnetic Microstructures, Springer, \nBerlin, 1998 \n[2]  V. Bokov and V. Volkov, Physics of the Solid S tate 50 (2008)198 \n[3]  L. Shuvalov, Sov. Phys. Crystallogr.   4(1959)399 \n[4]  L. Shuvalov, Modern Crystallography IV : Physi cal Properties of Crystals, Springer, Berlin, 1988 \n[5]  V. Baryakhtar, V. Lvov and D. Yablonsky,  JETP   87(1984)1863 \n[6] V. Baryakhtar, E. Krotenko and D. Yablonsky, JE TP  91(1986)921 \n[7]  B. Lilley, Phil.Mag.  41(1950)792  \n[8] A. Andreev and V. Marchenko, JETP 70(1976)1522 \n[9]  L. Landau, E. Lifshitz and L. Pitaevskii, Cour se of Theoretical Physics, vol.8. Electrodynamics o f \nContinuous Media, Pergamon Press, London, 1984  \n[10] V. A. Kopcik, Xubnikovskie Gruppy: Spravoqnik po simmetrii i fiziqeskim svostvam. kristalliqeskih  \nstruktur [Shubnikov’s groups: Handbook on the symme try and physical properties of crystalline \nstructures, in Russian],   Izdatel’stvo Moskovskogo Universiteta, Moscow, 196 6 \n         A.V. Shubnikov and N.V. Belov, Colored sym metry, Pergamon Press, London, 1964 \n        B. Tavger and V. Zaitzev, JETP 3(1956)430 \n[11]   B. Vanshtein, Modern Crystallography 1: Symm etry of Crystals, Methods of Structural \nCrystallography, Springer, Berlin, 1994 \n[12] E. Wigner, Group Theory and its Application to  the Quantum Mechanics of Atomic Spectra, \nAcademic Press, New York, 1959 \n[13] L. Heyderman, H. Niedoba, H. Gupta and I. Puch alska, J. Magn. Magn. Mater  96(1991)125. \n        R. Vakhitov, A Yumaguzin, J. Magn. Magn. Ma ter. 215-216(2000)52 \n[14] L.Landau and E.Lifshitz, Sov.Phys. 8(1935)153  10  \n[15] L. Neel, Compt.rend. 2419(1955)533 \n[16] A. Paoletti,  Physics of Magnetic Garnets,  Esevier, Amsterdam, 1978 \n[17] U. Atzmony and M. Dariel, Phys. Rev. B13(1976) 4006 \n[18] K.P. Belov, A.K. Zvezdin, R.Z. Levitin, A.S. M arkosyan, B.V. Mill’, A.A. Mukhin and A.P.Perov, \nJETP 41(1975)590 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  11  \nTable 1.  Magnetic symmetry classes of the plane α2-DWs with °≠180 2α. \nMSC \nnumb.  \nk Mutual \norientations  \nof the vectors   \n1m, 2m and Wn  \nSymmetry \nelements Coordinate dependences  \nof ()z~mcomponents    \nDW \ncenter International \nMSC \nsymbol ()zy~~m ()zx~~m ()zz~~m \n2 0===ΣΔΔaba  n2 , 2 , 2 , 12 1′′ (-) (A,S) (-) -  mm′2′ \n6 0===ΔΣ Caaa \n22 , 1 (-) (A,S) (-) - m \n7 0==ΔΣaa  1, 12′,22,n2′ (A) (S) (-) 0~=z 2 2′2′ \n8 0==ΔΣaa  1, n2′ (A,S) (A,S) (-) - 2′ \n9 0===ΣΔbaaC   1, 12′,22′,n2  (A) (-) (S) 0~=z m m′2′ \n10  0=Δa 1, 12′ (A) (S) (S) 0~=z 2′ \n11 0===ΣΔbaaC   1, n2 (A) (A) (S) 0~=z m \n12  0=Ca 1, 12′ (-) (A,S)  (A,S) - m′ \n13  0=Σa 1, 22 (A) (S) (A) 0~=z 2 \n16 Arbitrary 1 (A,S) (A,S) (A,S) - 1 \n17  0===ΔΣbaaC  1, 12′,22,n2′ (-) (S) (A) 0~=z m′m′2 \n18  0===ΔΣbaaC  1, n2′ (S) (S) (A) 0~=z m′ \n19 0==ΣΔbb  1, n2 (-) (-) (A,S) - 2 \n22  0==ΣΔbb  1, 12′,22′,n2 (-) (-) (A,S) - m′m′2 \n24  0==ΣΔbb  n3 (-) (-) (A,S) - 3 \n26  0==ΣΔbb  12 , 3′n (-) (-) (A,S) - m 3′ \n30  0==ΣΔbb  n4 (-) (-) (A,S) - 4 \n32  0==ΣΔbb  12 , 4′n (-) (-) (A,S) - mm4′′ \n37  0==ΣΔbb  n6 (-) (-) (A,S) - 6 \n39  0==ΣΔbb  12 , 6′n (-) (-) (A,S) - mm 6′′ \n43 0===ΣΔΔaba  ()()1 , 12 , 2 , 2 , 12 1×′′n (-) (S) (-) 0~=z mmm′′ \n44  0===ΣΔΔaba  n2 , 2 , 2 , 12 1′′ (-) (S) (-) 0~=z  mm′2′ \n45  0===ΣΔΔaba  1, 1,22,22 (-) (S) (-) 0~=z 2/m \n46  0===ΣΔΔaba  1, 1,n2′,n2′ (S) (S) (-) 0~=z m / 2′′ \n47  0==ΔΔba  1, 1,12′,12′ (-) (S) (S) 0~=z m / 2′′ \n48 0==ΔΔba  1, 1 (S) (S) (S) 0~=z 1  12  \nTable 1.  Magnetic symmetry classes of the plane α2-DWs with °≠180 2α (continue).  \nMSC \nnumb.  \nk Mutual \norientations  \nof the vectors   \n1m, 2m and Wn  \nSymmetry \nelements Coordinate dependences  \nof ()z~mcomponents    \nDW \ncenter International \nMSC \nsymbol ()zy~~m ()zx~~m ()zz~~m \n49 0===ΣΔΔbba  1, 1,n2,n2 (-) (-) (S) 0~=z 2/m \n50  0===ΣΔΔbba  1, 12′,22′,n2 (-) (-) (S) 0~=z 2 2′2′ \n51 0===ΣΔΔbba  ()()1 , 12 , 2 , 2 , 12 1×′ ′n (-) (-) (S) 0~=z mmm′′ \n52 0===ΣΔΔbba  n6 (-) (-) (S) 0~=z 6 \n53  0===ΣΔΔbba  12 , 3′n (-) (-) (S) 0~=z 2 3′ \n54  0===ΣΔΔbba  12 , 6′n (-) (-) (S) 0~=z 2m 6′ ′ \n55  0===ΣΔΔbba  12 , 3′n (-) (-) (S) 0~=z m 3′ \n56  0===ΣΔΔbba  nn2 , 4 (-) (-) (S) 0~=z 4/m \n57  0===ΣΔΔbba  12 , 4′n (-) (-) (S) 0~=z 224′′ \n58 0===ΣΔΔbba  nn2 , 2 , 41′ (-) (-) (S) 0~=z mm/m 4′′ \n59 0===ΣΔΔbba  n4 (-) (-) (S) 0~=z 4 \n60  0===ΣΔΔbba  12 , 4′n (-) (-) (S) 0~=z m24′ ′ \n61  0===ΣΔΔbba  nn2 , 6 (-) (-) (S) 0~=z 6/m \n62  0===ΣΔΔbba  12 , 6′n (-) (-) (S) 0~=z 2 2 6′′ \n63  0===ΣΔΔbba  nn2 , 2 , 61′ (-) (-) (S) 0~=z mm/m 6′′ \n64  0===ΣΔΔbba  n3 (-) (-) (S) 0~=z 3 \n \n \n \n \n \n \n \n  13  \nTable.2.  Number k (degeneracy Bq) of MSC of boundary conditions of arbitrary orient ed plane α2-DW \n(°>90 2α) in the cubic m 3 mcrystals at selected domain magnetization direction s. \n \nDW \nplane α2-DW boundary conditions \n180°-DW  \n[100],[]00 1 180°-DW \n[110], []0 1 1 180°-DW \n[111], []1 1 1 120°-DW \n[110], []1 1 0 109°-DW \n[111], []1 1 1 \n(100) 34 (6) 14  (24)  14 (24) 16 (96) 9 (24) \n(010) 1 (12)  14 (24)  14 (24) 13 (48) 13 (48) \n(001) 1 (12)  1 (12)  14 (24) 16 (96) 13 (48) \n(111) 14  (24) 14  (24)  29  (8) 16 (96) 12 (48) \n()11 1 14  (24) 4 (24)  14 (24) 13 (48) 12 (48) \n()1 1 1 14  (24) 4 (24)  14 (24) 16 (96) 10 (48) \n()111  14  (24) 14 (24)  14 (24) 13 (48) 10 (48) \n(110) 14  (24) 23  (12)  14 (24) 16 (96) 16 (96) \n(101) 14  (24) 15  (48)  14 (24) 11 (48) 16 (96) \n(011) 1 (12)  15  (48)  14 (24) 16 (96) 17 (24) \n()10 1 14  (24) 1 (12)  5 (24) 16 (96) 16 (96) \n()01 1 14  (24) 15  (48)  5 (24) 13 (48) 16 (96) \n()1 1 0 1 (12)  15  (48)  5 (24) 16 (96) 7 (24) \n(hhl ) 15  (48)  14  (24)  14 (24) 16 (96) 16 (96) \n(hkh ) 15  (48)  15  (48)  14 (24) 16 (96) 16 (96) \n(hkk ) 14  (24)  15  (48)  14 (24) 16 (96) 12 (48) \n()hl h 15  (48)  4  (24)  15  (48) 16 (96) 16 (96) \n()kh h 15  (48)  15  (48)  15  (48) 13 (48) 16 (96) \n()k k h 14  (24)  15  (48)  15  (48) 16 (96) 10 (48) \n(hk 0), ()0k h 14  (24)  14 (24)  15  (48) 16 (96) 16 (96) \n(h0l), ()l h0 14  (24)  15  (48)  15  (48) 16 (96) 16 (96) \n(0 kl ), ()l k0 4 (24)  15  (48)  15  (48) 16 (96) 13 (48) \n(hkl ), ()kl h,()l k h,()l hk  15  (48)  15  (48)  15  (48) 16 (96) 16 (96) \n \n \n  14  \nTable.3.  Number k (degeneracy Bq) of MSC of boundary conditions of arbitrary orient ed plane α2-DW \n(°≤90 2α) in the cubic m 3 mcrystals at selected domain magnetization direction s. \n \nDW \nplane α2-DW boundary conditions \n90°-DW \n[100], []0 1 0 90°-DW \n[110], []0 1 1 71°-DW  \n[111], []11 1 60°-DW  \n[110], [011]  \n(100) 12 (48) 9 (24) 17 (24) 16 (96) \n(010) 12 (48) 17 (24) 10 (48) 10 (48) \n(001) 7 (24) 7 (24) 10 (48) 16 (96) \n(111) 13 (48) 16 (96) 12 (48) 10 (48) \n()11 1 10 (48) 16 (96) 12 (48) 16 (96) \n()1 1 1 10 (48) 16 (96) 13 (48) 10 (48) \n()111  13 (48) 16 (96) 13 (48) 16 (96) \n(110) 17 (24) 12 (48) 16 (96) 16 (96) \n(101) 16 (96) 10 (48) 16 (96) 10 (48) \n(011) 16 (96) 13 (48) 9 (24) 16 (96) \n()10 1 9 (24) 12 (48) 16 (96) 16 (96) \n()01 1 16 (96) 10 (48) 16 (96) 18 (48) \n()1 1 0 16 (96) 13 (48) 7 (24) 16 (96) \n(hhl ) 13 (48) 16 (96) 16 (96) 16 (96) \n(hkh ) 16 (96) 16 (96) 16 (96) 10 (48) \n(hkk ) 16 (96) 16 (96) 12 (48) 16 (96) \n()hl h 10 (48) 16 (96) 16 (96) 16 (96) \n()kh h 16 (96) 16 (96) 16 (96) 16 (96) \n()k k h 16 (96) 16 (96) 13 (48) 16 (96) \n(hk 0), ()0k h 12 (48) 12 (48) 16 (96) 16 (96) \n(h0l), ()l h0 16 (96) 10 (48) 16 (96) 16 (96) \n(0 kl ), ()l k0 16 (96) 13 (48) 10 (48) 16 (96) \n(hkl ), ()kl h,()l k h,()l hk  16 (96) 16 (96) 16 (96) 16 (96) \n "    },    {        "title": "2210.01441v1.Local_density_of_states_as_a_probe_for_tunneling_magnetoresistance_effect__application_to_ferrimagnetic_tunnel_junctions.pdf",        "content": "Local density of states as a probe for tunneling magnetoresistance e\u000bect: application\nto ferrimagnetic tunnel junctions\nKatsuhiro Tanaka,1Takuya Nomoto,1and Ryotaro Arita1, 2\n1Research Center for Advanced Science and Technology,\nUniversity of Tokyo, Komaba, Meguro-ku, Tokyo 153-8904, Japan\n2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan\n(Dated: October 5, 2022)\nWe investigate the tunneling magnetoresistance (TMR) e\u000bect using the lattice models which\ndescribe the magnetic tunnel junctions (MTJ). First, taking a conventional ferromagnetic MTJ as\nan example, we show that the product of the local density of states (LDOS) at the center of the\nbarrier traces the TMR e\u000bect qualitatively. The LDOS inside the barrier has the information on\nthe electrodes and the electron tunneling through the barrier, which enables us to easily evaluate\nthe tunneling conductance more precisely than the conventional Julliere's picture. We then apply\nthis method to the MTJs with collinear ferrimagnets and antiferromagnets. We \fnd that the\nTMR e\u000bect in the ferrimagnetic and antiferromagnetic MTJs changes depending on the interfacial\nmagnetic structures originating from the sublattice structure, which can also be captured by the\nLDOS. Our \fndings will reduce the computational cost for the qualitative evaluation of the TMR\ne\u000bect, and be useful for a broader search for the materials which work as the TMR devices showing\nhigh performance.\nI. INTRODUCTION\nUtilizing the close connection between the spin and\ncharge degrees of freedom of electrons in solids, spintron-\nics has developed various phenomena that are novel from\nthe viewpoint of fundamental physics and promising\nfor industrial use [1{5]. Among those, the tunneling\nmagnetoresistance (TMR) e\u000bect [6, 7] is one of the\nrepresentative phenomena in its wide application [8{12].\nThe TMR e\u000bect is observed in the magnetic tunnel junc-\ntion (MTJ), which consists of two magnetic electrodes\nand the insulating barrier in between. The electrons\ncan tunnel through the MTJ as a quantum mechanical\ncurrent, and the tunneling resistances become di\u000berent\nwhen the magnetic moments of the two electrodes\nalign parallelly or antiparallelly. The set of these two\nalignments with di\u000berent tunneling resistances corre-\nsponds to a bit taking a binary 0 or 1, which has been\nutilized to the magnetic head and the magnetic random\naccess memory devices for the storages and the readout.\nAs well as the theoretical approaches [13{16], large\nTMR ratios have been experimentally observed in the\nMTJs such as the Fe(Co)/Al 2O3/Fe(Co) [17, 18],\nFe(Co)(001)/MgO(001)/Fe(Co) [19, 20] and\nCoFeB/MgO/CoFeB systems [21, 22]. Ferromag-\nnetic Heusler compounds have also been utilized as the\nelectrodes thanks to their half-metalicity [23{26].\nWhile the main target of the spintronics was ferromag-\nnets, recent spintronics has been extended to antiferro-\nmagnets and ferrimagnets owing to their superiorities to\nferromagnets; the smaller stray \feld and the faster spin\ndynamics [27{35]. The antiferromagnetic version of the\nspintronic phenomena, e.g., the giant magnetoresistance\ne\u000bect [36{38] and the anomalous Hall e\u000bect [39{42], has\nbeen developed. Along with these advances, the TMR\ne\u000bect using antiferromagnets has also been intensivelyinvestigated [43{48]. While most of the studies have\nbeen theoretical attempts, experiments have also been\ndeveloped; the TMR e\u000bect is observed in the MTJ whose\ntwo electrodes are the ferromagnet and the ferrimagnetic\nHeusler compound [44]. However, for more practical ap-\nplication of the MTJs with antiferromagnets and ferri-\nmagnets to the devices, we should search for materials\nconstructing the MTJs which show a large TMR ratio,\nand handy methods for the search are required.\nIn this paper, we examine the TMR e\u000bect using the\nlattice models mimicking the MTJs whose electrodes are\nmade of collinear ferrimagnets, including the antiferro-\nmagnets. Motivated by the studies indicating that the\ninterfacial electronic structures a\u000bect the TMR e\u000bect\nand they can be probed by the local density of states\n(LDOS) [49{53], we particularly focus on the LDOS to\nanalyze the TMR e\u000bect. We \fnd that the product of the\nLDOS at the center of the barrier usually reproduces the\ntransmission properties qualitatively in the ferrimagnetic\nMTJs as well as the ferromagnetic ones. The LDOS has\nthe information both on the magnetic properties of elec-\ntrodes and on the tunneling electrons. Besides, from the\nphysics point of view, we show that multiple con\fgura-\ntions can be realized in the ferrimagnetic MTJs due to the\nsublattice structure for each of the parallel and antipar-\nallel magnetic con\fgurations. The resultant TMR e\u000bect\nchanges depending on the con\fgurations, which suggests\nthat the magnetic con\fgurations should be carefully ex-\namined when we deal with the ferrimagnetic MTJs.\nConsidering the above qualitative estimation in terms\nof the LDOS, we present a hierarchy for evaluating the\nTMR e\u000bect in Fig. 1. To quantitatively estimate the\nTMR e\u000bect, we have to calculate the conductance itself\nthrough the Landauer{B uttiker formula [54{57]. Tech-\nnically, this method can be applied to any system and\ngives us highly accurate results, whereas its numerical\ncost is often expensive, particularly in calculating fromarXiv:2210.01441v1  [cond-mat.mes-hall]  4 Oct 20222\nLocal density of states\nAppearance: \nbulk density of states / spin configurationTransport calculation:\nLandauer–Büttiker formulaCoverage / Computational cost\nHigh\nLow\nFIG. 1. Hierarchy for the evaluation of the tunneling mag-\nnetoresistance e\u000bect. The upper has a broader coverage and\ngives more quantitative results, with the higher calculation\ncost. Symbols in the right side, G(E),N\u001b(E;x; y), and\nN\u001b(E), denote the conductance, the local density of states,\nand the density of states of the bulk, respectively, which are\nthe quantities obtained in each calculation.\n\frst-principles. By contrast, the Julliere's picture, which\nclaims that the density of states of the bulk electrodes\ndetermines the e\u000eciency of the MTJ [6, 58], is a sim-\nple and convenient picture to predict the TMR e\u000bect.\nHowever, the picture is valid only for limited cases, and\ncurrently it is found that the electronic states of the\ntunneling electrons are signi\fcant rather than those of\nthe bulk electrodes. Our results can be placed between\nthese two methods. Calculating the LDOS of the MTJs\nis much easier than the Landauer{B uttiker calculation.\nAdditionally, the estimation from the LDOS can cover a\nbroader range of the MTJs with higher reliability than\nthe prediction from the electronic structures of the bulk\nelectrodes.\nThe remainder of this paper is organized as follows. In\nSec. II, we introduce the model describing the MTJ. We\nsimulate the TMR e\u000bect using the ferromagnetic elec-\ntrodes in Sec. III, and see how the LDOS works for pre-\ndicting the tunneling conductance. In Sec. IV, we calcu-\nlate the TMR e\u000bect in the ferrimagnetic and antiferro-\nmagnetic MTJs applying the prediction from the LDOS.\nWe discuss in Sec. V the hierarchy shown in Fig. 1 and the\ncorrespondence between our models and the MTJ with\nreal materials. Section VI is devoted to the summary and\nperspective of this study.\nII. MODEL AND METHOD\nWe construct the two-dimensional square lattice MTJ\nusing two semi-in\fnite lattices which work as electrodes\nand the barrier in between, which is schematically shown\nin Fig. 2. We treat the tight-binding Hamiltonian withthes{dcoupling on this system, which is given as\nH=H0+Ht+Hs\u0000d; (1)\nH0=X\ni\"ini; (2)\nHt=\u0000tX\nhi;ji;\u001b\u0010\ncy\ni;\u001bcj;\u001b+ h.c.\u0011\n; (3)\nHs{d=\u0000JX\ni2electrode(si\u0001\u001b)\u000b\fcy\ni;\u000bci;\f: (4)\nHere,cy\ni;\u001b(ci;\u001b) is the creation (annihilation) opera-\ntor of an electron with spin- \u001bon thei-th site, and\nni=P\n\u001bcy\ni;\u001bci;\u001bis the number operator. The on-site\npotential is denoted as \"i, and the electron hopping be-\ntween two sites is written as t. The summation in Htis\ntaken over the neighboring two sites, which is expressed\nbyhi;ji. The e\u000bect of the magnetism of the electrodes\nis introduced by the s{dcoupling,Hs{d, where the local-\nized spin moment, si, and the conducting electrons cou-\nple each other with the magnetic interaction constant, J.\nThe spin degrees of freedom of the conducting electrons\nare expressed by \u001b, which is the vector representation of\nthe 2\u00022 Pauli matrices. We take the two-dimensional\ncartesian coordinate, ( x;y), for the MTJ, where the x-\naxis is parallel to the conducting path which in\fnitely\nextends, and the y-axis is perpendicular to the conduct-\ning path (see Fig. 2(a)). The width of the barrier in\nthex-direction is L, and the width of the MTJ in the\ny-direction is W. The lattice constant is taken to be\nunity. We hereafter impose the open boundary condition\non they-direction, while we con\frmed that the periodic\nboundary condition does not change the overall results.\nThe calculations of the transmissions are performed\nusing the kwant package [59], in which the quantum\ntransport properties are computed based on the scatter-\ning theory and the Landauer{B uttiker formula.\nIII. TUNNELING MAGNETORESISTANCE\nEFFECT WITH FERROMAGNETIC\nELECTRODES\nLet us \frst recall the conventional TMR, namely the\nTMR using the ferromagnetic electrodes as shown in\nFig. 2(a). We set the localized spin moments as si=\nt\u00000 0 1\u0001\nfor all sites in the left electrode. For the\nsites in the right electrodes, we set si=t\u00000 0 1\u0001\nand\nt\u00000 0\u00001\u0001\nfor the parallel and antiparallel con\fgura-\ntions, respectively. The schematics of these two con\fg-\nurations are shown in Figs. 3(a) and 3(b). We set the\nhoppingtas a unit,t= 1. The on-site potential is set\nas\"i= 0 for the electrodes, and \"i= 10 for the barrier\nregion. The system size is set as L= 8 andW= 160.3\nelectrode\n(ferrimagnetic)electrode\n(ferrimagnetic)barrier\n(nonmagnetic)(a)\nelectrode\n(ferromagnetic)electrode\n(ferromagnetic)barrier\n(nonmagnetic)\n(b)\nFIG. 2. Schematics of the two-dimensional magnetic tun-\nnel junctions (MTJ) used in our calculations. (a) MTJ with\nthe ferromagnetic electrodes. (b) MTJ with the ferrimagnetic\nelectrodes. Arrows represent the localized spin moments on\nthe electrodes.\nA. Bulk properties\nBefore discussing the properties of the tunneling con-\nductance, we see the properties of the bulk ferromagnetic\nmetals used as the electrodes, namely, the energy bands\nand the density of states (DOS). The DOS is given as\nN\u001b(E) =R\nBZdk\u000e(E\u0000Ek;\u001b), whereEk;\u001bis the energy\nband with spin- \u001b. For the energy bands of the bulk elec-\ntrodes, we consider the two-dimensional square lattice\ndescribed byHgiven in Eq. (1). The energy bands are\nfound to be E\u0006\nk=\u00002t(coskx+ cosky)\u0006J. The DOS of\nthe ferromagnet is shown in Fig. 3(c) at J= 1, where the\nright-hand side and the left-hand side are the ones with\nthe up-spin and down-spin, respectively. By introducing\na \fniteJ, the energy band with up-spin gains the energy\n\u0000J, while that with down-spin shifts by + J.\nB. Transmission and local density of states\nIn Fig. 3(d), we show the TMR ratio de\fned as\nrt=TP\u0000TAP\nTP+TAP; (5)\non the plane of JandE. Here,TP/AP denotes the trans-\nmission for the parallel/antiparallel con\fguration. We\nnote that the de\fnition above is slightly di\u000berent from\nthe conventional optimistic/pessimistic ones for the rea-\nson on normalization. At J= 0, the whole system is\nnonmagnetic and has the degeneracy on the spin degrees\nof freedom, and thus TP=TAPholds at each energy.\nNamely,rtis zero. When we introduce a \fnite magnetic\ninteraction J, the degeneracy is lifted, and TPstarts totake a larger value than TAP; a \fnite TMR ratio is ob-\nserved. Due to the asymmetric structure of the barrier in\nthe energy, rtis also asymmetric with respect to E= 0.\nTheJ-dependence of TPandTAPis shown in Figs. 3(e)\nand 3(f). At E=\u00002, both ofTPandTAPdecrease when\nJincreases, and TAPreaches zero at J= 2 [60]. When\nE= 0,TPincreases with J, whileTAPdecreases to zero.\nTo better understand the transmission properties, we\nexamine the LDOS in addition to the bulk DOS. In the\nnaive Julliere's picture, the product of the bulk DOS,\ng(E), de\fned as\ng(E) =X\n\u001bNL;\u001b(E)NR;\u001b(E); (6)\ndescribes the transmission [6, 58], where NL/R;\u001b(E) is\nthe bulk DOS with spin- \u001b,N\u001b(E), of the left/right elec-\ntrodes. However, g(E) does not consider the barrier, and\nthus this picture holds only for the limited cases. In-\nstead, the LDOS has been utilized to capture the details\non the MTJs. In particular, it has been proposed that\nthe transmission can be described using the LDOS at the\ninterfaces of the MTJ. In fact, when the potential of the\nbarrier is high enough, the conductance derived by the\nKubo formula [14] is proportional to the product of the\nLDOS at the left and right interfaces, and to the expo-\nnential function, e\u0000\u0014L, representing the decay inside the\nbarrier [49, 50]. Here, \u0014stands for the decaying property,\nnamely, 1=\u0014means the spin di\u000busion length [61]. Hence,\nwe consider the product of the LDOS at the interfaces,\ngi(E) given as\ngi(E) =1\n2X\n\u001b\u0014\nN\u001b\u0012\nE; 1;W\n2\u0013\nN\u001b\u0012\nE;L;W\n2\u0013\n+N\u001b\u0012\nE; 1;W\n2+ 1\u0013\nN\u001b\u0012\nE;L;W\n2+ 1\u0013\u0015\n;(7)\nwhereN\u001b(E;x;y) (1\u0014x\u0014L, 1\u0014y\u0014W) is the LDOS\nof the barrier at ( x;y). Since we impose the open bound-\nary condition in the y-direction, we take the average over\nthe sites at y=W=2 andW=2 + 1 in Eqs. (7) and (8) to\nreduce the e\u000bects of the oscillation due to the boundary,\nand we implicitly assume that the spins do not \rip in-\nside the barrier [62]. However, in general, it is not easy\nto precisely evaluate the exponent \u0014; we should need the\ntransmission coe\u000ecients [63] in addition to \fnding the\nelectronic structures.\nAlternatively, we here utilize the LDOS at the center of\nthe barrier. Expecting that it contains both the details\nof the MTJ and the decay, we consider the product as\nfollows;\ngc(E) =1\n2X\n\u001b\u0014\nN\u001b\u0012\nE;L\n2;W\n2\u0013\nN\u001b\u0012\nE;L\n2+ 1;W\n2\u0013\n+N\u001b\u0012\nE;L\n2;W\n2+ 1\u0013\nN\u001b\u0012\nE;L\n2+ 1;W\n2+ 1\u0013\u0015\n:\n(8)4\n(b)(e)\n(h) (i)(f) (a)\n(c)\n00\n-6-4-2246\n0.6 -0.6\ndensity of states013\n0 2 1 002 1 0 2 1120246\n0123\n0 200.3\n000.5\n22Parallel\nAntiparallel(g)(d)\n-404\n2\n-2\n01\n01\n-404\n2\n-2\nFIG. 3. (a), (b) Schematic pictures of (a) the parallel and (b) antiparallel con\fguration of the magnetic tunnel junction (MTJ)\nwith ferromagnetic electrodes. (c) Spin-resolved density of states of the electrode with J= 1. Broken lines are the energies\nwhere we show the results in (e), (f), (h), (i). (d){(i) Results of the calculation on the ferromagnetic MTJ. (d) TMR ratio rt\n(Eq. (5)) on the plane of JandE. (e), (f) Transmissions as a function of the magnetic interaction, J, at (e) E=\u00002 and (f)\n0. (g) Ratio rc(Eq. (9)). (h){(i) Magnetic interaction, J, dependence of gc(E).\nIn this scheme, we only have to \fnd the electronic struc-\ntures to estimate the transmission. While we now con-\nsider the case with even- Land calculate the product of\nthe LDOS at x=L=2 andL=2 + 1, we should replace\nboth of them with the one at x= (L+1)=2 for the odd- L\ncase.\nSimilarly to the TMR ratio rt, we calculate the ratio,\nrc, de\fned as\nrc=gc;P(E)\u0000gc;AP(E)\ngc;P(E) +gc;AP(E); (9)\nwheregc;P(E) andgc;AP(E) aregc(E) for the parallel\nand antiparallel con\fgurations, respectively. Figure 3(g)\nshowsrcon the plane of JandE. We \fnd that rcquali-\ntatively reproduces the TMR ratio rtshown in Fig. 3(d).\nWe show the J-dependence of gc(E) in Figs. 3(h) and\n3(i), and that of g(E) in the insets. At E=\u00002,gc(E)\nreproduces the overall properties of the transmission,\nwhileg(E) increases with Jand does not reproduce the\ntransmission properties. When we increase the energy to\nE= 0, the bulk DOS with the up and down spins take the\nsame values, and the estimation in terms of g(E) would\nlead to the absence of the TMR. Even in this case, the\nproduct of the LDOS at the center of the barrier well re-\nproduces the transmission properties; the J-dependence\nofgc(E) is qualitatively the same as the one of the trans-\nmissions.\nWe remark that we also examine the J-dependence of\ngi(E) following the previous proposals [14, 49, 50]. We\n\fnd thatgc(E) better reproduce the transmission prop-erties than gi(E) when we compare these two quantities\n(see Appendix for the detailed results of gi(E)), which\nis due to the absence of the decay e\u000bect in gi(E).\nIV. TUNNELING MAGNETORESISTANCE\nEFFECT WITH\nFERRIMAGNETIC/ANTIFERROMAGNETIC\nELECTRODES\nNext, we examine the tunneling conductance of the\nMTJ with ferrimagnetic electrodes, including the anti-\nferromagnet, and apply the analysis by means of the\nLDOS. We here concentrate on the ferrimagnet with two-\nsublattices, A and B, on the square lattice. We assume\nthat the ferrimagnet has the G-type structure; all spins\nat the nearest-neighbors of the spins in the A-sublattice\nare in the B-sublattice, and vice versa. In the ferrimag-\nnetic MTJ, we de\fne the parallel and antiparallel con-\n\fgurations by the alignments of the spins on the same\nsublattice. Due to the two-sublattice structure, there\nare two pairs of the parallel{antiparallel con\fgurations\nwhose schematic views are shown in Figs. 4(a){4(d). In\nthe \frst one, which we call con\fguration-1, two local-\nized spins next to the barrier layer on the left and right\nelectrodes with the same y-coordinates are in the di\u000ber-\nent sublattices. The parallel and antiparallel arrange-\nments of con\fguration-1 is shown in Figs. 4(a) and 4(b),\nrespectively. In the second one, which we refer to as\ncon\fguration-2, those two spins are in the same sublat-5\n(a)\n(b)Configuration-1\nParallel\nAntiparallel\n(d)Configuration-2\nAntiparallel(c) Parallel\nFIG. 4. Two con\fgurations of the magnetic tunnel junc-\ntions (MTJ) using the ferrimagnetic electrodes, con\fguration-\n1 and 2. (a) Parallel and (b) antiparallel alignments of\ncon\fguration-1. (c) Parallel and (d) antiparallel alignments\nof con\fguration-2.\ntices, whose parallel and antiparallel arrangements are\nrespectively shown in Figs. 4(c) and 4(d). The param-\neters in the Hamiltonian are taken in common with the\nferromagnetic MTJ; \"i= 0 and 10tfor the electrodes and\nthe barrier respectively, and L= 8 andW= 160.\nA. Bulk properties\nWe \frst see the bulk properties of the ferrimagnetic\nelectrodes; we calculate the energy band and the DOS\nof the system described by the Hamiltonian Hon the\nsquare lattice. The spins of A- and B-sublattices are set\nassA=t\u00000 0sA\u0001\nandsB=t\u00000 0sB\u0001\n, respectively.\nSince the ferrimagnet has two-sublattices, there are two\nenergy bands, E\u0006\nk;\u001b, for each spin degrees of freedom \u001b.\n(a) (b)\ndensity of states6\n0\n-66\n0\n-6-1 0 1\ndensity of states-1 0 1-4-224\n-4-224FIG. 5. Spin-resolved density of states of (a) the two-\ndimensional ferrimagnet with ( sA; sB) = (1 :0;\u00000:5), and (b)\nthe antiferromagnet with ( sA; sB) = (1 :0;\u00001:0). For both\ncases, J= 1:0.\nThe energy bands are written as\nE\u0006\nk;\"=\u0000J(sA+sB)\u0006q\nJ2(sA+sB)2\u00004 (J2sAsB\u0000\r2\nk)\n2;\n(10)\nE\u0006\nk;#=+J(sA+sB)\u0006q\nJ2(sA+sB)2\u00004 (J2sAsB\u0000\r2\nk)\n2;\n(11)\nwhere\rk=\u00002t(coskx+ cosky). In Figs. 5(a) and 5(b),\nwe show examples of the DOS of the ferrimagnet with\n(sA;sB) = (1:0;\u00000:5), and that of the antiferromagnet\nwith (sA;sB) = (1:0;\u00001:0), respectively.\nB. Transmissions and local density of states\n1. Ferrimagnetic electrode\nWe discuss the transmission properties of the ferrimag-\nnetic MTJ. First, we investigate the system where we\n\fx the magnetization of two sublattices as ( sA;sB) =\n(1:0;\u00000:5). Figures 6(a) and 6(b) show the TMR ratio,\nrt, on the plane of JandEfor con\fguration-1 and 2,\nrespectively. The J-dependence of the transmissions for\ncon\fguration-1 and 2 at E= 3:6 is respectively shown\nin Figs. 6(c) and 6(d). Without the s{dcouplingJ,\nthe localized magnetic moment does not a\u000bect the trans-\nmission, and TPandTAPtake the same values for both\ncon\fguration-1 and 2. When a small Jis introduced in\ncon\fguration-1, TAPis larger than TP, andTPbecomes\nlarger than TAPatJ'1:5. Meanwhile, TPis larger than\nTAPat \fniteJin con\fguration-2.\nAs well as the ferromagnetic MTJ discussed in Sec. IV,\nwe focus on gc(E) de\fned by Eq. (8) and calculate rc\ngiven in Eq. (9). In Figs. 6(e) and 6(f), we plot rc\nfor con\fguration-1 and 2, respectively, which indicates\nthatgc(E) qualitatively traces the TMR property also\nin the ferrimagnetic MTJs. We plot the J-dependence\nofgc(E) in Figs. 6(g) and 6(h) for con\fguration-1 and6\n(b) (c) (d)\n(e)(a)Configuration-1 Configuration-2 Configuration-1 Configuration-2\n(f) (g) (h)0123\n0123\n2\n00\n2 11\n0 2 1 0 2 12\n00\n2 11000.08\n2\nParallel\nAntiparallel1\n0\n-1\n1\n0\n-1 -404\n2\n-2-404\n2\n-2\n-404\n2\n-2-404\n2\n-2\nFIG. 6. Results of the transmission calculation for the ferrimagnetic tunnel junction with ( sA; sB) = (1 :0;\u00000:5) for con\fguration-\n1 and 2. (a), (b) TMR ratio rt(Eq. (5)) on the plane of JandE. (c), (d) Transmissions at E= 3:6 with respect to J. (e), (f)\nRatio rc(Eq. (9)). (g), (h) Product of the local density of states at the center of the barrier, gc(E), as a function of J. Inset\nin (g) is the J-dependence of g(E).\n2 atE= 3:6, respectively, together with g(E) given in\nEq. (6) in the inset of Fig. 6(g). We can see that the\ntransmission and gc(E) similarly changes with J, while\ng(E) does not reproduce the transmission. Here the av-\nerage is meaningful also for capturing the two-sublattice\nmagnetic structure in the y-direction in the ferrimagnetic\nMTJ, whereas in the ferromagnetic MTJ the average over\ny=W=2 andW=2 + 1 is important to take the open\nboundary conditions into account. We note that gc(E)\ntraces even the reversal of the transmission occurring in\ncon\fguration-1, whereas g(E) orgi(E) de\fned by the\ninterfacial DOS does not (see Fig. 10 for details).\n2. Antiferromagnetic electrode\nNext we discuss the antiferromagnetic limit, ( sA;sB) =\n(1:0;\u00001:0). In this case, con\fguration-1 and 2 are\ndegenerate; the parallel and antiparallel alignments of\ncon\fguration-1 correspond to the antiparallel and par-\nallel alignments of con\fguration-2, respectively. Hence,\nhere we consider con\fguration-1 only and de\fne the par-\nallel and antiparallel con\fgurations by con\fguration-1.\nWe show the TMR ratio rt(Eq. (5)) in Fig. 7(a), and\ntheJ-dependence of the transmission at in Fig 7(b) at\nE= 1. AtE= 1, both of TPandTAPincrease with J\natJ.1 as shown in Fig. 7(b). At J&1,N\u001b(E) is zero,\nnamely, there is no incidence from the electrodes. Thus,\nthe transmission of each con\fguration becomes zero.\nFigures 7(c) represents the ratio rc(Eq. (9)). We con-\frm thatrchas a parameter dependence qualitatively\nthe same as the one of rt. In fact, the J-dependence of\ngc(E) atE= 1 shown in Fig. 7(d) well reproduce the\nJ-dependence of the transmissions shown in Fig. 7(b).\nIn the inset of Fig. 7(d) we show g(E) for the paral-\nlel and antiparallel con\fgurations, which are degenerate\nand do not predict a \fnite TMR e\u000bect. We note that\ngi(E) de\fned by the interfacial LDOS has a parameter\ndependence qualitatively di\u000berent from the one of the\ntransmissions (see Fig. 11 for details).\nV. DISCUSSION\nA. Hierarchy in estimating the transmission\nproperties\nWe presented a hierarchy in the evaluation of the trans-\nmission properties in Fig. 1. Here we discuss how the\nappearance, namely, the bulk DOS and the spin con\fg-\nurations, and the LDOS are related to the transmission\nproperties. For the ferrimagnetic tunnel junction with\n(sA;sB) = (1:0;\u00000:5) atJ= 1:85, we show the energy\ndependence of the transmissions and gc(E) in Fig. 8. Fig-\nures 8(a) and 8(b) are the results for con\fguration-1, and\nFigs. 8(c) and 8(d) are for con\fguration-2. When either\nof the two spin-states has a \fnite DOS, TPtakes a larger\nvalue than TAP, which supports the Julliere's picture.\nWhen both spin-states have \fnite DOS, shown as the\nshaded regions in Fig. 8, TAPis sometimes larger than7\n(a) (b)\n(c) (d)\n048048\n0 1 2 0 1 2001\n2Parallel\nAntiparallel1\n0\n-1\n1\n0\n-1 -404\n2\n-2-404\n2\n-2\nFIG. 7. Results of the transmission calculation of the an-\ntiferromagnetic tunnel junction with ( sA; sB) = (1 :0;\u00001:0).\n(a) TMR ratio rt(Eq. (5)) on the plane of JandE. (b)\nTransmissions for the parallel and antiparallel con\fgurations\natE= 1. (c) Ratio rc(Eq. (9)). (d) Magnetic coupling, J,\ndependence of gc(E) atE= 1. Inset is the J-dependence of\ng(E).\nTP, e.g., atE\u00183 for con\fguration-1 (Fig. 8(a)). This\nmeans that the conventional Julliere's description breaks\ndown in these regions. Instead, the spin con\fgurations\nat the interface usually gives very rough estimation of\nthe transmission. Let us focus on the spin con\fgurations\nat the interface. As shown in Figs. 4(a) and 4(b), for\ncon\fguration-1, the interfacial spins of the two electrodes\nwith the same y-coordinates align antiparallelly in the\nparallel arrangement, and those spins in the antiparallel\narrangement align parallelly. For the con\fguration-2, the\nparallel arrangement have the interfacial spins with op-\nposite directions, and the antiparallel arrangement have\nthe interfacial spins pointing the same directions, which\nis represented in Figs. 4(c) and 4(d). Since the spins\nunlikely to \rip through coherent tunneling, the trans-\nmissions become larger when the interfacial spins of two\nelectrodes align parallelly. In fact, in the case where two\nspin-states have nonzero DOS, TP<T APbasically holds\nfor con\fguration-1, whereas TP> T APholds in a broad\nE-region for con\fguration-2 (see Figs. 8(a) and 8(c)).\nFrom the interfacial spin con\fgurations, we can\nroughly predict the TMR properties in many cases. As\nshown in Fig. 8(a), however, TPbecomes larger than TAP\ncontrary to the prediction from the interfacial spins at\nE\u00183:7 in con\fguration-1 with the bulk DOS consist-\ning of both spin-states. Still in this case, gc(E) for the\nparallel con\fguration takes a larger value than that for\nthe antiparallel con\fguration. Furthermore, gc(E) gives\n(a) (b)\n(c) (d)\n0 0Configuration-1\nConfiguration-2\n6\n4\n2\n0\n4 2 1 2Parallel\nAntiparallel\n0 0 4 2 1 26\n4\n2\n0FIG. 8. Results of the transmission calculations for the fer-\nrimagnetic tunnel junction with ( sA; sB) = (1 :0;\u00000:5) at\nJ= 1:85 for (a), (b) con\fguration-1 and (c), (d) 2. (a),\n(c) Transmissions and (b), (d) gc(E) for the parallel and an-\ntiparallel alignments. Shaded regions represent the energies\nwhere both spin-states have \fnite DOS (see Fig. 5(a)).\nus the detailed information on the parameter dependence\nas shown in Figs. 8(b) and 8(d), while we can only know\nfrom the appearance whether the parallel or antiparal-\nlel con\fguration gives the larger transmission. To com-\npletely understand the transmission properties, of course\nwe should calculate transmission itself, but we expect\nthat the estimation in terms of the LDOS is enough as\nan initial way.\nB. Details of the magnetic tunnel junctions\nWe have considered the simplest cases, where the elec-\ntronic orbitals are isotropic and the barrier is structure-\nless. In reality, we should consider the details of the\nMTJs. In the Fe(001)/MgO(001)/Fe epitaxial MTJ, for\nexample, the \u0001 1-symmetry state with a large spin po-\nlarization has less decay, which dominantly contributes\nto the large TMR ratio [15, 16]. If we focus on this \u0001 1\nBloch state, we can apply our treatment and estimate the8\nTMR e\u000bect in terms of the LDOS. Besides, due to struc-\ntural disorders or hybridization of orbitals, the electronic\nand magnetic states at the interfaces may be modulated.\nWe can trace the e\u000bect of the modulation with gc(E),\nwhich is indicated by the results that gc(E) reproduces\nthe transmissions with each of two di\u000berent interfaces,\ncon\fguration-1 and 2, for the ferrimagnetic MTJs.\nIn the typical metals used in the ferrimagnetic spin-\ntronics such as GdFeCo [64{66] or Mn 2RuxGa [67], the\nvalences of the ions carrying the magnetic moments of\nthe di\u000berent sublattices are di\u000berent. This charge in-\nequivalence determines the interfacial structure to keep\nthe charge neutrality at the interface. Namely, if we\nalso take the charge degrees of freedom into account,\nwe can in principle design the interfacial magnetic struc-\nture. On the other hand, when we cannot control the\ninterfacial structures precisely, the averaged structure of\ncon\fguration-1 and 2 seems to be realized, which can be\nregarded as the ferromagnetic MTJ of the net magnetic\nmoments. We have numerically con\frmed that the Jul-\nliere's picture with the bulk DOS holds like ferromagnets\nin that case.\nFor antiferromagnetic MTJs, when we use the antifer-\nromagnets with the macroscopic time-reversal symmetry,\ncon\fguration-1 and 2 are not distinguished. The TMR\ne\u000bect then vanishes since there is a degeneracy on the\ntransmission between con\fguration-1 and 2 as mentioned\nin Sec. IV. By contrast, the antiferromagnets macroscop-\nically breaking the time-reversal symmetry separate the\nMTJs with con\fguration-1 and 2, which enables us to ob-\nserve a \fnite TMR e\u000bect in the antiferromagnetic MTJs.\nActually, the MTJs using such antiferromagnets have\nbeen theoretically proposed [46{48].\nVI. SUMMARY AND PERSPECTIVES\nIn summary, we have numerically studied the tunnel-\ning magnetoresistance (TMR) e\u000bect modelizing the mag-\nnetic tunnel junction (MTJ) consisting of the ferrimag-\nnetic electrodes as well as the well-known ferromagnetic\nones. To grasp the transmission properties, we have fo-\ncused on the local density of states. We have shown that\nthe transmission properties can be qualitatively repro-\nduced by the product of the local density of states at the\ncenter of the barrier. In the physical aspect, there can be\nmultiple con\fgurations for the ferrimagnetic MTJs owing\nto the sublattice structure of the electrodes. Those mul-\ntiple con\fgurations give the di\u000berent transmission prop-\nerties, and thus we should be careful for the magnetic\ncon\fgurations in the ferrimagnetic TMR.\nOur approach can be applied to more complicated\ncases where the detailed structures are taken into ac-\ncount. When one performs a more realistic TMR calcu-\nlation, the electronic structures and the wave-functions\nshould be obtained from \frst-principles. To calculate the\ntransmission by using \frst-principles wave-functions, the\nmethods such as the nonequilibrium Green's function for-\n(c) (b)(a)\n048\n0 2 1 002 1123\nParallel\nAntiparallel-4-2024\n01\n0 2 1FIG. 9. Results of the simulation for ferromagnetic tunnel\njunctions (Sec. III). See also Fig. 3 as a comparison. (a) Ratio\nri(Eq. (A.1)) on the plane of JandE. (b), (c) Product of the\ninterfacial local density of states gi(E) (Eq. (7)) with respect\ntoJat (b) E=\u00002 and (c) 0.\nmalism [68{70] or the scattering problem approach [71{\n74] are widely adopted. However, these methods usually\ndemand huge numerical costs, which probably has pre-\nvented us from exploring the MTJ using various materi-\nals. The calculation of the local density of states is much\nless costly, so that the approach with the local density\nof states will serve an easy means to search for the MTJ\nwith high e\u000eciency.\nACKNOWLEDGMENTS\nThis work was supported by JST-Mirai Program (JP-\nMJMI20A1), a Grant-in-Aid for Scienti\fc Research (No.\n21H04437, No. 21H04990, and No. 19H05825), and JST-\nPRESTO (JPMJPR20L7).\nAppendix: Product of the local density of states at\nthe interfaces\nIn the main text, we have discussed the similarities\nbetween the transmissions and the product of the local\ndensity of states (LDOS) at the center of the barrier re-\ngion. Here we discuss the J-dependence of the product\nof the LDOS at the interface [49, 50]. Similarly to rc\nde\fned in Eq. (9), we de\fne the ratio for the interfacial9\n(c) (d)(a) (b)Configuration-1 Configuration-2\n0 1 2024\n0 1 2024\nParallel\nAntiparallel-4-2024\n0 2 1-4-2024\n0 2 11\n0\n-1\nFIG. 10. Results of the calculation for ferrimagnetic tun-\nnel junctions with ( sA; sB) = (1 :0;\u00000:5) (Sec. IV B 1). See\nalso Fig. 6 as a comparison. (a), (b) Ratio ri(Eq. (A.1))\non the plane of JandEfor (a) con\fguration-1 and (b) 2.\n(c), (d) Product of the interfacial local density of states gi(E)\n(Eq. (7)) as a function of JatE= 3:6 for (c) con\fguration-1\nand (d) 2.\n(a) (b)\n01\n0 1 2Parallel\nAntiparallel\n-4-2024\n0 2 1-101\nFIG. 11. Results of the simulation for antiferromagnetic tun-\nnel junctions with ( sA; sB) = (1 :0;\u00001:0) (Sec. IV B 2). See\nalso Fig. 7 as a comparison. (a) Ratio ri(Eq. (A.1)) on the\nplane of JandE. (b) Product of the interfacial local density\nof states gi(E) (Eq. (7)) at E= 1 as a function of J.LDOS,ri, as\nri=gi;P(E)\u0000gi;AP(E)\ngi;P(E) +gi;AP(E): (A.1)\nHeregi;P/AP isgi(E) for the parallel/antiparallel con\fg-\nuration. We plot riin Fig. 9(a). We \fnd that rialso\ndescribes the transmission in the large- or small- Ere-\ngions where rt'1 owing to the absence of the bulk DOS\nof either of the spin-states. However, ridoes not repro-\nduce the intermediate- Eregion. Actually, as shown in\nFig. 9(b),gi(E) changes similarly to the transmission at\nE=\u00002 (see Fig. 3(e)), whereas the J-dependence of\ngi(E) shown in Fig. 9(c) largely deviates from that of the\ntransmission at E= 0 (Fig. 3(f)).\nIn Figs. 10(a) and 10(b), we present rifor the ferrimag-\nnetic MTJ with ( sA;sB) = (1:0;\u00000:5) for con\fguration-1\nand 2, respectively. Figures 10(c) and 10(d) are the J-\ndependence of gi(E) atE= 3:6. 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Rev.\nB74, 045429 (2006)."    },    {        "title": "2006.12553v3.Insights_into_nature_of_a_magnetization_plateau_of_3_d__4_f__coordination_polymer__Dy__2_Cu__2____n__from_a_spin_1_2_Ising_Heisenberg_orthogonal_dimer_chain.pdf",        "content": "Condensed Matter Physics, 2020, Vol. 23, No 4, 43708: 1–11\nDOI: 10.5488/CMP.23.43708\nhttp://www.icmp.lviv.ua/journal\nInsights into nature of a magnetization plateau\nof 3𝒅-4𝒇coordination polymer [Dy 2Cu2]𝒏from\na spin-1/2 Ising-Heisenberg orthogonal-dimer chain\nJ. Strečka1, L. Gálisová2, T. Verkholyak3\n1Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. Šafárik University,\nPark Angelinum 9, 040 01 Košice, Slovakia\n2Institute of Manufacturing Management, Faculty of Manufacturing Technologies with the seat in Prešov,\nTechnical University of Košice, Bayerova 1, 080 01 Prešov, Slovakia\n3Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,\n1 Svientsitskii St., 79011 Lviv, Ukraine\nReceived June 22, 2020, in final form August 13, 2020\nThe ground state and magnetization process of an exactly solved spin- 12Ising-Heisenberg orthogonal-dimer\nchain with two different gyromagnetic factors of the Ising and Heisenberg spins are investigated in detail. It\nis shown that the investigated quantum spin chain exhibits up to seven possible ground states depending\non a mutual interplay of the magnetic field, intra- and inter-dimer coupling constants. More specifically, the\nfrustrated and modulated quantum antiferromagnetic phases are responsible in zero-temperature magneti-\nzation curves for a zero magnetization plateau. The intermediate 1/11- and 5/11-plateaus emerge due to the\nfrustrated and modulated quantum ferrimagnetic phases, while the intermediate 9/11- and 10/11-plateaus\ncan be attributed to the quantum and classical ferrimagnetic phases. It is conjectured that the magnetiza-\ntion plateau experimentally observed in a high-field magnetization curve of 3 𝑑-4𝑓heterobimetallic coordi-\nnation polymer [{Dy(hfac) 2(CH3OH)} 2{Cu(dmg)(Hdmg)} 2]𝑛(H2dmg=dimethylglyoxime; Hhfac =1,1,1,5,5,5-\nhexafluoropentane-2,4-dione) could be attributed to the classical and quantum ferrimagnetic phases.\nKey words: Ising-Heisenberg orthogonal-dimer chain, magnetization plateau, 3 𝑑-4𝑓coordination polymer\n1. Introduction\nThe frustrated spin-1\n2Heisenberg orthogonal-dimer (or dimer-plaquette) chain [1–4] has attracted\nconsiderable attention as it represents one-dimensional counterpart of the famous Shastry-Sutherland\nmodel [5], which is widely studied by virtue of elucidation a peculiar sequence of fractional plateaus\nexperimentally observed in low-temperature magnetization curves of SrCu 2(BO 3)2[6] and rare-earth\ntetraborides RB 4[7]. It has been argued by Schulenburg and Richter [8, 9] that a zero-temperature\nmagnetization curve of the spin-1\n2Heisenberg orthogonal-dimer chain displays an infinite series of\nfractional magnetization plateaus at rational numbers𝑛\n2𝑛¸2=1\n41\n31\n2, whereas the lowermost and\nuppermost plateaus from this series are the widest ones. Another interesting feature of the spin-1\n2\nHeisenberg orthogonal-dimer chain lies in its belonging to a prominent class of flat-band models, for\nwhich low-temperature thermodynamics can be elaborated by making use of an effective lattice-gas\ndescription as extensively discussed by Derzhko and coworkers [10, 11].\nRegrettably,thermodynamicpropertiesofquantumHeisenbergspinmodelsareingeneralinaccessi-\nblebyexactcalculationsatnonzerotemperature.ReplacementofsomeofthequantumHeisenbergspins\nby the classical Ising ones paves the way to an exact solution of the analogous Ising-Heisenberg models\nbyemployingexactmappingtransformationsandthetransfer-matrixmethod[12–14].Forinstanceafew\nexactly solved versions of the spin-1\n2Ising-Heisenberg orthogonal-dimer chain [15–19], to be further\nThis work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution\nof this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.43708-1arXiv:2006.12553v3  [cond-mat.str-el]  18 Jan 2021J. Strečka, L. Gálisová, T. Verkholyak\nFigure 1.(Colour online) (a) A crystal structure of 3 𝑑-4𝑓coordination polymer [Dy 2Cu2]𝑛(see the text\nforafullchemicalformula)adaptedaccordingtocrystallographicdatareportedinreference[21].Large\ncyanballsdeterminecrystallographicpositionsofDy3¸magneticions,whilesmallgreenballsstandfor\ncrystallographic positions of Cu2¸magnetic ions (a coloured scheme for atom labeling is presented in\nthelegend);(b)ThemagneticstructureofthecorrespondingIH-ODC,inwhichDy3¸magneticionsare\ntreated as the Ising spins while Cu2¸magnetic ions are treated as the Heisenberg spins. The coupling\nconstants𝐽I,𝐽0\nIand𝐽HareassignedtotheIsinginter-dimerinteractionbetweenDy3¸andCu2¸magnetic\nions (solid lines), the Ising intra-dimer interaction between Dy3¸magnetic ions (dashed lines) and the\nHeisenberg intra-dimer interaction between Cu2¸magnetic ions (dotted lines), respectively.\nabbreviated as IH-ODC, brought a deeper insight into the magnetization process [15, 17, 19], magne-\ntocaloric effect [15, 17], low-temperature thermodynamics [16–18] and bipartite thermal entanglement\n[16, 19] of this frustrated quantum spin chain. Besides, it was demonstrated that the exact solution for\nthe IH-ODC may serve as a useful starting point for the many-body perturbation treatment of the fully\nHeisenberg counterpart model within a more advanced strong-coupling approach [20].\nAlthough we are currently not aware of any experimental realization of the spin-1\n2Heisen-\nberg orthogonal-dimer chain, it surprisingly turns out that 3 𝑑-4𝑓heterobimetallic coordination poly-\nmer [{Dy(hfac) 2(CH 3OH)} 2{Cu(dmg)(Hdmg)} 2]𝑛(H2dmg=dimethylglyoxime; Hhfac =1,1,1,5,5,5-\nhexafluoropentane-2,4-dione) [21], hereafter abbreviated as [Dy 2Cu2]𝑛, provides an experimental real-\nizationoftheIH-ODC.Infact,thepolymericcompound[Dy 2Cu2]𝑛displaysapeculiarone-dimensional\narchitecturewithregularlyalternatingdimericunitsofDy3¸-Dy3¸andCu2¸-Cu2¸magneticionsstacked\nin an orthogonal fashion with respect to each other as illustrated in figure 1 (a) [21]. It should be\npointed out, moreover, that Dy3¸magnetic ion represents Kramers ion with the ground-state multiplet\n6H152, which is subjected to a relatively strong crystal-field splitting into eight well-separated Kramers\ndoublets[22,23].Inthisregard,themagneticbehaviourofDy3¸magneticionwiththetotalangularmo-\nmentum𝐽=152andtheassociated 𝑔-factor𝑔𝐽=43canbeapproximatedatlowenoughtemperatures\nbytheclassicalIsingspinwiththeeffectivegyromagneticfactor 𝑔Dy=20whenneglectingtheadmixture\nof all excited Kramers dublets [22, 23]. Getting back to a magnetic structure of the polymeric complex\n[Dy 2Cu2]𝑛,whichisschematicallydrawninfigure1(b),theverticaldimerofDy3¸-Dy3¸magneticions\nmay be approximated by a couple of the Ising spins, while the horizontal dimer of Cu2¸-Cu2¸magnetic\nions may be approximated by a couple of the Heisenberg spins.\nThe organization of this article is as follows. In section 2 we describe the studied IH-ODC and\nrecall basic steps of its exact analytical solution. The ground state and magnetization process of the\ninvestigated quantum spin chain are theoretically studied in section 3. The available experimental data\nfor the high-field magnetization curve of the polymeric compound [Dy 2Cu2]𝑛are interpreted by virtue\noftheIH-ODCinsection4.Finally,severalconclusionsandfutureoutlooksarementionedinsection5.\n43708-2Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\n2. Spin-1\n2Ising-Heisenberg orthogonal-dimer chain\nLet us start by introducing the IH-ODC in a magnetic field through the following Hamiltonian [see\nfigure 1 (b) for a schematic illustration]:\nˆH=𝐽H𝑁∑︁\n𝑖=1ˆS1𝑖\u0001ˆS2𝑖¸𝐽0\nI𝑁∑︁\n𝑖=1ˆ𝜎𝑧\n1𝑖ˆ𝜎𝑧\n2𝑖¸𝐽I𝑁∑︁\n𝑖=1\u0002ˆ𝑆𝑧\n1𝑖\u0000ˆ𝜎𝑧\n1𝑖¸ˆ𝜎𝑧\n2𝑖\u0001¸ˆ𝑆𝑧\n2𝑖\u0000ˆ𝜎𝑧\n1𝑖¸1¸ˆ𝜎𝑧\n2𝑖¸1\u0001\u0003\n\u0000𝑔H𝜇B𝐵𝑁∑︁\n𝑖=1\u0000ˆ𝑆𝑧\n1𝑖¸ˆ𝑆𝑧\n2𝑖\u0001\u0000𝑔I𝜇B𝐵𝑁∑︁\n𝑖=1\u0000ˆ𝜎𝑧\n1𝑖¸ˆ𝜎𝑧\n2𝑖\u0001 (2.1)\nwhere ˆS1¹2º𝑖\u0011 ¹ˆ𝑆𝑥\n1¹2º𝑖ˆ𝑆𝑦\n1¹2º𝑖ˆ𝑆𝑧\n1¹2º𝑖ºdenote standard spin-1\n2operators ascribed to Cu2¸magnetic\nions approximated by the notion of quantum Heisenberg spins and ˆ𝜎𝑧\n1¹2º𝑖refer to the𝑧-component of\nthe standard spin-1\n2operators ascribed to Dy3¸magnetic ions approximated by the notion of classical\nIsing spins. The coupling constants 𝐽Hand𝐽0\nIdetermine a strength of the Heisenberg and Ising intra-\ndimer interactions within the horizontal Cu2¸-Cu2¸and vertical Dy3¸-Dy3¸dimers, respectively, while\nthe coupling constant 𝐽Idetermines a strength of the Ising inter-dimer interaction between the nearest-\nneighbour Cu2¸and Dy3¸magnetic ions. The Zeeman’s terms ℎH=𝑔H𝜇B𝐵andℎI=𝑔I𝜇B𝐵take into\naccount a magnetostatic energy of magnetic moments relating to the Heisenberg and Ising spins in\npresenceoftheexternalmagneticfield 𝐵,whichdifferduetodifferentgyromagneticfactors 𝑔Hand𝑔Iof\nCu2¸and Dy3¸magnetic ions, respectively.\nIt is worthwhile to remark that the partition function, Gibbs free energy and magnetization of the\nIH-ODC defined through the Hamiltonian (2.1) was exactly calculated under the periodic boundary\nconditions ˆ𝜎𝑧\n1¹2º𝑁¸1\u0011ˆ𝜎𝑧\n1¹2º1inourprecedingpaper[19].Thecalculationprocedureusedtakesadvan-\ntage of splitting the total Hamiltonian (2.1) into commuting six-spin cluster Hamiltonians involving one\nhorizontalCu2¸-Cu2¸dimerandtwoenclosingverticalDy3¸-Dy3¸dimers,whichallowastraightforward\nfactorizationofthepartitionfunctionintoaproductoftherespectiveBoltzmannfactors.Aftertracingout\nspin degrees of freedom of the horizontal Cu2¸-Cu2¸dimer (Heisenberg dimer), the partition function\nis in fact expressed in terms of four-by-four transfer matrix depending on spin states of two adjacent\nvertical Dy3¸-Dy3¸dimers (Ising dimers) and the whole magnetothermodynamics can be elaborated\nby making use of the transfer-matrix method (the readers interested in further calculation details are\nreferredtoreference[19]).However,allnumericalresultspresentedinreference[19]wererestrictedjust\ntotheparticularcase ℎH=ℎI,whichcorrespondstosettingthesameLandé 𝑔-factorsforCu2¸andDy3¸\nmagnetic ions which is contrary to the expected (typical) values of the gyromagnetic factors 𝑔H\u00192for\nCu2¸magnetic ions and 𝑔I\u001920for Dy3¸magnetic ions. Therefore, in the present article we adapt the\nexact solution for the IH-ODC reported in reference [19] in order to investigate the effect of different\n‘localmagneticfields’ ℎH≠ℎIarisingfromthedifferenceofthegyromagneticfactorsofCu2¸andDy3¸\nmagnetic ions 𝑔H≠𝑔I.\n3. Theoretical results\nIn this section we examine in detail the ground state and magnetization process of the IH-ODC by\nassuming the gyromagnetic factors 𝑔H=2and𝑔I=20, which are close to typical values of the Landé\n𝑔-factors for Cu2¸and Dy3¸magnetic ions, respectively. To reduce the number of free parameters, a\nsizeofthecouplingconstant 𝐽I¡0correspondingtotheantiferromagneticIsinginter-dimerinteraction\nbetween Dy3¸-Cu2¸magnetic ions serves as an energy unit when defining a relative strength of the\nHeisenberg intra-dimer interaction 𝐽H𝐽Iwithin the horizontal dimers, the Ising intra-dimer interaction\n𝐽0\nI𝐽Iwithin the vertical dimers and the magnetic field 𝜇B𝐵𝐽I.\n43708-3J. Strečka, L. Gálisová, T. Verkholyak\n3.1. Ground state\nThe IH-ODC with the gyromagnetic factors 𝑔H=2and𝑔I=20may display, in presence of the\nmagnetic field, up to seven different ground states depending on a mutual interplay of the coupling\nconstants𝐽H𝐽I,𝐽0\nI𝐽Iand the magnetic field 𝜇B𝐵𝐽I. The typical ground-state phase diagrams are\nreportedinfigure2inthe 𝐽H𝐽I\u0000𝜇B𝐵𝐽Iparameterplaneforfourrepresentativevaluesoftheinteraction\nratio𝐽0\nI𝐽I=\u000005,05,20and25. It is quite evident from figure 2 (a) that the ground-state phase\ndiagram for the particular case with the ferromagnetic Ising intra-dimer coupling 𝐽0\nI𝐽I=\u000005involves\njustfourdifferentgroundstates,morespecifically,theclassicalferrimagneticphase jIiwithanantiparallel\nspin arrangement of the Ising and Heisenberg dimers characterized through the following eigenvector\nand corresponding eigenenergy\njIi=𝑁Ö\n𝑖=1\f\f\f\"\n\"E\n𝑖\nj##i𝑖 𝐸 I=𝑁\n4\u0000𝐽H¸𝐽0\nI\u00004𝐽I\u00004ℎI¸4ℎH\u0001 (3.1)\nthemodulatedquantumantiferromagneticphase jIIiwithalternatingcharacteroftheIsingdimersanda\nsinglet-like state of the Heisenberg dimers\njIIi=𝑁2Ö\n𝑖=1\f\f\f\"\n\"E\n2𝑖\u00001\n\u0010\nsin𝜑2j\"#i 2𝑖\u00001\u0000cos𝜑2j#\"i 2𝑖\u00001\u0011\n\n\f\f#\n#E\n2𝑖\n\u0010\ncos𝜑2j\"#i 2𝑖\u0000sin𝜑2j#\"i 2𝑖\u0011\n\n𝐸II=\u0000𝑁\n4\u0012\n𝐽H¸2√︃\n4𝐽2\nI¸𝐽2\nH\u0000𝐽0\nI\u0013\n 𝜑 2=1\n2arctan\u0012𝐽H\n2𝐽I\u0013\n (3.2)\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s40/s97/s41/s57/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73\n/s40/s49/s56\n/s66/s41\n/s48/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73\n/s40/s48\n/s66/s41/s49/s48/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73/s73\n/s40/s50/s48\n/s66/s41/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s124/s73/s86\n/s40/s50/s50\n/s66/s41\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s124/s86/s73/s66/s66 /s32/s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32\n/s66/s66 /s32/s47 /s32/s74\n/s73\n/s74\n/s72/s32 /s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s74\n/s72/s32 /s47/s32/s74\n/s73/s40/s98/s41/s40/s49/s48\n/s66/s41/s57/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73\n/s40/s49/s56\n/s66/s41\n/s48/s45/s112/s108/s97/s116/s101/s97/s117\n/s40/s48\n/s66/s41/s124/s73/s73/s49/s48/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73/s73\n/s40/s50/s48\n/s66/s41/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s124/s73/s86\n/s40/s50/s50\n/s66/s41\n/s124/s86\n/s40/s48\n/s66/s41\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s57/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s66/s66 /s32/s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32\n/s66/s66 /s32/s47 /s32/s74\n/s73\n/s74\n/s72/s32 /s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s74\n/s72/s32 /s47/s32/s74\n/s73/s40/s99/s41/s124/s73\n/s40/s49/s56\n/s66/s41/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s124/s73/s86\n/s40/s50/s50\n/s66/s41\n/s49/s48/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73/s73\n/s40/s50/s48\n/s66/s41\n/s48/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86\n/s40/s48\n/s66/s41/s83/s65/s70/s53/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86/s73\n/s40/s49/s48\n/s66/s41\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s83/s65/s70\n/s40/s100/s41/s57/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73\n/s40/s49/s56\n/s66/s41/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s124/s73/s86\n/s40/s50/s50\n/s66/s41\n/s49/s48/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73/s73\n/s40/s50/s48\n/s66/s41\n/s48/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86\n/s40/s48\n/s66/s41/s83/s65/s70/s53/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86/s73\n/s40/s49/s48\n/s66/s41\n/s83/s65/s70/s49/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86/s73/s73\n/s40/s50\n/s66/s41\nFigure 2.Asemi-logarithmicplotoftheground-statephasediagramoftheIH-ODCwiththegyromagnetic\nfactors𝑔H=2and𝑔I=20in the𝐽H𝐽I\u0000𝜇B𝐵𝐽Iplane for four representative values of the interaction\nratio: (a)𝐽0\nI𝐽I=\u000005; (b)𝐽0\nI𝐽I=05; (c)𝐽0\nI𝐽I=20; (d)𝐽0\nI𝐽I=25. The numbers in round brackets\ndetermine the total magnetization in units of Bohr magneton 𝜇Band the fractions represent its relative\nsize with respect to the saturation magnetization.\n43708-4Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\nthequantumferrimagneticphase jIIIiwithfullypolarizedIsingdimersandaperfectsinglet-dimerstate\nof the Heisenberg dimers\njIIIi=𝑁Ö\n𝑖=1\f\f\f\"\n\"E\n𝑖\n1p\n2¹j\"#i𝑖\u0000j#\"i𝑖º 𝐸 III=\u0000𝑁\n4\u00003𝐽H\u0000𝐽0\nI¸4ℎI\u0001 (3.3)\nand finally, the saturated paramagnetic phase jIViwith a fully polarized nature of the Ising as well as\nHeisenberg dimers\njIVi=𝑁Ö\n𝑖=1\f\f\f\"\n\"E\n𝑖\nj\"\"i𝑖 𝐸 IV=𝑁\n4\u0000𝐽H¸𝐽0\nI¸4𝐽I\u00004ℎI\u00004ℎH\u0001 (3.4)\nNote that all eigenvectors are written as a tensor product over states of the vertical Ising dimers (former\nstate vector) and the horizontal Heisenberg dimers (the latter state vector), respectively. At low enough\nmagnetic fields, the classical ferrimagnetic phase jIidominates in the parameter region with the fer-\nromagnetic Heisenberg intra-dimer coupling 𝐽H0, while the modulated quantum antiferromagnetic\nphasejIIiand the quantum ferrimagnetic phase jIIIidominate in the parameter region with the antifer-\nromagnetic Heisenberg intra-dimer coupling 𝐽H¡0. Of course, the saturated paramagnetic phase jIVi\nrepresentstheactualgroundstateathighenoughmagneticfieldsregardlessofwhethertheferromagnetic\nor antiferromagnetic Heisenberg intra-dimer coupling is considered.\nOn the other hand, the ground-state phase diagram of the IH-ODC with a relatively weak antiferro-\nmagnetic Ising intra-dimer coupling additionally involves the other two ground states [see figure 2 (b)\nfor𝐽0\nI𝐽I=05], which could be identified as the frustrated quantum antiferromagnetic phase jViwith\na two-fold degenerate antiferromagnetic state of the Ising dimers and a perfect singlet-dimer state of the\nHeisenberg dimers\njVi=𝑁Ö\n𝑖=1\f\f\f\"\n#E\n𝑖\u0010\nor\f\f#\n\"E\n𝑖\u0011\n\n1p\n2¹j\"#i𝑖\u0000j#\"i𝑖º 𝐸V=\u0000𝑁\n4\u00003𝐽H¸𝐽0\nI\u0001 (3.5)\nand the highly degenerate modulated quantum ferrimagnetic phase jVIiwith alternating ferro-\nantiferromagnetic character of the Ising dimers and a singlet-like state of the Heisenberg dimers\njVIi=𝑁2Ö\n𝑖=1\f\f\f\"\n\"E\n2𝑖\u00001\n\u0010\nsin𝜑6j\"#i 2𝑖\u00001\u0000cos𝜑6j#\"i 2𝑖\u00001\u0011\n\n\f\f\"\n#E\n2𝑖\u0010\nor\f\f#\n\"E\n2𝑖\u0011\n\n\u0010\ncos𝜑6j\"#i 2𝑖\u0000sin𝜑6j#\"i 2𝑖\u0011\n\n𝐸VI=\u0000𝑁\n4\u0012\n𝐽H¸2√︃\n𝐽2\nI¸𝐽2\nH¸2ℎI\u0013\n 𝜑 6=1\n2arctan\u0012𝐽H\n𝐽I\u0013\n (3.6)\nItisquiteclearfromfigure2(b)thattheground-statephasediagramhasnotchangedintheparameterspace\nwith the ferromagnetic Heisenberg intra-dimer coupling 𝐽H0, while two novel ground states emerge\natlow(uptomoderate)magneticfieldsintheparameterspacewithsufficientlystrongantiferromagnetic\nHeisenberg intra-dimer coupling 𝐽H¡0. It is noteworthy that the frustrated quantum antiferromagnetic\nphasejVisuppresses the modulated quantum antiferromagnetic phase jIIiupon increasing the relative\nstrengthoftheantiferromagneticIsingintra-dimercoupling 𝐽0\nI𝐽Iuntilthislattergroundstatecompletely\nvanishes from the ground-state phase diagram as evidenced by figure 2 (c) for 𝐽0\nI𝐽I=20. Last but not\nleast, the sufficiently strong antiferromagnetic Ising intra-dimer coupling 𝐽0\nI𝐽I¡2may additionally\ncauseanupriseofthenovelgroundstatealsointheparameterspacewiththeferromagneticHeisenberg\nintra-dimercoupling 𝐽H0andsmallenoughmagneticfields,whichcouldbeidentifiedasthefrustrated\nferrimagneticphase jVIIiwithatwo-folddegenerateantiferromagneticstateoftheIsingdimersandfully\npolarized Heisenberg dimers given by\njVIIi=𝑁Ö\n𝑖=1\f\f\f\"\n#E\n𝑖\u0010\nor\f\f#\n\"E\n𝑖\u0011\n\nj\"\"i𝑖 𝐸 VII=𝑁\n4\u0000𝐽H\u0000𝐽0\nI\u00004ℎH\u0001 (3.7)\n43708-5J. Strečka, L. Gálisová, T. Verkholyak\n3.2. Zero- and low-temperature magnetization curves\nThe gapped ground states (3.1)–(3.7) should be manifested in zero- and low-temperature magneti-\nzation curves of the IH-ODC as intermediate plateaus emergent at fractional values of the saturation\nmagnetization.Byconsideringspecificvaluesofthegyromagneticfactors 𝑔H=2and𝑔I=20oneshould\naccordinglydetectazeromagnetizationplateauinastabilityregionofthemodulatedquantumantiferro-\nmagneticphasejIIiandthefrustratedquantumantiferromagneticphase jVigivenbyequations(3.2)and\n(3.5),theintermediate1/11-plateaumayemergeduetothefrustratedferrimagneticphase jVIIigivenby\nequation (3.7), the intermediate 5/11-plateau can be ascribed to the modulated quantum ferrimagnetic\nphasejVIigiven by equation (3.6), the intermediate 9/11-plateau corresponds to the classical ferrimag-\nneticphasejIigivenbyequation(3.1)andfinally,theintermediate10/11-plateaurelatestothequantum\nferrimagneticphase jIIIigivenbyequation(3.3).Fromthisperspective,theIH-ODCexhibitsasubstan-\ntial diversity of the magnetization curves with nine possible magnetization scenarios as exemplified in\nfigure 3.\nInagreementwiththereportedground-statephasediagrams,onefindsthreedifferentmagnetization\nscenarios with either a single field-driven phase transition jIi\u0000jIVi[figure 3 (a)], three field-induced\nphase transitionsjIIi\u0000jIi\u0000jIIIi\u0000jIVi[figure 3 (b)] or two field-driven phase transitions jIIi\u0000jIIIi\u0000jIVi\n/s49/s69/s45/s51 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s66 /s32/s66/s32 /s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32\n/s66 /s32/s66/s32 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/s49 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s32/s61/s32/s50/s46/s53\n/s74\n/s72/s32 /s47/s32/s74\n/s73/s32/s61/s32/s48/s46/s53\n/s40/s105/s41/s49/s48/s47/s49/s49/s57/s47/s49/s49\n/s53/s47/s49/s49\nFigure 3.(Colouronline)Asemi-logarithmicplotofallrepresentativeisothermalmagnetizationcurvesof\ntheIH-ODCwiththegyromagneticfactors 𝑔H=2and𝑔I=20calculatedatthreedifferenttemperatures\n𝑘B𝑇𝐽I=0(red solid lines), 0.05 (blue dashed lines) and 0.15 (green dotted lines). Different panels\ndemonstrate a diversity of the magnetization process depending basically on a choice of the coupling\nconstants𝐽H𝐽Iand𝐽0\nI𝐽Iquoted in the respective panels.\n43708-6Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\n[figure3(c)]ontheassumptionthattheIsingintra-dimercouplingisferromagnetic 𝐽0\nI0.Theparticular\ncase with a weak antiferromagnetic Ising intra-dimer coupling 𝐽0\nI𝐽I&0displays the other three types\nof magnetization processes due to the presence of the phases jViand/orjVIi: one type involves a\nsequence of four field-driven phase transitions jIIi\u0000jVIi\u0000jIi\u0000jIIIi\u0000jIVi[figure 3 (d)] and the other two\ntypes include a sequence of three field-induced phase transitions jIIi\u0000jVIi\u0000jIIIi\u0000jIVi[figure 3 (e)]\norjVi\u0000jVIi\u0000jIIIi\u0000jIVi[figure 3 (f)], respectively. Finally, the specific case with a sufficiently strong\nantiferromagneticIsingintra-dimercoupling 𝐽0\nI𝐽I\u001d0mayexhibitthreeothermagnetizationscenarios,\nwhichconsecutivelyincludeasequenceoftwofield-drivenphasetransitions jVIIi\u0000jIi\u0000jIVi[figure3(g)],\nfive field-induced phase transitions jVi\u0000jVIIi\u0000jVIi\u0000jIi\u0000jIIIi\u0000jIVi[figure 3 (h)] or four field-induced\nphase transitionsjVi\u0000jVIi\u0000jIi\u0000jIIIi\u0000jIVi[figure 3 (i)], respectively.\nNow,afewcommentsareinorderasfarasathermalstabilityoftheindividualmagnetizationplateaus\nisconcerned.Itisquiteobviousfromfigure3thatsomeintermediateplateausarequiterobustwithrespect\ntothermalfluctuationsandtheycanbeclearlydiscernedintherespectiveisothermalmagnetizationcurves\natlow(𝑘B𝑇𝐽I=005)orevenatmoderate( 𝑘B𝑇𝐽I=015)temperatures.Thisistypicallythecaseforthe\nzero magnetization plateau and for the intermediate 9/11- and 10/11-plateaus emergent at higher values\nofmagnetization.Ontheotherhand,theintermediate1/11-and5/11-plateausemergentatsmallervalues\nofthemagnetizationaretypicallysubjectedtoaconsiderablesmoothinguponincreasingthetemperature\nandtheycannotbeclearlydiscernedintherespectiveisothermalmagnetizationcurvesevenatarelatively\nlow temperature ( 𝑘B𝑇𝐽I=005). This striking finding can be related to a substantial difference of the\nenergygapoftheindividualgroundstatesascribedtotherelevantmagnetizationplateaus.Asamatterof\nfact, the gapped ground states, for instance the classical and quantum ferrimagnetic phases jIiandjIIIi,\nduetotheiruniquenaturepossessahighenergygapinanexcitationspectrum.Onthecontrary,thehighly\ndegenerate ground states, such as the frustrated ferrimagnetic phase jVIIiand the modulated quantum\nferrimagnetic phase jVIi, possess only a tiny energy gap and hence, they are subjected to a substantial\nthermal smoothing owing to their macroscopic degeneracies.\n4. Magnetization curve of the polymeric compound [Dy 2Cu2]𝒏\nIn this part we compare available experimental data for the magnetization curve of the 3 𝑑-4𝑓\nheterobimetallic coordination polymer [Dy 2Cu2]𝑛measured in pulsed magnetic fields up to 10 T at\ntemperature 𝑇=05K [21] with a relevant theoretical prediction based on the IH-ODC. It should be\nmentioned that the magnetization data recorded in pulsed magnetic fields show a small hysteresis in\na low-field region 𝐵.05T, which was ignored for simplicity because it is beyond the scope of the\nintroduced IH-ODC. It actually follows from the inset of figure 4 (a) of reference [21] that the magnetic\nhysteresis basically depends on a field scan rate and it may be thus attributed to a quantum tunneling of\nthemagnetizationtypicallyobservedinsingle-chainmagnetsduetolevelcrossingamongexcitedstates.\nItisworthnoticingthatallfivemodelparametersoftheIH-ODCdefinedthroughtheHamiltonian(2.1)\nwere supposed to be free fitting parameters in order to get the best theoretical fit of the experimental\nmagnetization data (see figure 4 for a comparison). The best theoretical fit was accordingly obtained for\nthefollowingsetofthemodelparameters: 𝐽I𝑘B=802K,𝐽0\nI𝑘B=1735K,𝐽H𝑘B=173K,𝑔H=228\nand𝑔I=1854.\nThe reported value of the gyromagnetic factor 𝑔H=228is quite typical for Cu2¸magnetic ions,\nwhile the other reported value 𝑔I=1854is only by a few percent (cca. 7%) lower than the value\n𝑔Dy=20theoretically expected for the effective Ising-spin description of Dy3¸magnetic ions with the\ntotal angular momentum 𝐽=152and the respective 𝑔-factor𝑔𝐽=43[22, 23]. It should be pointed\nout, moreover, that the values of the Ising inter- and intra-dimer coupling constants 𝐽Iand𝐽0\nIshould be\nalso rescaled by the factors of 15 and 225 in order to get true values of the coupling constants between\nDy3¸-Cu2¸and Dy3¸-Dy3¸magnetic ions when passing away from the effective Ising-spin description\nof Dy3¸magnetic ions. The actual values of the three considered coupling constants consequently read:\n𝐽Dy-Cu𝑘B=053K,𝐽Dy-Dy𝑘B=008K and𝐽Cu-Cu𝑘B=173K, whereas the predicted values of the\ncouplingconstants 𝐽Dy-Cuand𝐽Dy-Dyfallintoareasonablerangeforthecouplingconstantsbetween3 𝑑-4𝑓\nand 4𝑓-4𝑓magnetic ions, respectively [22, 23]. Note, furthermore, that the former exchange constant is\ncomparable withthe meanvalue ofthe exchangecoupling 𝐽Dy-Cu𝑘B=048K, whichcan becalculated\n43708-7J. Strečka, L. Gálisová, T. Verkholyak\nFigure 4. (Colour online) A comparison between the magnetization curve of the polymeric compound\n[Dy2Cu2]𝑛measured in pulsed magnetic fields up to 10 T at temperature 𝑇=05K (a black solid line\nwithopencirclesadaptedfromreference[21])andthebesttheoreticalfitobtainedbyusingoftheIH-ODC\nwith the following fitting set of the parameters: 𝐽I𝑘B=802K,𝐽0\nI𝑘B=1735K,𝐽H𝑘B=173K,\n𝑔H=228and𝑔I=1854. The theoretical results for the isothermal magnetization curves are also\npresented for very low temperature 𝑇=005K (light blue dotted line) and slightly higher temperature\n𝑇=10K(reddashedline)inadditiontothetemperature 𝑇=05K(greensolidline)correspondingto\nthe displayed experimental data.\nfromthecouplingconstants 𝐽A𝑘B=0895Kand𝐽B𝑘B=0061Kassignedpreviouslytotwodifferent\nexchangepathwaysbetweenDy3¸-Cu2¸magneticionswithinthesimplifiedtetranuclearmodel[21].Itis\nworthwhile to remark that the reported value for the coupling constant 𝐽Cu-Cu𝑘B=173K is relatively\nsmallwithrespecttotheshortestdistancebetweenCu2¸-Cu2¸magneticions,butthissmallvaluecanbe\nattributed to a rather inefficient exchange pathway involving a double-oxygen bridge between magnetic\n(3𝑑𝑥2-𝑦2) and nonmagnetic ( 3𝑑𝑧2) orbitals of Cu2¸magnetic ions in an elongated square-pyramidal\nenvironment [see figure 1 (a)].\nThe predicted values of the coupling constants of the IH-ODC are consistent with the following\nvalues of the interaction ratio 𝐽0\nI𝐽I\u001922and𝐽H𝐽I\u001902, which should cause a magnetization sce-\nnario quite analogous to that shown in figure 3 (i) with a sequence of four field-driven phase transitions\njVi\u0000jVIi\u0000jIi\u0000jIIIi\u0000jIVi. A low-field part of the magnetization curve should accordingly display two\nnarrow plateaus at zero and approximately 5/11 of the saturation magnetization (note that the gyro-\nmagnetic factors slightly deviate from the ideal values 𝑔H=20and𝑔I=200), which are, however,\ncompletely smeared out by thermal fluctuations at temperatures as low as 𝑇&05K due to tiny energy\ngaps of the phases jViandjVIi. As a matter of fact, the zero magnetization plateau corresponding\nto the frustrated quantum antiferromagnetic phase jViis restricted to the magnetic fields smaller than\n017T, while the 5/11-plateau related to the modulated quantum ferrimagnetic phase jVIiis limited to\nthe magnetic-field range 017\u0000028T.\nOntheotherhand,thehigh-fieldpartofthemagnetizationcurveshouldexhibittwowiderintermediate\nplateaus roughly at 9/11- and 10/11 of the saturation magnetization. The 9/11-plateau pertinent to the\nclassical ferrimagnetic phase jIiis stable within the magnetic-field range 028\u0000412T, while the 10/11-\nplateau ascribed to the quantum ferrimagnetic phase jIIIiis stable within the magnetic-field range\n412\u0000637T. In this regard, it seems quite puzzling that the field-driven transition between the classical\nand quantum ferrimagnetic phases cannot be evidently seen from the magnetization data recorded for\nthe polymeric compound [Dy 2Cu2]𝑛at a relatively low temperature 𝑇=05K as both ground states jIi\nandjIIIiwithasubstantialenergygapshouldbequiteresistantwithrespecttoathermalsmoothing.Itis\nplausibletoconjecturetwopossiblereasonsforthediscrepancybetweentheexperimentalmagnetization\ndatarecordedattemperature 𝑇=05Kandtherespectivetheoreticalfit:eitherthesampleofthepolymeric\ncompound[Dy 2Cu2]𝑛wasnotkeptduringthemagnetizationprocessunderaperfectisothermalcondition\n43708-8Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\nsince small heating (e.g. due to the magnetocaloric effect) could resolve this discrepancy as evidenced\nby a theoretical curve calculated for a slightly higher temperature 𝑇=10K, or the discrepancy is of\nintrinsic origin and it comes from two different exchange pathways between Dy3¸-Cu2¸magnetic ions\nneglected within the investigated quantum spin chain.\n5. Conclusion\nIn the present article we have scrupulously investigated diverse ground-state phase diagrams and\nmagnetization curves of the IH-ODC by assuming two different gyromagnetic factors of the Ising\nand Heisenberg spins. The investigated quantum spin chain including two different Landé 𝑔-factors\nwas inspired by the polymeric coordination compound [Dy 2Cu2]𝑛, whose magnetic structure shows a\npeculiar one-dimensional architecture with regularly alternating dimeric units of Dy3¸-Dy3¸and Cu2¸-\nCu2¸magnetic ions stacked in an orthogonal fashion [21]. The vertical dimer of Dy3¸-Dy3¸magnetic\nionswasapproximatedbyacoupleoftheIsingspins,whilethehorizontaldimerofCu2¸-Cu2¸magnetic\nions was approximated by a couple of the Heisenberg spins.\nIt has been found that the IH-ODC exhibits a rich variety of the classical and quantum ground\nstates, which, besides the fully saturated paramagnetic phase emergent at sufficiently high magnetic\nfields,involvesixmoregroundstates:thefrustratedandmodulatedquantumantiferromagneticphase,the\nfrustratedandmodulatedquantumferrimagneticphase,aswellas,thequantumandclassicalferrimagnetic\nphase. These remarkable ground states are responsible for the presence of magnetization plateaus in\nzero- and low-temperature magnetization curves, which are manifested at 0, 1/11, 5/11, 9/11 and/or\n10/11ofthesaturationmagnetization.Astabilityoftheintermediatemagnetizationplateauswithrespect\nto thermal fluctuations was investigated in detail. It has been verified that the rather narrow 1/11-\nand 5/11-plateaus ascribed to the frustrated and modulated quantum ferrimagnetic phases with a high\nmacroscopicdegeneracyandatinyenergygapareeasilydestroyeduponasmallincreaseoftemperature,\nwhile the relatively wide 9/11- and 10/11-plateaus ascribed to the nondegenerate classical and quantum\nferrimagneticphaseswitharobustenergygaparequiteresistantwithrespecttothermalfluctuationsand\nmay be also discernible at relatively low (up to moderate) temperatures.\nConsequently, we have successfully applied the IH-ODC with two different gyromagnetic factors\nof the Ising and Heisenberg spins for a theoretical modelling of the high-field magnetization data\nrecordedpreviouslyforthepolymericcoordinationcompound[Dy 2Cu2]𝑛atasufficientlylowtemperature\n𝑇=05K [21]. The respective low-temperature magnetization curve evidently displays intermediate\nmagnetizationplateau(s),whichstartsatarelativelyhighvalueofthemagnetizationbeingapproximately\n78%ofthesaturatedvalue.ThebesttheoreticalfitoftheavailableexperimentaldatabasedontheIH-ODC\nsuggeststhatthemagnetizationplateau(s)observedexperimentallycouldbeascribedtotheclassicaland\nquantum ferrimagnetic phases given by equations (3.1) and (3.3) inherent to the intermediate 9/11- and\n10/11-plateau, respectively. It should be mentioned, however, that the magnetization data measured for\nthepolymericcomplex[Dy 2Cu2]𝑛atlowenoughtemperature 𝑇=05Kdonotprovideanyexperimental\nevidencefortherelevantfield-inducedphasetransitionbetweentheclassicalandquantumferrimagnetic\nphase [21]. This discrepancy could be resolved either by a small deviation from a perfect isothermal\ncondition during the experimental measurement caused by heating of the sample (e.g., due to the\nmagnetocaloriceffect)oritmayrelatetotheoversimplifiednatureoftheintroducedIH-ODCneglecting\ntwo structurally inequivalent exchange pathways between Dy3¸-Cu2¸magnetic ions. In our future work\nwe plan to examine the magnetization process of the IH-ODC taking into consideration two different\nexchange constants between Dy3¸-Cu2¸magnetic ions in order to clarify this issue.\nAcknowledgements\nTheauthorsaredeeplyindebtedtoOlegDerzhkoforalotofinsightfulandinspiringscientificdiscus-\nsions, from which they have substantially benefited over their whole careers. This work was financially\nsupportedbySlovakResearchandDevelopmentAgencyprovidedunderthecontractNo.APVV-16-0186\n43708-9J. Strečka, L. Gálisová, T. Verkholyak\nand by The Ministry of Education, Science, Research and Sport of the Slovak Republic provided under\nthe grant No. VEGA 1/0105/20.\nReferences\n1. Ivanov N.B., Richter J., Phys. Lett. A, 1997, 232, 308–312, doi:10.1016/S0375-9601(97)00374-5.\n2. Richter J., Ivanov N.B., Schulenburg J., J. Phys.: Condens. Matter, 1998, 10, 3635–3649,\ndoi:10.1088/0953-8984/10/16/015.\n3. Koga A., Okunishi K., Kawakami N., Phys. Rev. B, 2000, 62, 5558–5563, doi:10.1103/PhysRevB.62.5558.\n4. Miyahara Sh., In: Introduction to Frustrated Magnetism, Lacroix C., Mendels Ph., Mila F. (Eds.), Springer\nSeries in Solid-State Sciences, Vol. 164, Springer-Verlag, Berlin, Heidelberg, 2011, 513–536.\n5. Shastry B.S., Sutherland B., Physica B ¸C, 1981, 108, 1069–1070, doi:10.1016/0378-4363(81)90838-X.\n6. Matsuda Y.H., Abe N., Takeyama S., Kageyama H., Corboz P., Honecker A., Manmana S.R., Foltin G.R.,\nSchmidt K.P., Mila F., Phys. Rev. Lett., 2013, 111, 137204, doi:10.1103/PhysRevLett.111.137204.\n7. 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Matter, 2013, 25, 306003,\ndoi:10.1088/0953-8984/25/30/306003.\n17. Verkholyak T., Strečka J., Phys. Rev. B, 2013, 88, 134419, doi:10.1103/PhysRevB.88.134419.\n18. Verkholyak T., Strečka J., Acta Phys. Pol. A, 2014, 126, 22–23, doi:10.12693/APhysPolA.126.22.\n19. Gálisová L., Strečka J., Verkholyak T., Havadej S., Physica E, 2021, 125, 114089,\ndoi:10.1016/j.physe.2020.114089.\n20. Verkholyak T., Strečka J., Phys. Rev. B, 2016, 94, 144410, doi:10.1103/PhysRevB.94.144410.\n21. Okazawa A., Nogami T., Nojiri H., Ishida T., Chem. Mater., 2008, 20, 3110–3119, doi:10.1021/cm703530n.\n22. De Jongh L.J., Miedema A.R., Adv. Phys., 1974, 23, 1–260, doi:10.1080/00018739700101558.\n23. Jensen J., Mackintosh A.R., Rare Earth Magnetism, Oxford University Press, Oxford, 1991.\n43708-10Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\nПояснення природи плато намагнiчення 3d-4f\nкоординацiйного полiмера [Dy 2Cu2]𝒏на основi спiн-1/2\nортогонально-димерного ланцюжка Iзiнґа-Гайзенберґа\nЙ. Стречка1, Л. Ґалiсова2, Т. Верхоляк3\n1Iнститут фiзики, Факультет природничих наук, Унiверситет iменi П. Й. Шафарика, парк Ангелiнум 9,\nКошицi 04001, Словаччина\n2Iнститут управлiння виробництвом, Факультет виробничих технологiй у Прешовi,\nТехнiчний унiверситет в Кошицях, вул. Баєрова 1, Прешов 08001, Словаччина\n3Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна\nДетально дослiджено основний стан i процес намагнiчення точно-розв’язного спiн- 12ортогонально-\nдимерного ланцюжка Iзiнґа-Гайзенберґа з двома рiзними гiромагнiтними факторами Iзiнґових та Гай-\nзенберґових спiнiв. Показано, що дослiджений квантовий спiновий ланцюжок виявляє до семи можли-\nвих основних станiв в залежностi вiд взаємної дiї магнiтного поля, констант внутрiшньо- i мiж-димерної\nвзаємодiї. А саме, фрустрована та модульована антиферомагнiтнi фази вiдповiдальнi за нульове пла-\nто намагнiчення при нульовiй температурi, промiжнi 1/11- та 5/11-плато виникають у фрустрованiй\nта модульованiй квантових ферiмагнiтних фазах, в той час як промiжнi 9/11- та 10/11-плато можна\nвiднести до квантової i класичної ферiмагнiтних фаз. Запропоновано, що плато намагнiчення експе-\nриментально спостереженi при високих полях у 3d-4f гетеробiметалiчному координацiйному полiмерi\n[{Dy(hfac) 2(CH3OH)} 2{Cu(dmg)(Hdmg)} 2]𝑛(H2dmg = dimethylglyoxime; Hhfac = 1,1,1,5,5,5-hexafluoropentane-\n2,4-dione) можна вiднести до класичної та квантової феромагнiтних фаз.\nКлючовi слова: ортогонально-димерний ланцюжок Iзiнґа-Гайзенберґа, плато намагнiчення, 3d-4f\nкоординацiйний полiмер\n43708-11"    },    {        "title": "1907.01850v1.Dynamic_magnetic_features_of_a_mixed_ferro_ferrimagnetic_ternary_alloy_in_the_form_of_AB__p_C___1_p__.pdf",        "content": "mehmet.bati@erdogan.edu.tr  Dynamic magnetic features of a mixed ferro -ferrimagnetic ternary alloy in the \nform of AB pC1-p \nMehmet Batı1,a   Mehmet Ertaşb \naDepartment of Physics, Recep Tayyip Erdoğan University, Rize, Turkey  \nbDepartment of Physics, Erciyes University, Kayseri, Turkey  \n \nAbstract  \nDynamic magnetic features of a mixed ferro -ferrimagnetic ternary alloy in the form of ABpC1-p, \nespecially . The effect of Hamiltonian parameters on the dynamic magnetic features of the system \nare investigated. F or this aim, an ABpC1-p ternary alloy system was simulated within the mean - \nfield approximation based on a Glauber type stochastic dynamic and for simplicity, A, B and C \nions as SA = 1/2, SB = 1 and SC = 3/2, were chosen respectively.  It was found that in our dynamic \nsystem the critical temperature was always dependent on the concentration ratio of the ternary \nalloy.  \nKeywords:  ABpC1-p ternary alloys; Dynamic magnetic features ; Mean -field approximation, \nGlauber type stochastic  \n \n1- Introduction  \n \nMixed -spin Ising systems provide a good model for studying a ferro -ferrimagnetic ternary \nalloy. Ternary alloy in the form of ABpC1-p composed of Prussian blue analogs [1] have been the \ntopic of much research  because of their interesting magnetic behaviors,  such as  photo induced \nmagnetization, charge -transfer -induced spin tr ansitions, the existence of compensation \ntemperatures and h ydrogen storage capacity [2 -11]. It  has been widely studied in literature by \nvarious methods in equilibrium statistical physics,  such as exact recursion relations (ERR) on the \nBethe lattice, mean -field approximation (MFA), Monte Carlo (MC) simulations and effective -field \ntheory (EFT) with correlations [12 -29]. In the studies, ternary alloy in the form of ABpC1-p has \nbeen modelled w ith different magnitudes of spins, namely A, B and C. For instance, the magnetic \nfeatures of a mixed ferro -ferrimagnetic ternary alloy in the form of ABpC1-p consist of three \ndifferent metal ions with ternary Ising spins (1/2, 1, 3/2) [12 -16]; (3/2, 1, 1/2 ) [17]; (1/2, 1, 5/2) \n[18]; (1/2, 3/2, 5/2) [19], (1, 3/2, 5/2) [20 -23], (3/2, 1, 5/2) [24 -26] and with Ising spin s (3/2, 2, \n5/2) [27 -29]. It is worth noticing that there are many experimental studies on Prussian blue analogs \n[30–36].   \n Generally, dynamic p roperty  investigations of ABpC1-p ternary alloy s are difficult because of \ntheir structural complexity. Despite, the equilibrium magnetic properties of a mixed ferro -\nferrimagnetic ABpC1-p ternary alloy having  been studied in detail, there are only two studies  within \nour best knowledge about the dynamic properties of the ABpC1-p ternary alloy system contain ing \nspin (3/2, 1, 5/2) [37, 38]. The aim of this study was focused on studying the dynamic magnetic \nproperties of a ternary alloy system. Dynamic phas e transitions (DPT) were  obtained and the effect \nof Hamiltonian parameters on the dynamic properties of the system were  investigated. For this \naim, an ABpC1-p ternary alloy system  was simulated  by utilizing the MFA based on Glauber type \nstochastic dynamic and for simplicity, A, B and C ions as SA = 1/2 SB = 1 and SC = 3/2  were chosen, \nrespectively.  It is worth noting  that the DPT has been extensively studied theoretically [39 -45] \nand experimentally [46 -50] for different systems during the last decades.  \nThe paper is organized in the following way: Section II describe s the models and its \nformulations.  Section III presents  the numerical results. Finally, the conclusion is given in Sec. IV.   \n \n2- Model formulation  \n \nA mixed ferro -ferrimagnetic ABpC1-p ternary al loy system was considered on two \ninterpenetrating square sublattices. One of the sublattices only has spins SA = ± 1/2 as the other \nsublattices have spins  SB = ±1, 0 or SC = ± 3/2, ± 1/2. A sketch of the present model can be seen in \nFig. 1. The Hamiltonian  can be written as follows:  \n \n    1 1 , (1)\n          ΗA B C A B C\ni AB j j AC j j i j j j j\nij i jS J S J S h(t) S S S ξ ξ ξ ξ\n  \n \nwhere < ij> shows a summation over all pairs of the nearest -neighboring sites of different \nsublattices and JAB > 0 and JAC < 0 (model the ferro -ferrimagnetic interactions ) are the nearest -\nneighbor exchange constants. h(t) is the oscillating external magnetic field and is described by \nℎ(𝑡)=ℎ0cos(𝑤𝑡), where h0, w and t are the time, amplitude and angular frequency. 𝜉𝑗 is \ndistributed random variables and it takes the value of unity or zero, according to whether site j is \nfilled by an ion of B or C, respectively. So, 𝜉𝑗 is described by  \n \n                                                      \n 1 1 , (2)   j j jP p pξ ξ ξ                  \n    \n where p and (1−𝑝) are the concentration of B and C ions, respectively. A mixed ferro -\nferrimagnetic ABpC1-p ternary alloy system is in contact with an isothermal heat bath at an absolute \ntemperature Tabs and evolves according to the Glau ber-type stochastic process at a rate of 1/ τ. From \nthe master equation associated to the stochastic process, it follows that the average magnetization \nsatisfies the following equation [39 -45], \n \n              \n11 , (3a)2    A A B C\nAB j AC j 0\njjdτ S S tanh β J p S +J S p h cos wtdt\n          \n                  \n\n3\n, (3b)\n21\n  \nA\nAB i 0\ni BB\nA\nAB i 0\nisinhβ J p S +ph cos wt\ndτ S Sdtcoshβ J p S +ph cos wt                                 \n         \n \n 3 1.5 1 0.5 1, (3c)2 1.5 1 2 0.5 1                 CCsinh p βx sinh p βx dτ S Sdt cosh p βx cosh p βx                              \n \nwhere 𝑥=𝐽𝐴𝐶∑𝑆𝑖𝐴\n𝑖 +ℎ0cos(𝑤𝑡). Using  the mean -field theory; the dynamic mean -field \napproximation equations are obtained as follows  \n \n       \n  11 , (4a)2     A A AB AB A AC AC C 0dm m tanh β J z m p+J z m p h cosd                      \n  \n  3\n, (4b)\n21   AB BA A 0\nBB\nAB BA A 0sinhβ J z m p+ph cos dmmd coshβ J z m p+ph cos\n \n \n 3 1.5 1 0.5 1, (4c)2 1.5 1 2 0.5 1                 CCsinh p βy sinh p βy dmmd cosh p βy cosh p βy\n \n \nwhere 𝑦=𝐽𝐴𝐶𝑧𝐶𝐴𝑚𝐴+ℎ0cos(𝜉),  𝜉=𝑤𝑡, Ω=𝜏𝑤 and Ω=2𝜋,  zAB, zBA, zAC and zCA are taken \n4 for  a square lattice.  \n                                                                         𝑀𝑖=1\n𝜏∫𝑚𝑖(𝜉)𝑑𝜉                                                                                   (5) \n  \n where 𝑖=𝐴,𝐵 and 𝐶. In other words, 𝑀𝐴, 𝑀𝐵 and 𝑀𝐶 correspond to the dynamic order parameters \nof the magnetic components A, B and C. The total magnetization of the system is  \n \n                                                                        𝑀𝑇=(𝑀𝐴+𝑀𝐵+𝑀𝐶)\n2                                                                                   (6) \n \nThe physical parameters have been scaled in terms of 𝐽𝐴𝐵. For example, reduced temperature and \nfield amplitude are respectively   defined as 𝑇=𝑘𝐵𝑇𝑎𝑏𝑠\n𝐽𝐴𝐵, and ℎ=ℎ0\n𝐽𝐴𝐵,  throughout the  \n  \n3- Results and Discussion  \n \nThe effects of the concentration ratio p and the exchange interaction ratio 𝑅 (|𝐽𝐴𝐶|\n𝐽𝐴𝐵) on dynamic \nmagnetization and DPT  of the ternary alloy have been examined.  It should be noted that the p = 0 \ncase corresponds to a ferrimagnetic mixed spin -1/2 and spin -3/2 system while for p = 1, \ncorresponds to  a mixed spin -1/2 and spin -1 ferromagnetic system.  The phase diagram of the \nternary alloy in ( 𝑅−𝑇𝐶) and ( 𝑝−𝑇𝐶) planes  are shown  in Fig . 2 and Fig . 3, respectively. In these \nfigures, upper graphs are plotted for ℎ=0.1 and the lower ones are plotted for ℎ=0.5. The \ncritical temperature value (phase transition temperature)  is a little decrease d with increasing h. \n(𝑇𝐶=2.71 for h=0.1, 𝑇𝐶=2.67 for h=0.5)  The r eason for this situation  is that , the higher field \namplitude becomes dominant against the ferromagnetic and antiferromagnetic nearest -neighbor \nbonds.  It can be seen  that from the figures  that 𝑇𝐶 increases as 𝑅 increases and  the 𝑇𝐶 values do \nnot change with 𝑅 for 𝑝= 1.0. Because the system become  an AB alloy, there is no  AC \ninteraction. Therefore t he system becomes independent from 𝑅.  \n \nIn this section, the effects of p and R on the magnetization of a ternary alloy of the type ABpC1-p \nare discussed.   Fig. 4 shows the total magnetization  chancing with scaled temperature  for R=0.5, \n1.0 and R=2.0 values.  It is again seen  from the  figures that a ll the total magnetization curves merge \nat a unique transition temperature for p = 1.0.  For the p =0.0 case, Tc=0 at  R =0.0 (see Fig . 1 and \n2) and the dynamic critical temperature of the system increase with an increasing  of R. For p=0.25, \nthe antiferromagnetic exchange interaction between  the A and C magnetic components becomes \neffective in the system for larger R values . In other words , the 𝐽𝐴𝐶  interaction  becomes dominant \nand because  𝐽𝐴𝐶  is negative,  the 𝑀𝑇  results are negative.  It is noted  that the saturation values of \nmagnetization increase for p=0.25 and decrease for p=0.75 with the increasing of R. A second \norder phase transition occurs in the system for R=0.5 and R=1.0. But for R=2, first the  system gives  \n the first order phase transition and then the second order phase transition occurs.  𝑀𝑇 decreases as \nR increases in the range 0.5 < R <1 at p = 0.5 and after 𝑅>1 (AC interaction is more  dominant), \nas R increases, 𝑀𝑇 orientation increases by changing.  \n \nIn Fig. 5 the results have been depicted for R=1.0.  As expected, t he sign  of the 𝑀𝐴 magnetization  \nis  negative ( 𝑀𝐴=−1/2) 𝑀𝐵=0.0 and 𝑀𝐶=3/2 at T=0 for p=0.0.  For 0 .0<p<1.0 the sign  of \nthe 𝑀𝐶 magnetization is  negative  (𝑀𝐶=−3/2),  𝑀𝐴=0.5 and 𝑀𝐵=1.0 at T=0. Lastly, for \np=1.0 𝑀𝐴=1/2,  𝑀𝐵=1.0  and 𝑀𝐶=0 at T=0.  There are no dynamic magnetic properties \ninvestigation of Ternary alloy for spin (½,1,3/2). However, there is an investigation for spin \n(3/2,1,5/2) [37]. Very interesting behaviors with the variations of scaled temperature, \nconcentrations of the spins and ratio of the bilinear interactions are found in agreement with the \nliterature.  An interesting feature that has been observed in equilibrium systems is the existence of \na special interact ion ratio 𝑅𝐶=|𝐽𝐴C|/𝐽𝐴B where the critical temperature of the system becomes \nindependent of the concentration ratio 𝑝 of the ternary alloy [25]. However,  in the dynamic system \nit has been  observed  that critical temperature  is always dependent  on the  concentration ratio 𝑝 of \nthe ternary alloy.  Theoretically, t his behavior was first  reported  firstly  in literatu re by Vatansever \n[37]. Dynamic behaviors of ternary alloys have not yet been studied experimentally.  Actually, \nthere  are only a few  experimen tal studies  about dynamic phase transition in literature  [51-53], such \nas, time dependent magnetic behavior of uniaxial cobalt films in the vicinity of the dynamic phase \ntransition (DPT) as a function of an oscillating magnetic field experimentally studied  by Berger \nand coworkers  [51].   \n \n4-  Conclusions  \n \nIn this paper , the dynamic magnetic features of the  mixed ferro -ferrimagnetic  ternary alloy of the \ntype ABpC1-p on a square  lattice, composed  of spins 1/2, 1, and 3/2 has been studied . A ternary \nalloy system  was simulated within the mean - field approximation based on Glauber type stochastic \ndynamic s. The concentration ratio was obtained as well as the exchange interaction ratio \ndependence of critical temperature.  It was found that the critical temperature cou ld be controlled \nby the concentration ratio and the exchange interaction ratio. 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Phys. 81 (2018) 054001  \n  \n List of the Figure Captions  \n \nFig. 1: (color online) Schematic  representation of ternary alloy spin model the type of ABpC1-p \ndefined on a square lattice.  A, B and C represented in yellow, blue and red respectively . \nInteraction between A and B sites is ferromagnetic (FM) and A and C antiferromagnetic \n(AFM).   \nFig. 2: Dynamic phase diagrams of the system in a ( 𝑅−𝑇𝐶) plane for the different values of p \nwith ℎ=0.1 and ℎ=0.5. \nFig. 3: Dynamic phase diagrams of the system in a ( 𝑝−𝑇𝐶) plane for the different values of R \nwith ℎ=0.1 and ℎ=0.5. \nFig. 4: (color online) Scaled  temperature dependence of total magnetization for different values of \n𝑝 with 𝑅=0.5 (upper figure),  𝑅=1.0 (middle figure) and  𝑅=2.0 (lower figure) . \nFig. 5: (color online)  Scaled temperature dependence of magnetization ( 𝑀𝐴,  𝑀𝐵  and 𝑀𝐶) for \ndifferent values of 𝑝 with 𝑅=1.0. \n \n \n \n \n \nFig. 1: \n \nSA = 1/2  \n JAB > 0 (FM)  \nJBC < 0 (AFM)  \nSB=1 \nSC = 3/2   \n \nh=0.1\nR0.0 0.5 1.0 1.5 2.0 2.5 3.0TC\n02468\np=0.00 \np=0.25 \np=0.50 \np=0.75 \np=1.00 \nh=0.5\nR0.0 0.5 1.0 1.5 2.0 2.5 3.0TC\n02468\np=0.00 \np=0.25 \np=0.50 \np=0.75 \np=1.00  \n \nFig. 2:   \n  \nh=0.1\np0.0 0.2 0.4 0.6 0.8 1.0TC\n02468\nR=0.0 \nR=0.5\nR=1.0\nR=1.5\nR=2.0\nR=2.5\nR=3.0\nh=0.5\np0.0 0.2 0.4 0.6 0.8 1.0TC\n0246R=0.0 \nR=0.5\nR=1.0\nR=1.5\nR=2.0\nR=2.5\nR=3.0\n \n \nFig. 3:  \n  \nT0 1 2 3 4MT\n0.00.20.40.60.8\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\nR=JAC/JAB=0.5\nR=JAC/JAB=1.0\nT0 1 2 3 4MT\n-0.4-0.20.00.20.40.60.8\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\nR=JAC/JAB=2.0\nT0 1 2 3 4MT\n-0.6-0.4-0.20.00.20.40.60.81.0\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\n \nFig. 4:  \n  \np=0.00\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-1.0-0.50.00.51.01.52.0\nMA\nMB\nMC\np=0.25\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=0.50\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=0.75\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=1.00\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n0.00.20.40.60.81.01.2\nMA\nMB\nMC\n \n \nFig. 5: \n \n "    },    {        "title": "2012.14911v2.Spin_polarized_imaging_of_strongly_interacting_fermions_in_the_ferrimagnetic_state_of_Weyl_candidate_CeBi.pdf",        "content": "Spin-polarized imaging of strongly interacting fermions\nin the ferrimagnetic state of Weyl candidate CeBi\nChristian E. Matt,1Yu Liu,1Harris Pirie,1Nathan C. Drucker,2Na Hyun Jo,3, 4Brinda Kuthanazhi,3, 4\nZhao Huang,5Christopher Lane,5, 6Jian-Xin Zhu,5, 6Paul C. Can\feld,3, 4and Jennifer E. Ho\u000bman1, 2,\u0003\n1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA\n2School of Engineering & Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA\n3Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA\n4Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA\n5Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n6Center for Integrated Nanotechnology, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: April 12, 2022)\nCeBi has an intricate magnetic phase diagram whose fully-polarized state has recently been sug-\ngested as a Weyl semimetal, though the role of fstates in promoting strong interactions has remained\nelusive. Here we focus on the less-studied, but also time-reversal symmetry-breaking ferrimagnetic\nphase of CeBi, where our density functional theory (DFT) calculations predict additional Weyl nodes\nnear the Fermi level EF. We use spin-polarized scanning tunneling microscopy and spectroscopy\nto image the surface ferrimagnetic order on the itinerant Bi pstates, indicating their orbital hy-\nbridization with localized Ce fstates. We observe suppression of this spin-polarized signature at\nEF, coincident with a Fano line shape in the conductance spectra, suggesting the Bi pstates par-\ntially Kondo screen the fmagnetic moments, and this p\u0000fhybridization causes strong Fermi-level\nband renormalization. The pband \rattening is supported by our quasiparticle interference (QPI)\nmeasurements, which also show band splitting in agreement with DFT, painting a consistent picture\nof a strongly interacting magnetic Weyl semimetal.\nI. INTRODUCTION\n1 Merging strong electron interactions with topology\nis a new frontier for fundamental research and advanced\ntechnology [1{3]. Kondo lattice systems are a promis-\ning platform for strongly correlated topological phenom-\nena, exempli\fed by the recent observation of strongly-\nrenormalized Dirac surface states in the Kondo insula-\ntor SmB 6[4, 5], and the proposal for a Weyl-Kondo\nsemimetal phase [6]. In general, a Weyl semimetal [7]\narises when a bulk Dirac point is split into two Weyl\nnodes by breaking inversion or time-reversal symmetry\n(TRS). However, a crystal structure that breaks inver-\nsion symmetry is typically not tunable, while an applied\nmagnetic \feld Bthat breaks TRS yields only a small\nZeeman energy of \u00181 Kelvin/Tesla for a typical g-factor\nof 2. Materials with intrinsic magnetic order may have\nlarger energy scales that drive the Weyl nodes farther\napart and protect their well-de\fned chirality [8]. Such\nTRS-breaking Weyl semimetals were recently discovered\nin ferromagnets [9{11] and antiferromagnets [12]. The\nultimate goal is to combine the higher energy scales and\nstrong correlations with the practicality of external tun-\nability [13]. This goal motivates the search for topologi-\ncal phases in Kondo lattice compounds, which often host\nlarge spin-orbit coupling, strongly interacting electrons,\nand proximate \feld-tunable magnetic order [14{16].\n2 Cerium-monopnictides (Ce X,X= As, Sb, Bi) are\ncorrelated low-carrier-density Kondo lattice systems [17]\n\u0003jho\u000bman@physics.harvard.eduwith cascades of magnetic phase transitions, as shown for\nCeBi in Fig. 1 [18{20]. Though bulk magnetic phase di-\nagrams have been measured by neutron scattering, sur-\nface magnetic order has not been studied. Meanwhile,\nnon-trivial band topology was predicted in CeSb [21, 22]\nand CeBi [23, 24], and signatures of Weyl fermions were\nobserved in transport experiments in the fully-polarized\nmagnetic phase of CeSb [22]. However, the Weyl fermion\nbands have not yet been directly resolved in any phase\nof CeX, because the TRS-broken phases exist only un-\nder external magnetic \feld, which precludes the use of\nangle-resolved photoemission spectroscopy (ARPES). In\ncerium monopnictides, it remains crucial to measure the\nsurface magnetic order, its associated band splitting, and\nits orbital contributions, which could in\ruence the Fermi\narcs and their connectivity to Weyl cones [9]. Further-\nmore, characterizing the interplay between magnetic or-\nder and Kondo physics is essential to understand the pos-\nsible emergence of heavy fermions and \rat bands [25].\n3 Here we use spin-polarized (SP) scanning tunneling\nmicroscopy (STM) and spectroscopy (STS) to image the\nenergy-resolved surface magnetic order on CeBi, at low\ntemperature with applied magnetic \feld B. We extend\nprevious density functional theory (DFT) calculations of\nWeyl nodes in the high- Bfully-polarized phase [24], to\npredict additional Weyl nodes near the Fermi level EF\nin the intermediate- Bferrimagnetic phase. We therefore\nfocus our experiments at B= 3 T, where we image the\nexpected (+++\u0000) pattern of the spin orientation of the\nforbitals on the Ce sites, but only above EF. Surpris-\ningly, we observe the same magnetic pattern on the Bi\nsites below EF. The induced magnetic moments on the\nBipstates, co-aligned with the adjacent Ce fstates,arXiv:2012.14911v2  [cond-mat.str-el]  9 Apr 20222\nPM\nTemperature [K]0 10 20 30B [T]\n012345(a)\nFully polarized\nFerrimagnetic\nQ2a\nQa\nQb\n-1\n0\n1\n1\n0\n-1\n-1\n0\n1\n1\n0\n-1\ntip\n(d)(b)\n(c)\nz\nx\ny\nSP-STM\nB\nqz [2π/c]\n-1\n0\n1\n1\n0\n-1\nQlat\nqx [2π/a]\nc\na\nFIG. 1. (a) Magnetic phase diagram of bulk CeBi, with dots\nfrom magneto-transport measurements [19] marking an intri-\ncate cascade of transitions between magnetic orders. (b-d)\nReal-space structure and simulated Fourier transforms of the\nx\u0000zplane of CeBi in the (+ \u0000+\u0000), (++ \u0000\u0000), and (+++ \u0000)\nmagnetic phases. Here, \\+\"and \\ \u0000\" indicate the direction\nof the Cefnet magnetic moments, which are ferromagneti-\ncally aligned in each x\u0000yplane with varying order along the\nzdirection. Qlatindicates the wavevector of the Ce (or Bi)\nsublattice, which would appear in spin-averaged STM images.\nSpin-polarized STM would be sensitive to additional mag-\nnetic Bragg peaks. (b) Antiferromagnetic (+ \u0000+\u0000) phase for\nT.25 K. (c) Antiferromagnetic (++ \u0000\u0000) phase for T.12:5\nK. (d) Ferrimagnetic (+++ \u0000) phase with magnetic \feld B\napplied along z.\nindicatep\u0000forbital hybridization, commonly referred\nto asp\u0000fmixing [26, 27]. For energies closer to EF\nwe observe suppression of the (+++ \u0000) spin polarization,\ncoinciding with a Fano resonance in our measured con-\nductance (dI=dV ), further supporting p\u0000fhybridiza-\ntion. These observations suggest a competition between\nthe mechanism inducing the co-aligned (i.e. ferromag-\nnetically aligned) moments on the Bi pstates and the\nantiferromagnetic Kondo screening. Finally, we present\nquasiparticle interference (QPI) measurements showing\na\u0018100 meV splitting of the Bi p-band, validating our\nDFT calculations that show the crossing of the Bi pand\nCedbands to form Weyl nodes close to EF. Our QPI\nsuggests a \rattening of the mixed-character p\u0000dband\nthat forms the Weyl cones.\nII. METHODS\n4 We calculated the bulk band structure of CeBi using\nthe generalized-gradient approximation (GGA) as imple-\nmented in the all-electron code WIEN2K [28], with the\naugmented-plane-wave + local-orbitals (APW+lo) basis\nkzkx\nkyZXEnergy [eV]0.2\n-0.2\n-0.40.00.4\nZ ZFully polarizedParamagnetic (PM)Energy [eV]0.2\n-0.2\n-0.40.00.4\nZ Z\nW1Source\nSinkBi 6p\nCe 5d\nXW1 W2lower T\nincrease B\nBi p band bottom\n(b)\n(a)\n(c)\n(d)\n(e)FIG. 2. DFT band structure along the \u0000 \u0000Zdirection in the\n(a) paramagnetic; (b) antiferromagnetic (++ \u0000\u0000); (c) fully-\npolarized (++++); and (d) ferrimagnetic (+++ \u0000) phases.\nLight purple and dark green circles indicate locations of Weyl\nnodes. W1 and W2 indicate the Weyl nodes closest to the\nmeasured Fermi energy ( EF) in the ferrimagnetic phase. The\ncalculated bands in each panel have been rigidly shifted by (a)\n+110 meV, (b) \u0000120 meV, (c) \u0000110 meV, and (d) \u000055 meV\nto better match our QPI experiment. (e) Full 3-dimensional\nBrillouin zone (BZ) in the (+++ \u0000) phase.\nset. In the paramagnetic phase we treated the Ce 4 for-\nbitals as core electrons while in the magnetic phases we\nincorporated the Hubbard Coulomb interaction on the Ce\n4felectrons, with U= 7:9 eV andJ= 0:69 eV chosen\nto make the Ce 4 fenergy level qualitatively consistent\nwith ARPES measurements on CeBi [27]. We included\nspin-orbit coupling in all calculations.\n5 Single crystals of CeBi were grown by the self-\rux\nmethod [19]. We cooled the crystals in zero \feld, cleaved\nthem in cryogenic ultra-high vacuum at \u001830 K to expose\na neutral (010) surface before imaging them at T= 4:6\nK. We prepared non-magnetic PtIr STM tips by ex situ\nmechanical sharpening, then in situ \feld emission on Au\nfoil. We obtained spin-polarized tips by gently dunking\nthem into the sample to pick up a few atoms of magnetic\nmaterial [29{31].\nIII. DENSITY FUNCTIONAL THEORY\nCALCULATIONS\n6 Figure 2 shows the folding and splitting of the two\nBipbands and the Ce dband that cross EF, asTis\nlowered and Bis increased [32]. Our calculations predict\nan induced magnetic moment of \u00180:01\u0016Bon the Bip\nstates in both the ferrimagnetic and the fully-polarized\nphase. We also computed the Berry curvature at each\nband crossing in the fully-polarized and ferrimagnetic\nphases, and circled the sinks and sources that constitute3\n(d)\n(f)\n1.0 pm\n0\n2.5 pm\n0\nqz [2π/c]\nQ2a\nQ2a\n(a)\n(c)\n(e)\n(b)\nB = 0 T\nB = 0 T\nBz = 3 T\nqx [2π/a]\n1\n-1\n1\n-1\n0\nQlat\n1\n-1\n1\n-1\n0\n0\n1\n-1\n1\n-1\n0\n0\nx\nz\nx\nz\n2 nm\n2 nm\n2 nm\nx\nz\nBi\nqx [2π/a]\nqz [2π/c]\nqz [2π/c]\nqx [2π/a]\nTopo.\n1.1 pm\n0\nCe\nCe\ntip\ntip\ntip\nFIG. 3. (a) Topography of CeBi measured with spin-\ndegenerate PtIr tip at zero \feld. (Sample bias Vs= 400 mV,\ncurrent setpoint Is= 200 pA.) (b) Fourier transform (FT) of\n(a) shows Ce sublattice peaks at Qlat, but no magnetic peaks.\n(c,d) Topography and FT with spin-polarized tip (obtained\nby dunking to pick up magnetic material) show new struc-\nture and dominant Bragg peaks at Q2a(red circle), consis-\ntent with the expected bulk antiferromagnetic (++ \u0000\u0000) phase\nthat breaks the cubic symmetry even in zero applied \feld.\n(Vs= 400 mV, Is= 150 pA.) (e,f) By applying a horizontal\n\feld ofBz= 3 T orthogonal to the (++ \u0000\u0000) wavevector of\n(c,d), the spin-polarized topography and FT show reoriented\nstructure and Bragg peaks at Q2a(red circle) consistent with\nthe expected bulk ferrimagnetic (+++ \u0000) phase (Vs= 100\nmV,Is= 500 pA, recorded in vicinity of (c) and with the\nsame spin-polarized tip). Insets in panels (a),(c) and (e) show\noverlaid Bi (cyan) and Ce (yellow and red) atoms, with circles\ndenoting expected non-magnetic orbitals and triangles denot-\ning the expected spin orientations of the Ce forbital, from\nneutron scattering [18].\nWeyl nodes (see also Fig. 11) [24]. In the fully-polarized\nphase we \fnd a band splitting of \u0018100 meV that gen-\nerates Weyl nodes near the calculated EF, however we\ncaution that the true EFmay be signi\fcantly shifted in\nlow-carrier-density Kondo materials. The ferrimagnetic\nphase shows comparable band splitting, but is advanta-\ngeous because its band folding generates additional Weyl\nnodes . The predicted ferrimagnetic-phase Weyl nodes are\nwell-spaced over a larger energy range, making the Weyl\nphase and its low energy signatures more robust to band\nbending. This circumvents the problem of ill-de\fned chi-\nrality that arises from scattering between multiple de-\ngenerate Fermi arcs [33] or from Fermi surfaces encom-\npassing multiple Weyl points [8]. We therefore focus our\nattention experimentally on the ferrimagnetic phase for\nthe remainder of this work.\nIV. EXPERIMENTAL RESULTS\n7Figure 3(a) shows a topography acquired with a non-\n(a)\n(b)\n(c)\nx\nz\n2 nm\n2 nm\n2 nm\n-80 mV\nEF\n200 mV\nSpan: 439 pS\nSpan: 76 pS\nSpan: 105 pS\ndI/dV\ndI/dV [a.u.]\nBias [mV]\n-50\n0\n25\n-25\n50\n0\n1\n0\n1\n0\n1\n(e)\nB = 0 T\nBz = 3 T\nBz = 9 T\n(010) surface\n(001) surface\n200\n100\n0\n-300\n-200\n-100\nBias [mV]\ndI/dV [pS]\n0\n40\n120\n80\nQ2a\nQa\nQb\n(d)FIG. 4. Spin-polarized conductance maps at Bz= 3 T of the\n(a) \flled, (b) Fermi level, and (c) empty states. ( Vs= 300\nmV,Is= 4 nA, bias modulation Vrms= 7:1 mV.) The ener-\ngies of the three maps highlight the \flled and Fermi level Bi\nporbitals, and the unoccupied Ce dorbitals, respectively. As\nin Fig. 3(a,c,e), insets show overlaid Bi (cyan) and Ce (yellow\nand red) atoms, but here the triangles denote the observed in-\nduced spin orientations on the Bi p(a,b) and Ce d(c) orbitals.\n(d) Energy-dependent intensity of magnetic Bragg peaks (as\nde\fned in Fig. 1) on the (010) surface of the (+++ \u0000) phase\nof CeBi. The dip in Q2aaround \u0000150 mV corresponds to\nthe bottom of the outer Bi pband, which is marked by a\nred arrow in Fig. 2(d). (e) Background-subtracted, spatially-\naverageddI=dV curves, \ft to Fano line shape (see Figs. 9 and\n10).\nmagnetic PtIr tip. The corresponding Fourier transform\n(FT) in Fig. 3(b) shows four peaks at Qlat= (\u00061;\u00061),\narising from the Ce sublattice (see Fig. 6 ). After gen-\ntly dunking the tip into the sample, the new topogra-\nphy in Fig. 3(c) shows additional structure, suggesting\nthat the tip has picked up magnetic material, leading to\na spin-polarized tunneling current [30, 36]. The mag-\nnetic structure manifests in the FT in Fig. 3(d) as two\ndominant new peaks at Q2a= (0;\u00061\n2), consistent with\nthe expected bulk antiferromagnetic (++ \u0000\u0000) phase that\nbreaks the cubic symmetry of CeBi even in the absence\nof appliedB[see also Fig. 1(c)]. To verify the SP na-\nture of the tip, we applied an in-plane \feld Bz= 3 T,\nperpendicular to the zero-\feld (++ \u0000\u0000) order, to rotate\nthe magnetic ordering vector of the sample by ninety de-\ngrees into the expected (+++ \u0000) phase, as shown in Fig.\n3(e)-3(f) [see also Fig. 1(d)]. Although the tip spin may\nalso realign due to applied B, our observations in Fig.\n3(c) and 3(e) show that the tip has a component that is\nco-aligned with the sample magnetization in both cases.\nThis con\frms two essential requirements for our study:\nOur tip is sensitive to the expected magnetic order of\nCeBi, and we can tune that magnetic order by applying\nmagnetic \feld.\n8 To determine the energy-resolved orbital charac-4\nEnergy [eV]\nΓ\nZ\nΓ\nZ\n0.1\n-0.2\n-0.3\n0.0\n-0.1\n0.2\n0.3\n(b)\nEF\nEF\nBias [V]\n0.1\n-0.2\n-0.3\n0.0\n-0.1\n0.2\n0\n0.5\n-0.5\nqz [2π/c]\n0.3\n(a)\nhigh\nlow\nα\nβ\nγ\n(c)\nCe d2\npd\ndI/dV [a.u.]\n(d)\nCe f\nα\nβ\nγ\np1\np2\npd\np4\nd1\nd2\nd4\nd3\np3\nFIG. 5. (a) Quasiparticle interference (QPI) intensity along\nthe \u0000\u0000Zdirection of the (+++ \u0000) phase atBz= 3 T (see\nFig. 12 for details). (b) DFT band structure from Fig. 2(d)\nin an extended zone scheme. Predicted Weyl nodes are cir-\ncled in light purple and dark green. The dominant features\nin the QPI data, labeled by three white arcs \u000b,\f, and\rin\n(a), match the three intraband scattering processes p1,p2,\nandp3 marked as gray arrows on the DFT in (b). The pre-\ndominance of these three p-orbital band segments in the QPI\nsignal is consistent with a prior observation that porbitals\nextend farther from the surface than dorbitals, and are more\naccessible to the STM tip [34]. Our QPI data also shows\nthat thep3 segment of this folded outer pband is shifted\nup by \u001870 meV (black arrow) with respect to the p1 and\np2 segments, consistent with band \rattening due to strong\nelectron correlations. (c) DFT band structure close to tilted\n(type II) Weyl cone at EF. (d) Schematic showing the renor-\nmalization (\rattening) of the Bi 6 pband upon hybridization\nwith the Ce 4 fstates [35]. Dashed red line denotes the non-\nrenormalized pband. Gray shading denotes the renormalized\nfstate forming our observed Kondo resonance, in agreement\nwith ARPES [25]. Dashed black line indicates experimental\nEF.\nter of the spins in the ferrimagnetic (+++ \u0000) phase,\nwe mapped the spin-polarized di\u000berential conductance,\ndI=dV . Figure 4(a)-4(c) shows SP- dI=dV maps from\nenergies below, at, and above EF. Away from EF, we\n\fnd a high SP-conductance for three neighboring verti-\ncal columns and a low SP-conductance for one column, as\nexpected in the (+++ \u0000) ferrimagnetic phase. Surpris-ingly, comparison of the simultaneously-acquired images\nin Figs. 4(a) and 4(c) reveals that the magnetic contrast\nhas shifted from the Ce lattice sites above EFto the\nBi lattice sites below EF. These induced magnetic mo-\nments on the Bi psites are co-aligned with the underlying\nmagnetic pattern of the localized, unrenormalized Ce f\nstates, which have been observed \u00183 eV below EFby\nARPES [27]. Our evidence of p\u0000forbital hybridiza-\ntion agrees with our DFT prediction of a Bi pmagnetic\nmoment of\u00180:01\u0016B.\n9 We plot the intensity of the magnetic Bragg peaks\nvs. energy in Fig. 4(d). The Q2apeak is dominant at neg-\native energies, but strongly suppressed at EF, and recov-\ners only weakly above EF. This evolution is also appar-\nent in the colorscale spans of the SP- dI=dV maps. While\nthe (+++\u0000) pattern of induced magnetic moments is\ndominant at Vs=\u000080 and +200 meV in Figs. 4(a) and\n4(c), the map at EFin Fig. 4(b) shows a di\u000berent motif\nwith maxima on every second Bi atom. We speculate\nthat this (+\u0000+\u0000) structure may be caused by a residual\nout-of-plane ordering of surface spins (see Fig. 8). The\nsuppression of the Q2aintensity also coincides with a\n180\u000ephase \rip of the real-space pattern (see Fig. 6).\n10 Kondo screening can arise from p\u0000fhybridization,\nas suggested by previous Hall e\u000bect and ARPES measure-\nments [27, 37], so we search for its possible signatures\nindI=dV spectroscopy. Fig. 4(e) shows three spatially-\naverageddI=dV spectra around EFin the (++\u0000\u0000),\n(+++\u0000), and fully-polarized phases. All spectra have\na similar shape with a shoulder around \u000018 meV and a\ndip nearEF, characteristic of the asymmetric Fano line\nshape describing a Kondo resonance,\nF(E)/[q+ (E\u0000E0)=\u0000]2\n1 + [(E\u0000E0)=\u0000]2; (1)\nwhereE0is the energy of the localized many-body reso-\nnance (which we \fnd to be consistent with EF; see Table\nI),qis the Fano factor that describes the tunnelling ratio\nbetween the localized fstate and the itinerant conduc-\ntion electrons, and \u0000 is proportional to the hybridiza-\ntion between the localized and itinerant states [38]. Sim-\nilar line shapes have been observed in other Kondo lat-\ntice systems such as YbRh 2Si2[39], URu 2Si2[40], and\nSmB 6[5]. In all three magnetic phases of CeBi, we \fnd\n\u0000\u001810 meV, consistent with a resistivity upturn at \u0018100\nK, above the N\u0013 eel temperature [19]. However, the large\nresidualdI=dV at the Fano minimum (see Fig. 9) sug-\ngests that only 5 \u000010% of the conduction electrons par-\nticipate in Kondo screening in CeBi, consistent with the\nlocal-moment-like behavior of the Ce sublattice [18, 19].\n11 In CeBi the itinerant electrons closest to EFare\nof both Ce 5 dand Bi 6pcharacter, so we do not know\na priori which itinerant states participate in the partial\nKondo screening of Ce fmoments. However, the disap-\npearance of (+++ \u0000) order from the Bi sites at EFis\nconsistent with the involvement of Bi pstates in Kondo\nsinglet formation. Furthermore, the Kondo resonance is\nfacilitated by the shared symmetry of the Bi 6 porbitals5\nand the Ce f5=2\u00008multiplet [35, 41], while d\u0000fhy-\nbridization is forbidden on-site. We thus conclude that\nthe Bi 6pstates are the primary conduction electrons\nthat couple to the Ce 4 fmoments and participate in the\nformation of Kondo singlets.\n12 Fig. 4 highlights several competing interactions\nin CeBi. First, the Fano lineshape in our dI=dV spec-\ntra suggests a Kondo resonance in which some conduc-\ntion electrons anti-align with and partially screen the\nlocal moments. However, the observed shift of the same\n(+++\u0000) order from the Ce sites in Fig. 4(c) to the Bi\nsites in Fig. 4(a) demonstrates that the induced mag-\nnetic moments on the Bi pstates are co-aligned with\nthe localfmoments. These opposite magnetic inter-\nactions compete for the same pstates. Second, long-\nrange order can arise when localized moments couple\nto each other via the polarized conduction electrons,\nthrough the Ruderman-Kittel-Kasuya-Yosida (RKKY)\ninteraction. The competition between the screening of\nthe local moments (Kondo) and formation of long-range\nmagnetic order (RKKY) is typically determined by both\nthe conduction ( c) electron density and the coupling\nJf\u0000c[14, 16]. However, we observe long-range order in\nFigs. 4(a) and 4(c) coexisting with the Kondo resonance\nin Fig. 4(e). Our data suggests that these competing in-\nteractions in CeBi each dominate at separate energies ,\nin contrast with CeSb where the simultaneous observa-\ntion of Kondo screening and long-range order has been\nexplained by phase separation in momentum space [25].\n13 To determine the e\u000bect of p\u0000fhybridization on\nthe predicted Weyl fermion bands in the CeBi (+++ \u0000)\nphase, Fig. 5(a) shows our quasiparticle interference\n(QPI) measurement of the band dispersion along \u0000 \u0000Z.\nThe dominant QPI features show excellent agreement\nwith the calculated bands of majority Bi porbital char-\nacter shown in Fig. 5(b). From this band assignment,\nwe make several observations. First, the QPI-observed\n\u000band\fbands support the DFT-predicted \u0018100 meV\nsplitting of the hole-like Bi pband around \u0000 into p1\n(\u000b) at 230 meV and p2 (\f) at 120 meV, consistent\nwith the formation of induced magnetic moments on\nthe Bipsites by orbital hybridization. Second, the\nQPI-observed \rband can be attributed to scattering\nacross the BZ boundary between the folded p3 portion\nof the same Bi pband. However, the QPI-observed\n\rband is\u001870 meV higher than the corresponding\nDFT-predicted p3 band, supporting a scenario of p\u0000f\nhybridization that renormalizes (\rattens) the pband\nand the band of mixed pdcharacter that makes up half\nof the Fermi-level Weyl cone. We thus infer that the\nWeyl cones of CeBi are strongly renormalized compared\nto the DFT calculations, consistent with the enhanced\ne\u000bective mass of 4 :3meobserved in quantum oscillation\nexperiments on CeSb [42]. DFT calculations notoriously\nunderestimate band-renormalization e\u000bects in strongly\ninteracting materials, so our QPI experiment serves as a\ncrucial reality check.V. CONCLUSION\n14 Weyl cones are expected to be robust under renor-\nmalization, which can spread their position in momen-\ntum space but not lift the degeneracy of the requisite\ncrossings [43]. Neither p\u0000fnord\u0000fhybridization\nyields an exact realization of the proposed Weyl-Kondo\nsemimetal [6], in which the felectrons are directly in-\nvolved in the formation of the Weyl cones. However,\nour work shows ( i) induced magnetic moments on the Bi\npstates, co-aligned with the local Ce fmoments, from\nspin-polarized dI=dV images of the (+++ \u0000) order on\nthe Bi sites; ( ii)p\u0000fhybridization from dI=dV spec-\ntroscopy of the Fano resonance; ( iii)\u0018100 meV band\nsplitting from QPI that con\frms p\u0000fhybridization and\nTRS breaking; ( iv) and band \rattening from QPI. This\ncomprehensive evidence supports a consistent picture of\nCeBi as a strongly interacting magnetic Weyl semimetal.\nACKNOWLEDGEMENTS\nWe thank Jason Ho\u000bman, Robert Jan-Slager, Daniel\nMazzone for insightful discussions. Experimental and\ntheoretical work was supported by the Center for the\nAdvancement of Topological Semimetals (CATS), an\nEnergy Frontier Research Center funded by the U.S.\nDepartment of Energy (DOE), O\u000ece of Science, Basic\nEnergy Sciences (BES) through the Ames Laboratory\nunder its Contract No. DE-AC02-07CH11358. H.P. was\nfunded by the Gordon and Betty Moore Foundation's\nEPiQS Initiative through Grant GBMF4536. C.E.M.\nwas supported by the Swiss National Science Foundation\nunder fellowships P2EZP2 175155 and P400P2 183890.\nThe theory work was carried out under the auspices of\nthe U.S. DOE National Nuclear Security Administration\nunder Contract No. 89233218CNA000001. C.L. was\nsupported by Los Alamos National Laboratory (LANL)\nLDRD Program. The theory was also supported in\npart by the Center for Integrated Nanotechnologies, a\nDOE BES user facility, in partnership with the LANL\nInstitutional Computing Program for computational\nresources.\nAll data underlying Figs. 1-13 can be accessed in\nRef. [32].\nAppendix A: Surface atom and spin identi\fcation\nTo identify the atomic sublattice imaged in Figs. 3 and\n4, we investigated La-doped CeBi samples, where La is\nexpected to replace Ce. Measurements and DFT calcu-\nlations in Fig. 6 show that at large positive bias volt-\nage we are tunneling predominantly into Ce sites, while\nbelow\u00180:1 V we are tunneling predominantly into Bi\nsites. Fig. 7 shows the full energy dependence of the\nspin-polarized dI=dV maps in Fig. 4. Fig. 8(a) shows6\ntopo (300 mV)\ndI/dV (200 mV)\ndI/dV (-200 mV)\n0.3\n0.2\n0.1\n0.00.1Bias [V]\n(g)\n(h)\n(i)\n(j)\nhigh\nlow\n1 nm\n1 nm\nBi\nCe\n(b)\n(c)\n(e)\n(d)La-doped CeBi\n(a)\n1 nm\n1 nm\n4\n1\n3\n2\n1\n2\n3\n4\nLa\nLa\nvacancy\nLa\nTopo (800 mV)\nTopo \nhigh\nlow\ndI/dV \n-4 -3 -2 -1 0 1 20481216DOS [a.u.]\nEnergy  [eV]\nVswitchVswitch\nCe\nBi\n(f)\n0 -0.5 0.5\nFIG. 6. (a-e) Topography of CeBi with 2.5% (nominal) La dopants expected at the Ce sites, recorded with sample bias Vs= 800\nmV and tunneling current setpoint Is= 600 pA. The spacing of the visible atomic lattice (periodic bright spots) in a clean\nregion is 4.6 \u0017A, consistent with x-ray di\u000braction measurements of the lattice constant [20]. (b) Zoom around a typical La\nadatom that is centered between bright lattice spots. (c) Two La dopants in the top surface layer, located on the bright lattice\nsites. (d) La in the subsurface layer, laterally centered on a dark lattice site of the top surface layer. (e) Vacancy of two bright\nlattice sites of the top surface layer. All four observations in (b-e) suggest that at the large positive sample bias of Vs= 800\nmV, we observe the Ce sublattice. (f) DFT-calculated density of states (DOS) of bulk CeBi in the ferrimagnetic (+++ \u0000)\nstate, resolved according to elemental contribution. At negative energies, the DOS of Bi dominates while on the positive side,\nCe dominates. (g-j) Topography and dI=dV maps of nominally pristine (undoped) CeBi, to identify the shift of dominant\nsublattice depending on sample bias voltage. (g) Topography recorded at Vs= 300 mV. (h-j) Simultaneously recorded dI=dV\nmaps at +200 mV and \u0000200 mV, respectively. (j) dI=dV linecut spatially averaged between the vertical dotted lines in (i),\nillustrating the spatial switching of highest-conductance location at Vswitch\u00180:1 V. Average magnitude of horizontal rows in\n(j) has been normalized for visual purposes. All maps in (g-j) have been simultaneously recorded at Bz= 3 T with Vs= 300\nmV,Is= 4 nA, and Vrms= 7:1 mV.\na second spin-polarized dI=dV map at the Fermi level,\nacquired with identical measurement parameters as Fig.\n7(a11) but with di\u000berent tip termination. Together, Figs.\n7(a11) and 8(a) suggest a canting of the surface spins.\nThe Fermi level dI=dV map in Fig. 8(a) shows a pat-\ntern with maxima on every second Bi atom, which is\ndi\u000berent from the (+++ \u0000) pattern of the bulk magnetic\nmoments expected from the phase diagram of Fig. 1(d).\nThis Fermi level dI=dV corresponds to dominant Qaand\nQbmagnetic Bragg peaks, as shown in Fig. 8(b). Fig-\nure 7(c) shows a corresponding suppression of the Q2a\nBragg peak associated with the (+++ \u0000) order for ener-\ngies close to the Fermi level. The suppression of the Q2a\nintensity also coincides with a 180\u000ephase \rip of the real-\nspace pattern in Fig. 7(a). The suppression of Q2aat\nthe Fermi level implies that the (+ \u0000+\u0000) pattern seen at\nthat energy is not connected to the (+++ \u0000) order, but\nof di\u000berent origin. We suggest that this (+ \u0000+\u0000) struc-ture might be caused by a residual out-of-plane ordering\nof surface spins, as sketched in Fig. 8(e), and simulated\nin Figs. 8(c)-(d).\nAppendix B: Kondo lineshape\nFigure 9(a) shows the spatially-averaged raw dI=dV\nspectra around EFin the (++\u0000\u0000), (+++\u0000), and fully-\npolarized phase from which we subtracted the back-\nground (gray dashed lines) to obtain the Fano lineshapes\nshown in Fig. 4(e), and re-displayed here in Fig. 9(b).\nThe qualitative features of the Fano lineshape, with a\nshoulder around \u000018 meV and a dip near EF, are ap-\nparent in the raw spectra. But to quantify the Fano\nlineshape parameters in Table I, we \ft Eqn. 1 added to\na polynomial background. By comparing the residual\nconductance at EFwith the shoulder around \u000020 meV7\n200\n100\n0\n-300\n-200\n-100\nBias [mV]\ndI/dV [pS]\n0\n40\n120\n80\nQ2a\nQa\nQb\n(c)\nPhase [degree]\n360\n180\n0\n-180\n-360\n(d)\nQlat\n-1\n0\n1\n1\n0\n-1\nQ2a\nQa\nQb\nQlat\nqz [2π/c]\nqx [2π/a]\n(b)\n2 nm\ndI/dV\nlow\nhigh\nFIG. 7. (a1-a25) All energy layers of the conductance map presented in Fig. 4, showing the (010) surface of the (+++ \u0000)\nphase atBz= 3 T. Sample bias is indicated in top right corner of each layer. Setup parameters are Vs= 300 mV, Is= 4 nA,\nVrms= 7:1 mV. (b) Simulated Fourier transform of (+++ \u0000) magnetic order on the (010) surface with magnetic ( Qa,Qb,Q2a)\nand structural lattice ( Qlat) Bragg peaks as indicated. (c-d) Energy-dependent (c) intensity and (d) phase of magnetic and\nstructural Bragg peaks, calculated by Fourier transforming each measured conductance map in panels (a1-a25).8\nSpan: 172 pS\ndI/dV\n(a)\ndI/dV at E F\nQb\nQa\nQlat\nQb\nQa\nQlat\n(b)\n(c)\nSimulation\n(d)\n(e)\nCe\nBi\nFIG. 8. (a) Spin-polarized dI=dV map atBz= 3 T, at the\nFermi level, shows alternating magnetic (+ \u0000+\u0000) contrast,\nwhich is di\u000berent from the expected in-plane (+++ \u0000) mag-\nnetic order from the phase diagram of Fig. 1(d). This map\nwas extracted from the same dataset presented in Fig. 6(g-j),\nwithVs= 300 mV, Is= 4 nA,Vrms= 7:1 mV. Although\nthis map and Fig. 7(a11) were recorded with identical pa-\nrameters, the tip termination had a di\u000berent direction of the\nmagnetic moment (spin-DOS), so the real space images do not\nappear identical, but QaandQbare prominent in the Fourier\ntransform in both cases. (b) Corresponding Fourier transform\nshows that the intensity of the Q2amagnetic Bragg peaks is\nsuppressed, indicating suppression of the bulk (+++ \u0000) mag-\nnetic order on the Bi pstates due to the (partial) formation\nof Kondo singlet states. (c-d) Simulation of the (+ \u0000+\u0000)\nmagnetic order. (e) Possible scenario to explain the (+ \u0000+\u0000)\norder atEF: spins on top (010) surface may have a residual\nout-of-plane component, which would produce the intensity\npattern in our simulation (c-d).\nin Fig. 9(a), it can be seen that only 5 \u000010% of the\nconduction electrons participate in the Kondo screening.\nFurthermore, the point spectra in Fig. 10 recorded in the\nantiferromagnetic ground state ( B= 0 T) and in the\nferrimagnetic state ( Bz= 3 T) all show a Fano-like line-\nshape with only slight spatial variation at larger binding\nenergies due to the magnetic order.\nTABLE I. Fano line shape parameters determined by a least-\nsquare \ft. The Kondo temperature is estimated as TK=\n\u0000=kBwherekBis the Boltzman constant.\n\u0000 [meV] E0[meV]qTK[K]\nB= 0 T 10.2\u00064\u00001:9\u00064\u00000:3118.37\nBz= 3 T 8.8\u000610\u00001:6\u000610\u00000:6102.12\nBz= 9 T 7.7\u00062.5 0.8\u00062.5\u00000:498.35\n(010)  surface\n100 50 0 50 100\nBias [mV]0.00.51.01.52.02.53.0(a)\n50 25 0 25 50\nBias [mV]010101(b)dI/dV  [a.u.]dI/dV  [nS]Bz = 3 T\nBz = 9 TB = 0 T\n(001)  surfacex0.25\nBz = 0 T\nBz = 3 T, (010) sur face\nBz = 9 T, (001) sur faceFIG. 9. Kondo resonance in CeBi (a) Spatially-averaged\ndI=dV spectra around EFwith varying applied B. Ac-\nquisition parameters are: (blue, B= 0 T)Vs= 50 mV,\nIs= 1 nA,Vrms= 2:82 mV; (green, Bz= 3 T)Vs= 300 mV,\nIs= 2 nA,Vrms= 7:1 mV; (orange, Bz= 9 T),Vs= 200 mV,\nIs= 0:4 nA,Vrms= 1:8 mV. Blue and green curves are spa-\ntially averaged spectra from DOS maps recorded on the (010)\nsurface with a spin-polarized tip, while orange curve is nom-\ninally a point spectrum with a non-spin-polarized tip on the\n(001) surface, it was recorded over an extended time, so it\nis e\u000bectively spatially averaging due to lateral piezo drift in\nnm range. Black lines show the \ft of a Fano lineshape (Eqn.\n1) on top of a polynomial background (parabolic for 0 T and\n9 T, but third order polynomial for 3 T data), depicted by\ngray dashed lines. (b) Background-subtracted dI=dV , over-\nlaid with \fts to Fano lineshape, as shown in Fig. 4(e).9\n(b) (c)\n(d)(e) (f)\nSpan: 390 pS\ndI/dV\nSpan: 240 pS\ndI/dV\n1.21.41.61.82.02.22.42.62.83.0dI/dV [nS]\naverage\nFano+backg. fit\nbackground\n0246810 dI/dV [a.u.]\n2 nm\n10121416182022dI/dV [nS]\n0246810 dI/dV [a.u.]\n2 nm-50 mV(a)\n-50 mVBz = 3 TB = 0 Taverage\nFano+backg. fit\nbackground\n-50 -50 50 50 0 0\nBias [mV] Bias [mV]\n-50 -50 50 50 0 0\nBias [mV] Bias [mV]\nFIG. 10. Point conductance spectra at 0 T and 3 T in\nCeBi (a) Conductance map recorded at -50 mV bias ( B= 0\nT,Vs= 50 mV,Is= 1 nA,Vrms= 2:82 mV) indicating the\nlocation of the point spectra presented in (b). (b) Raw point\nspectra vertically o\u000bset for clarity. Blue spectrum is spatial\naverage within the \feld of view in panel (a), gray dashed\nline is polynomial background \ft, and black line is Fano line-\nshape \ft plus background. (c) Point spectra vertically o\u000bset\nfor clarity after background subtraction, indicating Fano-like\nlineshape at all locations. (d)-(f) Similar image and point\nspectra recorded at Bz= 3 T (Vs= 300 mV, Is= 2 nA,\nVrms= 7:1 mV).\nAppendix C: Comparing DFT to QPI\nFigure 11(a) shows our DFT calculation of the band\nstructure in the ferrimagnetic (+++ \u0000) phase of CeBi,\nover a larger energy range than in Fig. 2(d). To \fnd the\nWeyl nodes, we calculate the Berry curvature at each\ncrossing, and circle the sources (pink) and sinks (green).\nIn Figs. 11(c-e) we plot the Berry curvature of the Weyl\npoints W1 and W2 closest to the Fermi level.\nWe investigate the band structure experimentally by\nimaging quasiparticle interference (QPI), which probes\nelastic momentum transfer, predominantly originating\nfrom intra-band scattering, as shown in Fig. 12(a).\nTherefore, all scattering vectors appear around q= 0\nin a QPI measurement, as shown in Fig. 12(b). Figure\n12(c) and 12(d) show energy vs. qdispersion along the\nqzdirection measured in two di\u000berent energy ranges with\nsimilar setup conditions. The two datasets are combined\nin Figs. 5(a) and 12(e), where we overlay the calculated\nband dispersion ( \u000b,\f,\r) from the Bi porbitals. After a\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00\nZ ZW2 W1Energy [eV]\nkyZX\nXW1 W2\nZ Z\nW2 W1\nkx\nkz\n(c)\nW2 W1Sink Source\n(b)\n(d)\n(e)\n(a)kx\nkz Source\nSinkBi 6p\nCe 5dFIG. 11. Prediction of Weyl points in the (+++ \u0000)\nphase of CeBi (a) Band structure along the \u0000 \u0000Zdirection\nin the ferrimagnetic (+++ \u0000) phase, calculated using density\nfunctional theory (DFT). Open circles indicate the locations\nof Weyl nodes, colored to indicate sources (pink) and sinks\n(green) of Berry curvature. (b) Three-dimensional Brillouin\nzone in the (+++ \u0000) phase of CeBi, showing the Weyl nodes\nW1 and W2 closest to the Fermi level, around 50 meV [gray\ndashed line in (a)]. (c) Berry curvature \feld in kx\u0000kzplane of\nthe gray highlighted band in (a), con\frming the source (W1)\nand sink (W2) of Berry curvature. (d)-(e) Zoom on Berry\ncurvature around W1 and W2.\nslight upshift of the \rband, presumably caused by elec-\ntron correlations, we \fnd a good agreement between our\nDFT calculations and our QPI measurements.10\nα\nβ\nγ\nEner gy [eV]\nΓ\nΓ\nZ\n0.1\n-0.2\n0.0\n-0.1\n0.2\n0.3\n(a)\nα\nβ\n(b)\nγ\n0\n0.5\n0.5\nZ\nq-space  (mom entum tran sfer)\nk-space ( absolute momentum )\n0\n0.5\n-0.5\n1\nQPI\nDFT\n-kα\nkα\n-kβ\nkβ\nqγ = kγ - (-kγ) = 2kγ\nqα = 2kα\nqβ = 2kβ\nkγ\nqz [2π/c]\nkz [π/c]\n-kγ\nhigh\nlow\ndI/dV  [a.u.]\nBias [V]\n0.1\n-0.2\n0.0\n-0.1\n0.2\n0\n0.5\n-0.5\nqz [2π/c]\n0.3\n(c)\nα\nβ\nγ\n0\n0.5\n-0.5\nβ\nγ\n(d)\n0\n0.5\n-0.5\n(e)\n-0.3\nqz [2π/c]\nqz [2π/c]\nFIG. 12. Quasiparticle interference (QPI) in the\n(+++ \u0000) phase at Bz= 3T.(a-b) Relation between k-\nspace andq-space in QPI measurements. (a) DFT-calculated\nsegments of the Bi outer pband, replicated from Fig. 2(d), but\nwith the\rband shifted up by \u001870 meV to match our mea-\nsured data. Intra-band quasiparticles scatter from \u0000kto +k\nwith a total momentum transfer of q= 2k, shown as gray ar-\nrows. (b) The momentum transfer vectors ( q) originate from\nzero at the tops of all three bands, \u000b,\f, and\r. Therefore,\npockets around Zink-space appear around zero momentum\ntransfer inq-space. (c-e) Experimental QPI data and compar-\nison to DFT. (c) QPI data set #1 along the \u0000 \u0000Zdirection,\nreproduced in Fig. 5(a) for bias voltages above 80 mV. Setup\nparameters: Vs= 300 mV, Is= 2 nA,Vrms= 7:1 mV. (d)\nQPI data set #2 along the \u0000 \u0000Zdirection, reproduced in\nFig. 5(a) for bias voltages below 80 mV. Setup parameters:\nVs= 300 mV, Is= 4 nA,Vrms= 7:1 mV. (e) Direct overlay\nof DFT-calculated band structure on combined QPI measure-\nment. The bands with Bi porbital character, which de\fne the\nintra-band scattering vectors \u000b,\f, and\r, are consistent with\nthe observed QPI intensity. Here the calculated \rbands are\nshifted up by \u001870 meV with respect to the calculated \u000band\n\fbands from Fig. 2(d), indicating the presence of electron\ncorrelations.\nAppendix D: Zero padding\nIn order to improve atomic visibility, the following\nmaps were interpolated by Fourier-transforming, zero-\npadding the FT, then inverting the FT: Fig. 3(a), Fig.\n4(a)-(c), Fig. 6(f)-(j), Fig. 7(a), Fig. 8(a), Fig. 10(a),\nand Fig. 10(d). The comparison between raw and inter-\npolated data is shown in Fig. 13.\n(c)\n(d)\n(e)\nx\nz\n2 nm\n2 nm\n2 nm\n-80 mV\nEF\n200 mV\n(f)\n(g)\n(h)\nx\nz\n2 nm\n2 nm\n2 nm\n-80 mV\nEF\n200 mV\nSpan: 439 pS\nSpan: 76 pS\nSpan: 105 pS\ndI/dV\n(b)\n2 nm\nx\nz\nBi\nCe\nTopo.\n1.1 pm\n0\ntip\n(a)\nB = 0 T\n2 nm\nx\nz\nBi\nCe\ntip\ntip\nBz = 3 T\nFIG. 13. 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FC-ZFC measur ement of\nbulk at different applied magnetic field has revealed the existence of a ferri-\nmagnetic transition around 66K followed by an antiferromagnetic tr ansition\nclose to 30K. However, its nano counterpart has shown remarkab le change in\nmagnetic properties - a suppression of ferrimagnetic transition ac companied\nby strengthening low temperature magnetic phase and observatio n of a new\ntransition at 90K ( TP), which is weakly magnetic in nature. The frequency\ndependent ac susceptibility data of nanoparticle have been fitted t o the well\nknown de Almedia-Thouless equation and a H2/3dependence of the low tem-\nperature peak is observed with a resulting zero field freezing tempe rature (T0\nf)\nequal to 10.1K. Further, the dynamical behaviour near freezing t emperature\nhasbeenanalysedintermsofcriticalbehaviourandthe obtainedfit ted param-\neters values being as τ0(relaxation time constant) = 3 .6X10−6s,T0\nf= 8.7K\nandzν= 11.1. Moreover, Vogel-Fulcher law has been used to understand the\nH. Singh, T. Chakraborty, K. Srikanth, C. Mitra, U. Kumar\nIndian Institute of Science Education and Research (IISER) Kolkata, Mohanpur Campus,\nPO: BCKV Campus Main Office, Mohanpur 741252, Nadia, West Beng al, India.\nTel.: +91-33-25873121\nFax: +91-33-25873020\nE-mail: udayphy@iiserkol.ac.in\nT. Ono\nDepartment of Physical Science, Osaka Prefecture Universi ty, Gakuen-cho 1-1, Naka-ku,\nSakai, Osaka 599-8531, Japan.\nA. Venimadhav\nCryogenic Engineering Centre, Indian Institute of Technol ogy, Kharagpur-721302, India.\nR. Chandra\nNano Science Laboratory, Institute Instrumentation Centr e and Centre of Nanotechnology,\nIndian Institute of Technology Roorkee, Roorkee 247667, Ut tarakhand, India.2 H. Singh\nnature of freezing transition and the parameter after fitting are obtained as\nEa/kB= 58.9K,τ0= 5.22×10−8andT0= 8.03K.Finally,thespin-glassphase\nis concluded. Moreover, in contrast to bulk, the H2/3dependence of freezing\ntemperature of nanoparticle sample (75h) does support the 2D su rface like\nspin glass nature.\nKeywords Nanoparticles ·Spin Glass ·Antiferromagnetism\nPACS75.75.Fk ·75.50.Lk ·75.47.Lx\n1 Introduction\nOne of the interesting family of magnetic materials is spinel oxides, ma inly\nrepresented by the formula AB2O4[1]. Here the tetrahedral ‘A’ sites were\noccupied by divalent ion and octahedral ‘B’ site were equally occupied by di-\nvalent and trivalent ions. Among normal spinel compounds, nickel c hromite\nhas recently received considerable interest because of its classific ation into dy-\nnamically spin frustrated systems, based upon the realization of re sonance\nlike magnetic excitation observed in neutron scattering experiment [2]. How-\never, despite the presence of pyrochlore lattice which is formed by Cr3+ion,\nthe occurrence of Jahn-Teller distortion nullify the possibility of NiCr2O4to\nbe considered as geometrically frustrated magnet. A number of po tential ap-\nplication (light or heat sensitive micromechanical device, catalytic ma terials,\ngas sensors) has provided technological recognition to this mater ial [3,4]. Sin-\ngle crystals of NiCr2O4has been extensively studied from the perspective of\nneutron scattering [2], heat capacity [3], magnetodielectric [5] and m agnetic\n[6] measurements. However, magnetic studies of nanoparticles of this com-\npound have not been reported so far. Below a critical physical dime nsion (in\nnanometer range) magnetic nanoparticles become single domain as c ompared\nto the normal multi-domain structure of the bulk counterpart. Th ese single\ndomain nanoparticle are quite interesting owing to unusual phenome non they\nexhibit like superparamagnetism [7], quantum tunnelling of magnetizat ion [8]\nand large coercivities [9]. The magnetic studies investigated have sho wn the\noccurrenceoftwotransitiontemperatures TCandTS, correspondingto the on-\nsetofferrimagnetic(longitudinal) componentand spiral(transve rse)antiferro-\nmagnetic component. Recently , Tomiyasu et. al.has reported a new magnetic\nstructure in which for transverse and longitudinal components, B -sites were\nsorted into two sublattices [6]. NiCr2O4undergoes structural transition from\ncubic to tetragonal structure below 310K owing to Jahn-Teller dist ortion. The\nc\n(c−a)ratio is 4% at 4.2K [10]. However,Ishibashi et. al.has confirmed another\nstructural transition to orthorhombic structure at 65K and is co rrelated with\nthe onset of ferrimagnetic ordering [11]. In the present study, we reported the\nmagnetic study of NiCr2O4nanoparticles prepared by mechanical milling of\npolycrystallinepowder.FC-ZFCmeasurementweredoneat0.005an d0.1Tap-\nplied magnetic field to investigate the effect of magnetic field on the be haviour\nof the nanoparticle samples. The observed magnetic transition are in good2D spin-glass phase in NiCr2O4nanoparticles 3\nagreement with the earlier reported results. However, a new tran sition (TP),\nat 90Khasbeen reportedfor the firsttime in bulk NiCr2O4. In high field mea-\nsurement this transition disappears. But in nanoparticle samples, s oftening of\nTCwith decrease in particle size was found, whereas TPis distinctly visible.\nThe decrease in magnetization values in the magnetic nanoparticle sy stem in\ncomparison with its bulk phase is well known but still the subject matt er of\ndebate. This could be seen in the present study. The reason for th is reduction\nin magnetization could be associated with canted spin arrangement o r spin\ndisorder on the surface or finite size effect [12,13,14]. The result o f these phe-\nnomenon leads to superparamagnetism or spin-glass phases. We ha ve explored\nthe low temperature regime (down to 0.4K) as a function of magnetic field to\ninvestigate in detail the magnetic behaviour of the low temperature anomaly\n(TS). Frequency dependent ac susceptibility was also carried out to re veal any\nfrequency dependent magnetic phase. A 2D spin-glass like behaviou r has been\nestablishedfromdeAlmeida-Thouless(AT) analysiswhichisfurthers upported\nbyfrequency dependent acsusceptibility study. Finally, wetried to understand\nthe mechanism responsible for the existence of2D spin glasslike phas e and the\nsurface spin disorder is concluded as the reason behind such an obs ervation.\n2 Experimental Details\nNiCr2O4bulk was prepared by decomposition of NiCr2O4obtained by re-\nactingNiCl2.6H2Owith (NH4)2.Cr2O7. In a typical preparation, 10 mmol\nofNiCl2.6H2Oand 10 mmol of ( NH4)2Cr2O7were mixed in 40 mL distilled\nwater. To this mixture, 90 mmol of ethylene glycol mixed in 40 mL of dis -\ntilled water was added. The resulting solution was stirred for 12 hour s after\nwhich the solution was heated at 60C to evaporate water and obtain a brown\ncoloured gel which on further heating results in a greenish-black po wder pre-\ncursor. The greenish-black powder was heated at 350C to remove ethylene\nglycol and ammonium chloride which was finally heated at 1200C for 72 h ours\nto obtain NiCr2O4bulk crystalline form. This bulk powder form was used\nfor the synthesis of different nano size NiCr2O4sample using high energy ball\nmill machine Fritsch PlanetaryMono Mill Pulverisette 6. For this purpo se, the\nbulk powder sample was kept in an 80 ml agate bowl with 10 mm agate ba ll\nin 1: 8 sample and ball weight ratio and milled up to different milling time.\nAll the nano size samples were annealed at 400C for 4 hours and coole d very\nslowly to room temperature (40C/h) to reduce the possible existing strain in\nthe nano size sample. The phase purity and crystal structure (fr om 320 K\nto 10 K) of all the powder samples were studied by using Rigakus Smar tLab\nX-ray diffractometer using Cu K lines in parallel beam geometry mode.\nThe X-ray diffraction (XRD) pattern of all the samples were scanne d from\n15 to 70 at the step angle of 0.02 and anode power 9 kW. To know the e xact\nparticle size distribution, the samples were subjected to transmiss ion electron\nmicroscopy study and micrographs were collected under 200 kV ano de volt-\nage using model. A set of three sample 0 (Bulk), 33, 75h were prepar ed and4 H. Singh\nsubsequently studied. Zero field cool (ZFC) and field cool (FC) tem perature\ndependent magnetic moment measurement was done at 0.005 and 0.1 T field\nusing Magnetic Property Measurement System (MPMS). A systema tic mini-\nmization of the trapped magnetic field in the superconducting coil of MPMS,\nis usually performed before commencing any measurement. To stud y the low\ntemperature transition, ZFC measurement up to very low tempera ture of0.4K\nwere carried out at applied magnetic field of 0.005, 0.01, 0.05 and 0.1T. These\nlow temperature measurements were performed at Osaka, Japan . The fre-\nquency dependent ac susceptibility curves as a function of temper ature were\nalso collected at 7, 73 and 143 Hz in the MPMS at small oscillating magnet ic\nfield of 1Oe amplitude.\n3 Results and Discussion\nThe room temperature XRD pattern of bulk, 33h, and 75h milled samp les are\nshown in Fig. 1. The diffraction pattern of the bulk NiCr2O4powder sample\nis identical to the earlier reported results [5] confirming the phase purity of\nthe sample. The profile fitting for bulk diffraction patterns was carr ied out\nusing Fullprof Suite version 2009 and structural parameters were determined.\nAbove 310 K, the bulk NiCr2O4exits in cubic spinel structure with space\ngroup Fd(-3m) and lattice parameter 8.3194 ˚A. However, it adopts a tetrago-\nnal cubic spinel structure with space group I 41/amd and lattice pa rameters\nof a = b = 5.8369 ˚Aand c = 8.4312 ˚Abelow 310 K. The splitting of Bragg\npeaks in Fig. 1 can be seen as an indication of cubic tetragonal struc ture and\nis because of Jahn-Teller distortion. These lattice parameters for cubic and\ntetragonal structure are in good agreement with earlier reporte d values [5,6].\nA clear and systematic base broadening in the XRD pattern of 75h an d 33h\nsamples compared to bulk one can be seen in Fig. 1 which is associated t o\ndecreasing particle size effect. The dominant peak at nearly 36 [(211 )d plane]\nis taken under consideration for particle size determination using st andard\nDebye-Scherrer equation. The average particle size of 33h and 75 h ball milled\nsamples was 14.5 and 22.15 nm respectively. To confirm these particle sizes,\nthe 75h sample were subjected to TEM study as shown in Fig. 2. Distin ctly\nvisible lattice fringes confirms the high quality of the sample as is usually\nobserved in single crystalline nanostructures.\nFC-ZFC measurement of bulk and nanoparticle samples were perfor med at\ndifferent applied magnetic field (0.005, 0.01, 0.1 and 2T). For 0.005T ap plied\nmagneticfield, in caseofbulk sample, the observedtransitionsat67 Kand 29K\ncorresponding to TCandTSis well reproducible and is close to the reported\nresults in this system [6] as shown in Fig. 3. We have zeroed the magne tic\nfield of the superconducting magnet of MPMS by following a standard zeroing\nprotocol to ensure that the field was zero at the time of cooling. In terestingly,\nthe ZFC measurement at 0.005T of this system has shown a new tran sition at\n90K (named as TPhere) as shown in the inset of Fig. 3. ZFC measurement\nat 0.005T has not been reported so far in the literature for this sys tem. We2D spin-glass phase in NiCr2O4nanoparticles 5\nFig. 1 X-ray pattern of bulk and nanoparticle samples prepared wit h milling time of 33\nand 75h, shown along with bulk sample.\narguethat there is development of a weakmagnetic phase in the sys tem at this\ntransition temperature, however, the real nature of interactio n might be more\ncomplex. Study performed by Ishibashi et. al., has shown that the high tem-\nperature magnetic transition is occurring simultaneously with the st ructural\ntransition from tetragonal to orthorhombic at TC[11]. The temperature de-\npendent XRD measurement (not shown here) performed over bulk sample did\nnot revealed any structural transition happening at 90K. Hence, we verified\nthat this new transition at TPis solely magnetic.\nThe variation of temperature dependent magnetic behaviour at 0.0 05T ap-\nplied magnetic field for all the samples is shown in Fig. 3. The decrease in\nmagnetization at TCis observed, with decrease in particle size. This soften-\ning ofTCindicates the breaking of spin-lattice coupling owing to particle size\nreduction. The presence of spin-lattice coupling is common among ch romium\nbased spinels. The weakening of spin-lattice coupling is also supporte d by a\nshift in of spiral AFM ordering at TSin our samples towards lower temper-\nature, as the particle size decreases and can be seen in the insets o f Fig. 3\n[15]. The decrease in particle size destroys the long range AFM corre lation,\nresulting in a shift in TStowards lower temperatures [16]. As the particle\nsize decreases, a reduction in the magnetization value is observed. Lowering of\nmagnetization can be explained through increased surface spin can ting, spin\ndisorder or finite size effect at nanoparticle surface [17]. All the sam ples were\nalso investigated at 0.01T magnetic field as shown in Fig. 4. Apart from in-\ncreasein magnetization value no significanteffect ofmagnetic field is o bserved.\nHowever, the new transition TPcould not be observed distinctly. This could\nbe due to the formationof ferrimagneticspin clusters which is overc omeby the\napplication of higher fields causing a disappearance of TPat higher fields. On\nthe other hand, TPis clearly visible in the nanoparticlesamples. The exchange\ncoupling between magnetic ions ( Ni2+andCr3+) is directly proportional to\nCurie-Weiss temperature ( θCW) and can be expressed by relation,6 H. Singh\nFig. 2TEM image of the 75h milled sample revealing the lattice grat ing lines. In the inset\nis the particle size distribution of the same sample with ave rage particle size close to 12nm .\nJ=A|θCW| (1)\nwhere,A=3kB\nZS(S+1),kBis the Boltzmann constant, Z is the number of\nnearest neighbour interaction and S is the total spin. The high temp erature\nregion (T >150K) of ZFC magnetization data at 0.005 and 0.01T was fitted\nwith the Curie-Weiss law and subsequently θCWvalue was calculated for both\nbulk and nanoparticles. For 0.005T field value, the θCWfor bulk, 33 Hrs and\n75h samples are found to be −346.6±2,−295.05±2 and−238.26±2K re-\nspectively. However, for 0.01T field value, it varies as −607.23±2,−295.4±2\nand−236.3±2K. The negative value of θCWindicates the presence of antifer-\nromagnetic correlation interaction. One can see that for both 0.00 5 and 0.01T\nfield,θCWdecreases with decrease in particle size. Now, according to eq. 1,\nwith decreasein θCW, the exchangeinteractionamongmagnetic ionsdecreases\nand this signifies the weakening of TCin our nanoparticle samples. According\nto mean field theory the frustration parameter is defined as f = θCW/TN. For\nbulk, ‘f’ is nearly 20 and for 75h nanoparticle sample it is 23.6. The gene ral\ncondition for the existence of geometrically frustrated magnets, the ground\nstate should be either antiferromagnetic or spin-glass in nature [18 ].\nTo investigate the low temperature behaviour of TS, magnetization mea-\nsurement down to 0.4K at different applied dc magnetic fields were car ried\nout as shown in Fig. 5. With increase in applied dc magnetic field, the pea k\ntemperature ( TS) is found to be shifted to lower temperatures which is a char-\nacteristic feature of a spin-glass. In such a situation, the peak te mperature\n(TS) is identified as the freezing temperature represented as Tfin the ensuing\nanalysis. The small kinks seen in the main panel of Fig. 5, in low tempera ture\nregion (T <6K) are may be due to experimental errors. We fit the observed\nfreezing temperature to the power law relation which is well known as De\nAlmeidaThouless (AT) equation [19], given by2D spin-glass phase in NiCr2O4nanoparticles 7\n/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s52/s53/s48/s46/s48/s48/s57/s48/s48/s46/s48/s49/s51/s53/s48/s46/s48/s48/s48/s48/s48/s48/s46/s48/s48/s48/s50/s57/s48/s46/s48/s48/s48/s53/s56/s48/s46/s48/s48/s48/s56/s55\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s46/s48/s48/s48/s48/s46/s48/s51/s57/s48/s46/s48/s55/s56/s48/s46/s49/s49/s55/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s49/s46/s53/s51/s46/s48/s52/s46/s53\n/s32/s32/s77/s32 /s40/s109\n/s66/s47/s70/s46/s85/s46 /s41/s32/s88/s32/s49/s48/s45/s52\n/s84/s32/s40/s75/s41/s84\n/s83\n/s84\n/s80/s90/s70/s67 /s32\n/s84\n/s67\n/s51/s51/s104/s32/s77/s32 /s40\n/s66/s47/s70/s46/s85/s46 /s41\n/s32/s32\n/s84\n/s83/s84\n/s67/s84\n/s80/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s32\n/s32 /s84/s32/s40/s75/s41/s77/s32 /s40\n/s66/s47/s70/s46/s85/s46 /s41/s32/s88/s32/s49/s48/s45/s52\n/s32/s32\n/s90/s70/s67\n/s84\n/s83\n/s84\n/s67/s84\n/s80/s32\n/s84/s32/s40/s75/s41\n/s32/s55/s53/s104/s32\n/s84\n/s83\n/s84\n/s67/s84\n/s80\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s50/s52/s54/s56\n/s32/s32\n/s84/s32/s40/s75/s41/s77/s32 /s40\n/s66/s47/s70/s46/s85/s46 /s41/s32 /s88/s32/s49/s48/s45/s51\n/s32/s32\n/s90/s70/s67\n/s84\n/s83/s84\n/s67\n/s84\n/s80\n/s32/s32\n/s66/s85/s76/s75/s32\n/s84\n/s83\n/s84\n/s67/s84\n/s80\nFig. 3 Magnetization FC-ZFC measurement of 75h, 33h and bulk NiCr2O4sample in\n0.005T cooling field. Inset shows the ZFC of respective sampl es in 0.005T cooling field. All\nthe three transitions can be clearly seen.\nHAT(Tf)\n∆J∝/parenleftBigg\n1−Tf\nT0\nf/parenrightBigg3/2\n(2)\nandis shown in the inset ofFig. 5. Here, ∆Jis the width ofthe distribution\nof exchange interaction and T0\nfis the freezing temperature at zero magnetic\nfield. The fitting parameter we used is T0\nf. The zero field freezing point was\nobtained by extrapolating the AT line to the temperature axis and is f ound\nto beT0\nf=10.1K as shown in the inset of Fig. 5. The error in determining\nfreezing temperatures is very small (barely visible) and is evaluated by the\ntemperature step size for the measurement of ∼0.2K. It can be seen that\nthe freezing temperature Tf, corresponding to which the magnetization value\nMFC−MZFC=∆Mbecomes non-zero (Fig. 3, 4) indicating the onset of\nfreezing temperature, decreases (low temperature shift) with in crease in ap-\nplied dc magnetic field (H). It shows a Tf∝H2/3dependence, which reflects\n2D surface like spin-glass behaviour. It is worth mentioning here tha t, in bulk\nthe system reflects 3D ferrimagnetic behaviour and in nanoparticle regime\nit reflects a 2D spin-glass like behaviour with Tf= 10.1K. This 2D spin-\nglass like nature is associated with the surface spin character of th eNiCr2O4\nnanoparticles. However, core of the nanoparticle is still ferrimagn etic in na-\nture. In this regard, the temperature dependent response of m agnetization\nfrom bulk to nanoparticle could be more informative (Fig. 3, 4). The w eak-\nening of ferrimagnetic transitions ( TC) and strengthening of low temperature\npeak (TN/Tf) could be seen from bulk to nanoparticle regime with the evolu-\ntion of a new peak ( TP∼90K) towards higher temperature side. Obviously,\nthe surface (shell) spins have dominating role/effect compared to b ulk (core)8 H. Singh\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s48/s46/s48/s49/s50/s48/s46/s48/s49/s54/s48/s46/s48/s50/s48\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s84\n/s67/s84\n/s83\n/s32/s32\n/s84/s32/s40/s75/s41/s77/s32/s40\n/s66/s47/s70/s46/s85/s46/s41\n/s32/s32\n/s90/s70/s67/s32\n/s84/s32/s40/s75/s41/s77/s32 /s40\n/s66 /s47/s70/s46/s85/s46 /s41\n/s32/s32\n/s66/s85/s76/s75/s32\n/s84\n/s83/s84\n/s67/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51/s48/s46/s48/s48/s52/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56\n/s84\n/s80/s84\n/s67/s32\n/s32/s84/s32/s40/s75/s41/s77/s32/s40\n/s66/s47/s70/s46/s85/s46/s41\n/s32/s32\n/s90/s70/s67\n/s84\n/s83\n/s84\n/s80\n/s32/s32 /s32/s32\n/s51/s51/s104/s32\n/s84\n/s83/s84\n/s67/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51/s48/s46/s48/s48/s52/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55\n/s84\n/s67/s32\n/s32/s77/s32/s40\n/s66/s47/s70/s46/s85/s46/s41\n/s84/s32/s40/s75/s41\n/s32/s32\n/s32/s90/s70/s67\n/s84\n/s83\n/s84\n/s80\n/s32/s32 /s32/s55/s53/s104/s32\n/s84\n/s83\n/s84\n/s67/s84\n/s80\nFig. 4FC-ZFC magnetization measurement of 75h, 33h and bulk NiCr2O4sample in 0.1T\ncooling field. Inset shows the ZFC of respective samples. All the three transitions can be\nclearly seen.\nspins in the nanoparticles which leads to the conclusion that ferrimag netically\nordered spins are quite less in number in comparasion to the surface spins.\nThe origin and nature of the TPwill be discussed latter in the letter.\nThe presence of spin-glass like phase in the reported system might b e due\nto superparamagnetism and AT line analysis alone is not sufficient. Hen ce it is\nnecessarytoimplementanotherexperimentalevidencetoestablis htheclaimof\nspin-glassphaseinoursystem.Thus,thefrequencydependenta csusceptibility\nanalysis was done to establish this claim. Fig. 6, shows the real part o f the ac\nsusceptibility ( χ′) as a function of temperature measured at frequencies 7, 73,\nand143Hz.Forthismeasurement,thesamplewasfirstcooleddown to4Kfrom\n300K. Then a probing ac magnetic field of amplitude 1 Oe was applied for the\nac susceptibility measurement. The dc biasing field was set to zero du ring data\naccusation.The freezingtemperature( Tf) isidentified asthe peak in the curve\nand found to be shifted towards high temperature with increase in f requency.\nIt is believed that superparamagnetism (SPM) can also give rise to pe aks in ac\n‘χ’ measurementsaccompaniedby frequencydependent shift in pea k (blocking\ntemperaturein SPM)positions canbe seen,similartothe resultssho wninFig.\n6. The two magnetic phases (SPM and spin-glass) can be distinguishe d by the\nempirical quantity ∆Tf/[Tflog(f)], with values varying from 0.004-0.018 for\nspin-glass to as large as 0.3 for SPM [20]. Here, ∆Tf=Tf−T0\nf, is the shift in\nfreezing temperature for spin-glass phase from T0\nf. For 75h sample, the value\nof the empirical quantity is found to be 0.05 which is close to the spin-g lass\nphase. To understand the dynamical behaviour of the spin glass ph ase near\nfreezing temperature, two different theoretical approaches ha ve been adopted.\nThe first approach assumes that a phase transition takes place at the freezing2D spin-glass phase in NiCr2O4nanoparticles 9\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48 /s50/s50/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48\n/s55/s46/s53 /s56/s46/s48 /s56/s46/s53 /s57/s46/s48 /s57/s46/s53 /s49/s48/s46/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s32/s32/s72/s50/s47/s51\n/s32/s40/s79/s101/s32/s50/s47/s51\n/s41\n/s84\n/s102/s32/s40/s75/s41/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s32/s65/s84/s45/s76/s105/s110/s101/s32/s70/s105/s116\n/s84\n/s102/s48/s77 /s32/s40/s101/s109 /s117/s47/s77 /s111/s108/s101/s41\n/s32/s32\n/s84/s32/s40/s75/s41/s32/s48/s46/s48/s48/s53/s84\n/s32/s48/s46/s48/s49/s84\n/s32/s48/s46/s48/s53/s84\n/s32/s48/s46/s49/s84\nFig. 5ZFC curves for 75h sample, measured at different cooling field of 0.005, 0.01, 0.05,\n0.1T. The inset shows the field field dependence of freezing te mperature and the solid line\nrepresents a fit with De- Almedia Thouless equation.\ntemperature in the vicinity of which a critical behaviour in the temper ature\ndependence ofrelaxationtime ( τ) isexpected. This is knownascriticalslowing\ndown model. In this model, the critical relaxation time, defined as τ= 1/f,\nnear the transition point is related to the correlation length (Tf\nT0\nf−1) of the\nspins and is expressed in the form of a power law as given below [21,22, 23].\nThe freezing temperatures obtained from Fig. 6 were fitted with th e power\nlaw equation,\nτ=τ0/parenleftBigg\nTf\nTf\n0−1/parenrightBigg−zν\n, (3)\nwhere,τ=1/f, f is the frequency of the ac χmeasurement, τ0is the relax-\nation time constant normally lying in the range of 10−9to 10−13sec andzνis\nthe critical exponent. The fitting shown in the inset (a) of Fig. 6, ha s yielded\nparameter values as τ0= 3.6×10−6,T0\nf= 8.66K and zν=11.1. Here, the\nvalue of relaxation time constant τ0= 3.6×10−6is larger at least by a fac-\ntor of 103than the normal value for spin glass phase. The reason for this is\nnot understood. However, the value of T0\nf(= 8.7K) is close to the T0\nfvalue\n10.1K obtained from AT line analysis. The obtained value of critical exp onent\n‘zν’ (11.1K) matches well with the typical value of ‘ zν’ for spin-glass system\nranging from 4 to 12 [20,24], suggesting a spin-glass ground state in NiCr2O4\nnanoparticles.\nThe second approach assumes that the freezing phase transition is a non-\nequilibrium phenomenon and the dynamical properties of spin glass ph ase can\nbe explored by Vogel-Fulcher law. This law takes into account the inte racting\nproperty of spin-glass clusters. The Vogel-Fulcher law is expresse d as,10 H. Singh\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54/s48/s46/s50/s48/s48/s46/s50/s52/s48/s46/s50/s56/s48/s46/s51/s50\n/s45/s53 /s45/s52 /s45/s51 /s45/s50/s49/s50/s46/s48/s49/s50/s46/s50/s49/s50/s46/s52/s49/s50/s46/s54/s49/s50/s46/s56/s49/s51/s46/s48/s69/s120/s112/s46/s32/s100/s97/s116/s97\n/s32/s70/s105/s116\n/s32/s32\n/s32/s32/s40/s98/s41\n/s108/s110/s32/s40/s49/s47/s102/s41/s84\n/s102/s32/s40/s75/s41/s49/s50/s46/s48 /s49/s50/s46/s51 /s49/s50/s46/s54 /s49/s50/s46/s57/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57/s48/s46/s49/s50/s48/s46/s49/s53\n/s32/s32/s40/s115/s101/s99/s46/s41\n/s84\n/s102/s32/s40/s75/s41/s32/s69/s120/s112/s46/s32/s100/s97/s116/s97\n/s32/s70/s105/s116/s40/s97/s41\n/s32/s32/s39/s32/s88/s32/s49/s48/s45/s51\n/s32/s40/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41/s32/s55/s72/s122\n/s32/s55/s51/s72/s122\n/s32/s49/s52/s51/s72/s122\nFig. 6 A plot of AC susceptibility as a function of temperature meas ured at different\nfrequencies of 7, 73 and 143 Hz for 75h sample. The inset (a) an d (b) shows the fitting as\nper eq. 3 and 4.\nτ=τ0exp/bracketleftbiggEa\nkB(Tf−T0)/bracketrightbigg\n, (4)\nhere,Earepresents the energy barrier or activation energy and T0is the\nphenomenological parameter describing interaction between clust ers. The plot\nofTfvs. ln(1/f) is plotted together with the fitted curve given in eq. 4 an d is\nshown in the inset (b) of Fig. 6. The fitted parameters are Ea/kB= 58.9K,\nτ0= 5.22×10−8andT0= 8.03K. The non-negative value of T0= 8.03K\nindicates the presence of interacting spin-glass clusters. Moreov er, forT0= 0,\neq.4reducestoArrheniuslawgivenby τ=τ0exp(Ea/kBTf).Inthissituation,\ntheinter-clusterinteractionisnegligible.Inviewofthisrelation,the fitteddata\ngive unreasonably low value of τ0(∼10−40sec), which seems much faster than\nthe atomic relaxation of spins ( ∼10−14sec) which is physically not possible.\nNow our basic concern is to understand the mechanism responsible f or the\ngeneration of spin glass like phase in NiCr2O4nanoparticle which shows a\ndifferent behaviour as compared to its bulk counterpart. The 2D na ture of\nthe spin glass phase established using AT-line fitting indicates a surfa ce effect.\nThis may be associated with either spin canting or surface spin disord er. Spin\ncanting exists due to two competing magnetic exchanges (one is isot ropic and\nthe other is asymmetric in nature). The marginal weakening of ferr imagnetic\ntransition in 75h sample is an indication of weakening of isotropic excha nge.\nFurther, the bulk NiCr2O4does not show glassy feature in our measurements\nconfirming existing reports to the best of our knowledge. Therefo re, we ruled\nout spin canting as the reason for existing surface effect in NiCr2O4nanopar-\nticle. Actually, surface effects result from the lack of translationa lsymmetry at\nthe boundaries of the particle because of the lower coordination nu mber there\nand the existence of broken magnetic exchange bond which leads to surface2D spin-glass phase in NiCr2O4nanoparticles 11\nspin disorder and frustration [27]. Another important effect is the fi nite size\neffect which is observed in nanoparticle sample. This effect originates from the\ncut off of some characteristic length due to the purely geometric co nstraint on\nfinite volume. It results in superparamagnetic behaviour, which is no t present\nin our sample (already pointed out in earlier discussion). Finally, it is obs erved\nthat nanoparticle sample has higher value of mean field frustration p arame-\nter ‘f’ (23.6) as compare to its bulk sample (f=20). Also, weakening o fTCin\nnanoparticlesampleindicatesabsenceoflongrangeferrimagnetico rderingand\nin fact exhibits spin-glass behaviour. This fulfils the general conditio n for the\nsystem to be geometrically frustrated magnet (GFM). Hence, the NiCr2O4\nnanoparticle sample might be retaining geometrical frustration.\n4 Conclusion\nFrom the entire study, the following points can be concluded: 1. The nanopar-\nticle ofNiCr2O4shows 2D spin glass like character at low temperature with\nTf= 10.1K. Vogel-Fulcher law analysis confirms the existence of interac ting\nspin clusters.\n2. The surface spin disorder seems to be the reason for the existe nce of 2D\nspin glass feature.\n3. The decrease in magnetization value of NiCr2O4nanoparticle compared to\nits bulk counterpart may be due to the surface spin disorder.\n4. The existence of new transition ( TP) may be due to the presence of weak\nferrimagnetic spin clusters.\n5. TheNiCr2O4nanoparticles may be considered as a GFM due to higher\nvalue of ‘f’ parameter, absence of long range ordering and the pre sence of spin\nglass ground state.\n5 Acknowledgement\nWewouldliketothankMinistryofHumanResourceandDevelopment(M HRD),\nGovt. of India, for funding.\nReferences\n1. S. Krupika, and P. Novak, ( Ferromagnetic Materials ), edited by E. P. Wolfarth, Vol 3,\np.189. North-Holand, Amsterdam (1982).\n2. K. Tomiyasu, H. Hiraka, K. Ohoyama, and K. Yamada, Resonan ce-Like Magnetic Exci-\ntations in Spinel Ferrimagnets FeCr2O4andNiCr2O4Observed by Neutron Scattering,\nJ. Phys. Soc. Jpn., 77, 12 (2008).\n3. S. Klemme, and J. C. Miltenburg, Thermodynamic propertie s of nickel chromite\n(NiCr2O4) based on adiabatic calorimetry at low temperatur es, Phys. Chem. Miner.,\n29, 663 (2002).\n4. O. Crottaz, F. Kubel, and H. Schmid, Jumping crystals of th e spinels NiCr2O4and\nCuCr2O4, J. Mater. Chem., 7, 143 (1997).12 H. Singh\n5. N. Mufti, A. A. Nugroho, G. R. Blake, and T. T. M. Palstra, Ma gnetodielectric coupling\nin frustrated spin systems: the spinels MCr2O4(M = Mn, Co and Ni), J. Phys. Cond.\nMat., 22, 075902-07 (2002).\n6. K. Tomiyasu, and I. Kagomiya, Magnetic Structure of NiCr2O4Studied by Neutron\nScattering and Magnetization Measurements, J. Phys. Soc. J pn., 73, 2539-2542 (2004).\n7. J. Tejada, R. F. Ziolo, and X. X. Zhang, Quantum Tunneling o f Magnetization in Nanos-\ntructured Materials, Chem. Mater., 8, 1784 (1996).\n8. J. Tejada, X. X. Zhang, E. del Barco, J. M. Hernandez, and E. M. Chudnovsky, Macro-\nscopic Resonant Tunneling of Magnetization in Ferritin, Ph ys. Rev. Lett., 79, 1754 (1997).\n9. E. F.Kneller,and F.E.Luborsky, Particle Size Dependenc e of Coercivity and Remanence\nof Single Domain Particles, J. Appl. Phys., 34, 656 (1963).\n10. E. Prince, Structure of Nickle Chromite*, J. Appl. Phys. , 32, 68S (1961).\n11. H. Ishibashi, and T. Yasumi, Structural transition of sp inel compound NiCr2O4at\nferrimagnetic transition temperature, J. Magn. Magn. Mate r., 310, e610-e612 (2007).\n12. J. M. D. Coey, Noncollinear Spin Arrangement in Ultrafine Ferrimagnetic Crystallites,\nPhys. Rev. Lett., 27, 1140 (1971).\n13. A. H. Morrish, K. Hanada, and P. J. Schurer, J. Phys. (Pari s), Colloq. 37, C6 (1976).\n14. F. T. Parker, M. W. Foster, D. T. Margulies, and A. E. Berko witz, Spin canting, surface\nmagnetization, and finite-size effects in γ−Fe2O3particles, Phys. Rev. B, 47, 7885 (1993).\n15. H. Ueda, H. Mitamura, T. Goto, and Y. Ueda, Successive fiel d-induced transitions in a\nfrustrated antiferromagnet HgCr2O4, Phys. Rev. B., 73, 094415 (2006).\n16. X. H. Chen, H. T. Zhang, C. H. Wang, X. G. Luo, and P. H. Li, Eff ect of particle size\non magnetic properties of zinc chromite synthesized by solg el method, Appl. Phys. Lett.,\n81, 4419 (2009).\n17. R. H. Kodama, A. E. Berkowitz, E. J. McNiff, Jr., and S. Fone r, Surface Spin Disorder\ninNiFe2O4Nanoparticles, Phys. Rev. Lett., 77, 394 (1996).\n18. K. H. J. Buschow, Handbook of magnetic material , edited by K. H. J. Buschow, p-423.,\nVol. 13, North Holland, Amsterdam (2001).\n19. J. R. L. Almeida, and D. J. Thouless, Stability of the Sher rington-Kirkpatrick solution\nof a spin glass model, J. Phys. A, 11, 983 (1978).\n20. J. A. Mydosh, Spin glasses: An Experimental Introductio n, Taylor and Francis, London\n(1993).\n21. J. Souletie and J. L. Tholence, Critical slowing down in s pin glasses and other glasses:\nFulcher versus power law, Phys. Rev. B, 32, 516(R) (1985).\n22. E. A. Edwards and P. W. Anderson, Theory of spin glasses, J . Phys. F: Met. Phys., 5,\n965 (1975).\n23. M. D. Mukadam, S. M. Yusuf, P. Sharma, S. K. Kulshreshtha, and G. K. Dey, Dynamics\nof spin clusters in amorphous Fe2O3, Phys. Rev. B, 72, 174408 (2005).\n24. S. Bedanta, and W. Kleemann, Supermagnetism, J. Phys. D: Appl. Phys., 42, 013001\n(2009).\n25. D. N. H. Nam, R. Mathieu, P. Nordblad, N. V. Khiem, and N. X. Phuc, Spin-glass\ndynamics of La0.95Sr0.05CoO3, Phys. Rev. B, 62, 8989 (2000).\n26. K.Gunnarsson,P.Svedlindh, P.Nordblad,L.Lundgren,H .Aruga,andA.Ito, Dynamics\nof an Ising Spin-Glass in the Vicinity of the Spin-Glass Temp erature, Phys. Rev. Lett.,\n61, 754 (1988).\n27. X. Batlle, and A. Labarta, Finite-size effects in fine part icles: magnetic and transport\nproperties, J. Phys. D: Appl. Phys., 35, R15-R42 (2002)."    },    {        "title": "1911.02207v3.Enhancement_of_domain_wall_mobility_detected_by_NMR_at_the_angular_momentum_compensation_temperature.pdf",        "content": "arXiv:1911.02207v3  [cond-mat.mtrl-sci]  3 Jul 2020Enhancement of domain-wall mobility detected by NMR at the a ngular momentum\ncompensation temperature\nMasaki Imai,1Hiroyuki Chudo,1Mamoru Matsuo,1,2,3Sadamichi Maekawa,1,2,3and Eiji Saitoh1,4,5,6\n1Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan\n2Riken Center for Emergent Matter Science (CEMS), Wako 351-0 198, Japan\n3Kavli Institute for Theoretical Sciences, University of Ch inese\nAcademy of Sciences,19 Yuquan Road, Beijing 100049, P.R.Ch ina\n4Advanced Institute for Materials Research, Tohoku Univers ity, Sendai 980-8577, Japan\n5Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n6Department of Applied Physics, The University of Tokyo, Hon go, Bunkyo-ku, Tokyo, 113-8656, Japan\n(Dated: July 6, 2020)\nThe angular momentum compensation temperature TAof ferrimagnets has attracted much at-\ntention because of high-speed magnetic dynamics near TA. We show that NMR can be used to\ninvestigate domain wall dynamics near TAin ferrimagnets. We performed57Fe-NMR measurements\non the ferrimagnet Ho 3Fe5O12withTA= 245 K. In a multi-domain state, the NMR signal is en-\nhanced by domain wall motion. We found that the NMR signal enh ancement shows a maximum at\nTAin the multi-domain state. The NMR signal enhancement occur s due to increasing domain-wall\nmobility toward TA. We develop the NMR signal enhancement model involves domai n-wall mobility.\nOur study shows that NMR in multi-domain state is a powerful t ool to determine TA, even from\na powder sample and it expands the possibility of searching f or angular momentum-compensated\nmaterials.\nI. INTRODUCTION\nTheangularmomentumcompensationinferrimagnets,\nwhere angular momenta on different sublattices cancel\neach other out, has attracted much attention because of\nits unique character[1–6]. In terms ofan angularmomen-\ntum, ferrimagnets at an angular momentum compensa-\ntion temperature, TA, can be regarded as antiferromag-\nnets, even though they have spontaneous magnetization.\nMagnetic dynamics in ferrimagnets at TAis also anti-\nferromagnetic and much faster than in ferromagnets. In\nferrimagnetic resonance (FMR), for example, the Gilbert\ndamping constant was predicted to be divergent at TA\n[7]. The resonance frequency of the uniform mode aris-\ning from a ferromagnetic character increases and merges\nwith that of an exchange mode arising from an antiferro-\nmagnetic character at TA[1, 8], and the Gilbert damping\nparameter estimated from the linewidth of the uniform\nmode shows an anomaly near TA[1]. Due to this fast\nmagnetic dynamics, the high-speed magnetization rever-\nsal wasrealized in the amorphousferrimagnet ofGdFeCo\nalloy atTA[1].\nMoreover, in GdFeCo alloy, the domain wall mobility\nis enhanced at TA[4]. Domain wall motion occurs due\nto the reorientation of magnetic moments. Angular mo-\nmentum accompanied by a magnetic moment prevents\nthe magnetic moment from changing its direction due to\nthe inertia of the angular momentum, and the domain-\nwall mobility is suppressed. At TA, however, magnetic\nmoments can easily change their direction because of the\nlack of inertia. As a result, the domain-wall mobility is\nenhanced. Thus, the angular momentum compensation\nof ferrimagnets may be useful for next-generation high-\nspeed magnetic memories, such as racetrackmemories[9].\nThe rare-earthiron garnet R3Fe5O12(RIG, where Risarare-earthelement) isaferrimagnetaccompaniedby TA\n[8, 10, 11]. However, RIG does not show any anomaly in\nFMR at TAbecause the angular momentum of R3+ions\nweaklycoupleswiththatofFe3+ionsandbehavesalmost\nas a free magnetic moment. As a result, the magnetic re-\nlaxationfrequencyofthemagneticmomentof R3+ionsis\nmuch higher than that of the magnetic moment of Fe3+\nions or the exchange frequency between R3+and Fe3+\nions [12–15]. In this case, the magnetic moment of R3+\nions adiabaticallyfollows the motion of the magnetic mo-\nment of Fe3+ions. Hence, R3+ions contribute to the\nmagnetization but not to the angular momentum due to\nheavy damping of the R3+site [15].\nAlthough TAinRIGcannotbedeterminedusingFMR,\nthe mobility of the bubble domains formed in the epitax-\nial thin film of the substituted RIG increases at a certain\ntemperature, which is regarded as TA[16]. Furthermore,\nrecently, it has become possible to directly and exactly\nmeasurethenet angularmomentumregardlessofthema-\nterial and its shape by using the Barnett effect, in which\nmagnetization is induced by mechanical rotation due to\nspin–rotation coupling, HSR=−J·Ω, whereJandΩ\nare the angular momentum of an electron and the an-\ngular velocity of the rotation [5, 17]. When a sample is\nrotated, angular momenta of electrons in a magnetic ma-\nterial align along with the rotational axis, and, then, the\nmaterial is magnetized without any external magnetic\nfields. In this method, TAis determined as the tem-\nperature where magnetization induced by the mechani-\ncal rotation vanishes because of the disappearance of the\nnet angular momentum. Consequently, TAof Ho3Fe5O12\n(HoIG) was determined to be 245 K [5]. With the focus\non magnetic dynamics at TA, a microscopic method was\nrequired to investigate the spin dynamics at TAregard-\nless of materials and their shape.2\nFIG. 1. Schematic illustration of enhancement of the NMR\nsignal in a domain wall. (a) An input RF magnetic field H1\ncauses the domain wall to move. The electron spins in the\ndomain wall rotate, exciting the nuclear resonance through\nthe hyperfine coupling. As a result, H1appears to be ηin\ntimes for the nuclear spins. (b) The domain wall moves in\naccordance with the precession of nuclear spins, and the bul k\nmagnetization oscillates with NMR frequency. The NMR sig-\nnal becomes ηouttimes.\nHere, we propose an NMR method to explore the spin\ndynamics at TA. In a magnetic ordered state such as in\nferromagnets and ferrimagnets, an NMR signal can be\nobserved without any external magnetic field due to an\ninternal field, which enables us to observe domain walls\nat zero or low magnetic fields. Furthermore, the macro-\nscopic magnetization of electrons enhances the NMR sig-\nnal via hyperfine interactions. Particularly, the NMR\nsignal from nuclei in domain walls is strongly enhanced\ndue to the magnetic domain wall motion, as shown in\nFig. 1. An input radio frequency (RF) magnetic field\nH1used for NMR can move domain walls, thereby rotat-\ning magnetic moments in the walls and generating the\ntransverse component of a hyperfine field in synchroniza-\ntion with the RF field. As a result, H1is enhanced to\nbecomeηinH1, whereηinis the enhancement factor for\nthe input process. In the reverse process, the Larmor\nprecession of nuclear spins causes domain wall motion,\nbecause the electronic system feels an effective magnetic\nfieldHefffrom the nuclearmagnetizationthroughthe hy-\nperfine interaction, which leads to the oscillation of the\nbulk magnetization; thus, a much stronger voltage is in-\nduced in the NMR pickup coil than the precession of nu-\nclear magnetic moment mn, and the output NMR signal\nis enhanced to be ηoutmn, whereηoutis the enhancement\nfactor for the output process. This enhancement effect\nenables us to selectively observe the NMR signal from\nnuclei in domain walls, even though the volume fraction\nof domain walls is much smaller than that of domains.\nIn this paper, we report results of an NMR study of\nHoIG under magnetic fields of up to 1.0 T. For a multi-\ndomain state below 0.3 T, the temperature dependenceof the NMR intensity shows a maximum at TA. On the\nother hand, for a single-domain state above 0.5 T, the\ntemperature dependence of the NMR intensity does not\nshow any anomalies at TA. These results indicate the en-\nhancement of the domain wall mobility at TA. Extend-\ning a simple conventional model for describing ηout[18],\nwe formulated the modified enhancement factor η′\noutby\ntaking the domain wall mobility into account. This en-\nhancement of the NMR intensity at TAenables us to\nestimate the domain wall mobility to determine TA, even\nin a powder sample.\nII. EXPERIMENTAL METHOD\nWe synthesized HoIG by solid-state reaction for this\nstudy[5, 17]. We ground the sample in a mortar to cre-\nate a fine powder with a typical particle diameter of 5\nµm. The sample was packed in the NMR coil, which was\nperpendicular to the external magnetic field. The NMR\nmeasurements of57Fe nuclei at the dsite in Ho 3Fe5O12\nwere carried out using a standard phase-coherent pulsed\nspectrometer. The NMR signals were obtained using the\nspin-echo method, with the first and second pulse du-\nrations of 1.0 and 2.0 µs, respectively. During the mea-\nsurements, the pulse width waskeptconstantand the RF\npowerwasvariedtomaximizetheNMR signal. Thespin-\necho decay time T2was measured by varying the interval\ntimeτbetween the first and second pulses. The value of\nT2is defined such that I(2τ) =I(0)exp(−2τ/T2), where\nI(2τ) andI(0) are the NMR intensity at 2 τandτ= 0,\nrespectively. The nuclear spin-lattice relaxation time T1\nwas measured using the inversion recovery method.\nIII. RESULTS\nFigure 2(a) shows the temperature variation in the\nNMR spectra of57Fe at the dsite without external fields.\nEach NMR spectrum shows a single peak, and the peak\nshifts to higher frequencies with decreasing temperature.\nThe NMR intensity shows the maximum at 245 K. The\ntop panel of Fig. 2(b) shows integrated NMR intensi-\nties. Generally, the NMR intensities need to be cali-\nbrated when comparing them under different conditions.\nThe NMR intensity Iis proportional to the voltage in-\nduced in a pickup NMR coil by the precession of the\nnuclear magnetization mn. Thus, Iis proportional to\ndmn(t)/dt. Because mn(t) rotates at the Larmor fre-\nquencyν,Iis proportional to νmn. The size of mn\ndepends on the polarization of the nuclear spin derived\nfrom the Boltzmann distribution function. Thus mnis\nproportional to ν/T, whereTis the temperature. As a\nresult,Iis proportional to ν2/T. Moreover, the NMR\nintensity measured by the spin echo method depends on\nT2. Therefore, we calibrated the NMR intensity by mul-\ntiplyingTν−2exp(2τ/T2).\nThe calibrated NMR intensity is retained to show a3\nFIG. 2. The57Fe NMR results for the dsite and the magnetic properties in Ho 3Fe5O12. (a) Temperature dependence of the\nNMR spectra. (b) In the upper panel, the red open and filled cir cles show the bare integrated signal intensity and calibrat ed\nintensity by multiplying by Tν−2exp(2τ/T2), respectively. In the bottom panel, the blue cross shows MΩobtained by the\nBarnett effect. The blue curve is a guide to the eye. The orange curve shows the magnetization obtained under the magnetic\nfieldof 1000 [Oe]. Inbothpanels, theblack solid anddashed l ines showthemagnetization and angular momentumcompensat ion\ntemperatures of Ho 3Fe5O12, respectively. (c) Temperature dependence of resonance fr equency (top), 1 /T1, and 1/T2(bottom).\nmaximum at 245 K, which coincides with TAdetermined\nby the Barnett effect in which mechanical rotation in-\nduces magnetization MΩdue to spin–rotation coupling\n[5]. The blue crossin the bottom panel ofFig. 2(b) shows\nthe temperature dependence of MΩunder a rotation of\n1500 Hz without any external magnetic field. MΩbe-\ncomes zero at two temperatures: The lower temperature\ncoincides with the magnetization compensation temper-\natureTMdetermined by a conventional magnetization\nmeasurement asshownbytheorangecurveinthebottom\npanel of Fig. 2(b). AtTM, spin–rotationcoupling is effec-\ntive, but MΩbecomes zero due to the disappearance of\nbulk magnetization. In contrast, the higher temperature\ncan be assigned to TA, where the bulk magnetization re-\nmains but the spin–rotation coupling is not effective due\nto the disappearance of the net angular momentum [5].\nUnlike the temperature dependence of the NMR inten-\nsity, there are no anomalies in the temperature depen-\ndence of ν, 1/T1, and 1/T2as shown in Fig. 2(c). These\nresults indicate that the maximum NMR intensity can\nbe attributed to an anomaly in the enhancement factor.\nTo perform NMR experiments for the single-domain\nstate, we characterized the magnetic field dependence of\nHoIG as shown in Fig. 3. The top panel of Fig. 3 shows\nthe NMR frequency in magnetic fields ranging from 0\nto 1 T at 300 K. With the increase in the magnetic\nfield the resonance frequency decreases because the mag-\nnetic moment at the dsite aligns with the magnetic field\naboveTM, and the hyperfine coupling constant is nega-\ntive. The line in the top panel of Fig. 3 shows a slope of\n−57γ=−1.3757 MHz/T. In the multi-domain state at\nFIG. 3. The NMR results in the magnetic fields ranging from\n0 to 1 T at 300 K. The top panel shows the field dependence\nof the resonance frequency. The solid line shows the slope of\nthe gyromagnetic ratio of a57Fe nucleus. The bottom panel\nshows the field dependence of optimized RF power.4\nFIG. 4. (a) The temperature dependence of the NMR in-\ntensity in the magnetic fields ranging from 0 to 1 T. The\nNMR intensity of 0 and 0.3 T is calibrated by multiplying\nbyTν−2exp(2τ/T2). The NMR intensity of 0.5 and 1 T is\ncalibrated by multiplying by Tν−3exp(2τ/T2). The solid and\ndashed lines show the magnetization and angular momentum\ncompensation temperatures, respectively. (b)Schematic i llus-\ntration of the domain wall motion induced by an effective RF\nmagnetic field Heffthrough the hyperfine coupling. Here, R,\ndandxare a particle radius, a domain wall thickness, and do-\nmain wall displacement, respectively. The domain wall move s\nl= 4xin one cycle of t= 1/ν.\nlow fields, the rate of decrease in the NMR frequency by\napplying external field is smaller than −57γuntil all the\ndomain walls disappear because the external field at nu-\nclear positions is canceled out by the demagnetizing field\ncaused by domain wall displacement due to the external\nmagnetic field[19]. In the single-domain state above 0.6\nT, the NMR frequency decreases with the ratio of57γby\nthe magnetic field.\nThe optimized RF input power is shown in the bottom\npanel of Fig. 3. At low magnetic fields, the RF input\npower is small due to the large ηin, suggesting that the\nNMR signal from the domain walls, which is more en-\nhanced than that from domains, dominates the NMR in-\ntensity. The input power sharply increases in the region\nbetween 0.4 and 0.5 T and saturates above 0.6 T. This\nresult indicates that the domain structure changes from\nmulti-domain to single domain between 0.4 and 0.5 T.\nThis is consistent with the result of the field dependence\noftheNMR frequency. Athighmagneticfields, the NMR\nsignal from the domain dominates the NMR intensity.\nFigure 4(a) shows the temperature dependence of the\ncalibrated NMR intensity in various magnetic fields. In\nthe multi-domain state at 0 and 0.3 T, the NMR in-\ntensity shows a maximum at 245 K and then decreases\ntowardTM. On the other hand, the calibrated values ofthe NMR intensity at 0.5 and 1.0 T do not show any\nanomalies around TA. These results indicate that the\nmaximum NMR intensity is attributed to the domain\nwalls. The drop in the NMR intensity at various mag-\nnetic fields around TMresults from the decrease in sig-\nnal enhancement, which is proportional to the magneti-\nzation. Notably, in the ferromagnetic or ferrimagnetic\nstate, the enhancement factor ηoutis proportional to the\nhyperfinefield Hn, whichisalsoproportionaltotheNMR\nfrequency ν[18, 19]. Therefore, the NMR intensity in the\nsingle-domain state above 0.5 T is calibrated by multi-\nplying by Tν−3exp(2τ/T2). In the multi-domain state of\nthissamplebelow0.3T,however,theenhancementfactor\ndoes not depend on νso that the NMR intensity below\n0.3 T is calibrated by multiplying by Tν−2exp(2τ/T2).\nThe temperature at which the NMR intensity shows a\nmaximum at 0.3 T decreaseslightly. It is speculated that\nTAdecreases under magnetic fields in RIG, because the\nexpectation value of the angular momentum of R3+de-\ncreases above TMin a magnetic field due to the decrease\nin molecular field at the Rsite [17].\nIV. DISCUSSION\nFirst, we introduce the conventional model describ-\ning NMR enhancement due to domain-wall motion [18].\nIn this model, the domain wall displacement xis lim-\nited by a demagnetizing field Hd(x). The maximum dis-\nplacement xmaxis determined from a position in which\nHd(xmax) is balanced with an oscillating effective field\nHeff, which is created by the precession of the nu-\nclear magnetic moment through the hyperfine interac-\ntion,Hd(xmax) =Heff. Because the sample used for\nthe NMR measurement in the present study is a pow-\nder, each particle in it is assumed to be spherical with\nradiusR, as shown in Fig. 4(b). Hdis expressed as\nHd(xmax) = 2πxmax\nRm. The net electron magnetic mo-\nmentmis tilted by the effective field of Heff=mn\nmHn,\nwhereHnis the hyperfine field, and the tilt angle θofm\ncan be described by θ=πx/d, wheredis the domain-\nwall thickness. Then, the bulk magnetization induced by\nnuclear magnetization is expressed as\nRHnmn\n2dm=ηoutmn, (1)\nwhereηoutis the enhancement factor of the NMR signal\nfor the output process and is defined as ηout=RHn\n2dm.\nThis model assumes that the velocity of the domain-wall\nmotionvis fast enough to move 4 xmaxduring one cycle\nof oscillating effective field, i.e. v=µHeff>4xmaxν,\nwhereµis the domain wall mobility.\nThe conventional derivation of NMR enhancement in-\nduced by domain wall motion does not include the mobil-\nity of the domain-wall. Herein, we consider that vis not\nfast enough to follow the oscillating effective field, i.e.,\nv <4xmaxν. In this case, the displacement xis limited\nbyµ. Then,xis expressed to be v/4ν. The enhancement5\nfactorηoutis modified such that\nη′\nout=v\n4xmaxνηout=π\n4dγµ. (2)\nThis formula indicates that η′\noutin the slow limit of do-\nmain wall motion is proportional to µ, andη′\noutin the\nfast limit of domain-wall motion is continually connected\nto the conventional ηout. It is noted that η′\noutdoes not\ndepend on the NMR frequency ν.\nThe domain-wall mobility of HoIG has not been re-\nported, but it can be estimated from the reported damp-\ning parameters [20, 21]. The domain wall mobility of\nGd3Fe5O12(GdIG) is 225 m ·sec−1Oe−1at 298 K [20].\nThe magnitude of the damping is inversely proportional\nto the domain wall mobility because the damping pa-\nrameter of HoIG is 80 times as great as that of GdIG\n[21], the domain wall mobility of HoIG at room tempera-\nture is estimated to be 2.8 m ·sec−1Oe−1. However, the\ndomain-wall mobility required for motion xmaxis defined\nsuch that 4 xmaxν/Heff= 2Rν/πM, which is estimatedto be 4 m ·sec−1Oe−1for 4πM∼500 G,R∼5µm, and\nν∼50 MHz. Thus, this evaluation indicates that, in\nHoIG, the displacement of domain walls induced by nu-\nclear precession is limited by µ. Therefore, in the multi-\ndomainstateinHoIG,weusedthemodifiedenhancement\nfactorη′\noutin Eq. (2). We estimate the value of µatTA\nto be 3.5 m ·sec−1Oe−1usingµ= 2.8 m·sec−1Oe−1at\n300 K. When we assume dto be 0.1–1.0 µm,ηoutis es-\ntimated to be 102–103, which is comparable to typical\nenhancement factors [18, 22]. Thus, the NMR method\nis very sensitive to detect such a small enhancements at\nTA.\nACKNOWLEDGMENTS\nWethankH. Yasuokaforfruitful discussion. This work\nwas supported by JST ERATO Grant Number JPM-\nJER1402, JSPS Grant-in-Aid for Scientific Research on\nInnovative Areas Grant Number JP26103005, and JSPS\nKAKENHI Grant Numbers JP16H04023, JP17H02927.\nM. I and H. C contributed equally to this work.\n[1] C. D. Stanciu, A. V. Kimel, F. Hansteen,\nA. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Ras-\ning, Phys. Rev. B 73, 220402 (2006).\n[2] C. D. Stanciu, A. Tsukamoto, A. V. Kimel,\nF. Hansteen, A. Kirilyuk, A. Itoh, and T. Rasing,\nPhys. Rev. Lett. 99, 217204 (2007).\n[3] M. Binder, A. Weber, O. Mosendz, G. Woltersdorf,\nM. Izquierdo, I. Neudecker, J. R. Dahn, T. D. Hatchard,\nJ.-U. Thiele, C. H. Back, and M. R. Scheinfein,\nPhys. Rev. B 74, 134404 (2006).\n[4] K.-J. Kim, S. K. Kim, Y. Hirata, S.-H. Oh, T. Tono, D.-\nH. Kim, T. Okuno, W. S. Ham, S. Kim, G. Go, et al.,\nNature materials 16, 1187 (2017).\n[5] M. Imai, Y. Ogata, H. Chudo, M. Ono, K. Harii,\nM. Matsuo, Y. Ohnuma, S. Maekawa, and\nE. Saitoh, Appl. Phys. Lett. 113, 052402 (2018),\nhttps://doi.org/10.1063/1.5041464.\n[6] Z. Zhu, X. Fong, and G. Liang,\nPhys. Rev. B 97, 184410 (2018).\n[7] R. K. Wangsness, Phys. Rev. 91, 1085 (1953).\n[8] T. R. McGuire, Phys. Rev. 97, 831 (1955).\n[9] S. S. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n[10] R. C. LeCraw, J. P. Remeika, and\nH. Matthews, J. Appl. Phys. 36, 901 (1965),\nhttps://doi.org/10.1063/1.1714259.\n[11] C. Borghese, R. Cosmi, P. De Gasperis, and R. Tappa,\nPhys. Rev. B 21, 183 (1980).\n[12] G. P. Rodrigue, H. Meyer, and R. V.Jones, J. Appl. Phys. 31, S376 (1960),\nhttps://doi.org/10.1063/1.1984756.\n[13] N. Ohta, T. Ikeda, F. Ishida, and\nY. Sugita, J. Phys. Soc. Jpn. 43, 705 (1977),\nhttps://doi.org/10.1143/JPSJ.43.705.\n[14] C. M. Srivastava, B. Uma Mahesh-\nwar Rao, and N. S. Hanumantha Rao,\nBulletin of Materials Science 7, 237 (1985).\n[15] C. Kittel, Phys. Rev. 115, 1587 (1959).\n[16] V. V. Randoshkin, V. A. Polezhaev, N. N. Sysoev, and\nY. N. Sazhin, Physics of the Solid State 45, 513 (2003).\n[17] M. Imai, H. Chudo, M. Ono, K. Harii,\nM. Matsuo, Y. Ohnuma, S. Maekawa, and\nE. Saitoh, Appl. Phys. Lett. 114, 162402 (2019),\nhttps://doi.org/10.1063/1.5095166.\n[18] A. M. Portis and A. C. Gos-\nsard, J. Appl. Phys. 31, S205 (1960),\nhttps://doi.org/10.1063/1.1984666.\n[19] H. Yasuoka, J. Phys. Soc. Jpn. 19, 1182 (1964),\nhttps://doi.org/10.1143/JPSJ.19.1182.\n[20] G. Vella-Coleiro, D. Smith, and L. Van Uitert,\nIEEE Transactions on Magnetics 7, 745 (1971).\n[21] G. VellaColeiro, D. Smith, and\nL. Van Uitert, Appl. Phys. Lett. 21, 36 (1972),\nhttps://doi.org/10.1063/1.1654209.\n[22] J. Dho, M. Kim, S. Lee, and W.-\nJ. Lee, J. Appl. Phys. 81, 1362 (1997),\nhttps://doi.org/10.1063/1.363872."    },    {        "title": "1208.4615v1.Gap_Generation_in_Topological_Insulator_Surface_States_by_non_Ferromagnetic_Magnets.pdf",        "content": "arXiv:1208.4615v1  [cond-mat.mtrl-sci]  22 Aug 2012Gap Generation in Topological Insulator Surface States by n on-Ferromagnetic\nMagnets\nL´ aszl´ o Oroszl´ any1and Alberto Cortijo2\n1Department of Physics of Complex Systems, Eotvos Universit y,\nH-1117 Budapest, P´ azm´ any P´ eter s´ et´ any 1/A, Hungary\n2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantobl anco, 28049 Madrid, Spain.\nIt is shown that, contrary to the naive expectation, single p article spectral gaps can be opened\non the surface states of three dimensional topological insu lators by using commensurate out- and\nin-plane antiferromagnetic or ferrimagnetic insulating t hin films.\nPACS numbers: 75.70-i, 75.30.GW, 73.20.-r, 85.75.-d\nIntroduction. One of the most remarkable proper-\nties of a three dimensional topological insulator is the\npresence of a topologically quantized magnetoelectric\nterm (TMET) in its electromagnetic response. This\nterm has far reaching consequences since it constitutes a\ncondensed-matter realization of axion electrodynamics[1,\n2]. Experimental signatures of the TMET include the\nquantized Kerr angle and Faraday rotation[3–5], Casimir\nrepulsion[6], inverse spin galvanic effect[7], monopole\nimages[8], surface half integer Hall effect[9], topological\nviscoelastic response[10] just to name a few.\nThe keypoint for the observabilityofthis topologically\nquantized response is the breakdown of the time reversal\nsymmetry in the surface of the otherwise time reversal\ninvariant TI[9]. In terms of the electric and magnetic\nfields, the TMET in the electromagnetic action takes the\nform\nSθ=α\n4π2ˆ\nd3rdtθE·B, (1)\nwhereαis the fine structure constant and θis the so\ncalled axion parameter which takes the value of 0 or\n(2n+1)πwithn∈Nin trivial and topological insulators,\nrespectively[9]. Alternatively to the above description in\nterms of the electromagnetic fields, one can understand\nthe TMET as a Chern Simons (CS) term induced in the\nelectromagnetic response of the insulator by the gapped\nsurface states of a TI that are described by usual massive\nDirac Hamiltonian:\nHD=v(σ×k)·ˆz+mσz, (2)\nwherevis the Fermi velocity, and mis the induced mass\nof the Dirac states. In this case, the value θ=πcorre-\nsponds to the value σ=1\n2sign(m) for the Hall coefficient\nin the corresponding CS term. In short, breaking time\nreversal symmetry opens a gap in the TI surface states,\nthus making the TMET observable.\nWithin the effective low energy approximation de-\nscribed by (2) there are several proposals in the liter-\nature for opening a gap in the helical metal by means\nof weak magnetic fields through a Zeeman term HZ=\nFIG. 1: (Color online). Low energy spectrum of the surface\nstates of a TI in homogeneous in-plane magnetization. Red\narrows show the direction of the magnetization field.\ngµBσ·B[11], or through exchange coupling to ferromag-\nnetic thin films Hexc=JM·σ[9], and magnetic im-\npuritiesHimp=J/summationtext\njSj·σδ(r−Rj)[12–14]. The ex-\nchange coupling between magnetic thin films and TI is\nthe most appealing from the theoretical point of view\nbecause it not only gives a simple mechanism to develop\nthe theory of the TMET but it allow us to look for unex-\npected effects that can alter the thin film magnetization\ndynamics[7, 15]. However, this proposal is experimen-\ntally challenging, and also it poses some questions. First\nof all, it is not so easy to find insulating ferromagnetic\nmaterials. Some candidate materials like GdN and EuO\nhave been theoretically suggested[3, 7] but to the best of\nour knowledge so far there is no experimental evidence\nsupporting this claim. Also, even if ferromagnetic in-\nsulating thin films were available, it is not guaranteed\nthat the thin film magnetization would point in the out-\nof-plane direction[16]. There is the problem of possible\nmismatch between the TI surface lattice structure and\nthe thin film lattice structure and even the issue of the\ntwo lattices not being commensurate. These problems\nare in the heart of the experimental difficulties for imple-\nmenting this mechanism.\nDirectly using eq.(2) implicitly forces us to consider\na continuum medium approach for the magnetization\n[17].The question is then how to construct this effective2\n00.40.81.21.600.20.40.60.8mFerro\nAntiferro\nFerri\n00.010.020.030.040.05\n00.20.40.60.811.21.41.6\n-0.200.20.40.60.81\nSeff0.9\n0.5\n0.10\n-0.1-0.18\na) b) c)\nFIG. 2: a): Exchange induced gap m(in units of B11) on the TI surface states vs. the lattice mismatch δy. Blue (solid),\nred (dashed), and green (dashed-dotted) lines corresponds to ferromagnetic, antiferromagnetic, and ferrimagnetic l attices,\nrespectively For the ferrimagnetic configuration S1=−5S2andβ= 0.02a2. b): The effective Zeeman field on the surface as\nthe function of the relative displacement δybetween the magnetic and the TI lattices and the localizatio n parameter β. c):\nReal space configurations for the TI surface. Black dots repr esent lattice Se positions and the arrows correspond to a hex agonal\nantiferromagnetic lattice.\ndescription of the exchange coupling between the sur-\nface electronic spin and the magnetization starting from\na microscopic model. For ideal insulating ferromagnets\nthe most naive way would be to couple the electron spin\nwith the averaged magnetization in the magnetic unit\ncell. However when more realistic magnetic insulators\nare considered we immediately run into difficulties. For\ninstance this approach automatically rules out the possi-\nbility of considering antiferromagnetic insulators as mag-\nnetic material candidates. Also it is not at all clear what\nthe correct form for a continuum description of the mag-\nnetization of a ferrimagnetic insulator is. Motivated by\nthese experimental and theoretical issues, we address in\nthis work the problem of coupling a magnetically active\nthin film with the surface electronic states of a TI em-\nploying an tight binding approach.\nThe model. In order to study qualitatively the ways\nin which the gap can be microscopically induced in the\nsurface spectrum we will employ a tight binding model\nvalid for the topological insulators of the form Bi 2X3\nwhich include the prototypical examples of TI’s Bi 2Se3\nand Bi 2Te3. We will then follow references [18, 19] and\nwill consider a Bi 2Se3sample made of Nquintuple lay-\ners (QL) grown in the (111) direction and terminated\nin Se planes. The surface will thus have a triangular\nlattice structure. We are interested in the bandstruc-\nture around the Fermi level so the tight-binding basis\nset will be made of linear combinations of the atomic\norbitals/parenleftbig\n|p+\nBi,↑/angbracketright,|p−\nSe,↑/angbracketright,|p+\nBi,↓/angbracketright,|p−\nSe,↓/angbracketright/parenrightbig\n. The super-\nscript reflect the parity of the state and the second index\nis the spin polarization. The tight binding Hamiltonian\nin real space can be written in this basis in the following\nway[18, 19]:\nH=/summationdisplay\nnC†\nnˆǫCn+/summationdisplay\nn,ai/biC†\nnˆtai/biCn+ai/bi+H.c.(3)Here the lattice vectors aiandbiconnect unit cell posi-\ntions within the same QL and of different QL, respec-\ntively and nlabels the lattice positions as defined in\n[18, 19]. We use a=|ai|as the lateral spatial length\nscale. The on-site energy ˆ ǫand hopping terms ˆtai/biare\n4×4matrices which can be written aslinear combination\nof Γimatrices which are matrix products of spin σand\nparityτPauli matrices:\nˆǫ=ǫ0Γ0+mΓ5,\nˆta1=A0Γ0−i(A12Γ3−A14Γ2)+A11Γ5,\nˆtb1=B0Γ0+i(B12Γ4−B14Γ1)+B11Γ5,(4)\nΓ1,2=τ1⊗σ1,2,Γ3=τ1⊗σ3,\nΓ4,5=τ2,3⊗σ0,Γ0=τ0⊗σ0.\nThe remaining hopping matrices ˆta2,3/b2,3can be ob-\ntained from (4) by applying the rotation operation R3=\nexp(iπ\n3σ3⊗τ0). The Hamiltonian (3) is thus made of\nintra-QL hopping terms and on-site energies and hop-\nping terms coupling different QL. In all calculations pre-\nsented here we use B11= 1,A14= 1.4,A12=B12= 3,\nA11= 2,m=−10,B14=A0=B0= 0, for modelling\na bulk TI[19].Next we add a exchange term coupling to\nthe Se atomic orbitals in (3) of the first QL of the form\nHexc=J/summationdisplay\nnS(Rn)C†\nnΣCn. (5)\nwhere The matrices Σare of the form Σ=\n1\n2(τ0−τ3)⊗σand now Rnrepresent the lattice posi-\ntions of the Se atoms in the outer part of the first QL.\nThe important observation here is that the magnetic and\nsurface lattices do not need to be the same for generic\nmagnetic layers so the magnetic moment of the magnetic\nlayerS(Rn) atRnwill not be the magnetic moment of3\neach magnetic position. Usually the magnetic moments\nrepresent the magnetic moment associated to a bounded\natomic orbital with a short spatial extension so not all\nthe magnetic moments will couple in the same manner to\nthe electronic spins on the surface and the coupling will\nbe stronger for nearer atoms. The two previous observa-\ntions lead us to define S(Rn) as\nS(Rn)≡/summationdisplay\niS(ˆRi)Φ/parenleftBig\nRn−ˆRi/parenrightBig\n, (6)\nwhere now the sum is performed over the magnetic lat-\ntice positions. The function Φ encodes the information\nabout the short range characterof the localized magnetic\norbitals. In our calculations we have chosen a Gaussian\nprofile, Φ( r) =e−r2/βparametrized by the parameter β\nwhich has the meaning of the (squared) mean size of the\nspatial profile of the magnetic orbital. We have checked\nthat any other choice for Φ does not modify the qualita-\ntive results presented in this work.It is important to note\nthat for a given Se position, nearby moments will con-\ntribute to S(Rn) but not equally if there is a mismatch\nbetween the two sublattices. This key observation is in-\nteresting because it opens the possibility of considering\nnot just ferromagtetic, but also other types of magnetic\nordering as a candidate for inducing gaps in the TI sur-\nface states by the exchange coupling mechanism.\nResults. In order to show the ideas explained above at\nwork, let us consider first the case where the positions\nof the magnetic lattice lie in the middle of the triangles\nformedbythesurfacelatticeasitisshowninFig.(2c)and\ncalculatethespectrumwitheqs(3-6). Wewillcontrolthe\nmismatch between the lattices by displacing a magnetic\nbipartite lattice a distance δywith respect the center\nof the triangle in the OYdirection, and we will consider\nthe three casesof ferromagnetic( S1=S2), antiferromag-\nnetic (S1=−S2) and ferrimagnetic ( S1=−5S2) out of\nplane configurations. In Fig.(2a) we show the value of\nthe gap defined as m=|min[Ec(k)]−max[Ev(k)]|/2.\nFor the ferromagnetic case, the system always develops\na non zero gap, as expected, irrespective of the relative\nposition of the two lattice sites. The modulation in the\nvalue of the gap is understood in terms of the differ-\nent contribution of the magnetic moments to Seff(Rj).\nMuch more interesting are the cases of ferrimagnetic and\nantiferromagnetic lattice structures. The first important\nobservation is that in both cases a gap is opened when\nvarying the relative position of the lattices, showing that\nin principle one can open gaps in the TI surface states\nby the interaction with ferri- and antiferromagnetic lay-\ners. In principle, nothing guarantees that the magnetic\nlattice sites must lie on the exact center of the triangles\nformed by the surface positions, but the gap might be\nstill open. Moreover, if during the fabrication process\nit were possible to control the lattice mismatch, the gap\ncould be tuned. Another important observation is that\nalthough mis a positive definite quantity by construc-tion, the value of the effective Zeeman coupling is not.\nIndeed it will change its sign, as it is shown in fig.(2b),\nwhere the effective Zeeman term is plotted as a function\nof the lattice displacement δyand the value of β. This\nresult means that for the caseofferrimagnet different rel-\native displacements might lead to different values of the\ncoupling, which is the signature of a topological phase\ntransition, controlled by δy. This is also true for generic\nantiferromagnetic configurations. As can be readily seen\nin fig.(2b) there is always a change of sing of Seffirre-\nspective of how tight are the magnetic atomic orbitals to\ntheir lattice sites.\nSo far, we have considered magnetic configurations in\nthin layers with the magnetization being out-of-plane. It\nis well known that when thin film geometries are consid-\nered for ferromagnets it is more energetically favorable\nfor the system to havethe magnetization in plane[16, 20].\nFrom the form of (2) an in-plane homogeneous magneti-\nzation would not induce any gap since an in plane mag-\nnetic moment would just shift the position of the Dirac\npoint. Actually this is not the case and a gap can be\ninduced when lattice effects are considered in addition to\n(2) as it was shown by Fu[21]. We can add to (2) the two\nnext to leading terms in the expansion in momenta[21]:\nHw=k2\n2m0σ0+αk2(σ×k)·ˆz+λ/parenleftbig\nk3\nx−3kxk2\ny/parenrightbig\nσz,(7)\nwherem0,α, andλcome from the comparison between\nthe band structure calculated from the tight binding\nand ARPES measurements[22]. When the Hamiltonian\nHD+Hwis considered together with Hexc=J/bardblσ/bardblm/bardbl, it\nis apparent that a gap of value ˜ m=λJ3\n/bardbl\nv3/parenleftbig\nm3\ny−3mym2\nx/parenrightbig\nappears at kg=J/bardbl\nvm/bardbl׈zas it is shown in fig.(1). Apart\nfrom the mass generation due to the hexagonal terms\nthere is a self-doping effect defined through the param-\neterµ=|min[Ec(k)] + max[ Ev(k)]|/2 due to the first\nterm in (7). The effective model HD+Hw+Hexccan\nbe considered as a good description for the interaction\nbetween the TI surface states and smooth varying ferro-\nmagnetic in-plane magnetization. However, as we argued\nbefore there are many materialswhere the magnetization\nvaries at the order of the lattice spacing and the above\neffective description cannot be directly applied. Even in\nsuch cases, a non vanishing gap can be found if one goes\nto the microscopic description of the system.\nWe will exemplify this situation considering the\nKagome lattice with classical planar magnetic configu-\nrations. The magnetism on this frustrated lattice is a\ncurrent subject of research[23]. In particular we have\nchosen the q= 0 and q=√\n3×√\n3 ground state spin\nconfigurations as plotted in the insets of Fig.(3). In or-\nder to monitor the evolution of the spectral properties of\nthe system we have chosen the lateral displacement δx\nbetween lattices in the OXdirection as a control param-\neter. In this way the magnetic moment sitting on the4\n00.20.40.60.8 100.050.10.150.2m\n00.20.40.60.8 100.10.20.30.4µa)\nxy\nb)\nFIG. 3: Evolution of the gap a) and the self doping b) as\na function of the mismatch δxbetween the two sublattices,\nas it is explained in the text. Insets correspond to magnetic\nconfigurations with δx= 0.\nhorizontal axis will play a dominant role. The results\nfor the gap and for the self doping are also displayed in\nFig.(3). We can easily understand the results by keeping\nin mind that the exchange interaction between magnetic\nmoments and electron spins is short ranged and the be-\nhavior of the gap with the magnetic moment components\naccording to the Fu’s model. In the inset of fig.(3a) the\nred sublattice magnetization points along the OYdirec-\ntion so the gap will open when this sublattice magneti-\nzation is closest to the atomic lattice. On the contrary,\nin the inset of Fig.(3b) the red sublattice magnetization\npoints along the OXaxis so according Fu’s model, no\ngap will be generated. Also, it is expected that both\nconfigurations will give rise to a non vanishing self dop-\ning effect when the effective magnetization is non zero as\nit is shown in Fig.(3b). Another important observation\nis that different in-plane magnetic configurations in adja-\ncent space regions (N´ eel domain walls[24]) might induce\na mass with opposite sign which would generate chiral\n1D fermionic states.\nExperimental feasibility. One of the proposed ferro-\nmagnetic insulators is the EuO[25]. It possesses a gap\nof the order of 1 .2eVand it crystallizes in the simple\ncubic structure, not commensurate to the triangular lat-\ntice structure of the surface, introducing further com-\nplexity in the problem. In contrast to ferromagnetic in-\nsulators, ferrimagnetic insulators offer more reliable ex-\nperimental opportunities. The ferrimagnetic insulating\nstate is present in Nature in many compounds and in\nmany crystalline structures, and ferrimagnetic thin filmscan be manufactured in many ways[26]. Among them we\nhighlight the hexagonal ferrites of which PbFe 12O19is\nthe archetypal material. They crystallize in the hexago-\nnal magnetoplumbite structure having a rather complex\natomic configuration. Many other ferrites grow in the\nspinel structure, like the magnetites (Fe 3O4) and Cobalt\nferrites (CoFe 2O4) that might be grown in thin films\nwith appreciable out-of-plane magnetization[27]. Al-\nthough CoFe 2O4have a strong mismatch between the\nmagneticand the Selattice structure andthe resultspre-\nsented here are not directly applicable we suggest it as\nprospect candidate for experimentally analyzing the ef-\nfect of ferrimagnetism on the surface states of a TI. Con-\ncerningin-planemagneticconfigurations,wecanmention\nKagome systems with different planar spin ground states\nlike SrCr 9Ga3O19, herbersmiththite, jarosite and many\nothers[28].\nConclusions. In the present paper we have addressed\nthe question if the effective Hamiltonian (2) is valid when\nthe helical surface states of a TI are coupled to magnet-\nically active layers, . By using a tight-binding model for\nboth the TI and the magnetization, we have shown that\ncontrary to the (perhaps too) naive expectation that the\nhelical spin couples to the total magnetization present in\nthe unit cell, it couples to a weighted average of the mag-\nnetic moments present in the unit cell. This result tell us\nthatinprinciple, thereisnophysicalreasonforrulingout\nantiferromagnetic insulating thin films as candidates for\ninducing gaps in TI surface states. We have considered\nalso the possibility of ferrimagnetic insulating thin films.\nIn all the cases, we have shown that the magnetic ex-\nchange mechanism induces a gap in these surface states.\nAs a result, we have found that the gap is sensitive to\nthe mismatch between the magnetic and surface lattices.\nWe have considered also the realistic situation where the\nfilm magnetization is in-plane and homogeneous. In this\ncase, a gap might be opened due to hexagonal warping\neffects[21] even for materials whose thin film magnetiza-\ntion is textured at the scale of the lattice spacing, induc-\ning a zero average magnetic moment per unit cell.\nTheauthorsgratefullyacknowledgeconversationswith\nCarlos Pecharrom´ an, Andr´ as P´ alyi and J´ ozsef Cserti. A.\nC. aknowledges the CSIC JAE-doc fellowship program\nfor financial support. O. L. aknowledges the support of\nthe EU grant NanoCTM.\n[1] F. Wilczek, Phys. Rev. Lett. 58, 1799 (1987).\n[2] A. Karch, Phys. Rev. Lett. 103, 171601 (2009).\n[3] W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. 105,\n057401 (2010).\n[4] J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang,\nPhys. Rev. Lett. 105, 166803 (2010).\n[5] G. S. J. et al., Phys. Rev. B 82, 125120 (2010).\n[6] A. G. Grushin and A. Cortijo, Phys. Rev. Lett. 106,5\n020403 (2011).\n[7] I. Garate and M. Franz, Phys. Rev. Lett. 104, 146802\n(2010).\n[8] X. L. Qi, R. Li, J. Zang, and S. C. 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Middelhoek, J. Appl. Phys. 34, 1054 (1963).\n[25] P. G. S. et al., Phys. Rev. Lett. 88, 047201 (2002).\n[26] H. Kronmuller and S.Parkin, Handbook of Magnetism\nand Advanced Magnetic Materials (Willey, 2007).\n[27] A. L. et al., Phys. Rev. B 76, 054405 (2007).\n[28] J. E. Greedan, J. Mater. Chem. 11, 37 (2001)."    },    {        "title": "1206.6656v2.Ferrimagnetic_spin_1_2_chain_of_alternating_Ising_and_Heisenberg_spins_in_arbitrarily_oriented_magnetic_field.pdf",        "content": "arXiv:1206.6656v2  [cond-mat.stat-mech]  25 Dec 2012Condensed Matter Physics, 2012, Vol. 15, No 4, 43002: 1–11\nDOI: 10.5488/CMP.15.43002\nhttp://www.icmp.lviv.ua/journal\nFerrimagneticspin-1/2chainofalternatingIsing\nandHeisenbergspinsinarbitrarilyoriented\nmagneticfield\nJ. Strečka1, M. Hagiwara2, Y. Han2, T. Kida2, Z. Honda3, M. Ikeda2\n1Department of Theoretical Physics and Astrophysics, Facul ty of Science, P.J. Šafárik University,\nPark Angelinum 9, 040 01 Košice, Slovak Republic\n2KYOKUGEN (Center for Quantum Science and Technology under E xtreme Conditions), Osaka University,\n1-3 Machikaneyama, Toyonaka, Osaka 560–8531, Japan\n3Department of Functional Materials Science, Graduate Scho ol of Science and Engineering,\nSaitama University, 255 Simookubo Saitama, Saitama 338–85 70, Japan\nReceived July 3, 2012\nThe ferrimagnetic spin-1/2 chain composed of alternating I sing and Heisenberg spins in an arbitrarily oriented\nmagneticfield is exactly solved using the spin-rotation transformati on and the transfer-matrix method. It is\nshown that the low-temperature magnetization process depe nds basically on a spatial orientation of the mag-\nneticfield. A sharp stepwise magnetization curve with a marked inte rmediate plateau, which emerges for the\nmagneticfield applied along the easy-axis directionof the Ising spins , becomes smoother and the intermediate\nplateau shrinks if the external field is tilted from the easy-axis direction. The magnetizati on curve of a polycrys-\ntalline system is also calculated by performing powder aver aging of the derived magnetization formula. The\nproposed spin-chain model brings an insight into high- field magnetization data of 3d-4fbimetallic polymeric\ncompound Dy(NO 3)(DMSO) 2Cu(opba)(DMSO) 2, which provides an interesting experimental realization o f the\nferrimagnetic chain composed of two different but regularl y alternatingspin-1/2 magnetic ions Dy3+and Cu2+\nthat are reasonably approximatedby the notion of Ising and H eisenberg spins, respectively.\nKeywords:ferrimagneticspinchain,exactresults,magnetizationpl ateau, 3d-4fbimetalliccompound\nPACS:05.30.-d,05.50.+q,75.10.Hk,75.10.Jm,75.10.Pq,75.40. Cx\n1.Introduction\nExactlysolvedquantumspinchainsbelongtothemostattrac tingissuestodealwithinthecondensed\nmattertheory,becausetheyarecapableofprovidingadeepe runderstandingintomanyunconvential\nquantumcooperativephenomena[1].Recently,theparticul arresearchinteresthasbeenturnedtowards\nsophisticatedIsing-Heisenbergchains,whichareaimedat describinghybridspinsystemscomposedof\n‘classical’IsingandquantumHeisenbergspins[2–10].Amo ngothermatters,theIsing-Heisenbergchains\nhavebecomehelpfulinprovidingtheevidenceforseveralno velandunexpectedquantumstates[2–6],\nfractionalmagnetizationplateausinthelow-temperature magnetizationprocess[4–6],enhancedmagne-\ntocaloriceffectduringtheadiabaticdemagnetization[5] ,thermalentanglement[7],etc.Itis,therefore,\nquitechallengingtosearchforsuitableexperimentalreal izationsoftheIsing-Heisenbergchainstesti-\nfyingtotheaforementionedtheoreticalfindings,butonlya fewexperimentalsystemssatisfyavery\nspecificrequirementofaregularalternationoftheIsingan dHeisenbergspins.Uptonow,themag-\nneticbehaviourofthreedifferentpolymericchainsCu(3-C lpy) 2(N3)2[8],[(CuL) 2Dy][Mo(CN) 8][9]and\n[Fe(H 2O)(L)][Nb(CN) 8][Fe(L)][10]wassuccessfullyinterpretedwithinthefram eworkoftheIsing-Heisen-\nbergchains.\nThemaingoalofthepresentworkistoexaminethemagnetizat ionprocessinthespin-1\n2chainof\nalternatingIsingandHeisenbergspins,whichbringsanins ightintoferrimagnetismof 3d-4fbimetallic\n©J. Strečka, M. Hagiwara, Y. Han, T. Kida, Z. Honda, M. Ikeda, 2 012 43002-1J. Strečkaetal.\ncoordinationcompoundDy(NO 3)(DMSO) 2Cu(opba)(DMSO) 2[11].Theorganizationofthispaperisasfol-\nlows.Exactresultsforthetotalandsublatticemagnetizat ionsoftheinvestigatedspin-chainmodelare\nderivedinsection2.Themostinterestingtheoreticalresu ltsarepresentedinsection3,wheretheyare\nalsoconfrontedwiththerelevantexperimentaldata.Thepa perendsupwithseveralconcludingremarks\ngiveninsection4.\n2.Spinalternatingchain\nLetusconsiderthespin-1\n2chaincomposedofalternatingIsingandHeisenbergspinsin anexternal\nmagneticfieldofarbitraryspatialdirection.Itisquitepl ausibletosupposethatmagneticpropertiesof\nthis’classical-quantum’spin-chainmodelwillbehighlya nisotropicduetothepresenceoftheIsingspins.\nInthisrespect,itisofparticularinteresttoexaminehowt hemagnetizationprocessdependsonaspatial\norientationoftheexternalmagneticfield,whichcanbeunam biguouslygivenbythedeviationangle θ\nreferredwithrespecttoauniqueeasyaxisoftheIsingspins .Theinvestigatedspin-chainmodelcanbe\ndefinedthroughthefollowingHamiltonian\nH=− JN/summationdisplay\ni=1Sz\ni(σz\ni+σz\ni+1)−gz\n1µBBcosθN/summationdisplay\ni=1σz\ni−gx\n2µBBsinθN/summationdisplay\ni=1Sx\ni−gz\n2µBBcosθN/summationdisplay\ni=1Sz\ni.(1)\nHere, σz\niand Sα\ni(α=x,z)denotestandardspatialcomponentsofthespin-1\n2operator,whereastheformer\n(latter)operatorsapparentlyrefertotheIsing(Heisenbe rg)spins.Thefirstsummationtakesintoaccount\ntheIsing-typeexchangeinteraction Jbetweenthenearest-neighbourHeisenbergandIsingspins, the\nsecondtermdeterminestheZeeman’senergyoftheIsingspin sintheexternalmagneticfieldwiththe\nprojection Bcosθtowardstheireasy( z)axis,whilethelasttwoZeeman’stermsdeterminetheovera ll\nmagnetostaticenergyoftheHeisenbergspinsaffectedboth bythetransverse( Bsinθ)andlongitudinal\n(Bcosθ)componentoftheexternalmagneticfield.Thequantities gα\n1and gα\n2(α=x,z)labelspatialcom-\nponentsofLandé g-factorsoftheIsingandHeisenbergspins,respectively, µBisBohrmagnetonand B\nstandsfortheexternalmagneticfield.NoticethattheHamil tonian(1)isbuiltontheassumptionthatthe\ntransversecomponentofLandé g-factoroftheIsingspinsisnegligible( gx\n1≈0),i.e.,thesituation,which\nisrealisticonlyforthehighlyanisotropic(theso-called Ising-type)magneticions[12,13].\nTakingadvantageofa‘classical’natureoftheIsingspins, whichrepresentabarrierforlocalquantum\nfluctuationsinducedbythetransversecomponentoftheexter nalfieldactingontheHeisenbergspins,\nonemayrewritethetotalHamiltonian(1)asasumofcommutin gsiteHamiltonians\nH=N/summationdisplay\ni=1Hi, (2)\nwhereaseachsiteHamiltonians Hiinvolvesalltheinteractiontermsofthe ithHeisenbergspinandthe\nZeeman ’senergyofitstwonearest-neighbourIsingspins\nHi=− JSz\ni(σz\ni+σz\ni+1)−Hx\n2Sx\ni−Hz\n2Sz\ni−Hz\n1\n2/parenleftbig\nσz\ni+σz\ni+1/parenrightbig\n. (3)\nForsimplicity,wehaveintroducedherethree ‘effective fields’Hz\n1=gz\n1µBBcosθ,Hx\n2=gx\n2µBBsinθ,and\nHz\n2=gz\n2µBBcosθinordertowritetheHamiltonian(3)andallsubsequentexpr essionsinamoreab-\nbreviatedform.Owingtothevalidityofthecommutationrel ation [Hi,Hj]=0betweendifferentsite\nHamiltonians,thepartitionfunctionoftheconsideredspi n-chainmodelcanbepartiallyfactorizedinto\nthefollowingproduct\nZ=/summationdisplay\n{σi}N/productdisplay\ni=1TrSiexp/parenleftbig\n−βHi/parenrightbig\n=/summationdisplay\n{σi}N/productdisplay\ni=1T(σz\ni,σz\ni+1), (4)\nwhere β=1/(kBT),kBisBoltzmann ’sconstant, Tistheabsolutetemperature,thesymbolTr Sidenotes\natraceovertwospinstatesofthe ithHeisenbergspinandthesummation/summationtext\n{σi}runsoverallavailable\n43002-2Spin alternating chain in arbitrarily oriented field\nconfigurationsoftheIsingspins.Toproceedfurtherwithacalcu lation,thepartialtraceoverspindegrees\noffreedomoftheHeisenbergspinsmustbeperformedbefores ummingoverspinstatesoftheIsingspins.\nItis,therefore,quiteconvenienttodiagonalizethesiteH amiltonian(3)bymakinguseofthespin-rotation\ntransformation\nSx\ni=Sx\ni′cosφi+Sz\ni′sinφi, Sz\ni=−Sx\ni′sinφi+Sz\ni′cosφi, (5)\nwhichbringsthesiteHamiltonian(3)intothediagonalform\nHi′=−Sz\ni′/radicalBig/bracketleftbig\nJ/parenleftbig\nσz\ni+σz\ni+1/parenrightbig\n+Hz\n2/bracketrightbig2+/parenleftbig\nHx\n2/parenrightbig2−Hz\n1\n2/parenleftbig\nσz\ni+σz\ni+1/parenrightbig\n(6)\nprovidedthatthespinrotation(5)isperformedbythespeci ficangle\nφi=arctan/bracketleftBigg\nHx\n2\nJ/parenleftbig\nσz\ni+σz\ni+1/parenrightbig\n+Hz\n2/bracketrightBigg\n. (7)\nTheeffectiveBoltzmann ’sweight,whichentersthefactorizedformofthepartitionf unction(4),can\nbenowsimplyevaluatedbyemployingthetraceinvariancean dthediagonalizedformofthesiteHamil-\ntonian(6)\nT(σz\ni,σz\ni+1)=TrSiexp/parenleftbig\n−βHi/parenrightbig\n=TrSi′exp/parenleftbig\n−βHi′/parenrightbig\n=2exp/bracketleftbiggβ\n2Hz\n1/parenleftbig\nσz\ni+σz\ni+1/parenrightbig/bracketrightbigg\ncosh/braceleftbiggβ\n2/radicalBig/bracketleftbig\nJ/parenleftbig\nσz\ni+σz\ni+1/parenrightbig\n+Hz\n2/bracketrightbig2+/parenleftbig\nHx\n2/parenrightbig2/bracerightbigg\n.(8)\nTheeffectiveBoltzmann ’sfactor(8)apparentlydepends,aftertracingoutthespind egreesoffreedomof\nthe ithHeisenbergspin,onlyuponitstwonearest-neighbourIsi ngspins σiandσi+1.Thus,theexpres-\nsion(8)canalternativelybeviewedastheeffectivetwo-by -twotransfermatrix\nT/parenleftbig\nσz\ni,σz\ni+1/parenrightbig\n=/parenleftbigg\nT/parenleftbig\n+1\n2,+1\n2/parenrightbig\nT/parenleftbig\n+1\n2,−1\n2/parenrightbig\nT/parenleftbig\n−1\n2,+1\n2/parenrightbig\nT/parenleftbig\n−1\n2,−1\n2/parenrightbig/parenrightbigg\n=/parenleftbiggT+ T0\nT0T−/parenrightbigg\n(9)\nwiththreedifferentmatrixelementsde finedas\nT±≡T/parenleftbigg\n±1\n2,±1\n2/parenrightbigg\n=2exp/parenleftbigg\n±β\n2Hz\n1/parenrightbigg\ncosh/bracketleftbiggβ\n2/radicalBig/parenleftbig\nJ±Hz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig2/bracketrightbigg\n,\nT0≡T/parenleftbigg\n±1\n2,∓1\n2/parenrightbigg\n=2cosh/bracketleftbiggβ\n2/radicalBig/parenleftbig\nHz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig2/bracketrightbigg\n. (10)\nSubstitutingthematrix(9)intotherelation(4)onemaycon sequentlyemploythestandardtransfer-\nmatrixapproachinordertoobtaintheexactresultforthepa rtitionfunction\nZ=/summationdisplay\n{σi}N/productdisplay\ni=1T(σz\ni,σz\ni+1)=TrTN=λN\n1+λN\n2, (11)\nwhichiswrittenintermsoftworespectiveeigenvaluesofth etransfermatrix(9)\nλ1,2=1\n2/bracketleftbigg\nT++T−±/radicalBig\n(T+−T−)2+4T2\n0/bracketrightbigg\n. (12)\nNow,letusproceedtothecalculationofthemostimportantq uantities,whicharerelevantforour\nsubsequentanalysisofthemagnetizationprocess.TheGibb sfreeenergycaneasilybecalculatedfrom\ntheexactexpression(11)forthepartitionfunction.Inthe thermodynamiclimit N→∞,oneobtainsthe\nfollowingpreciseanalyticalresultforthefreeenergyper elementaryunit\nf=−kBTlim\nN→∞1\nNlnZ=kBTln2−kBTln/bracketleftbigg\nT++T−+/radicalBig\n(T+−T−)2+4T2\n0/bracketrightbigg\n.(13)\n43002-3J. Strečkaetal.\nSubsequently,onemayreadilycalculatethesublatticemag netizationsoftheIsingandHeisenbergspins\nbydifferentiatingthefreeenergy(13)withrespecttothea ppropriateeffective fields.Thesublatticemag-\nnetizationoftheIsingspinsintwomutuallyorthogonaldir ectionsoftheexternalmagnetic fieldoriented\neitherperpendicularorparallelwithrespecttotheeasyax isread\nmx\n1≡gx\n1µB〈σx\ni〉=0, mz\n1≡gz\n1µB〈σz\ni〉=gz\n1µB\n2T+−T−/radicalBig\n(T+−T−)2+4T2\n0. (14)\nSimilarly,thesublatticemagnetizationoftheHeisenberg spinsintwoaforementionedorthogonaldirec-\ntionsoftheexternalmagnetic fieldcanbecalculatedfromthefollowinguniqueformula\nmα\n2≡gα\n2µB〈Sα\ni〉=gα\n2µB\n2(T+−T−)/parenleftbig\nUα\n+−Uα\n−/parenrightbig\n+4T0Uα\n0+/parenleftbig\nUα\n++Uα\n−/parenrightbig/radicalBig\n(T+−T−)2+4T2\n0\n(T+−T−)2+4T2\n0+(T++T−)/radicalBig\n(T+−T−)2+4T2\n0,(15)\nwhichisvalidforbothspatialdirections α=xand zifthecoe fficients Uα\n±and Uα\n0aredefinedas\nUx\n±=Hx\n2/radicalBig/parenleftbig\nJ±Hz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig22exp/parenleftbigg\n±βHz\n1\n2/parenrightbigg\nsinh/bracketleftbiggβ\n2/radicalBig/parenleftbig\nJ±Hz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig2/bracketrightbigg\n,\nUx\n0=Hx\n2/radicalBig/parenleftbig\nHz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig22sinh/bracketleftbiggβ\n2/radicalBig/parenleftbig\nHz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig2/bracketrightbigg\n, (16)\nand\nUz\n±= ±/parenleftbig\nJ±Hz\n2/parenrightbig\n/radicalBig/parenleftbig\nJ±Hz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig22exp/parenleftbigg\n±βHz\n1\n2/parenrightbigg\nsinh/bracketleftbiggβ\n2/radicalBig/parenleftbig\nJ±Hz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig2/bracketrightbigg\n,\nUz\n0=Hz\n2/radicalBig/parenleftbig\nHz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig22sinh/bracketleftbiggβ\n2/radicalBig/parenleftbig\nHz\n2/parenrightbig2+/parenleftbig\nHx\n2/parenrightbig2/bracketrightbigg\n. (17)\nItshouldbepointedoutthattheexactanalyticalformulafo rthefreeenergy(13)permitsastraightfor-\nwardderivationofthemagnetizationformulaforthemostge neralcaseofarbitrarilyorientedexternal\nmagnetic fieldaswell.The finalmagnetizationformulaintheexternal fieldofanarbitraryspatialdi-\nrectioncanbeconvenientlyexpressedintermsoftheformer lyderivedsublatticemagnetizations(14)\nand(15)oftheIsingandHeisenbergspins\nm(θ)=/parenleftbig\nmz\n1+mz\n2/parenrightbig\ncosθ+mx\n2sinθ. (18)\nItisworthnotingthatthemagnetizationformula(18)might beoffundamentalimportancefortheanal-\nysisoftheangulardependenceofthemagnetizationprocess inasingle-crystalsamplerelatedtothe\nconsideredspin-chainmodel.Ontheotherhand,themagneti zationofapowdersamplepertinentto\ntheinvestigatedspin-chainmodelcanalsobeformallyobta inedbyintegratingthemagnetizationfor-\nmula(18)overonehemisphereyielding\nmp=π\n2/integraldisplay \n0m(θ)sinθdθ, (19)\nbuttheinvolvedintegralprecludesderivationoftheclose d-formanalyticalexpressionduetotoocompli-\ncatedfunctionsinvolvedinthesublatticemagnetizations (14)and(15)oftheIsingandHeisenbergspins,\nrespectively.Ofcourse,theintegralenteringtherelatio n(19)forthemagnetizationofpowdersamples\ncanbeevaluatednumericallyandhence,itmaybeofpractica limportanceforaninvestigationofthe\nmagnetizationprocessintherelatedpolycrystallinesyst ems.\n43002-4Spin alternating chain in arbitrarily oriented field\n3.Resultsanddiscussion\nLetusproceedtoadiscussionofthemostinteresting findingsacquiredfortheferrimagneticspin-1\n2\nchainofalternatingIsingandHeisenbergspinscoupledthr oughtheantiferromagneticnearest-neigh-\nbourinteraction J=−| J|<0.First,wewillpresentacomprehensivesurveyoftheoretic alresultswith\ntheaimtoshedlightontypicalmagnetizationfeaturesofth eproposedspin-chainmodelandthen,high-\nfieldmagnetizationdataofonespeci ficpolymericcoordinationcompoundwillbeclari fiedwithinthe\nframeworkofthemodelunderinvestigation.\n3.1.Surveyoftheoreticalresults\nLetusbeginwiththeanalysisofthegroundstate.Thediagon alformofthesiteHamiltonian(6)allows\nonetogetthelowest-energyeigenstateoftheinvestigated spinalternatingchain\n|FRI〉=N/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingleσz\ni=1\n2/angbracketrightBig/parenleftbigg\nc−/vextendsingle/vextendsingle/vextendsingleSz\ni=−1\n2/angbracketrightBig\n+c+/vextendsingle/vextendsingle/vextendsingleSz\ni=1\n2/angbracketrightBig/parenrightbigg\n, (20)\nwhichindicatesthe quantumferrimagneticordering withaperfectalignmentoftheIsingspinstowards\ntheireasyaxisand,respectively,thequantumsuperpositi onoftwospinstatesofeachHeisenbergspin\nthatbasicallydependsonamutualinterplaybetweentheexc hangeconstant,Landé g-factor,sizeand\nspatialorientationoftheexternal fieldviatheoccurrenceprobabilities\nc2\n±=1\n2\n1∓|J|−gz\n2µBBcosθ\n/radicalBig/parenleftbig\n|J|−gz\n2µBBcosθ/parenrightbig2+/parenleftbig\ngx\n2µBBsinθ/parenrightbig2\n. (21)\nTheperfectalignmentoftheIsingspinsisalsocon firmedbythefollowingzero-temperaturevaluesofthe\nrelevantsublatticemagnetizationintwoconspicuousorth ogonaldirections(parallelandperpendicular)\nwithrespecttotheeasyaxis\nmz\n1=1\n2gz\n1µB, mx\n1=0. (22)\nContrarytothis,bothspatialcomponentsofthesublattice magnetizationoftheHeisenbergspinsare\nsubjecttothequantumreductionofmagnetizationonbehalf oflocalquantum fluctuationsarisingfrom\nthetransverse field(i.e.,perpendicularprojectionoftheexternalmagnet icfieldwithrespecttotheeasy\naxisoftheIsingspins)\nmz\n2= −gz\n2µB\n2\n|J|−gz\n2µBBcosθ\n/radicalBig/parenleftbig\n|J|−gz\n2µBBcosθ/parenrightbig2+/parenleftbig\ngx\n2µBBsinθ/parenrightbig2\n,\nmx\n2=gx\n2µB\n2\ngx\n2µBBsinθ\n/radicalBig/parenleftbig\n|J|−gz\n2µBBcosθ/parenrightbig2+/parenleftbig\ngx\n2µBBsinθ/parenrightbig2\n. (23)\nItisquiteobviousfrom(22)and(23)[oralternativelyfrom (20)and(21)]thatthequantumferrimagnetic\norderchangestotheclassicalferrimagneticorderwheneve rthetransverseprojectionoftheexternal\nmagnetic fieldvanishes.However,thetotalmagnetizationoftheferri magneticchainofalternatingIsing\nandHeisenbergspinsissubjecttothequantumreductionofm agnetizationforanyotherspatialorienta-\ntionoftheexternalmagnetic field\nm=µBcosθ\n2\ngz\n1−gz\n2|J|−gz\n2µBBcosθ\n/radicalBig/parenleftbig\n|J|−gz\n2µBBcosθ/parenrightbig2+/parenleftbig\ngx\n2µBBsinθ/parenrightbig2\n\n+gx\n2µBsinθ\n2\ngx\n2µBBsinθ\n/radicalBig/parenleftbig\n|J|−gz\n2µBBcosθ/parenrightbig2+/parenleftbig\ngx\n2µBBsinθ/parenrightbig2\n, (24)\n43002-5J. Strečkaetal.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/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/s109/s120\n/s50/s109/s122\n/s50/s109/s122\n/s49\n/s176/s107\n/s66/s32/s84/s32/s47/s32/s124/s74/s124/s32 /s61/s32/s48/s46/s48/s49\n/s40/s97 /s41\n/s66/s32/s66/s32/s47/s32/s124/s74/s124/s32/s109/s122 /s49/s91/s103/s122 /s49\n/s66/s93/s32/s44/s32 /s109/s122 /s50/s91/s103/s122 /s50\n/s66/s93/s32/s44 /s32/s109/s120 /s50/s91/s103/s120 /s50\n/s66/s93\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/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/s40/s98 /s41\n/s66/s32/s66/s32/s47/s32/s124/s74/s124/s32/s109/s120\n/s50\n/s109/s122\n/s50/s109/s122\n/s49\n/s176/s107\n/s66/s32/s84/s32/s47/s32/s124/s74/s124/s32 /s61/s32/s48/s46/s48/s49\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/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/s40/s99 /s41\n/s66/s32/s66/s32/s47/s32/s124/s74/s124/s32/s109/s120\n/s50\n/s109/s122\n/s50/s109/s122\n/s49\n/s176/s107\n/s66/s32/s84/s32/s47/s32/s124/s74/s124/s32 /s61/s32/s48/s46/s48/s49\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/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/s40/s100 /s41\n/s66/s32/s66/s32/s47/s32/s124/s74/s124/s32/s109/s120\n/s50\n/s109/s122\n/s50/s109/s122\n/s49\n/s176/s107\n/s66/s32/s84/s32/s47/s32/s124/s74/s124/s32 /s61/s32/s48/s46/s48/s49/s109/s122 /s49/s91/s103/s122 /s49\n/s66/s93/s32/s44/s32 /s109/s122 /s50/s91/s103/s122 /s50\n/s66/s93/s32/s44 /s32/s109/s120 /s50/s91/s103/s120 /s50\n/s66/s93\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/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/s40/s101 /s41\n/s66/s32/s66/s32/s47/s32/s124/s74/s124/s32/s109/s120\n/s50\n/s109/s122\n/s50/s109/s122\n/s49\n/s176/s107\n/s66/s32/s84/s32/s47/s32/s124/s74/s124/s32 /s61/s32/s48/s46/s48/s49\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/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/s40/s102/s41\n/s66/s32/s66/s32/s47/s32/s124/s74/s124/s32/s109/s120\n/s50\n/s109/s122\n/s49/s32/s61/s32 /s109/s122\n/s50\n/s176/s107\n/s66/s32/s84/s32/s47/s32/s124/s74/s124/s32 /s61/s32/s48/s46/s48/s49\nFigure1.(Coloronline)Longitudinalandtransverseprojectionsof thesublatticemagnetizationofthe\nIsing( mz\n1,mx\n1=0)andHeisenberg( mz\n2,mx\n2)spinsasafunctionofthemagnetic fieldatlowenough\ntemperature kBT/|J|=0.01,oneparticularchoiceof g-factors gz\n1=6,gx\n1=0,gx\n2=gz\n2=2,andseveral\nspatialorientationsoftheexternalmagnetic field.\nwhichappearsduetothequantumentanglementoftwospinsta tesofeachHeisenbergspinarisingfrom\nthetransverse field.Inaccordancewiththisstatement,thequantumreducti onoftotalmagnetization\nbecomesgreater,thehigherthetransversecomponentofthe external fieldis.\nNow,letusturnourattentiontoadiscussionofthelow-temp eraturemagnetizationprocessserving\ninevidenceoftheaforedescribedground-statefeatures.F orsimplicity,wewillfurtherassumethatthe\nHeisenbergspinshavethecompletelyisotropicLandéfacto rgx\n2=gy\n2=gz\n2=2incontrasttothehighly\nanisotropicLandéfactorofIsingspins gz\n1≫2and gx\n1=gy\n1=0.Toprovideanin-depthunderstandingof\nthemagnetizationprocess, figure1illustratestypical fielddependencesfortwoorthogonalprojectionsof\nthesublatticemagnetizationoftheIsingandHeisenbergsp insunderdifferentspatialorientationofthe\nappliedmagnetic field.Iftheexternal fieldisappliedalongtheeasyaxisoftheIsingspins,oneobse rvesa\nsteepfield-inducedincreaseinthelongitudinalsublatticemagne tizationoftheHeisenbergspins mz\n2ata\ncritical fieldduetoanabruptreversalofallHeisenbergspinstowards theexternal- fielddirection(atlow\nfields,theHeisenbergspinsarealignedantiparallelwithre specttotheexternal field,becausetheypos-\nsesslower g-factorcomparedtothatoftheIsingspins).Thisclassical mechanismfortheformationofan\nintermediatemagnetizationplateauwillconsequentlylea dtoasharpstepwisepro fileinthetotalmagne-\ntizationversustheexternal- fielddependence(see figure2andthesubsequentdiscussion).Iftheexternal\nfieldistiltedfromtheeasy-axisdirection,therelevantbeh aviouroftheHeisenbergspinsbecomesmuch\nmoreintricateowingtoamutualcompetitionbetweentwodif ferentspatialdirectionsgivenbytheeasy\naxisoftheIsingspinsandthespatialorientationoftheapp liedmagnetic field.Asamatteroffact,the\nlongitudinalcomponentofthesublatticemagnetizationof theHeisenbergspins mz\n2apparentlyexhibits\nasmoothervariationiftheexternal fieldisgraduallytiltedfromtheeasyaxisandthemoregentle field-\ninducedreversaloftheHeisenbergspins(givenbythechang eofsignof mz\n2)simultaneouslyappearsat\nthehighercritical field\nµBBc\n|J|=1\ngz\n2cosθ. (25)\nItshouldbementionedthatthetransverseprojectionofthe sublatticemagnetizationoftheHeisenberg\nspins mx\n2becomesnon-zeroforthismoregeneralcaseand mx\n2exhibitsastrikingnon-monotonousde-\n43002-6Spin alternating chain in arbitrarily oriented field\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s49/s50/s51/s52\n/s40/s97 /s41\n/s66/s32/s66/s32/s47/s32/s124/s74/s124/s32/s176/s176/s176/s176/s176/s176\n/s107\n/s66/s32/s84/s32/s47/s32/s124/s74/s124/s32 /s61/s32/s48/s46/s48/s49/s109/s32 /s91\n/s66/s93/s176\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s176\n/s40/s98/s41\n/s66/s32/s66/s32/s47/s32/s124/s74/s124/s32/s176/s176/s176\n/s176/s176/s176\n/s107\n/s66/s32/s84/s32/s47/s32/s124/s74/s124/s32 /s61/s32/s48/s46/s48/s49/s109/s32 /s91\n/s66/s93\nFigure2.(Coloronline)Low-temperaturedependenceofthetotalmag netizationasafunctionofthe\nmagnetic fieldforseveralspatialorientationsoftheexternal fieldandtwodifferentsetsof g-factors:\n(a)gz\n1=6,gx\n1=0,gx\n2=gz\n2=2;(b) gz\n1=20,gx\n1=0,gx\n2=gz\n2=2.Brokenlinesshowmagnetizationcurves\nforthepowdersamples,whichwereobtainedbythenumerical solutionof(19).\npendencewithamoreorlesssharperglobalmaximumlocateda tthecritical field(25).Theparticular\ncaseoftheexternalmagnetic fieldorientedperpendicularwithrespecttotheeasyaxisdes ervesaspecial\nmention.Underthisspeci ficconstraint,theIsingspinsbecomecompletelyuncorrelat ed(disordered)and\ntheHeisenbergspinsgraduallytendtoaligntowardstheext ernal-fielddirectionuponthestrengthening\nofthetransverse field.\nInfigure2,thetotalmagnetizationisplottedagainstthemagne ticfieldforvariousspatialorientations\noftheexternal fieldandtwodifferentvaluesoftheLandé g-factoroftheIsingspins.Itisworthnoting\nthatthepro fileofthedisplayedmagnetizationcurvescanreadilybeunde rstoodfromtherelevantlow-\ntemperaturedependencesofthesublatticemagnetizations (cf.figure2with figure1).Inagreementwith\nourexpectations,theclassicalmechanismfortheformatio noftheintermediatemagnetizationplateau\nbecomesquiteevidentundertheparticularorientationoft heexternal fielddirectedalongtheeasyaxisof\ntheIsingspins.Underthisspeci ficcondition,oneactuallydetectsasharpstepwisemagnetiz ationcurve\nwithamarkedintermediateplateauatthefractionalvalue (gz\n1−gz\n2)µB/2perelementaryunit,which\nimpliestheexistenceoftheclassicalferrimagneticorder duetoanunequalmagneticmomentofthe\nnearest-neighbourspin-1\n2atomsdifferingintheir g-factors.Itcanbeclearlyseenfrom figure2thatthe\nintermediatemagnetizationplateaugraduallyshrinksand therelevantdependenceofthetotalmagne-\ntizationbecomessmootherastheexternal fielddeviatesfromtheeasy-axisdirectionoftheIsingspins .\nTheobservedbreakdownofintermediatemagnetizationplat eau,thegradualsmoothingofthemagne-\ntizationcurveaswellastheoverallquantumreductionofth etotalmagnetizationcanallbeattributed\ntolocalquantum fluctuations,whicharisefromthetransversecomponentofth eexternalmagnetic field\nactingontheHeisenbergspins.Forcomparison,thelow-tem peraturemagnetizationcurveofapolycrys-\ntallinesystemisalsodepictedin figure2byabrokenline.Interestingly,thepowderaveraging through\ntheformula(19)yieldsthemagnetizationcurveofapolycry stallinesystem,whichquitecloselyfollows\nthemagnetizationcurveofasingle-crystalsystemforthep articularorientationoftheexternalmagnetic\nfielddeviatingbytheangle θ=60◦fromtheeasy-axisdirection.\n3.2.High-fieldmagnetizationofDyCu\nOurtheoretical findingsforthemagnetizationprocesswillbenowconfronted withhigh- fieldmag-\nnetizationdataof 3d-4fbimetalliccoordinationpolymerDy(NO 3)(DMSO) 2Cu(opba)(DMSO) 2(DMSO=\ndimethylsulfoxide,opba =orthophenylenebisoxamato)tobefurtherabbreviatedasDy Cu.Thepolycrys-\ntallinesampleofDyCuwaspreparedaccordingtotheprocedu rereportedpreviouslybyCalvezandco-\nworkers[11].Eventhoughoureffortsaimedatpreparingasi ngle-crystalsamplesuitableforacrystal-\nstructurecharacterizationandmagnetizationmeasuremen tswasnotsuccessful,theelementalanaly-\nsishasprovidedastrongsupportthatthepreparedpolycrys tallinesampleshouldconsistofbimetallic\npolymericchainsrepresentingastructuralanalogueofLn( NO 3)(DMSO) 2Cu(opba)(DMSO) 2(Ln=Gd–Er),\nwhichisvisualizedin figure3bymakinguseofthecrystallographicdatareportedby Calvezetal.[11].\n43002-7J. Strečkaetal.\nFigure3.(Coloronline) Thevisualization ofLnCu polymericchainin bimetallic coordinationcom-\npoundsLn(NO 3)(DMSO) 2Cu(opba)(DMSO) 2(Ln=Gd–Er)byadoptingthecrystallographicdatafromref-\nerence[11]depositedatTheCambridgeCrystallographicDa taCentre.Coloringschemefortheatomla-\nbelling:Ln(magenta),Cu(green),O(red),N(blue),C(grey ),S(yellow).\nBearingthisinmind,itcouldbeexpectedthatthebis-chela tingbridgingligandopbamightmediatea\nrelativelystrongsuperexchangecouplingbetweenthenear est-neighbourDy3+andCu2+magneticions,\nwhichshouldconsequentlyformthebimetallicpolymericch ainrunningalongthecrystallographic c-\naxis.\nFirst,letusmakeafewcommentsontheconstituentmagnetic ionsofDyCu.Itisquitewellestablished\nthatthemagneticbehaviourofoctahedrallycoordinatedCu2+ionscanbequitefaithfullyrepresentedby\nthenotionofHeisenbergspinswithamoreorlessisotropicL andéfactor gx\n2≈gy\n2≈gz\n2/greaterorsimilar2[13],while\ntheeight-coordinatedDy3+ionsoftenobeymoresubtlerequirementsoftheIsingspinsn ecessitatinga\nhighlyanisotropic g-factor gz\n1≫2and gx\n1≈gy\n1≈0[9,12–14].Infact,Dy3+ionrepresentsKramersion\nwiththeground-statemultiplet6H15/2,whichusuallyundergoesaratherstrongcrystal- fieldsplitting\nintoeightwell-separatedKramersdoublets[12,13].Inthi srespect,themagneticbehaviourofDy3+ion\ncanbeofteninterpretedatlowenoughtemperaturesinterms oftheIsingspinwiththeeffectiveLandé\nfactor gz\n1≈20and gx\n1≈gy\n1≈0[9,12–14].Withallthisinmind,thecoordinationpolymerD yCucould\nprovideasuitableexperimentalrealizationofthespin-1\n2chainofthealternatingIsingandHeisenberg\nspins.However,oneshouldalsokeepinmindthatthissimpli ficationisreasonableonlyatsu fficientlylow\ntemperaturescomparedwiththeenergygapbetweenthelowes t-energyandexcitedKramersdoubletsto\navoidanydangerofover-interpretationinherentinthisap proximation.\nFigure4(a)showsthehigh- fieldmagnetizationcurveofthepowdersampleofDyCumeasure dattwo\nsufficientlylowtemperatures1.3Kand4.2K(seereference[15]f ormoredetailsontheset-upofmagne-\ntizationmeasurements).Thedisplayedmagnetizationcurv esexhibitaremarkablecrossingaround12T,\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s52/s54/s56\n/s32/s84/s32 /s61/s32/s49/s46/s51/s75\n/s32/s84/s32 /s61/s32/s52/s46/s50/s75/s109/s32 /s91\n/s66/s32/s47/s32/s102/s46/s117/s46/s93\n/s32/s40/s97 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s66/s32 /s91/s84/s93/s68/s121/s67/s117\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s52/s54/s56\n/s32/s40/s98 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s66/s32 /s91/s84/s93/s109\n/s84/s73/s80/s32/s61/s32/s48/s46/s48/s50/s57\n/s66/s66/s32/s109\n/s101/s120/s112\n/s32/s109\n/s101/s120/s112/s45/s32 /s109\n/s84/s73/s80\n/s84 /s32/s61/s32/s49/s46/s51/s75/s109/s32 /s91\n/s66/s32/s47/s32/s102/s46/s117/s46/s93/s68/s121/s67/s117\nFigure4.(Coloronline)(a)High- fieldmagnetizationdatarecordedforthepowdersampleofDyC uat\ntwosufficientlylowtemperatures1.3Kand4.2K;(b)High- fieldmagnetizationcurveofDyCuatthe\nlowestmeasuredtemperature1.3Kbeforeandaftersubtract ingtheparamagneticVanVleckcontribution\nestimatedfromthequasi-lineardependenceinthehigh- fieldrange 36÷52T.\n43002-8Spin alternating chain in arbitrarily oriented field\nwhereasthemagnetizationdatarecordedatahighertempera ture4.2Kbecomegreaterthantheones\nrecordedatalowertemperature1.3Kabovethiscrossing field.Apartfromthisfact,onemayclearly\nrecognizethreecharacteristicregionsintherelevantmag netizationprocessasillustratedin figure4(b)\nbythinsolidlinesservingasguidesforeyesonly.Themagne tizationinitiallyshowsaveryrapidincrease\nwiththemagnetic fieldinthelow- fieldrange 0÷3T,thenitexhibitsarathersteepincreaseintherange\nofmoderate fields 5÷32T,whichisconsecutivelyfollowedwithamoresteadyquasi- lineardependence\ninthehigh- fieldrange 34÷52T.Thesteadyquasi-lineardependenceobservedinthehigh- fieldrangeim-\npliesasigni ficantcontributionofthetemperature-independent(VanVle ck)paramagnetism[14],which\nwasevaluatedtobe mTIP=0.029µBT−1perDy3+ionfromthelinear fitofthehigh- fieldregion 36÷52T\nandsubsequentlysubtractedfromthemeasuredmagnetizati ondata.Eventhoughthisproceduremight\ngiveonlyaratherroughestimateof mTIP,theobtainedvalueisinarathergoodaccordwithtypical\nvaluesof mTIPreportedpreviouslyforothercompoundsinvolvingDy3+ion[9].\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s52/s54/s56\n/s74 /s32/s47/s32 /s107\n/s66/s32/s61/s32/s45/s50/s54/s75/s109/s32 /s91\n/s66/s32/s47/s32/s102/s46/s117/s46/s93\n/s32/s40 /s97 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s66/s32 /s91/s84/s93/s68/s121/s67/s117/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84 /s32/s61/s32/s49/s46/s51/s75\n/s103/s120\n/s67/s117/s32/s61/s32/s50/s46/s51/s50/s103/s122\n/s67/s117/s32/s61/s32/s50/s46/s49/s54/s103/s122\n/s68/s121 /s32/s61/s32/s50/s50/s46/s54\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s52/s54/s56/s109/s32 /s91\n/s66/s32/s47/s32/s102/s46/s117/s46/s93\n/s32/s40/s98 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s66/s32 /s91/s84/s93/s68/s121/s67/s117/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84 /s32/s61/s32/s52/s46/s50/s75\n/s74 /s32/s47/s32 /s107\n/s66/s32/s61/s32/s45/s50/s54/s75\n/s103/s120\n/s67/s117/s32/s61/s32/s50/s46/s51/s50/s103/s122\n/s67/s117/s32/s61/s32/s50/s46/s49/s54/s103/s122\n/s68/s121 /s32/s61/s32/s50/s50/s46/s54\nFigure5.(Coloronline)High- fieldmagnetizationdataofthepowdersampleofDyCuattwodif ferent\ntemperatures:(a) T=1.3K;(b) T=4.2K.Bluesolidlinesshowthebesttheoretical fitobtainedusingthe\nformulas(18)and(19)forthespin-1\n2chainofalternatingIsingandHeisenbergspins.\nHigh-fieldmagnetizationdataofthepowdersampleofDyCuaftersub tractingthetemperature-inde-\npendentparamagnetismarepresentedin figure5togetherwiththebesttheoretical fitobtainedbyusing\nthemagnetizationformulas(18)and(19)derivedforthefer rimagneticspin-1\n2chainofalternatingIsing\nandHeisenbergspins.Asonecansee,onegenerallyobtainsa quiteplausibleconcordancebetweenthe\nrecordedexperimentaldataandtherelevanttheoreticalpr edictionsexceptasmalldiscrepancyobserv-\nableinthelow- fieldregion,whereamoreabruptchangeinthemagnetizationi stheoreticallypredicted\nthantheoneexperimentallyobserved.Itshouldbeneverthe lessmentionedthatthedeterminedvalueof\nLandéfactorofDy3+ion gz\n1=22.6issomewhatgreaterthantheoreticallyexpected.Thismigh tindicate\nasmallbutnon-negligibleeffectofthetransversecompone ntof g-factor,whichwascompletelyignored\nforDy3+ionswithinourexacttreatment.Ifthetransversecomponen tgx\n1ofLandéfactoristakeninto\naccountforDy3+ionsatthemean- fieldlevel,themagnetizationformula(18)forasingle-crys talsample\ncanbeeasilycorrectedforthecontributionofthetransver semagnetizationoftheIsingspins(Dy3+ions)\nyielding\nm(θ)=(mz\n1+mz\n2)cosθ+(mx\n1+mx\n2)sinθ (26)\nwith\nmx\n1≡gx\n1µB〈σx\ni〉=gx\n1µBHx\n1/radicalBig/parenleftbig\n2Jmz\n2+Hz\n1/parenrightbig2+/parenleftbig\nHx\n1/parenrightbig21\n2tanh/bracketleftbiggβ\n2/radicalBig/parenleftbig\n2Jmz\n2+Hz\n1/parenrightbig2+/parenleftbig\nHx\n1/parenrightbig2/bracketrightbigg\n.(27)\nIndoingso,thecorrectedmagnetizationformula(26)canbe substitutedintotheformula(19)derivedfor\nthepolycrystallinesysteminordertoanalyzethemagnetiz ationcurveofthepowdersampleofDyCu.\nInthisway,oneactuallyresolvestheproblemwithatoohigh valueoftheLandé g-factorofDy3+ions\n43002-9J. Strečkaetal.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s52/s54/s56\n/s103/s120\n/s68/s121 /s32/s61/s32/s51/s46/s55/s74 /s32/s47/s32 /s107\n/s66/s32/s61/s32/s45/s50/s54/s75/s109/s32 /s91\n/s66/s32/s47/s32/s102/s46/s117/s46/s93\n/s32/s40 /s97 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s66/s32 /s91/s84/s93/s68/s121/s67/s117/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84 /s32/s61/s32/s49/s46/s51/s75\n/s103/s120\n/s67/s117/s32/s61/s32/s50/s46/s49/s55/s103/s122\n/s67/s117/s32/s61/s32/s50/s46/s50/s48/s103/s122\n/s68/s121 /s32/s61/s32/s50/s48/s46/s50\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s52/s54/s56/s109/s32 /s91\n/s66/s32/s47/s32/s102/s46/s117/s46/s93\n/s32/s40/s98 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s66/s32 /s91/s84/s93/s68/s121/s67/s117/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84 /s32/s61/s32/s52/s46/s50/s75\n/s103/s120\n/s68/s121 /s32/s61/s32/s51/s46/s55/s32/s74 /s32/s47/s32 /s107\n/s66/s32/s61/s32/s45/s50/s54/s75\n/s103/s120\n/s67/s117/s32/s61/s32/s50/s46/s49/s55/s103/s122\n/s67/s117/s32/s61/s32/s50/s46/s50/s48/s103/s122\n/s68/s121 /s32/s61/s32/s50/s48/s46/s50\nFigure6.(Coloronline)High- fieldmagnetizationdataofthepowdersampleofDyCuattwodif ferent\ntemperatures:(a) T=1.3K;(b) T=4.2K.Bluesolidlinesshowthebesttheoretical fitobtainedusingthe\nformulas(26)and(19)forthespin-1\n2chainofalternatingIsingandHeisenbergspinswiththemea n-field\ncorrectionforthetransversemagnetizationoftheIsingsp ins.\nasevidencedbythebest fittingdatasetindicatedin figure6.Itisalsoquiteevidentfrom figure6that\ntheacquiredtheoreticalpredictionfollowsmoreaccurate lytherelevantexperimentaldatainthelow-\nfieldregionandbesidethis,thevaluesof g-factorsgainedforDy3+andCu2+ionsfromthislatter fitting\nprocedurearesimultaneouslymuchmorereliable.\n4.Concludingremarks\nThepresentarticleisdevotedtoanexactstudyofthespin-1\n2chainofalternatingIsingandHeisenberg\nspinsinthemagnetic fieldofarbitraryspatialdirection.Thelow-temperaturema gnetizationcurveofthe\ninvestigatedspin-chainmodelwasscrupulouslyexaminedi nthedependenceonaspatialorientationof\ntheappliedmagnetic field.Ithasbeendemonstratedthatthemagnetizationcurveb ecomessmoother,\ntheintermediateplateaushrinks,andthetotalmagnetizat ionisreducedbyquantum fluctuationsas\ntheexternal fielddeviatesfromtheeasyaxisoftheIsingspins.According ly,thepowderaveraginginthe\nrelatedpolycrystallinesystemyieldsasmoothlow-temper aturemagnetizationcurve,whichquiteclosely\nfollowsthemagnetizationcurveofasingle-crystalsystem foraspatialorientationoftheexternal field\ndeviatingby θ=60◦fromtheeasy-axisdirection.\nTheproposedspin-chainmodelhasbeenemployedforinterpr etinghigh- fieldmagnetizationdata\nofpolymericcoordinationcompoundDyCu,whichprovidesan interestingexperimentalrealizationof\ntheferrimagneticspin-1\n2chainofregularlyalternatingDy3+andCu2+magneticionsreasonablyapprox-\nimatedbythenotionofIsingandHeisenbergspins,respecti vely.Fromthisperspective,experimental\nstudiesperformedonsingle-crystaland field-alignedsamplesofDyCurepresentaparticularlychall eng-\ningproblemforfutureinvestigations.\nAcknowledgements\nThisworkwaspartlysupportedbytheGlobalCOEProgram(Cor eResearchandEngineeringofAd-\nvancedMaterials-InterdisciplinaryEducationCenterfor MaterialsScience)(No.G10)fromtheMinistry\nofEducation,Culture,Sports,ScienceandTechnology(MEX T),Japan.J.S.acknowledgeswarmhospitality\nduringhisstayasvisitingresearchscholaratKYOKUGENcen tre.\n43002-10Spin alternating chain in arbitrarily oriented field\nReferences\n1. MattisD.C.,TheMany-bodyProblem,WorldScienti fic,Singapore,1993.\n2. ValverdeJ.S.,RojasO.,deSouzaS.M.,J.Phys.:Condens. Matter,2008,20,345208;\ndoi:10.1088/0953-8984/20/34/345208.\n3. OhanyanV.,Condens.MatterPhys.,2009, 12,343;doi:10.5488/CMP.12.3.343.\n4. AntonosyanD.,BellucciS.,OhanyanV.,Phys.Rev.B,2009 ,79,014432;doi:10.1103/PhysRevB.79.014432.\n5. ČanováL.,StrečkaJ.,Lučivjanský,Condens.MatterPhys .,2009,12,353;doi:10.5488/CMP.12.3.353.\n6. RojasO.,deSouzaS.M.,OhanyanV.,KhurshudyanM.,Phys. Rev.B,2011,83,094430;\ndoi:10.1103/PhysRevB.83.094430.\n7. AnanikianN.,AnanikyanL.,ChakmakhyanL.,RojasO.,J.P hys.:Condens.Matter,2012, 24,256001;\ndoi:10.1088/0953-8984/24/25/256001.\n8. StrečkaJ.,Ja ščurM.,HagiwaraM.,MinamiK.,NarumiY.,KindoK.,Phys.Rev .B,2005,72,024459;\ndoi:10.1103/PhysRevB.72.024459.\n9. VandenHeuvelW.,ChibotaruL.F.,Phys.Rev.B,2010, 82,174436;doi:10.1103/PhysRevB.82.174436.\n10. SahooS.,SutterJ.P.,RamaseshaS.,J.Stat.Phys.,2012 ,147,181;doi:10.1007/s10955-012-0460-7.\n11. CalvezG.,BernotK.,GuillouO. etal.,Inorg.Chim.Acta,2008, 361,3997;doi:10.1016/j.ica.2008.03.040.\n12. WolfW.P.,Braz.J.Phys.,2000, 30,794;doi:10.1590/S0103-97332000000400030.\n13. DeJonghL.J.,MiedemaA.R.,Adv.Phys.,1974, 23,1;doi:10.1080/00018739700101558.\n14. JensenJ.,MackintoshA.R.,RareEarthMagnetism,Oxfor dUniversityPress,Oxford,1991.\n15. HanY.,KidaT.,IkedaM.,HagiwaraM.,StrečkaJ.,HondaZ .,J.KoreanPhys.Soc.,2012(submitted).\nФеримагнiтнийспiн-1/2ланцюжокзпочергових\nГайзенберговихтаIзинговихспiнiвудовiльно\nорiєнтованомумагнiтномуполi\nЙ.Стречка1,М.Хагiвара2,Й.Ган2, T.Кiда2,З.Гонда3, M. Iкеда2\n1Природничийфакультет ,Унiверситет iм.П.Й.Шафарика ,Кошiце,Словацькареспубл iка\n2KYOKUGEN (Центрквантовихнаук iтехнолог iй),Унiверситетм .Осака, 560–8531Осака,Японiя\n3Факультетфункц iональногоматер iалознавства ,Вищашколаприродничихнаукта iнженер iї,\nУнiверситетм .Сайтама , 338–8570Сайтама ,Японiя\nФеримагн iтнийсп iн-1/2ланцюжок ,якийскладаєтьсязпочерговихГайзенберговихта Iзинговихсп iнiву\nдовiльноор iєнтованомумагн iтномупол i,розв’язуєтьсяточно ,використовуючиперетворенняповороту\nспiнiвтаметодтрансфер -матриц i.Показано ,щонизькотемпературнийпроцеснамагн iченнязалежить\nвосновномув iдпросторовоїор iєнтацiїмагнiтногополя .Гострасходинкопод iбнаформакривоїнамагн i-\nченостiзпомiтнимпром iжнимплато ,якез’являєтьсяумагн iтномупол iприкладеномувздовжнапрямку\nлегкоїос i Iзинговихсп iнiв,стаєгладшоютапром iжнеплатокоротшає ,якщозовн iшнєполев iдхиляється\nвiднапрямкулегкоїос i.Криванамагн iченостiполiкристал iчноїсистемитакожобчислюється ,здiйсню-\nючиконф iгурацiйнеусередненняотриманоїформулидлянамагн iченостi.Запропонованамодельсп i-\nновоголанцюжкадаєрозум iннянамагн iченостiусильнихполях 3d-4fбiметалiчноїпол iмеpноїсполу -\nкиDy(NO 3)(DMSO) 2Cu(opba)(DMSO) 2,якадопускаєц iкавуекспериментальнуреал iзацiюферимагн iтного\nланцюжка ,складеногоздвохр iзних,алерегулярнозм iннихсп iн-1/2магнiтнихiонiвDy3+таCu2+,що\nприйнятноапроксимуютьсяпоняттями IзинговихтаГайзенберговихсп iнiв,вiдповiдно.\nКлючовiслова:феримагнiтнийспiновийланцюжок,точнiрезультати,плато намагнiченостi, 3d-4f\nбiметалiчнасполука\n43002-11"    },    {        "title": "2209.10216v1.Effect_of_Co_Substitution_on_Ferrimagnetic_Heusler_compound_Mn3Ga.pdf",        "content": "Effect of Co Substitution on Ferrimagnetic Heusler compound Mn 3Ga\nQuynh Anh T. Nguyen1, Thi H. Ho1, Myung-Hwa Jung2, and Sonny H. Rhim1\u0003\n1Department of Physics and Energy Harvest Storage Research Center, University of Ulsan, Ulsan 44610, Republic of Korea\n2Department of Physics, Sogang University, Seoul, 04107, Republic of Korea\n(Dated: September 22, 2022)\nEffect of Co substitution on Mn 3Ga is investigated using first-principles study for structural and magnetic\nproperties. Without Co, ferrimagnetic Heusler compound Mn 3Ga is in tetragonal phase. With Co substitution,\ndepending on Co concentration ( x) Mn 3Ga prefers tetragonal (cubic) phase when x\u00140:5 (x\u00150:5). Ferrimag-\nnetism is robust regardless of xin both phases. While magnetic moments of two Mn do not vary significantly\nwith x, Co magnetic moment in two phases exhibit different behaviors, leading to distinct features in total mag-\nnetic moment (M tot). When x\u00140:5, in tetragonal phase, Co magnetic moment is vanishingly small, resulting in\na decrease of M totwith x. In contrast, when x\u00150:5, in cubic phase, Co magnetic moment is roughly 1 mB, which\nis responsible for an increase of M tot. Electronic structure is analyzed with partial density of states for various x.\nTo elucidate the counterintuitively small Co moment, the magnetic exchange interaction is investigated where\nexchange coefficient between Co and Mn is much smaller in x\u00140:5 case than x\u00150:5 one.\nI. Introduction\nFor decades, Heusler compounds with plethora of material\nclasses have drawn continuously renewed attentions [1, 2].\nRich physics has been platforms for various research. Numer-\nous functionalities and properties, such as superconductivity,\nmagnetism, multiferroicity, half-metallicity, topological insu-\nlators, and so forth [3–6], have attracted for applications in\nmany areas. In particular, in the context of rapid progress\nin magnetic devices and spintronics, tetragonal Heusler com-\npounds have been intensively and extensively explored. Prop-\nerties of tetragonal Heusler compounds, perpendicular mag-\nnetic anisotropy (PMA) and low saturation magnetization\n(MS) [7–9], are regarded advantageous in applications with\nlow switching current ( IS) [10–12].\nMn 3Ga belonging to Heusler family in tetragonal structure,\npossesses aforementioned magnetic properties: strong PMA\naround 0.89 MJ/m\u00003, high Curie temperature ( TC) of 740 K\n[13, 14], low MSaround 200 emu/cm3, and approximately\n58% of spin polarization [15]. Low MSstems from the intrin-\nsic ferrimagnetism of antiparallel alignment of two Mn mo-\nments with unequal magnitudes. Efforts have been devoted\nto enhance the magnetic properties of Mn 3Ga. One is to re-\nplace Mn by other transition metal (TM) [16] where V-shape\nmagnetization with respect to TM content ( x) is quite distinct.\nUpon substitution of nonmagnetic TM, M totdecreases with\nxas nonmagnetic TM replaces one Mn, moment is reduced\nbut the antiparallel moment with the other Mn is retained.\nMtot=0 is achieved for certain xabove which the survived\nantiparallel moment dominates hence M totincreases.\nThe V-shape of magnetization is also prominent with mag-\nnetic TM substitution such as Fe, Co, and Ni [17, 18]. How-\never, M totnearly vanishes above x=0:5 and is not compen-\nsated. Interestingly, Co substitution accompanies a structural\ntransition from tetragonal to cubic while the substitution by\nFe and Ni retains the tetragonal structure. Moreover, the mag-\nnetic behavior of magnetic TM in two structural phases is dif-\nferent. Magnetic moments in tetragonal phase are smaller than\n\u0003sonny@ulsan.ac.krthe bulk counterpart but those in cubic phase are comparable\nto the bulk ones. While the substitution of Fe results in hard\nmagnet and that of Ni exhibits the shape memory phenom-\nena, the property of Co substitution depends on the Co con-\ntent. It has been reported earlier that Co-poor (0 <x<0:5)in\ntetragonal alloy is well suited for spin-transfer torque (STT)\nwhereas Co-rich (0:5<x<1)favors cubic phase with half-\nmetallicity. As STT and half-metallicity are favorable compo-\nnents in spintronic application, further study would shed light\non uncovering Mn 3\u0000xCoxGa.\nIn this paper, employing ab initio calculations, the effect\nof Co substitution on Mn 3\u0000xCoxGa is investigated. Magnetic\nproperty of Mn 3\u0000xCoxGa is illustrated along with total as well\nas atomic resolved magnetic moments. Analysis follows to re-\nveal the electronic structure of Mn 3\u0000xCoxGa. To elucidate the\ncontrasting behavior of Co magnetism in two phases, tetrag-\nonal and cubic, the exchange coefficients are discussed in the\nframework of the magnetic force theorem using the Heisen-\nberg model.\nII. Structure and computational Methods\nTetragonal ( x=0) and cubic phases ( x=1) are presented\nin Fig. 1, which correspond to D022andL21structure, respec-\ntively. Both tetragonal and cubic phase have two inequivalent\nMn atoms distinguished by Mn(I) and Mn(II). Space group\nof tetragonal phase is I4=mmm (No. 139) and that of cubic\nphase is Fm¯3m(No. 225). Wyckoff positions of both phases\nare listed in Table I.\nTABLE I. Wyckoff positions of constituent atoms of tetragonal\nand cubic phases, whose space groups are I4=mmm (No:139)and\nFm¯3m(No:25), respectively. Co takes either 2 cor 2dsite.\nTetragonal Cubic\nMn(I) 2 b 4a\nMn(II) 4 d 4c\nGa 2 a 4b\nCo 2c/2d 4 darXiv:2209.10216v1  [cond-mat.mtrl-sci]  21 Sep 20222\nFIG. 1. Crystal structure of (a) tetragonal Mn 3Ga and (b) cubic\nMn2CoGa. Blue, red, green and grey spheres denote Mn(I), Mn(II),\nCo and Ga atoms, respectively.\nFirst-principles calculations are performed using Vienna\nAb initio Simulation Package (V ASP) [19, 20] with projec-\ntor augmented-wave basis [21]. The exchange-correlation\npotential is treated using generalized gradient approxima-\ntion (GGA) with Perdew, Burke, and Ernzerhof (PBE)\nparametrization [22, 23]. Cutoff energy of 500 eV is em-\nployed for wave function expansion. For Brillouin zone in-\ntegration, 15\u000215\u00027 and 15\u000215\u000211kmeshes are used\nfor tetragonal and cubic phases, respectively, wherep\n2=2\u0002p\n2=2\u00021 unit cell is adopted for the primitive cell.\nFrom lattice constant optimization, we obtain a=3:78˚A\n(c=a=1:88) for tetragonal phase and a= 4.12 ˚A for cubic\nphase, respectively, which are used for further calculations\nThe exchange coefficients are calculated using the Heisenberg\nmodel in the framework of the magnetic force theorem (MFT)\n[24]. To do so, additional self-consistent calculations are per-\nformed using OpenMX [25], where atomic basis sets for Mn,\nGa, and Co are s3p2d1,s2p2d1, and H- s3p2d1, respectively\nwith energy cutoff of 300 Ry and cutoff radii of 10.0 ˚A(Mn\nand Co) and 9.0 ˚A(Ga), respectively. Then, Jxpackage is em-\nployed [26–28] to extract the exchange coefficients ( J) in the\nframework of MFT.\nIII. Results and Discussion\nA. Magnetic Moments\nMn 3Ga is either tetragonal or cubic depending on Co con-\ncentration x. When x<0:5, tetragonal phase with c=a=1:88\nis favored whereas when x>0:5 cubic phase is preferred. As\nsuch, x=0:5 is a phase boundary of two phaes [29].\nTotal magnetic moment (M tot) as function of xis plotted\nin Fig. 2, where each moment of individual Co, Mn(I), and\nMn(II) is also shown. For all x, Mn 3\u0000xCoxGa is ferrimagnetic:\nMoments of Mn(I) and Mn(II) are antiparallel with different\nmagnitudes. Without Co, i.e. x=0, Mn(I) and Mn(II) have\nmoments of 2.87 and -2.32 mB, respectively, exhibiting indeed\nferrimagnetism. Total magnetic moment (M tot) is 1.72 mBper\nFIG. 2. (a) Magnetic moment (M tot) per unit cell with respect to x,\nwhere white and grey area are tetragonal and cubic phases, respec-\ntively. Right axis is in emu/cm3converted from mBon the left axis.\n(b) Atom-resolved magnetic moment: blue, red, and green circles\ndenote Mn(I), Mn(II), and Co, respectively. Open (solid) circles rep-\nresent unstable (stable) phase.\nformula unit, consistent with previous works [14, 30], which\ncorresponds to magnetization of 314 emu/cm3. As shown in\nFig. 2(a), M totdecreases (increases) with xin tetragonal (cu-\nbic) phase. Notably, at the phase boundary when x=0:5, M tot\nof cubic phase is larger by 0.4 mBthan tetragonal phase. This\ndifference is due to different Co magnetism in both phases.\nAcknowledging ferromagnetism of Co, Co moment in tetrag-\nonal phase is quite small [31], which is against to the conven-\ntional wisdom. This is discussed later in Sec.III B and III C.\nMagnetic moments of individual atoms are shown in\nFig. 2(b) [32]. For all x, Mn(I) and Mn(II) have almost con-\nstant moments with 2.90 and -2.32 mB, respectively. However,\nCo moment in tetragonal phase nearly vanishes while in cubic\nphase is around 1.0 mB. Consequently, M totdecreases with x\nin tetragonal phase: M tot=1:72mBforx=0 drops to 0.72 mB\nforx=0:5. On the other hand, in cubic phase, more Co sub-\nstitution results in increase of M totfrom 1.00 to 2.00 mB. The\nvanishingly small Co magnetic moment in tetragonal phase is\nalso reported in previous work which has been also reported in\nother theoretical work [33]. As mentioned earlier, this coun-\nterintuitive feature will be visited in III C.\nB. Electronic Structure\nPartial density of states (PDOS) is presented in Fig. 3.\nLeft (right) upper panel is tetragonal (cubic) phase for x=0\n(x=1). Lower panel is for x=0:5 of tetragonal and cubic\nphases, respectively. Important peaks are labelled with tri-\nangle marks: 1, 2, and Cstand for Mn(I), Mn(II), and Co,\nrespectively; + (-) for the majority (minority) spin state, re-\nspectively. Different peaks in the same spin state of particular\natom are distinguished by primes. PDOS’s of x=1=3 and 2 =3\nare shown in Fig. S3, where the vertical line serves to manifest\nthe overall trend upon x. The shift of EFby Co substitution is\nevidence as Co has two more electrons than Mn[34].\nIn tetragonal phase with x=0 [Fig. 3(a)], Mn(I) and Mn(II)\nexhibit distinct features. Mn(I) has two main peaks: 1+ (1-)\nin occupied (empty) states around -2.7 eV (+1.0 eV) in the3\nFIG. 3. Partial density of states (PDOS) of dorbitals. Left and\nright panels for tetragonal and cubic phases, respectively. Tetragonal\nphase with (a) x=0 and (b) x=0:5. Cubic phase with (c) x=1:0 and\n(d)x=0:5. Lower panel is comparison between tetragonal and cubic\nphases with x=0:5. Blue, red, and green lines denote dorbitals\nof Mn(I), Mn(II), and Co, respectively. Peaks are emphasized with\ntriangles. Fermi energy ( EF) is set to zero.\nmajority (minority) spin state. On the other hand, Mn(II) has\nrather broader peak with smaller height than Mn(I): 2+ around\n+1.3 eV in the majority spin state and 2-, 20-, and 200- around\n-2.5, -1.2 eV , and +1.0 eV in the minority spin state, respec-\ntively.\nForx=0:5 in tetragonal phase [Fig. 3(b)], Mn(I) and\nMn(II) PDOS do not change much with respect to x=0 but\nshifted by +0.20 eV due to Co. [See vertical line in Fig. S3].\nOn the other hand, several peaks emerge in Co PDOS. As Co\nreplaces Mn(II), most Co peaks overlap with Mn(II): C- co-\nincides with 20- and small hump just above EFoverlap with\n20+. Nonetheless, the overlap with Mn(I) in the majority spin\nstate around -2.8 eV (marked with double triangles) is evi-\ndent. Broad C+ appear between -1.4 and 0.4 eV owing to hy-\nbridization with Ga pstate [See Fig. S2]. In particular, there\nare small humps around +0.2 and +0.6 eV , where the former,\njust mentioned earlier, overlaps with 20+ but the latter has no\nsuch feature. As the sum of occupied PDOS of two spin states\nis almost equal, Co magnetic moment nearly vanishes, which\nis prominent in tetragonal phase.\nNow we switch to cubic phase with x=1 [Fig. 3(c)]. Com-\npared to x=0 of tetragonal phase, 1+ changes little but 1-\nshifts by +0.3 eV . However, changes in Mn(II) and Co are ap-\nparent. In the minority spin state, 2- shifts by 0.2 eV and 20-\nis replaced by C-. Overlaps between 200- (not labeled) and C0-\nare prominent around EF+0.8 eV . On the other hand, in the\nmajority spin state, C+ appears almost around the same place\nasC-. Overlaps between C0+ and 2+ around EF+0.6 eV are\nFIG. 4. Calculated exchange coefficients ( J) for nearest neighbor in-\nteractions for various x(0\u0014x\u00141).J0,J1, and J2, represent the ex-\nchange coefficients of Mn(I)-Mn(II), Co-Mn(I), Co-Mn(II) in black,\nblue, and red, respectively. Tetragonal and cubic regions are divided\nby white and grey regions. Solid and dashed lines for tetragonal and\ncubic phases, respectively.\nmuch closer to EFthan x=0. Notably, 20+ is occupied in\ncontrast to x=0.\nCubic phase with x=0:5 is shown in Fig. 3(d). Here, we\ndiscuss changes with respect to x=0:5 of tetragonal phase\nandx=1 of cubic phase. Hence in this paragraphs x=0:5\n(x=1:0) refers to that in tetragonal (cubic) phase. Mn(I) is lit-\ntle affected: 1+ and 1- change negligibly with respect to x=1\nwhereas 1- shifts by +0.3 eV with respect to x=0:5. On the\nother hand, Mn(II) is quite affected with negative (positive)\nshifts with respect to x=0:5 (x=1). More specifically, com-\npared to x=1, peaks of 2+ and 20+ shift by +0.30 and +0.20\neV , respectively; that of 2- by +0.35 eV . Compared to x=0:5,\nshifts of 2+ and 20+ are -0.60 and -0.40 eV , respectively; 2- by\n-0.50 eV . In particular, 20+ accompanies occupation change.\nFor Co PDOS, C+ and C- shift by -0.4 and -0.1 eV , respec-\ntively, with respect to x=1, and 0 and -0.1 eV with respect to\nx=0:5. Positive shift of Mn(II) with respect to x=1 is well\nexpected, as mentioned earlier, as Co has more electrons than\nMn.\nC. Magnetic Exchange Interaction\nSo far PDOS analysis is presented. While Co magnetic mo-\nment in cubic phase ( x\u00150:5) is around 1 mB, that in tetrag-\nonal phase x\u00140:5 nearly vanishes, which is against the con-\nventional wisdom. In order to clarify this puzzling Co mag-\nnetism, the exchange coefficients, J, are discussed, employing\nthe Heisenberg model, ˆH=\u0000åi jJi jSi\u0001Si, where Ji j, the ex-\nchange coefficients, with positive (negative) sign implies par-\nallel (antiparallel) magnetic coupling. As briefly mentioned4\nin II, J0s are calculated in the framework of MFT [24–26, 28].\nFig.4 shows calculations of the exchange coefficients for three\ncases, Mn(I)-Mn(II), Co-Mn(I), and Co-Mn(II), denoted as J0,\nJ1, and J2, respectively.\nJ0, the exchange coefficients for two Mn types, is nega-\ntive for all cases. This indicates the antiparallel magnetism\nbetween two Mn sites confirming the ferrimagnetism. More-\nover,jJ0jis larger thanjJ1jorjJ2jroughly by an order of mag-\nnitude. In tetragonal phase ( x\u00140:5),J0=\u00009:89 meV for\nx=0 whose magnitude increases with xranging 18 – 21 meV .\nIn cubic phase ( x\u00150:5),J0=\u000014:9 meV for x=0:5, whose\nmagnitude also increases with xup to 20 meV . From this, ad-\ndition of Co introduces stronger tendency of the antiparallel\nmagnetism between Mn(I) and Mn(II).\nThe exchange coefficients between Co-Mn(I) and Co-\nMn(II), namely J1and J2, exhibit opposite sign behavior\nwhere magnitudes in cubic phase are slightly larger. In con-\ntrast to J0, bothjJ1jandjJ2jare little influenced by Co. In\ntetragonal phase,jJ1jandjJ2jare approximately 1 and 5 meV ,\nrespectively, while in cubic phase jJ1jandjJ2jare approxi-\nmately 5 and 7 meV , respectively. The opposite signs of J1\nandJ2is due to the antiparallel magnetism between Mn(I)\nand Mn(II). From jJ2j>jJ1j, the antiparallel coupling be-\ntween Co-Mn(II) is stronger than Co-Mn(I). Moreover, J1and\nJ2change signs between tetragonal and cubic phases, hence-\nforth Co shows the opposite sign of magnetic moment in both\nphases. From the analysis of the exchange coefficients, Mn(I)\nand Mn(II) exhibit rather rigid magnetic moments. More-\nover, the exchange coefficients with Co in tetragonal phase are\nweaker than cubic phase. As a result, Co moments in tetrago-nal phase is smaller than cubic one.\nIV . Conclusion\nIn summary, we have investigated the effect of Co substi-\ntution on Mn 3Ga for structural and magnetic properties. The\nferrimagnetic feature is robust regardless of Co concentration\n(x). When x\u00140:5, tetragonal phase is favored over cubic and\nvice versa, where the magnetic behavior of each phase is dif-\nferent. In tetragonal phase, M totdecreases with xwhile in\ncubic phase M totincreases with x. Electronic structure upon\nCo substitution is investigated with PDOS, whose peak struc-\ntures are analyzed. Except Co, magnetic moments of indi-\nvidual atom change little with x. Co magnetic moment in\ntetragonal phase nearly vanishes in contrast to 1 mBof cubic\nphase. To get more insight on this contrasting Co moment in\ntwo phases, the exchange coefficients are estimated using the\nHeisenberg model. Indeed, the antiparallel magnetism, more\nprecisely, the ferrimagnetism between Mn(I)-Mn(II) with dif-\nferent magnitudes is well manifested. On the other hand, the\nexchange coefficients of Co-Mn(I) and Co-Mn(II) exhibit op-\nposite signs in tetragonal and cubic phase.\nAcknowledgments\nWe are grateful to Soon Cheol Hong for fruitful discussions\nand comments. This work was supported by the National Re-\nsearch Foundation (NRF) of Korea (2019R1I1A3A01059880,\n2020R1A2C3008044).\n[1] K. Manna, Y . Sun, L. Muechler, J. K ¨ubler, and C. Felser,\nHeusler, Weyl, and Berry, Nat. Rev. Mater. 3, 244 (2018).\n[2] L. Wollmann, A. K. Nayak, S. S. Parkin, and C. Felser, Heusler\n4.0: tunable materials, Annu. Rev. Mater. Res. 47, 247 (2017).\n[3] Y . Kurtulus, R. Dronskowski, G. D. Samolyuk, and V . P.\nAntropov, Electronic structure and magnetic exchange coupling\nin ferromagnetic full Heusler alloys, Phys. Rev. B 71, 014425\n(2005).\n[4] T. Graf, C. Felser, and S. S. P. Parkin, Simple rules for the un-\nderstanding of Heusler compounds, Prog. Solid State Chem. 39,\n1 (2011).\n[5] M. Sargolzaei, M. Richter, K. Koepernik, I. Opahle, H. 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Felser, Mn 3Ga, a\ncompensated ferrimagnet with high curie temperature and low\nmagnetic moment for spin torque transfer applications, Appl.\nPhys. Lett. 90, 152504 (2007).\n[15] H. Kurt, K. Rode, M. Venkatesan, P. Stamenov, and J. M. D.\nCoey, High spin polarization in epitaxial films of ferrimagnetic\nMn3Ga, Phys. Rev. B 83, 020405 (2011).\n[16] R. Sahoo, L. Wollmann, S. Selle, T. H ¨oche, B. Ernst,\nA. Kalache, C. Shekhar, N. Kumar, S. Chadov, C. Felser, S. S. P.\nParkin, and A. K. Nayak, Compensated ferrimagnetic tetrago-\nnal Heusler thin films for antiferromagnetic spintronics, Adv.\nMater. 28, 8499 (2016).5\n[17] C. Felser, V . Alijani, J. Winterlik, S. Chadov, and A. K. Nayak,\nTetragonal Heusler compounds for spintronics, IEEE Trans.\nMagn. 49, 682 (2013).\n[18] A. K. Nayak, M. Nicklas, S. Chadov, P. Khuntia, C. Shekhar,\nA. Kalache, M. Baenitz, Y . Skourski, V . K. Guduru, A. Puri,\nU. Zeitler, J. M. D. Coey, and C. 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Gubanov, Local spin density functional approach to the the-\nory of exchange interactions in ferromagnetic metals and alloys,\nJ. Magn. Magn. Mater. 67, 65 (1987).\n[25] T. Ozaki, Variationally optimized atomic orbitals for large-scale\nelectronic structures, Phys. Rev. B 67, 155108 (2003).\n[26] H. Yoon, T. J. Kim, J.-H. Sim, S. W. Jang, T. Ozaki, and M. J.\nHan, Reliability and applicability of magnetic-force linear re-\nsponse theory: Numerical parameters, predictability, and or-\nbital resolution, Phys. Rev. B 97, 125132 (2018).[27] H. Yoon, T. J. Kim, J.-H. Sim, and M. J. Han, Jx: An open-\nsource software for calculating magnetic interactions based on\nmagnetic force theory, Comput. Phys. Commun. 247, 106927\n(2020).\n[28] M. J. Han, T. Ozaki, and J. Yu, Electronic structure, magnetic\ninteractions, and the role of ligands in Mn n(n= 4, 12) single-\nmolecule magnets, Phys. Rev. B 70, 184421 (2004).\n[29] See Supplemental Material at http://link.aps.org/xxx for com-\nparison tetragonal and cubic phases for Mn (II) sites. DE=\nET\u0000ECis plotted in Fig. S2 as function of Co concentration\n(x).\n[30] W. S. Yun, G.-B. Cha, I. G. Kim, S. H. Rhim, and S. C. Hong,\nStrong perpendicular magnetocrystalline anisotropy of bulk and\nthe (001) surface of D0 22Mn3Ga: a density functional study, J.\nPhys.: Condens. Mater. 24, 416003 (2012).\n[31] See Supplemental Material at http://link.aps.org/xxx for mag-\nnetic moments using GGA+U calculations. The validity of ap-\nplying on-site Coulomb interaction to Mn 3Ga is not justified\nyet. However, two cases are considered: (i) Ufor all magnetic\natoms and (ii) only for Co atom, with U=0\u00183 eV . When\nU=2 eV on Co only, Co moment becomes -1.14 mBwhile\nother Uvalues Co magnetic moments are considerably smaller\nthan the bulk value.\n[32] See Supplemental Material at http://link.aps.org/xxx for PDOS\nand magnetic moment of Ga atom. Ga moments are less than\n0.1mB.\n[33] L. Wollmann, G. H. Fecher, S. Chadov, and C. Felser, A scheme\nfor spin-selective electron localization in Mn 3Ga Heusler mate-\nrial, J. Phys. D: Appl. Phys. 48, 164004 (2015).\n[34] See Supplemental Material at http://link.aps.org/xxx for PDOS\nof Mn(I), Mn(II), and Co for various xfor both cubic and tetrag-\nonal phases."    },    {        "title": "2305.02971v1.Effective_rectification_of_THz_electromagnetic_fields_in_a_ferrimagnetic_iron_garnet.pdf",        "content": "arXiv:2305.02971v1  [cond-mat.mtrl-sci]  4 May 2023Effective rectification of THz electromagnetic fields in a fer rimagnetic iron garnet\nT.G.H. Blank,1,2E.A. Mashkovich,3K.A. Grishunin,1C. Schippers,2\nM.V. Logunov,4B. Koopmans,2A.K. Zvezdin,5and A.V. Kimel1\n1Radboud University, Institute for Molecules and Materials , 6525 AJ Nijmegen, the Netherlands.\n2Department of Applied Physics, Eindhoven University of Tec hnology,\nP.O. Box 513, Eindhoven 5600 MB, the Netherlands.\n3University of Cologne, Institute of Physics II, Cologne D-5 0937, Germany.\n4Kotel’nikov Institute of Radioengineering and Electronic s, 125009 Moscow, Russia.\n5Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia.\n(Dated: May 5, 2023)\nIt is found that single-cycle THz electromagnetic fields effic iently excite a GHz spin resonance\nmode in ferrimagnetic Tm 3Fe5O12, despite the near absence of GHz spectral components in the\nexciting THz pulse. By analyzing how the efficiency of excitat ion depends on the orientation and\nstrength of the THz electric field, we show that it can be expla ined in terms of the nonlinear THz\ninverse Cotton-Mouton effect. Here, the THz electric field ge ts effectively rectified and acts on the\nferrimagnetic spins as a uni-polar effective magnetic field p ulse. This interpretation is confirmed by\na theoretical model based on the phenomenological analysis of the effective magnetic field, combined\nwith the equations of motion derived from the effective Lagra ngian for a ferrimagnet. Moreover,\nby using the outcome of two-dimensional THz spectroscopy, w e conjecture a quantum-mechanical\ninterpretation of the observed effect in terms of stimulated Raman scattering of THz photons by\nthe crystal-field split f-f electronic transitions of Tm3+.\nI. INTRODUCTION\nThe development of ultrafast magnetism opened up\na research field [ 1], which explores new regimes of spin\ndynamics triggered in ferro, ferri, and antiferromagnetic\nmaterials by stimuli much shorter than the time required\nto reach thermodynamic equilibrium ( ∼100 ps). Con-\nsequently, in such regimes of spin dynamics, the con-\nventional approximations of equilibrium thermodynam-\nics to describe magnetic phenomena fail, and the result-\ning spin motion is often counter-intuitive. For instance,\neven though Curie’s principle [ 2] of equilibrium thermo-\ndynamics predicts that “the symmetry of the causes are\nto be seen in the effects” and thus solely heating cannot\nresult in magnetization reversal, it was shown that ultra-\nfast heating induced by femto or picosecond laser pulses\nor picosecond electrical pulses is able to reverse the mag-\nnetization in ferrimagnetic materials [ 3–6]. Similarly, in\nequilibrium thermodynamics, the fastest and the least\ndissipative route of magnetization reorientation seem to\nbe always mutually excluding. However, this seemingly\nimpossible exclusive combination was achieved by em-\nploying the effect of photo-induced magnetic anisotropy\nby a femtosecond laser-pulse, demonstrating the simul-\ntaneously record-fast and least-dissipative writing of a\nmagnetic bit [ 7]. Naturally, this counter-intuitive but\nvery appealing regime of spin dynamics attracted the at-\ntention of researchers in applied magnetism, including\nthe fields of spintronics, magnonics, and magnetic data\nstorage. This interest, in turn, fuels the search for ever\nnew ultrashort stimuli to enable more ultrafast and even\nless dissipative writing of magnetic bits.\nWhile femto and picosecond laser pulses in the near-\ninfrared and the visible spectral range are the most pop-\nular stimuli in ultrafast magnetism, it was realized thatnearly single-cycle THz electromagnetic pulses [ 8], con-\nsisting of photons with a thousand times smaller energy,\ncan affect the spins in magnetic media in a more energy\nefficient non-thermal way. In particular, it was shown\nthat the magnetic component of such THz pulses can\ndirectly couple to spins via Zeeman torque [ 9]. At the\nsame time, such an approach is associated with a sig-\nnificant disadvantage. The time integral of the electro-\nmagnetic field of the THz pulses, as in the case of any\nother freely propagating electromagnetic wave in a neu-\ntral medium, is strictly zero and the net effect of such\na stimulus on spins is thus questionable. But recently,\nit was shown that THz electric fields can be effectively\nrectified and thus become much more efficient in control-\nling spins [ 10,11]. Such rectification of THz fields has so\nfar only been demonstrated in canted antiferromagnetic\nmedia [12]. Here, we fill the gap and demonstrate that a\nsimilar mechanism of effective field rectification can also\nbe realized in ferrimagnets.\nIn this article, we show that a single-cycle THz pulse\nin ferrimagnetic Tm 3Fe5O12is able to excite not only a\nTHz [13], but also GHz mode of spin resonance. The ex-\ncitation ofthe lattermode is surprisingasits frequencyis\nnearly absent in the spectrum of the THz pulse. By an-\nalyzing how the amplitude of the GHz mode depends on\nthe orientation and the strength of the THz electromag-\nnetic fields, we rule out heating and show that the fields\nare effectively rectified and that the GHz spin resonance\nis excited by an effective magnetic field generated in the\nmedium duetothe inverseCotton-Moutoneffect (ICME)\n[14,15]. These findings are supported by a phenomeno-\nlogicalanalysisofthe rectified fieldin combinationwith a\nLagrangian model of spin dynamics [ 16], which all fit the\nexperimental observations. Quantum mechanically, the\nprocess can be described in terms of stimulated Raman2\nscattering [ 17], where a first photon with frequency ω1\nbrings the electron to the excited state, while a second\nphoton with frequency ω2stimulates fast recombination\nto the Stokes-shifted ground state accompanied by an\nemission of a magnon with frequency ω=ω1−ω2. By\nemploying two-dimensional (2D) THz spectroscopy [ 18],\nwe show that the excited state is characterized by a life-\ntime shorter than 1 ps. This fact practically excludes\nthat the GHz mode is excited via the second THz mode\nof spin resonance.\nThe paper is organized as follows. Section IIprovides\ndetails about the sample and describes the experimental\nsetup. Section IIIpresents the main experimental find-\nings. These findings are supported by the theory pre-\nsented in section IV, where we employed a Lagrangian\napproachto describe the magnetizationdynamics, driven\nby an effective rectified magnetic field due to the ICME.\nWe suggest a possible microscopic mechanism of the lat-\nterinthediscussionofsection V,supportedbyadditional\nmeasurements using 2D THz spectroscopy. Our conclu-\nsions are summarized in the final section VI.\nII. SAMPLE AND EXPERIMENTAL SETUP\nThe particular ferrimagnet that we examined was a\n19µm thick film of bismuth and gallium substituted\nthulium iron garnet (TmIG) Tm 2BiFe4.2Ga0.8O12grown\nby liquid phase epitaxy on a 500 µm gadolinium gallium\ngarnet (Gd 3Fe5O12) substrate with (111) orientation.\nThe garnet structure has eight formula units per unit\ncell and has space-group symmetry Ia 3d (point group\nOh) [19]. In the parent compound, Tm 3Fe5O12, two in\nfive Fe3+ions can be found in octahedral surroundings\nof O2−and the other three in a tetrahedral environment\n[20]. The spins of the tetrahedrally surrounded Fe3+ions\ncouple antiferromagnetically to the spins of those in the\noctahedral environment. The two Fe3+magnetic sublat-\ntices have different magnetizations, this difference results\nin a net magnetization MFe. The spins of Tm3+couple\nantiferromagnetically with respect to MFe, resulting in\na net magnetization MTm. Because the exchange inter-\naction between the iron sublattices is large compared to\nany other exchange interaction between the three sub-\nlattices (Tm-Tm and Tm-Fe), it is sufficient to treat\nthe two iron sublattices as one [ 21,22]. Therefore, al-\nthough Tm 3Fe5O12is, in reality, a three-sublattice ferri-\nmagnet, it is effectively treated as a more conventional\ntwo-sublattice Tm-Fe ferrimagnet with net magnetiza-\ntionM=MFe+MTmand N´ eel vector L=MFe−MTm.\nIn the garnet studied here, a part of the Tm3+ions\nwere substituted by Bi3+to enhance the magneto-optical\nFaraday effect [ 23–25]. Moreover, some Fe3+ions on\nthe tetrahedral sites [ 26] were partly substituted with\nnon-magnetic Ga3+to reduce MFeand thus ensure that\na possible ultrafast spin reorientation is not hampered\nby the need to facilitate an ultrafast exchange of large\nangular momentum between the lattice and the spinsystem. Moreover, a sufficiently large Ga3+dilution\nshould grant a magnetization compensation temperature\nto Tm 3Fe5O12that usually has no compensation point\nduetothesmallmagneticmomentofTm3+[24], butsuch\na temperature was not observed in the present composi-\ntion. Finally, both the parent compound as well as the\nsubstituted versionsdisplayuniaxialmagneticanisotropy\nalong the out-of-plane [111] axis [ 24], as was confirmed\nby magneto-optical measurements [ 13].\nWe performed a pump and probe experiment on\nthe sample by employing single-cycle THz-pulses gener-\nated by tilted-pulse-front optical rectification in LiNbO 3\n[8,27], yielding pulses with a peak electric field strength\nup to∼1 MV/cm in focus as calibrated by electro-optic\nsampling in a slab of GaP-(110) (see Fig. 1(b)). The\nTHz pulses were brought to temporal and spatial over-\nlap with low-intensity near-infrared (NIR) probe pulses\nwith a central wavelength of 800 nm and pulse duration\nof 100 fs. The experimental scheme and coordinate sys-\ntem are depicted schematically in Fig. 1(a), indicating\nthe THz pump and NIR probe polarization angles αand\nβdefined with respect to the y-axis. The polarizations\nwere controlled by a set of wire-grid polarizers for the\nTHz pump and a half wave-plate for the probe pulse.\nThe probe pulse transmitted through the sample and its\nTHz-induced rotation in polarization was mapped by a\ncombination of a Wollaston prism and a set of balanced\nphotodiodes. Bymeasuringthe polarizationrotationasa\nfunction of the time-retardation tbetween the pump and\nprobe pulses, we traced the THz-induced spin dynamics.\nFIG.1. (a)Schematicillustration oftheexperimental sche me.\n(b) The calibrated waveform of the THz pulse measured by\nelectro-optic sampling in GaP. The THz polarization is ini-\ntially along the y-axis, but it could be rotated using wire-grid\npolarizers. (c) Typical THz-induced transient of the probe -\npolarization rotation, measured at T= 6 K with an applied\nexternal field of 110 mT, pump polarization α=−45◦and\nprobe polarization β= 0◦. The data at long timescales has\nbeen multiplied by a factor of 5 for visibility.3\nOur previous results showed that the magneto-optical\nsignal of the transmitted probe pulse originates exclu-\nsively from out-of-plane magnetization components [ 13].\nTherefore it can be expected that no dynamic magneto-\noptical signal due to magnetization precession will be de-\ntected when the equilibrium magnetization is out of the\nplane. Tothis end, a magnetic field wasapplied predomi-\nnantlyintheplaneinordertosaturatethemagnetization\nin the plane, slightly titled at a small angle δ∼3◦. The\ntilt was required to be able to excite THz spin dynamics\nfor every polarization of the THz pulse (see Ref. [ 13]),\nas is explained in more detail in the conclusions of Ap-\npendixA. Finally, XRD analysis of the sample (see Sup-\nplemental Material [ 28]) confirmed its [111] orientation.\nMoreover, the analysis provided us with the orientation\nof the crystallographic axes with respect to the experi-\nmental coordinate system x∝bardbl[112] andy∝bardbl[110].\nIII. RESULTS\nFigure1(c) shows a typical dynamical polarization ro-\ntation transient. In our previous article [ 13], we showed\nthat the ultrafast THz dynamics in the first 50 picosec-\nonds can be attributed to the ferrimagnetic THz Kaplan-\nKittel exchange mode [ 29]. Due to the unequal g-factors\nof the Tm and Fe sublattices, the Zeeman torque (or\nmagnetic dipole interaction) acts differently on each sub-\nlattice, rendering a relatively efficient resonant excita-\ntion of the mode. Moreover, the bismuth-substitution\nyields a strong magneto-optical Faraday effect, which\nmade detection with a good signal-to-noise ratio possi-\nble. In addition to the previously reported THz mode,\nthe transients also reveal oscillations at a much lower\n(GHz) frequency. So low frequencies are typical for the\nferromagnetic resonance (FMR) mode in a ferrimagnet,\nas was also predicted by the theory of Kaplan and Kittel.\nGiven that the duration of our THz pulse ∼1 ps is much\nshorter than the period of the mode ∼50 ps, a resonant\nexcitation by the THz magnetic field similar to what was\nseen with the exchange mode is unlikely. Instead, the\nexcitation must be impulsive in nature. In order to re-\nveal the excitation mechanism of this supposedly FMR\nmode, we measured the dynamics as a function of the\npump polarization angle α, the probe polarization an-\ngleβ, the strength of the THz pump electric field, the\nstrength of the external magnetic field as well as sample\ntemperature.\nA. Pump and probe polarization\nThe measured dynamics are strongly dependent on\nthe polarization angle αof the THz pump pulse (see\nFig2(a)), which rules out heating as the dominant mech-\nanism. The optimal excitation occurs for α=±45◦(see\nFig.2(b)). The extrema are slightly asymmetric, such\nthat the excitation is actually stronger for α=−45◦.\n/s70/s97/s114/s97/s100/s97/s121/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s109/s100/s101/s103/s41\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s40/s97/s41\n/s97 /s32/s40/s100/s101/s103/s41\n/s113\n/s70/s32/s40/s109/s100/s101/s103/s41\n/s70/s70/s84/s32/s97/s109/s112/s108/s46/s32/s40/s110/s111/s114/s109/s46/s41\n/s84/s72/s122/s32/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32 /s97 /s32/s40/s100/s101/s103/s41/s70/s77/s82\n/s70/s105/s116\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s48 /s49/s56/s48/s98 /s40/s100/s101/s103/s41/s40/s99/s41/s40/s98/s41\n/s97/s32 /s61/s32/s45/s52/s53/s111\nFIG. 2. (a) THz induced dynamics as a function of the pump\npolarization angle α, measured at a temperature of 6 K and\nexternal field of 110 mT. The data have been fitted by sine\nfunctions shown by the solid black lines. Two maxima at\napproximately ±45◦can be observed, with nearly opposite\nphases. (b) Normalized peak Fourier amplitudes of the mode\nfor different α, where positive and negative values indicate the\ndifferent phases of the dynamics. (c) THz-induced dynamics\nmeasured at the optimal THz polarization α=−45◦and\nfieldBext= 130 mT, for various angles of the initial probe\npolarization β, showing no variations.\nThe dynamics for negative and positive αare approxi-\nmately in opposite phases. These facts imply that the\nexcitation mechanism is π-periodic with respect to α.\nThis is in contrast with the 2 πperiodicity of the exci-\ntation of the exchange mode due to the Zeeman torque\n[13]. Instead, the results are more comparable to the\nnonlinear excitation of the quasi-ferromagnetic mode in\nantiferromagnetic FeBO 3, where the largest signals were\nalso found when the electric (or magnetic) field of the\npump pulses was polarized at ±45◦from the net magne-\ntization [ 12].\nNext, itcanbeseeninFig. 2(c)thatthesignaldoesnot\ndepend on the initial probe polarization angle β. This\nimplies that the observed dynamics are a result of the\nmagneto-optical Faraday effect, to which only the out-\nof-plane magnetization contributes θF(t)∝Mz(t) [13].\nTherefore, starting from Fig. 2, we referred to the in-\nduced polarization rotation as Faraday rotation θF(t).\nB. THz amplitude\nFigure3shows the FFT spectrum of the dynamics\n(red) measured at the optimal THz polarization angle\nα=−45◦, as well as the spectrum of the THz pump-\npulse (blue). The sharp red peak that corresponds to the\nGHz (FMR) mode lies completely out of the spectrum\nof the exciting THz pulse. This fact indicates that the4\n/s32/s84/s72/s122/s40/s97/s41 /s40/s98/s41\nFIG. 3. (a) Typical Fourier spectrum of the signal (red) and\nthe exciting THz pulse (blue). The sharp peak in the signal\nspectrum that corresponds to the FMR mode falls out of the\nTHz spectrum. (b) Peak FFT amplitude of the FMR mode\nas a function of THz electric field amplitude, measured at a\ntemperature of 6 K, THz polarization α=−45◦andBext=\n110 mT. When fitting the data with a power law dependence\nof the form Eγ\nTHzfor variable γ, the data is best fitted with a\nquadratic dependence γ= 2.\nexcitation is most likely nonlinear. This nonlinearity was\nconfirmed by measuring the peak Fourier amplitude of\nthe mode as a function ofthe strength of the THz electric\nfieldETHz(see Fig. 3(b)). The data could only be fitted\naccurately with a quadratic dependence E2\nTHz.\nC. External magnetic field and temperature\nThe assignmentofthe exactoriginofthe FMR mode is\nnot straightforward, because TmIG is, in reality, a three-\nsublattice ferrimagnet. This means that possible spin\nresonances are associated not only with the spins of Fe3+\nbut also with those of the Tm3+ions. To reveal the ori-\nginofthe mode, westudied the frequencyasafunction of\nthe external magnetic field and temperature. Figure 4(a)\nshows that small changes in the external magnetic field\nhave an enormous impact on the frequency of the mode.\nIt can be seen in Fig. 4(b) that the frequency initially\ndecays, but for larger fields >25 mT it increases approx-\nimately linearly. Theoretically, the linear slope can be\nrelated to the g-factor of the FMR mode [ 30]. In the\ncase of FMR involving only iron, which has a g-factor of\ng≈2,theslopeshouldamountto28GHz/T.However,in\nour case, the slope is approximately ∼90 GHz/T, which\nallows us to give an estimate of the effective g-factorgeff\nis about ˆ geff≈6.4.\nThe fact that the effective g-factor is so much larger\nthan that of iron, tells us that the thulium sublattice\nmust be involved. Let gTmandgFebe theg-factors of\nthe individual Tm and Fe sublattices, respectively. The\ntheory of Kittel predicts that the effective g-factor, in\nthat case, is given by [ 31]:\ngeff(T) =MFe(T)−MTm(T)\ng−1\nFeMFe(T)−g−1\nTmMTm(T).(1)\nThe effective g-factor can thus have a strong temper-ature dependence in the vicinity of the angular mo-\nmentum compensation point, i.e. when g−1\nFeMFe(T) =\ng−1\nTmMTm(T). Otherwise, no strong temperature depen-\ndenceoftheeffective g-factorisexpected. Hereweshould\nnote that the magnetization compensation point in this\nsample either does not exist or is slightly above 0 K.\nFigure4(c) shows that the frequency at a fixed mag-\nnetic field does drop significantly when increasing the\ntemperature. But at the same time, it can be seen in\nFig.4(d) that the slope of the frequency as a function\nof the field does not decrease as a function of tempera-\nture. It means that the decay in frequency observed in\nFig.4(c) is a result of a temperature-dependent effective\nfield such as magnetic anisotropy, and not of a decreas-\ning effective g-factor. In Fig. 4(e), we show the effective\ng-factors estimated from the slopes of the linear fits in\nFig.4(d). It reveals that the effective g-factor has a pe-\nculiar temperature dependence and remains to be large.\nLarge values of the effective g-factor have also been ob-\nserved in a Tm 3Fe5O12compound with La, Ca, and Ge\nsubstitutions [ 32]. Here, the result could be fitted by\nassuming that the Tm angular momentum is quenched,\ni.e.gTm= 0. It is therefore likely that the peculiar tem-\nperature dependence of geffis greatly influenced by the\ndilution of Bi and Ga. Other factors might also play a\nrole, such as the potential noncollinear arrangement of\nthe Tm spins [ 33,34] that were observed in these materi-\nals, which raises interesting questions for further studies.\nIn any case, the fact that the g-factor remains well above\nthat ofthe free electron( g= 2) in the entire temperature\nrangefrom 6 to 170K implies that the thulium sublattice\nis involved in the FMR mode.\nFinally, we measured how the dynamics depend on a\nchangein the polarityofthe externalmagnetic field. Fig-\nure5shows that for the excitation maxima α=±45◦,\nthe phase of the dynamics is independent of the change\noffield polarity ±Bext. This result is in contrastwith the\nexcitationoftheTHzmodebyZeemantorquereportedin\nRef. [13], where the detected magneto-optical transients\ndid reverse their phase upon a change of polarity of the\nexternalmagnetic field. This fact againconfirms that the\ndominatingmechanism ofexcitation ofthe FMR mode in\nthe studied TmIG is not due to the linear coupling of the\nTHz magnetic field to the spins via the Zeeman torque.\nOnly around α= 0◦, one can distinguish a very weak sig-\nnal that changes sign upon the field reversal. This signal\ncould be interpreted to be due to the linear excitation\nmechanism, but it is clear that this mechanism is quite\ninefficient and negligible in comparison with nonlinear\nexcitation.\nTo summarize, the obtained experimental dependen-\ncies reveal that THz electromagnetic pulse excites GHz\noscillations of the magneto-optical Faraday effect in\nthulium iron garnet, which probes the out-of-plane pro-\njection of the magnetization dynamics Mz(t). The exci-\ntation is π-periodic with respect to the THz pump polar-\nizationαand reaches a maximal efficiency for α=±45◦.\nThe dependency of the amplitude of the oscillations5\nFIG. 4. (a) THz-induced transients for several strengths of the applied magnetic field, measured at T= 6 K and α=−45◦.\nPart (b) shows the extracted frequencies, where the width of the error bars equals the FWHM of the fitted Gaussian in the\nFFT spectrum. The slope of the linear part of the curve provid es a reasonable estimate ˆ gefffor the effective g-factorgeff. (c)\nExtracted frequency as a function of temperature for a fixed m agnetic field of 130 mT, where the bars again depict the FWHM.\n(d) The frequency measured at three different external magne tic fields, for various temperatures, where the blurred area s depict\nthe 95% confidence bands of the fitted line. (e) The resulting e stimations of the effective g-factor based on the fitted slope in\n(d), the bars denote the standard error.\nclearly reveals that the mechanism of the excitation is\nnonlinear with respect to the strength of the THz field.\nThe dependency of the frequency of the oscillations on\nthe applied external magnetic field implies that the oscil-\nlations must be assigned to a spin resonance in the com-\npound. The unusually large effective g-factor deduced\nfrom the measurements and the typical GHz frequency\nsuggests that the oscillationsare associated with the low-\nfrequency FMR mode in the system of two macro spins\nformed by the magnetizations of iron MFeand thulium\nMTmsublattices. To further support this interpretation,\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48 /s52/s48/s48 /s52/s53/s48/s48/s50/s48/s52/s48/s54/s48/s70/s97/s114/s97/s100/s97/s121/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s109/s100/s101/s103/s41\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s97 /s32/s61/s32/s52/s53/s111\n/s97 /s32/s61/s32/s48/s111\n/s97 /s32/s61/s32/s45/s49/s53/s111\n/s97 /s32/s61/s32/s45/s51/s48/s111\n/s97 /s32/s61/s32/s45/s52/s53/s111\n/s97 /s32/s61/s32/s45/s54/s48/s111\n/s97 /s32/s61/s32/s45/s57/s48/s111/s43/s32/s49/s51/s48/s32/s109 /s84 /s45/s32/s49/s51/s48/s32/s109 /s84\nFIG. 5. THz-induced dynamics measured at T= 6 K, for\nseveralαand two opposite field polarities Bext=±130 mT.\nFor the angles αwhere the excitation is strong, the phase of\nthe mode remains unaltered under field reversal. However,\nit appears that the dynamics measured around the minimum\nα= 0◦is affected by the polarity of the field.we propose a theoretical model described in the next sec-\ntion.\nIV. THEORY\nThissectionisstructuredasfollows: insubsection IVA\nwe derived an expression for the FMR frequency. In the\nderivation, we employed an effective ferrimagnetic La-\ngrangian that was simplified to describe dynamics corre-\nsponding to the FMR mode only. A complete analysis\nwith the total effective Lagrangian is presented in Ap-\npendixA. From the derivations, it becomes clear that\nthe spin dynamics can be triggered if THz light acts as\nan effectively rectified magnetic field. Hence, in sub-\nsectionIVB, we phenomenologically describe the rec-\ntification in terms of the inverse Cotton-Mouton effect.\nWe derived the form of the effective rectified magnetic\nfield based on point-group symmetry. Finally, in subsec-\ntionIVC, we combine the Lagrangian with the ICME\ninteraction potential to obtain equations for the THz-\ninduced motion of the spins.\nA. Frequency of FMR\nLetθ,ϕbe the polar and azimuthal angles of\nthe net magnetization M=MFe+MTm≡\nm(sinθcosϕ,sinθsinϕ,cosθ), where m=MFe−MTm.\nThe total effective Lagrangian density describing magne-\ntization dynamicsofatwo-sublatticeferrimagnetis given\nin Appendix A. The equations of motion derived from6\nthis Lagrangian possess two eigenmodes, corresponding\nto the THz exchangemode and the GHz FMR mode [ 13].\nWhen considering only the FMR mode, the description\ncan be simplified using the fact that the involved mag-\nnetizations remain antiparallel during the dynamics [ 30],\nwhile for the exchange mode, the sublattices become mu-\ntually canted. Thewaytoimpose onthe Lagrangianthat\nwe only want to consider the FMR mode is, therefore, to\nenforce the sublattices to remain antiparallel. This can\nbe done by letting the antiferromagnetic exchange cou-\npling parameter λ, which defines the exchange energy\nUex=−λMFe·MTm, go to infinity λ→ −∞. Thisfer-\nromagnetic approximation does not influence the FMR\nmode and its frequency (see Appendix A). In this case,\nthe effective Lagrangian reduces to the low-frequency ef-\nfective Lagrangian density LFM:\nLFM=−m\nγeff˙ϕcosθ−U(θ,ϕ). (2)\nHere,γeffis the effective gyromagnetic ratio γeff≡\ngeffµB//planckover2pi1andU(θ,ϕ) is the static potential energy den-\nsity, which contains uniaxial anisotropy and Zeeman in-\nteraction with the external magnetic field:\nU(θ,ϕ) =−Ku(M·ˆz)2\nm2−M·Hext,(3)\nwhereKu>0 is the uniaxial anisotropy constant and\nHext= (Hx,0,Hz) the external magnetic field (in Tesla).\nSimilar to the experiment, we let the external magnetic\nfield be slightly tilted away from the sample plane, i.e.\nHz/Hx= tanδ(see Fig. 1). The ground state angles θ0,\nϕ0can be found by minimization of ( 3). Afterwards, the\nequationsofmotionthat aregivenbythe Euler-Lagrange\nequations can be linearized around the ground-state an-\nglesθ=θ0+θl,ϕ=ϕ0+ϕlwithθl,ϕl≪1:\n0 =d\ndt∂L\n∂˙θ−∂L\n∂θ≈ −m\nγeff˙ϕlsinθ0+U′′\nθ(θ0,ϕ0)θl,\n0 =d\ndt∂L\n∂˙ϕ−∂L\n∂ϕ≈m\nγeff˙θlsinθ0+U′′\nϕ(θ0,ϕ0)ϕl.(4)\nwhere we introduced the notation U′′\nθ(θ0,ϕ0)≡\n∂2\n∂θ2U(θ,ϕ)/vextendsingle/vextendsingle\nθ=θ0,ϕ=ϕ0. The FMR frequency is given by\nthe eigenfrequency of these coupled equations, which we\nseparately derived for a purely in-plane field and tilted\nfield configuration.\n1. In-plane field ( δ= 0◦)\nWhen ignoring the tilting of the external magnetic\nfield, the static energy potential becomes:\nU(θ,ϕ) =−Kucos2θ−mHxsinθcosϕ.(5)\nMinimization w.r.t. ϕyieldsϕ0= 0 for Hx>0 and\nϕ0=πforHx<0. For the minimization w.r.t. θ, weneed to consider two regimes: when the applied external\nmagnetic field is greater or smaller than the anisotropy\nfieldHa≡2Ku/m:\nθ0=/braceleftigg\nsin−1Hx\nHafor|Hx|< Ha,\nπ/2 for |Hx| ≥Ha.(6)\nUsing these ground-state angles, the eigenfrequency of\nthe equations of motion ( 4) can be found:\nωFM=/braceleftigg\nγeff/radicalbig\nH2a−H2xfor|Hx|< Ha,\nγeff/radicalbig\nHx(Hx−Ha) for|Hx| ≥Ha.(7)\nThe solution for this case ( δ= 0◦) is plotted in Fig. 6,\nwhich shows that the frequency drops to zero when\n|Hext| →Ha, and afterward approaches a linear trend.\nWe do observe the linear increase of the frequency in the\nexperiment, but we do not see such a significant drop at\nlow fields. Therefore, we need to include the small tilt of\nthe external magnetic field δ∝negationslash= 0◦.\n2. Tilted field ( δ/negationslash= 0◦)\nIn the case of a tilted magnetic field, the static poten-\ntial energy is given by:\nU(θ,ϕ) =−Kucos2θ−mHxsinθcosϕ−mHzcosθ.(8)\nAgain we have that ϕ0= 0 for Hx>0 andϕ0=π\nforHx<0, which means Hxcosϕ=|Hx|and we can\nminimize ( 8) only with respect to θ:\n1\nmU′\nθ(θ) = cosθ(Hasinθ−|Hx|)+Hzsinθ= 0.(9)\nTo be able to solve this equation, we treat the out-of-\nplane field as a perturbation with respect to the case of\n/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51/s52/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s103\n/s101/s102/s102/s72\n/s97/s41\n/s72\n/s101/s120/s116/s32/s47/s32/s72\n/s97/s32/s32/s32/s100/s32/s61/s32 /s48/s111\n/s32\n/s32/s100/s32/s61/s32 /s51/s111\n/s32/s100/s32/s61/s32 /s54/s111\n/s100\n/s100\n/s101/s120/s116/s32 \n/s101/s102/s102/s32 \nFIG. 6. Theoretical curves for the FMR frequency as a func-\ntion of the external magnetic field, for different tilting ang les\nδ.7\nan in-plane field. Therefore, we substitute θ=θ0−ǫ\nin Eq. (9) withθ0as in the in-plane case (Eq. ( 6)) and\nassumed ǫ≪1. We retained maximally cubic terms ∼\nǫ3and solved the third-degree polynomial for ǫ(Hx,Hz).The solutions gives the new ground-state angle θ′\n0=θ0−\nǫ(Hx,Hz). From here, we again linearized the Euler-\nLagrangeequations( 4) andcalculatedexpressionsforthe\nFMR resonance mode as presented in Eq. ( 10).\nωFM=\n\nγeff/radicalbigg\nH2a−H2x+/parenleftBig7\n2H2x−2H2a/parenrightBig\nǫ2+/parenleftBig\n3Hx(H2a−H2x)1\n2+HzHx/parenrightBig\nǫ+Hz(H2a−H2x)1\n2\n/radicalbigg\n1−ǫ(H2a/H2x−1)1\n2−1\n2ǫ2for|Hx|< Ha,\nγeff\n(1−1\n2ǫ2)1\n2/radicalig\nHx/bracketleftbig\n(Hx−Ha)+Hzǫ+(2Ha−1\n2Hx)ǫ2/bracketrightbig\nfor|Hx| ≥Ha.(10)\nWe solved the third-degree polynomial of ǫnumerically\nfor different external magnetic fields and tilt angles and\ncalculated the corresponding resonance frequencies. The\nresultisplotted in Fig. 6, whichshowsthat wecanobtain\na better agreement to our data of Fig. 4when account-\ning for the tilted field. We also see that the minimal\nfrequency occurs when the external magnetic field equals\ntheuniaxialanisotropyfield. Lookingbackatexperimen-\ntal data from Fig. 4(b), we see that the dip occurs at ap-\nproximately 25 mT which therefore allows us to estimate\nthe size of the uniaxial anisotropy field Ha≈25 mT.\nHowever, when inserting this value into Eq. 10, our the-\nory predicts ωFM=γeffHaat zero external field which\namounts to a frequency of (2 π)−1ωFM≈2.24 GHz, while\nexperimentally we observe ∼14 GHz (see Fig. 4(b)).\nThis unaccounted frequency offset of about 12 GHz sug-\ngests that we should include some effective biasing field\nwithamagnitudeofabout130mT.Atthispoint,wehave\nno definite answer to where this field stems from. Shape\nanisotropy should play a role given the clear domain pat-\nterns seen in this sample without an external magnetic\nfield [12]. Alternatively, the so-called “double umbrella\nstructure” - a noncollinear arrangement of the thulium\nions seen at helium temperatures in TmIG using neutron\ndiffraction [ 33,34] - might be involved. Also, strain in\nthe sample induced by the Gd 3Fe5O12substrate could be\nthe sourceofthe missingfield [ 35]. Fortunately, thisopen\nquestion does not obstruct the theoretical treatment of\nthe rectified effective magnetic field by the ICME which\nwill be held in the coming section.\nB. Energy considerations for the ICME\nInthissection, wederiveaneffectiveinteractionpoten-\ntialstartingfromthemostbasicprinciplesoflight-matter\ninteraction. This interaction potential will then enter the\nLagrangianin the next section. When an oscillating elec-\ntric field of light E(t) enters a non-absorbing medium,\nit interacts with the medium by inducing electric polar-\nization. The change in interaction energy density dW\ndue to the increase of electric polarization d Pis givenbydW=−E·dP. Here we ignored higher-order multi-\npole contributions as well as magnetic dipole interactions\nbecause they are expected to be weak. In the linear op-\ntical approximation, the amount of induced polarization\nis linear to the applied electric field Pi=χijǫ0Ej, with\nǫ0the dielectric permittivity of vacuum and χijthe elec-\ntric susceptibility tensor. Therefore, the total interaction\npotential energy density W, after integration, becomes:\nW=−1\n2χijǫ0EiEj. (11)\nThe electric susceptibility tensor is related to the dielec-\ntric permittivity ǫij= (1+χij)ǫ0. The presence of static\nmagnetization induces magneto-optical birefringence in\nthe medium, and modifies this dielectric permittivity\n[36]:\nδǫij=kijkMk+ξijklMkMl+···,(12)\nwhere the third-rank antisymmetric axial tensor kijkand\nfourth-rank symmetric polar gijkltensor describe the\nwell-known Faraday and Cotton-Mouton effects , respec-\ntively. The relevant interaction potential that describes\nthe interaction of light and magnetization can then be\nfound by substituting ǫ0(δχij) = (δǫij) from Eq. ( 12) in\n(14) to obtain [ 10]:\nW=−1\n2kijkEiEjMk−1\n2ξijklEiEjMkMl.(13)\nOn one hand, magnetization may therefore induce bire-\nfringence in the medium and modify the properties of\nlight via the well-known Faraday and Cotton-Mouton ef-\nfects. On the other hand, inversely, the presence of light\nmay induce magnetization. That is, by thermodynamics,\nthe electric field oflight acts asan effective magnetic field\nHeff=−δW\nδM[37]. The appearance of an effective field\noriginating from the two terms in Eq. ( 13) are known as\nthe inverse Faraday effect (IFE) and ICME, respectively.\nTo obtain an expression for the effective fields, we\nmake a Fourier expansion of the wave and assume it\nis monochromatic (the extension to non-monochromatic\nlight is straightforward) E(t) = Re[E(ω)exp(−iωt)] =8\n1\n2[E(ω)exp(−iωt)+E∗(ω)exp(iωt)], where E(ω) is the\n“Jones vector”. The Fourier components of the polariza-\ntionareagainrelatedtothoseofthe electricfield Pi(ω) =\nǫ0˜χij(ω)Ej(ω) where ˜χij(ω) is the optical susceptibility.\nBecause we consider a non-absorbingmedium, the tensor\n˜χij(ω) is required to be Hermitian ˜ χ∗\nij(ω) = ˜χji(ω) [38].\nUsing this fact, the interaction potential ( 14) becomes\n[39]:\nW=−1\n4ǫ0˜χijE∗\niEj+h.f., (14)\nwhere we neglect the high-frequency ( h.f.) terms as they\naverageout on the relevant timescales [ 38,40]. Similarly,\nEq. (13) becomes:\nW=−1\n4kijkE∗\niEjMk−1\n4ξijklE∗\niEjMkMl(15)\nBy Neumann’s principle [ 41], the tensors kijkandξijkl\nshould be invariant under the crystallographic point\ngroup operations. In an isotropic or cubic medium, kijk\nis an antisymmetric imaginary tensor with only a sin-\ngle nonzero tensor component kxyz=kzxy=kyzx=\n−kxzy=−kyxz=−kzyx=−ik[40,42,43], and the\ncorresponding effective field is:\nHIFE=ik\n4E(ω)×E∗(ω). (16)\nTherefore, a circularly polarized light pulse generates a\nrectified effective magnetic field with opposite directions\nforleft/right-handedpolarizedlight[ 40,42,44,45]. How-\never, for linearly polarized light, the effective field is zero.\nGiven that ourTHz pulses are linearlypolarized, we only\nconsider the inverse Cotton-Mouton field [ 15]:\n(HICME)l=ξijkl\n2E∗\ni(ω)Ej(ω)Mk.(17)\nUsing the crystallographic point group Oh, we can find\ntheminimalexpressionforthetensor ˆξ. Thenonzeroten-\nsor components in the standard cubic coordinate system\nx∝bardbl[100],y∝bardbl[010],z∝bardbl[001] are tabulated in Ref. [ 43],\nwhere it can be seen that the tensor has only two inde-\npendent components ξxxxxandξxxyy. We transformed\nthe tensor ξijklto our experimental coordinate system\n(see Fig. 1), where z∝bardbl[111],x∝bardbl[112] andy∝bardbl[110]. In\nthis coordinate system, the tensor can be expressed in\nVoight notation by a 6 ×6 matrix ˜ξ:\n˜ξ=\n3ξ1ξ1ξ1+ξ20−√\n2ξ20\nξ13ξ1ξ1+ξ20√\n2ξ20\nξ1+ξ2ξ1+ξ23ξ1−ξ20 0 0\n0 0 0 ξ1+ξ20√\n2ξ2\n−√\n2ξ2√\n2ξ20 0 ξ1+ξ20\n0 0 0√\n2ξ20ξ1\n\nwhereξ1≡ξxxxx\n6+ξxxyy\n2andξ2=ξxxxx\n6−ξxxyy\n2. Thisgives the ICME interaction energy from Eq. ( 13):\nWICME=−ξ1\n2/parenleftbig\n2(ExMx+EyMy)2+E2\nTHzm2/parenrightbig\n−ξ2\n2/parenleftig\nE2\nTHzM2\nz+2√\n2MxMz(E2\ny−E2\nx)\n+4√\n2ExEyMyMz/parenrightig\n.(18)\nwhere we used that m=/summationtext\ni/radicalbig\nM2\niconstant. Note that\nthe latter acts as a holonomic constraint on the system,\nmaking the expression for the ICME field as in Eq. ( 17)\na bit naive because it presumes no constraints on the\nvariables Mi. Although such a constraint complicates a\nNewtonian approach to describe the influence of HICME\nonM, in the Lagrangian approach treated in the next\nsubsection the problem is naturally circumvented.\nThe Newtonian approach is anyhow insightful and we\nbriefly treat it here before going back to the Lagrangian\napproach. We can obtain the approximate ICME field\nby assuming that |Hext| ≥Hawhile ignoring the tilt\nMz≪Mxsuch that M=Mxˆx. In that case, Mxcan\nbe regarded as constant (only changing sign for ±Hext),\nwhileMy,zare two independent variables. Furthermore,\nthe THz electric field ETHz(t) lies in the xyplane, and\ntherefore ETHz(t) =ETHz(t)(sinα,cosα,0). Then, an\napproximate expression for the inverse Cotton-Mouton\nfield derived from Eq. ( 13) is given by:\nHICME=E2\nTHzMx\n0\nξ1sin2α√\n2ξ2cos2α\n.(19)\nGiven that the THz pulse duration of about tTHz∼1 ps\nismuchshorterthanthatoftheperiodoftheFMRmode,\nwe could treat the effect of this field on magnetization M\nas an instantaneous (impulsive) torque τ=M×HICME,\nwhich triggers dynamics that can be described using the\nLandau-Lifshitz equationdM\ndt=−γM×H, with the ini-\ntial condition M(t= 0) = (Mx,0,0):\n˙M(t= 0) =γM2\nx\n0√\n2ξ2cos2α\n−ξ1sin2α.\n/integraldisplaytTHz\n0/bracketleftbiggd\ndtE2\nTHz/bracketrightbigg\ndt,\n(20)\nwhereETHz(t) is the THz pulse electric field. It can be\neasily seen that the torque fulfills the quadratic depen-\ndence with respect to the THz field amplitude ∼E2\nTHz.\nMoreover, in order to have maximal torque at α=±45◦,\nwe find out that ξ1must be dominant over ξ2, i.e.ξ1≪\nξ2. At the same time, it can be shown that the latter is\nalso required for the invariance of the phase of the ob-\nserveddynamicsunderfield reversalasweshowedexperi-\nmentally in Fig. 5. This can be derived fromthe fact that\nwe are only sensitive to the z-projection of the magneti-\nzation, while the precession of the magnetization around\nthe external magnetic field is always right-handed. If the\ntermξ2would have been dominant, magnetization pre-\ncession would have been launched with a z−projection9\nthat has an exactly opposite phase for ±Hext, unlike the\nexperiment. This invarianceofthe phaseon fieldpolarity\nwhenξ1≪ξ2can be seen directly in the coming section\nafter we solved the Lagrangian equations of motion.\nC. Lagrangian equations of motion driven by the\nICME\nTo include light-matter interaction in the Lagrangian,\nwe introduce the interaction energy f(θ,ϕ):\nLFM=−m\nγeffcosθ˙ϕ−U(θ,ϕ)+f(θ,ϕ),(21)\nwhich contains Zeeman interaction with a dynamic THz\nmagnetic field h(t) as well as the ICME of a THz electric\nfieldE(t) that is derived from the interaction potential\nin Eq. (18):\nf(θ,ϕ) =msinθ(hxcosϕ+hysinϕ)\n+m2ξ1sin2θ/bracketleftbig\nE2\nxcos2ϕ+2ExEycosϕsinϕ+E2\nysin2ϕ/bracketrightbig\n+m2ξ2\n2/bracketleftig\nE2cos2θ+2√\n2sinθcosθcosϕ/parenleftbig\nE2\ny−E2\nx/parenrightbig\n+4√\n2ExEysinθcosθsinϕ/bracketrightig\n. (22)\nNote that for the full Lagrangian (see Appendix A), the\ndrivingenergycontainsseveralothertermsthatareperti-\nnent to ferrimagnets (and absent for ferromagnets). The\nequations of motion derived from the Euler-Lagrange\nequations, after linearization, are given by:\nU′′\nθ(θ0,ϕ0)θl−sinθ0m\nγeff˙ϕl=fθ(t),\nU′′\nϕ(θ0,ϕ0)ϕl+sinθ0m\nγeff˙θl=fϕ(t),(23)\nwhere the generalized force terms fνare defined for ν=\nθ,ϕand are evaluated around the ground-state angles:\nfν(t) =−/bracketleftbiggd\ndt/parenleftbigg∂f\n∂˙ν/parenrightbigg\n−∂f\n∂ν/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nν=ν0=∂f\n∂ν/vextendsingle/vextendsingle/vextendsingle\nν=ν0,(24)\nThese generalized forces can be interpreted as torque.\nWhen ignoring the small canting angle δ, we obtain the\nfollowing driving terms evaluated for the case |Hx| ≥Ha:\nfθ(t) =−√\n2m2ξ2E2\nTHzcosϕ0cos2α,\nfϕ(t) =mhycosϕ0+m2ξ1E2\nTHzcos2ϕ0sin2α.(25)\nThe driving term fθ(t) proportional to ξ2is maximal\nwhenα= 0◦,90◦, in contrast with the experiment.\nTherefore, we again conclude that ξ2≪ξ1and essen-\ntially ignore this term. Next, as expected, it can be seen\ninfϕ(t) that the THz magnetic field component hy(t),\nwhich is perpendicular to the equilibrium magnetization,\ncould drive the precession. However, this term scales\nlinearly with the THz amplitude, which is also not thedominant mechanism in the experiment. Theoretically,\nthis term is also negligible as we will treat the excita-\ntion as instantaneous in which such linear driving terms\ndisappear because/integraltexttTHz\n0h(t)dt= 0. Therefore, we only\nconsider the ICME term in fϕthat is proportional to ξ1.\nExperimentally, we are only sensitive to the z-\ncomponent of the magnetization Mz(t) =mcosθ(t)≈\n−mθl(t)(for|Hx| ≥Hainwhichcase θ0=π/2). Inother\nwords, we only detect dynamics of θl(t). The equation\nof motion for θl(t), found by differentiation and mutual\nsubstitution of Eqns. ( 23), is given by:\n¨θl+ω2\nFMθl=mγeffξ1/bracketleftbiggd\ndtE2\nTHz/bracketrightbigg\ncos2ϕ0sin2α.(26)\nWe treat the excitation as instantaneous (see “photonic\nimpact” [ 46]), in which case the excited dynamics is fully\ndetermined by the initial condition θl(0) = 0 and:\n˙θl(0) =mγeffξ1cos2ϕ0sin2α/integraldisplaytTHz\n0/bracketleftbiggd\ndtE2\nTHz/bracketrightbigg\ndt.(27)\nAsϕ0= 0 for Hx>0 andϕ0=πforHx<0, we\nhave that cos2ϕ0= 1 in both cases such that a change in\nfield polarity does not change the phase of the dynamical\nout-of-plane magnetization ˙Mz(0)≈ −m˙θl(0), just as we\nobserved in the experiment (see Fig. 5). Moreover, the\nexcitation is maximal and has a mutually opposite sign\nforα=±45◦, and scales quadratically with the THz\nfield. Therefore, all experimentally observed features re-\ngarding the excitation are captured by this Lagrangian\napproach.\nV. 2D SPECTROSCOPY AND DISCUSSION\nThe predictions made by the phenomenological treat-\nmentoftheICMEfittheexperimentaldataverywell, but\nit does not yet clarify the underlying microscopic mecha-\nnism. The effect is conventionally seen as a consequence\nof impulsive stimulated Raman scattering (ISRS) involv-\ning either resonant or off-resonant excitation of a certain\nelectronic transition [ 17,42,47–51]. It is thus interesting\nif we are able to assign a certain transition that medi-\nates the ISRS process. Here, it is important to take into\naccount the possibility of the recently discovered mecha-\nnism of magnonic Raman scattering [ 52], where the GHz\nq-FM resonance in FeBO 3was excited through resonant\nRaman scattering of the q-AF THz resonance. A notable\nsimilarity between FeBO 3and the current experiment is\nthat the maxima of excitation both occur when the THz\npolarization is α=±45◦from the net magnetization.\nBut at the same time, the materials are largely differ-\nent since FeBO 3is an antiferromagnet with N´ eel vector\nL⊥M, while collinear ferrimagnets such as TmIG have\nL∝bardblM. In any case, if the exchange mode mediates\nthe excitation of the FMR mode through magnonic Ra-\nman scattering, it should become visible in 2D THz spec-\ntroscopy [ 52]. In this technique, instead of only one THz10\npulse, two THz pulses are applied at a mutual delay τ.\nThis allows us to track the Faraday rotation signal of the\nprobe pulse when both THz pulses arepresent θF,12(t,τ),\nbutalsothesignals θF,1(t,τ) andθF,2(t,τ)obtainedwhen\nonly oneofthe twopulses excited the sample. From here,\nwe obtained the nonlinear signal:\nθF,NL(t,τ)≡θF,12(t,τ)−θF,1(t,τ)−θF,2(t,τ).(28)\nIf the exchange mode mediates the excitation of the\nFMR mode, this should become apparent as periodicity\nin the excitation efficiency of the FMR mode with a pe-\nriod defined by the frequency of the exchange mode. We\nperformed this experiment and the result of θF,12(t,τ) is\nshown in Fig. 7. The measured signal does not show any\nperiodicity of the mode, and neither did we observea sig-\nnal in the nonlinear part calculated with Eq. ( 28). The\nonly logical explanation for the absence of a nonlinear\nsignal in 2D spectroscopy is that the nonlinear excita-\ntion is mediated by a state with a short lifetime <1 ps.\nIn this case, no information will be carried over between\ntwosubsequentTHzpulseswhichareseparatedbyatime\nlonger than their own pulse duration of ∼1 ps. In other\nwords, the second pulse delayed at a time τ >1 ps will\nnever be able to experience the presence of the first THz\npulse, meaningthatsuperpositionholdsandnononlinear\nsignal (see Eq. 28) will come up. Therefore, the THz ex-\nchange mode does not mediate the ISRS mechanism and\nis not responsible for the excitation of the FMR mode.\nFigure8illustrates the mechanism of ISRS via a\nshort-lived excited state. The process could occur com-\npletely off-resonantly via electronic or phononic transi-\ntions which lie beyond the THz excitation spectrum, in\nFIG. 7. Result of θF,12(t,τ) measured in 2D THz spec-\ntroscopy. Fast modulations shortly after the pulse arrival as-\nsociated with the exchange mode can be observed, as well as\nfaint contours which correspond to the FMR mode at longer\ntimescales. However, no modulations in the excitation effi-\nciency as a function of τof the latter mode were found.\nFIG. 8. Schematic illustration of ISRSfrom a short-lived el ec-\ntronic excited state, which is believed to be the microscopi c\nmechanism for the ICME.\nwhich case the excited state is virtual. However, a reso-\nnant excitation via a state that lies within the THz pulse\nspectrum is expected to be more likely. The only candi-\ndates for such an excited state are the crystal-field split\ntransitions of the Tm3+ground state. The dodecahe-\ndral crystal-field environment of the Tm3+(4f12) ions,\nwith local symmetry described by the dihedral point-\ngroup D 2, splits its ground-state multiplet3H6(J= 6)\ninto 2J+ 1 = 13 Stark levels. The energy level dia-\ngram for substituted Tm3+ions in dodecahedral D 2sites\nin Y3Al5O12was earlier obtained from experimental ab-\nsorption and emission spectra in [ 53,54] and more re-\ncently it was studied in [ 55]. The lowest electric-dipole-\nallowed transition has an energy of 27 cm-1correspond-\ning to about 0 .809 THz. This energy is in the vicinity of\na spectral feature we actually observed in this material\n[13], but which we couldn’t assign. Such a crystal-field\nsplit transition can be very short-lived and effectively act\nas an effective virtual electronic state from which light\ncan scatter. Moreover, since there are no other transi-\ntions in our spectrum, we conjecture by deduction that\nISRS of the lowest crystal-field split state of thulium is\nthe responsible microscopic mechanism of excitation (see\nFig.8for a schematic illustration of ISRS). In order to\nsubstantiate this hypothesis, one must develop a micro-\nscopic theory that takes into account both the Tm-Fe ex-\nchange interaction and the electronic structure of Tm3+.\nThis problem is beyond the scope of this article.\nVI. CONCLUSION\nWe showed that a single-cycle THz pulse is able to ex-\ncite a GHz magnon in ferrimagnetic TmIG. The experi-\nmental dependencies reveal that the excitation is a result\nof ICME, where the THz field of light becomes effectively\nrectified to generate a unipolar magnetic field pulse. The\nresultsaresupportedbytheequationsofmotionobtained\nfrom an effective ferrimagnetic Lagrangian, using a phe-\nnomenological expression for the rectified field. We dis-\ncussed the possible microscopic picture of ICME by con-11\nsidering the mechanism of ISRS. 2D spectroscopy ruled\nout magnon-magnonscattering, similar to what occurred\nin FeBO 3[52], to be responsible. Instead, we conjectured\nthat the effect is more similar to that in TmFeO 3[10],\nand is based on light-induced scattering from the crystal-\nfield split electronic states of Tm3+.\nOur results demonstrate that nonlinear THz optomag-\nneticeffectsdononotonlyplayaroleinantiferromagnets\nbut also in ferrimagnetic materials. In general, this non-\nlinearity facilitates a channel of energy transfer from the\nelectric field of light to the magnetic spin system. Such a\nchannel recently enabled coherent steering of spins overa\npotentialbarrierinantiferromagneticTmFeO 3[11]. Sim-\nilarly, our results, therefore, open a way for future data-\nwriting, spintronics, and magnonics applications basedon ferrimagnets.\nACKNOWLEDGMENTS\nThe authors thank Sergey Semin and Chris Berkhout\nfor their technical support. The work was supported by\nthe Dutch Research Council (NWO). The authors de-\nclare that this work has been published as a result of\npeer-to-peer scientific collaboration between researchers.\nThe provided affiliations represent the actual addresses\nof the authors in agreement with their digital identifier\n(ORCID) and cannot be considered as a formal collabo-\nration between the aforementioned institutions.\nAppendix A: Total Effective Lagrangian\nThe total effective Lagrangian density of a two-sublattice ferrima gnet in the vicinity of the compensation tem-\nperature can be written in terms of the polar and azimuthal angles o f the net magnetization M=MFe+MTm≡\nm(sinθcosϕ,sinθsinϕ,cosθ) [13,16]:\nLeff=χ⊥\n2\n/parenleftigg˙θ\nγ+Hxsinϕ/parenrightigg2\n+/parenleftbigg/parenleftbigg˙ϕ\nγ−Hz/parenrightbigg\nsinθ+Hxcosθcosϕ/parenrightbigg2\n−m˙ϕ\nγeffcosθ−U(θ,ϕ)+¯f(θ,ϕ),(A1)\nwhereχ⊥≡1/|λ|withλ <0 the exchange coupling parameter that defines the exchange ene rgy density Uex=\n−λMFe·MTm, andγ≡(MFe+MTm)/(MFe/γFe+MTm/γTm). The static potential Ucontains Zeeman interaction\nwith the external magnetic field and magnetic anisotropy as defined in Eq. (3). The interaction energy density ¯f(θ,ϕ)\nis equal to:\n¯f(θ,ϕ) =χ⊥\n2/bracketleftigg\ncos2θ(hxcosϕ+hysinϕ)2+2/parenleftbigg/parenleftbigg˙ϕ\nγ−Hz/parenrightbigg\nsinθ+Hxcosθcosϕ/parenrightbigg\ncosθ(hxcosϕ+hysinϕ)\n+2/parenleftigg˙θ\nγ+Hxsinϕ/parenrightigg\n(hxsinϕ−hycosϕ)+(hxsinϕ−hycosϕ)2/bracketrightigg\n+msinθ(hxcosϕ+hysinϕ)\n+m2ξ1sin2θ/bracketleftbig\nE2\nxcos2ϕ+2ExEycosϕsinϕ+E2\nysin2ϕ/bracketrightbig\n+m2ξ2\n2/bracketleftig\nE2cos2θ+2√\n2sinθcosθcosϕ/parenleftbig\nE2\ny−E2\nx/parenrightbig\n+4√\n2ExEysinθcosθsinϕ/bracketrightig(A2)\nDamping can be included through the Rayleigh function R=αM\n2γ/parenleftig\n˙θ+sin2θ˙ϕ/parenrightig\n, whereM ≡MFe+MTm. The\ngeneral equations of motion are defined by the Euler-Lagrange eq uations:\nd\ndt∂Leff\n∂˙θ−∂Leff\n∂θ+∂R\n∂˙θ= 0,d\ndt∂Leff\n∂˙ϕ−∂Leff\n∂ϕ+∂R\n∂˙ϕ= 0. (A3)\nThe result can be linearized around the ground-state angles θ=θ0+θl,ϕ=ϕ0+ϕl,θl,ϕl≪1 (where θ0,ϕ0are\nas in Sec IVA). This gives the linearized equations of motion:\n¨θl+ζ˙θl+γ2\nχ⊥U′′\nθ(θ0)θl−sinθ0γ2\nγeff|λ|m˙ϕl=γ2\nχ⊥¯fθ(t),¨ϕl+ζ˙ϕl+γ2U′′\nϕ(θ0,ϕ0)\nχ⊥sinθ0ϕl+γ2\nγeff|λ|m˙θl=γ2\nχ⊥¯fϕ(t)\nsinθ0,\n(A4)\nwhereζ=α¯γM\nχ⊥and the driving terms can be calculated from the interaction energy ¯ffrom (A2) using equation ( 24).\nThese equations have two eigenfrequencies, one corresponding t o the exchange mode ωex≈ |λ|(γTmMFe−γFeMTm),12\nand one corresponding to the FMR mode whose frequency is given by Eq. (10). The driving terms after linearization,\ntaking only the leading contributions within the first order of ǫinto account, are given by:\n¯fθ(t) =χ⊥γ−1˙hycosϕ0+√\n2m2ξ2cosϕ0/parenleftbig\nE2\nx−E2\ny/parenrightbig\n+ǫ/parenleftbig\nmhxcosϕ0+2m2ξ1E2\nxcos2ϕ0−m2ξ2E(t)2/parenrightbig\n¯fϕ(t) =/parenleftbig\nmhycosϕ0+2m2ξ1ExEycos2ϕ0/parenrightbig\n+ǫ/parenleftig\n−χ⊥γ−1˙hxcosϕ0+2√\n2m2ξ2ExEycosϕ0/parenrightig(A5)\nwhere we also ignored the terms −ǫχ⊥cos2ϕ0(h2\nx+2Hxhx) in¯fθ(t) and−ǫχ⊥Hzhycosϕ0in¯fϕ(t) because they are\nsmall. The expressions ( A5) display a weak field-derivative Zeeman torque ∝ǫ˙hx(whereǫ≪1) when the THz\nmagnetic field is along the x-axis (˙hy= 0,˙hx∝negationslash= 0), being even zero when there is no tilt of the field ǫ= 0. On the\nother hand, there is a strong field-derivative torque ∝˙hywhen the THz magnetic field is aligned along the y-axis,\nwhich is perpendicular to the equilibrium spin-direction and thus M. 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Takanashi, Observation of inverse spin Hall effect\nin ferromagnetic FePt alloys using spin Seebeck effect,\nApplied Physics Letters 107, 092401 (2015) .arXiv:2305.02971v1  [cond-mat.mtrl-sci]  4 May 2023Supplemental Material for “Effective rectification of THz\nelectromagnetic fields in a ferrimagnetic iron garnet”\nT.G.H. Blank,1,2E.A. Mashkovich,3K.A. Grishunin,1C. Schippers,2\nM.V. Logunov,4B. Koopmans,2A.K. Zvezdin,5and A.V. Kimel1\n1Radboud University, Institute for Molecules and Materials , 6525 AJ Nijmegen, the Netherlands.\n2Department of Applied Physics, Eindhoven University of Tec hnology,\nP.O. Box 513, Eindhoven 5600 MB, the Netherlands.\n3University of Cologne, Institute of Physics II, Cologne D-5 0937, Germany.\n4Kotel’nikov Institute of Radioengineering and Electronic s, 125009 Moscow, Russia.\n5Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia.\n(Dated: May 5, 2023)\nXRD ANALYSIS\nWe performed X-ray diffraction (XRD) analysis on the sample to obta in the in-plane crystallo-\ngraphic orientation. Fig. S1(a) depicts the experimental setup. The scattering plane of the s ample\ncan be tilted at an angle ψ, and can be rotated around its normal axis by an angle φ. Because cubic\ncrystals are self-dual, we can identify the Miller planes ( hkl) to be simply the planes perpendicular\nto the vector [ hkl].\nThe data for φ= 0◦, andψ= 0◦confirms the [111] orientation of the sample [ ?]. From the\nillustrations in Fig. S1(c), it can be seen that after tilting the sample at ψ= 54.74◦(angle between\n[111] and [100]), it should be possible to let the scattering plane coincid e with one of the base\nvector planes (100), (010) or (001) by rotating the sample over a certain angle φ. Therefore, we\nputψ= 54.74◦and set the radiation and detection angle at 2 θ= 58.86◦corresponding to a (800)\npeak to recognize a (100) plane. By scanning the rotation angle φ, we obtained three candidates\natφ= 0◦,33.3◦,66.6◦(see Fig. S1(d)). Due to the three-fold symmetric [111] axis, the peaks at\nφ+n·2π/3 are equivalent. The data in Fig. S1(e) establishes that only φ= 33.3◦(and equivalent\nangles) coincides with a (100) plane. For additional confirmation, we note that by tilting the sample\natψ= 19.47◦, the same rotation φ= 33.3◦should correspond to a (211) plane. This is confirmed\nin Fig.S1(f). Therefore, we conclude that the crystallographic axes in our experiment are oriented\nlike the dotted arrows in Fig. S1(g).\nNote that although the peaks at φ+n·2π/3 withn= 0,1,2 should be equivalent, it turns out\nthat the equivalent axes for φ= 33.3◦were rather φ= 150◦andφ= 270◦, meaning that the first\nrotation is slightly shifted. Similar results were obtained in Ref. [ ?] and were explained as a result\nof strain between the thulium iron garnet film and the Gd 3Fe5O12substrate.\n1Figure S1: (a) Schematic illustration of the angles involved in the XRD analysis of the sample, where x,\nyandzcoincide with the experimental axes from Fig. 1(a) of the mai n article. (b) XRD data for the\nsample plane, which confirms the [111]orientation of the sample [ ?]. (c) Illustration of the cubic crystal\naxes and the two-dimensional projection seen from the [111]direction. (d) Scattering intensity at\n2θ= 58.86◦((800)) for different sample rotations φ, where the sample tilt is set at ψ= 54.74◦which is\nequal to the angle between the [111]and[100]axes. This data suggests three possible candidates\nφ= 0◦,33.3◦,66.6◦which could coincide with a (100)plane. (e) Angular scattering intensity\nmeasurements for a tilting angle of ψ= 54.74◦and for the three different candidate angles φ. The data at\nφ= 33.3◦shows what is typical for a (100)plane. In (f), we perform similar measurements, but for a\ntilting angle ψ= 19.47◦, corresponding to the angle between the [111]and[112]axes. The result confirms\nthatφ= 33.3◦corresponds to a (211)plane. (g) The results allowed us to conclude that the project ions of\nthe cubic axes are rotated at an angle of 33.3◦compared to the drawing in (c), meaning that the\nexperimental xandyaxis correspond to the [112]and[110]axes, respectively.\n2"    },    {        "title": "2102.02116v2.Infinite_Series_of_Ferrimagnetic_Phases_Emergent_from_the_Gapless_Spin_Liquid_Phase_of_Mixed_Diamond_Chains.pdf",        "content": "arXiv:2102.02116v2  [cond-mat.str-el]  6 Apr 2021Journal of the Physical Society of Japan FULL PAPERS\nInfinite Series of Ferrimagnetic Phases Emergent from the Ga pless Spin\nLiquid Phase of Mixed Diamond Chains\nKazuo Hida∗\nProfessor Emeritus, Division of Material Science, Graduat e School of Science and Engineering,\nSaitama University, Saitama, Saitama, 338-8570\n(Received )\nThe ground-state phases of mixed diamond chains with ( S,τ(1),τ(2)) = (1/2,1/2,1), where Sis the\nmagnitude of vertex spins, and τ(1)andτ(2)are those of apical spins, are investigated. The apical spin s\nτ(1)andτ(2)are connected with each other by an exchange coupling λ. Other exchange couplings are set\nequal to unity. This model has an infinite number of local cons ervation laws. For large λ, the ground state\nis equivalent to that of the uniform spin 1 /2 chain. Hence, the ground state is a gapless spin liquid. For\nλ≤0, the ground state is a Lieb-Mattis ferrimagnetic phase wit h spontaneous magnetization msp= 1\nper unit cell. For intermediate λ, we find a series of ferrimagnetic phases with msp= 1/pwhereptakes\npositive integer values. The phases with p≥2 are accompanied by the spontaneous breakdown of the\np-fold translational symmetry. It is suggested that the phas e with arbitrarily large p, namely infinitesimal\nspontaneous magnetization, is allowed as λapproaches the transition point to the gapless spin liquid p hase.\n1. Introduction\nThe quantum effects in low-dimensional frustrated\nmagnets have been extensively studied in recent con-\ndensed matter physics.1,2)Various exotic quantum\nphases emerge from the interplay of quantum fluctuation\nand frustration. Among them, the diamond chain,3–9)\nwhose lattice structure is shown in Fig. 1, is known as a\nmodel with an infinite number of local conservationlaws.\nThe ground states can be classified by the correspond-\ning quantum numbers. If the two apical spins have equal\nmagnitudes, which is the case widely investigated, each\npair of apical spins can form a singlet dimer. It cuts the\ncorrelation between both sides and the ground state is a\ndirect product of the cluster ground states separated by\ndimers.\nThe ground states of spin-1/2 diamond chains have\nbeen investigated in Ref. 3. In addition to the spin clus-\nter ground states, the ferrimagnetic state with sponta-\nneous magnetization msp= 1/2 per unit cell is found.\nIn the latter phase, the apical spins form triplet dimers\nand all the spins collectively form a long-range ordered\nferrimagnetic state. Extensive experimental studies have\nbeen also carried out on the magnetic properties of nat-\nural mineral azurite that is regarded as an example of\ndistorted spin-1/2 diamond chains.8,9)\nThe ground states of spin-1 diamond chains have been\nalso investigated in Refs. 3 and 4. In addition to the spin\ncluster ground states, the nonmagnetic Haldane state\nand the ferrimagnetic states with spontaneous magne-\ntizationmsp= 1 and 1/2 are found. It should be noted\nthat the latter ferrimagnetic state is accompanied by a\nspontaneous translational symmetry breakdown.\n∗E-mail address: hida@mail.saitama-u.ac.jpOn the other hand, if the magnitudes of the two apical\nspins are not equal to each other, they cannot form a sin-\nglet dimer. Hence, all spins in the chain inevitably form a\nmany-body correlatedstate. In manycases, (quasi-)long-\nrange order evolves including the vertex spins. As a sim-\nple example of such cases, we investigate the mixed di-\namond chain with apical spins of magnitude 1 and 1/2,\nand vertex spins, 1/2 in the present work. Remarkably,\nwe find an infinite series of ferrimagnetic phases.\nThispaperisorganizedasfollows.InSect.2,themodel\nHamiltonian is presented. In Sect. 3, the ground-state\nphase diagram is determined numerically. The behavior\nof the spontaneous magnetization in each phase is pre-\nsented and analyzed. The last section is devoted to a\nsummary and discussion.\n2. Hamiltonian\nWe consider the Hamiltonian\nH=L/summationdisplay\nl=1/bracketleftBig\nSl(τ(1)\nl+τ(2)\nl)\n+(τ(1)\nl+τ(2)\nl)Sl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,(1)\nwhereSl,τ(1)\nlandτ(2)\nlare spin operators with magni-\ntudesSl=τ(1)\nl= 1/2 andτ(2)\nl= 1. The number of the\nunit cells is denoted by L, and the total number of sites\nis 3L. Here, the parameter λcontrols the frustration as\ndepicted in Fig. 1.\nThe Hamiltonian (1) has a series of local conservation\nlaws. To see it, we rewrite Eq. (1) in the form,\nH=L/summationdisplay\nl=1/bracketleftbigg\nSlTl+TlSl+1+λ\n2/parenleftbigg\nT2\nl−11\n4/parenrightbigg/bracketrightbigg\n,(2)\n1J. Phys. Soc. Jpn. FULL PAPERS\nSlλ\n11τl(1)\nSl+1\nτl(2)11 S=τ(1)=1/2\nτ(2)=1\nFig. 1. Structure of the diamond chain investigated in this work\nS=τ(1)= 1/2 andτ(2)= 1.\nwhere the composite spin operators Tlare defined as\nTl≡τ(1)\nl+τ(2)\nl(l= 1,2,···,L).(3)\nThen, it is evident that\n[T2\nl,H] = 0 (l= 1,2,···,L). (4)\nThus, we have Lconserved quantities T2\nlfor alll. By\ndefining the magnitude Tlof the composite spin Tlby\nT2\nl=Tl(Tl+1), we have a set of good quantum numbers\n{Tl;l= 1,2,...L}whereTl= 1/2 and 3/2. The total\nHilbertspaceoftheHamiltonian(2)consistsofseparated\nsubspaces, each of which is specified by a definite set of\n{Tl},i.e.,asequenceof1/2and3/2.Apairofapicalspins\nwithTl= 1/2 is called a doublet (hereafter abbreviated\nas d) and that with Tl= 3/2 a quartet (abbreviated as\nq).\n3. Ground-State Phase Diagram\n3.1 Ground states for λ≫1\nForλ≫1,∀l Tl= 1/2. Hence, this model is equiva-\nlent to the spin-1/2 antiferromagnetic Heisenberg chain\nwhose ground state is a gapless spin liquid.\n3.2 Ground states for λ≪1\nForλ≪1,∀l Tl= 3/2. Hence, this model is equiv-\nalent to the spin-1/2-3/2 alternating antiferromagnetic\nHeisenberg chain whose ground state is a ferrimagnetic\nstate with spontaneous magnetization msp= 1 per unit\ncell according to the Lieb-Mattis theorem.10)Here,msp\nis defined by\nmsp=1\nLL/summationdisplay\nl=1(/an}bracketle{tSz\nl/an}bracketri}ht+/an}bracketle{tTz\nl/an}bracketri}ht) (5)\nwhere/an}bracketle{t/an}bracketri}htdenotes the expectation value in the ground\nstate with an infinitesimal symmetry breaking magnetic\nfield inz-direction.\n3.3 Intermediate λ\nIn the absence of spontaneous translational symmetry\nbreakdown, only above two phases are allowed. To pur-\nsue the possibility of other phases, we employ the finite\nsize DMRG method with the geometry of Fig. 2. TheSlλ\n11τl(1)\nSl+1\nτl(2)11\nFig. 2. Lattice structure used for the finite size DMRG calcula-\ntion\ncorresponding Hamiltonian is given by\nH=L/summationdisplay\nl=1SlTl+L−1/summationdisplay\nl=1TlSl+1+L/summationdisplay\nl=1λ\n2/parenleftbigg\nT2\nl−11\n4/parenrightbigg\n.(6)\nThis geometry is chosen to allow for the nonmagnetic\nground state for λ≫1. Here and in what follows, the\nnumber of states χkept in each subsystem in the DMRG\ncalculation ranged from 240 to 360. We find that the\nresults with χ= 240 are accurate enough in the present\nwork.\nThe ground-state energies of the Hamiltonian (6) up\ntoL= 16 for all possible configurations {Tl}are calcu-\nlated. The configurations that give the lowest energy are\nidentified for each λ. The spontaneous magnetizationper\nunit cell is given by\nmsp=1\nL/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleL/summationdisplay\nl=1(Tl−Sl)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(7)\nfromtheLieb-Mattistheorem10)fortheHamiltonian(6).\nFigure 3 shows our numerical result for the λ-\ndependence of msp. In addition to the two phases de-\nscribed in Sect. 3.1 and Sect 3.2, a quantized ferrimag-\nnetic ground state with msp= 1/2 is clearly observed.\nThis state has the configuration with Tl= q for odd\nlandTl= d for even l. We denote this configuration\nas (qd)L/2. In the thermodynamic limit, this configura-\ntion tends to the (qd)∞configuration.This configuration\nis equivalent to the (dq)∞configuration in the thermo-\ndynamic limit. Hence, this phase is doubly degenerate\nand is accompanied by the spontaneous breakdown of\ntwofold translational symmetry. In what follows, similar\nnotations are employed for other configurations. Within\nthe finite-size DMRG calculation, the intermediate fer-\nrimagnetic phases are observed between this phase and\nthe nonmagnetic phase. Although two intermediate fer-\nrimagnetic phases are also observed near msp= 1, these\nareartifactsofthefinite sizeeffect, sincethey correspond\ntomsp= 1−1/Land 1−2/L.\nFor finite chains, however, the values of mspare con-\nstrainedbythe systemsize.Hence, itis notclearwhether\nmspis quantized or continuously varying with λfor\n0< msp<1/2 in the thermodynamic limit.\nHence, we resort to the infinite-size DMRG calcula-\ntion. However, it is not possible to carry out the calcu-\nlation for all possible configurations of {Tl}for infinite\n2J. Phys. Soc. Jpn. FULL PAPERS\n0.4 0.6 0.800.51\nλmsp\nL=8\nL=10\nL=12\nL=14L=14\nL=16(qd)∞\nFig. 3. λ-dependence of mspcalculated by the finite size DMRG\nmethod.\nS S S\nsingle q and (p−1) d’s\np S’s (S=1/2)1 segment = p unit cells\n=  qdp−1q d d q\nFig. 4. (qdp−1)∞configuration.\nchains. Hence, we analyze this regime in the following\nway: We start with plausible candidates of periodic con-\nfigurations of Tl’s and find the configuration that gives\nthe lowest energy ground state among them. This leads\nto a stepwise λ-dependence of msp. Then, we check the\nstability of these steps against the formation of defects.\nAs plausible candidates of the ground states, we consider\nthe configurations(qdp−1)∞withmsp= 1/pthat consist\nof an infinite array of segments qdp−1with length of p\nunit cells as depicted in Fig. 4. These configurations can\nbe obtained from the configuration d∞corresponding to\nthe nonmagnetic ground state by inserting a q every p\nunit cells periodically. It should be remarked that these\nstates are accompanied by the p-fold spontaneous trans-\nlational symmetry breakdown. Hereafter, this series of\nconfigurations is called the main series configurations.\nThe case p= 1 corresponds to the q∞configuration that\ncorresponds to the Lieb-Mattis type ferrimagnetic phase\nwithmsp= 1.\nTheλ-dependence of mspin the main series is shown\nin Fig. 5. The spontaneous magnetization msprises con-\ntinuously from msp= 0 at the critical value of λgiven0.6 0.7 0.800.51\nλmsp\np≤38λc(∞)−~0.807p=2p=1\np=3\np=4\nFig. 5. λ-dependence of mspfor the main series configurations\ncalculated by the infinite size DMRG method.\n0 0.05 0.1 0.150.780.790.80.81λ\n1/p\nFig. 6. Extrapolation scheme of λc(p,p−1) andλc(p+1,p) to\np→ ∞. The filled and open circles correspond to the fits by Eq.\n(9) and Eq. (10), respectively. The filled square is the extra polated\nvalueλc(∞).\nby\nλc(∞)≡lim\np→∞λc(p,p−1)≃0.807 (8)\nwhere the boundary between the (qdp−1)∞phase with\nmsp= 1/pand (qdp−2)∞phase with msp= 1/(p−1) is\ndenoted by λc(p,p−1). The extrapolation to p→ ∞is\ncarried out using the data for 38 ≥p≥11 assuming the\nfollowing two asymptotic forms\nλc(p,p−1) =λc(∞)+C1\np+C2\np2, (9)\n3J. Phys. Soc. Jpn. FULL PAPERS\n0.8 0.802 0.804 0.806 0.80800.020.04\nλmsp\np≤38λc(∞)−~0.807\nEqs. (9),(10)\nFig. 7. λ-dependence of mspfor the main series configurations\nnearλ=λc(∞). The open circles are the results of the infinite\nsize DMRG calculation. The small filled circles and solid lin es are\nextrapolation by Eqs. (9) and (10).\nqdp−1qdp−1qdp−2qdp−1qdp−2\nn segments=qdp−2+(n−1)×qdp−1(a)\nqdp−1qdp−1qdpqdp−1qdp\nn segments=qdp+(n−1)×qdp−1(b)\nFig. 8. (a) (qdp−2(qdp−1)n−1)∞configuration and (b)\nqdp((qdp−1)n−1)∞configuration.\nλc(p+1,p) =λc(∞)+C′\n1\np+C′\n2\np2.(10)\nThe extrapolation procedure is plotted in Fig. 6. The\nboth extrapolations give the same value for λc(∞) up\nto the above digit. It should be noted that Eq. (9) and\nEq. (10) correspond to the extrapolation of left and right\nends of the steps, respectively. Using Eqs. (9) and (10),\ntheλ-dependence of mspis plotted down to msp= 0 in\nFig. 7.\nThe remaining question is whether the intermediate\nconfigurations with 1 /p < m sp<1/(p−1) can be a\nground state. To answer this question, it is necessary to\ncalculatethe ground-stateenergiesforall possibleconfig-\nurations of {Tl}, which is impossible. Hence, we confine\nourselves to the following configurations that are plausi-\nble to compete with the (qdp−1)∞configurations.\n(1) (qdp−2(qdp−1)n−1)∞configuration( msp=n/(p(n−1) + (p−1))) depicted in Fig. 8(a): A qdp−1seg-\nment is replaced by a qdp−2segment per every n\nsegments in the (qdp−1)∞configuration. For n= 1,\nthis configuration reduces to the (qdp−2)∞configu-\nration with msp= 1/(p−1) that corresponds to the\nneighboring step of the main series with higher msp.\nForn= 2, this configuration reduces to that with\nalternating qdp−1and qdp−2segments.\n(2) (qdp(qdp−1)n−1)∞configuration ( msp=\nn/(p(n−1)+(p+1))) depicted in Fig. 8(b):\nA qdp−1segment is replaced by a qdpsegment per\neverynsegments in the (qdp−1)∞configuration.\nForn= 1, this configuration reduces to the\n(qdp)∞configuration with msp= 1/(p+1) that\ncorresponds to the neighboring step of the main\nseries with lower msp. Forn= 2, this configuration\nreduces to that with alternating qdp−1and qdp\nsegments.\n(3) The configurations with spatial periodicity less than\n12 and spontaneous magnetization msp≤1/4 have\nbeen thoroughly checked.\nFor configurations (1) and (2), the computational cost\nincreases with an increase of pandn. We have numeri-\ncally confirmed that these states with 2 ≤n≤4 are not\nthe ground state for 2 ≤p≤11. It is further confirmed\nthat those with 2 ≤n≤6 are not the ground state for\n2≤p≤8. Within our numerical accuracy, we find no\nconfigurations that give lower energy than the main se-\nries configurations. Within the available numerical data,\nthe configurations with higher nare even less favorable.\nHence, it is highly plausible that the state intermedi-\natemspis not a ground state at least for p≤11. Also,\nconfigurations (3) do not give the ground state except\nfor the main series configurations. Thus, we expect that\nonly the configurations in the main series are realized in\nthe ground state.\n4. Summary and Discussion\nThe ground-state phases of diamond chains (1) with\n(S,τ(1),τ(2)) = (1/2,1/2,1) are investigated. Between\nthe gapless spin liquid phase for large λand Lieb-Mattis\nferrimagnetic phase with msp= 1 forλ≤0, we find a\nseries of quantized ferrimagnetic phases with msp= 1/p\nwhereptakes all positive integer values.\nTheλ-dependence of the spontaneous magnetization\nmspis very different from other diamond chains with fer-\nrimagnetic ground states. Although the quantized ferri-\nmagneticphasesarepresenteveninundistorteddiamond\nchains, the allowed values of spontaneous magnetization\nare limited to several simple rational values.3,4)\nThere are some examples of ferrimagnetic ground\nstates of diamond chains that are induced by the lat-\ntice distortion.5,6)In these cases, the ground state of\nthe undistorted chain is a paramagnetic state consist-\ning of clusters with finite magnetic moments. The sizes\n4J. Phys. Soc. Jpn. FULL PAPERS\nof the clusters are limited even in the absence of dis-\ntortion. The distortions induce ferromagnetic interac-\ntions between the cluster spins leading to the quantized\nferrimagnetic phases. The quantum fluctuations of the\nlengths of clusters are also induced by distortion leading\nto the partial ferrimagnetic phases.5)\nIn the present case, the ferrimagnetic phases are\npresent even in the absence of distortion. In contrast to\nthe cases of Refs. 3–6, arbitrarily large segments are al-\nlowed leading to the infinitesimally small steps around\nλ=λc(∞). However, no fluctuations of the lengths of\nthe segments are allowed in the present model, since the\nmagnitudes of the composite spins Tlremain good quan-\ntum numbers. Hence, partial ferrimagnetic phases are\nabsent in contrast to the cases of Refs. 5 and 6.\nIn the spin-1 alternating bond diamond chain with\nbond alternation δ, the nonmagnetic phase is equivalent\ntothegroundstateofthespin-1alternatingbondHeisen-\nberg chain with bond alternation δ.7)In this model, an\nintermediate ferrimagnetic phase is observed in the tiny\nregion close neighborhood of the point ( λ,δ) = (λc,δc)≃\n(1.0832,0.2598) that corresponds to the endpoint of the\nHaldane-dimer critical line.11–14)In Ref. 7, it has been\nspeculated that this region is the partial ferrimagnetic\nphase. However, considering the similarity of the gapless\nspin-liquid phase of the present model and the Haldane-\ndimercriticallineofthespin-1alternatingbonddiamond\nchain, it would be more reasonable to speculate that the\ninfinite series of quantized ferrimagnetic phases similar\nto those discussed in the present work is realized also in\nthis case. Unfortunately, the numerical confirmation is\ndifficult due to the smallness of the width of this region.\nAs mentioned above, in some examples of the quan-\ntized ferrimagnetic phases in undistorted diamond\nchains, the allowed values of spontaneous magnetization\nare limited to several rational values.3,4)In these exam-\nples, the nonmagnetic phases neighboring the ferrimag-\nnetic phases are spin-gap phases. On the other hand, the\nnonmagnetic phase neighboring the ferrimagnetic phase\nin the present model with infinitesimal step is the gap-\nless spin liquid phase. This seems to suggest that the\ninfinitesimal energy scale of the gapless spin liquid phase\nhelps the emergence of the exotic ferrimagnetic phase\nwith infinitesimal spontaneous magnetization. A further\nanalytical approach would be required to get insight into\nthe physical implication of the present phenomenon.So far, the infinite series of ferrimagnetic phases pro-\nposed in this work have not been found in real mate-\nrials. However, since the gapless spin liquid phases are\ngeneric critical states in quantum spin chains, it would\nbe possible that these series of phases are realized in\nthe presence of appropriate frustrating exchange inter-\nactions. Nevertheless, in more realistic cases, the pertur-\nbation that does not preserve the conservation laws (4)\nis inevitable. In such cases, the infinitesimal structure\nof spontaneous magnetization might be smeared. In this\ncontext, it would be an interesting problem to investi-\ngate the effect of lattice distortion in the present model.\nThese studies are left for future investigation.\nA part of the numerical computation in this work has\nbeencarriedoutusingthe facilitiesofthe Supercomputer\nCenter, Institute for Solid State Physics, University of\nTokyo,andYukawaInstituteComputerFacilityatKyoto\nUniversity.\n1)Introduction to Frustrated Magnetism: Materials, Experi-\nments, Theory , ed. C. Lacroix, P. Mendels, and F. Mila\n(Springer Series in Solid-State Sciences, Springer, Heide lberg,\n2011).\n2)Frustrated Spin Systems , ed. H. T. Diep, (World Scientific,\nSingapore, 2013) 2nd ed.\n3) K. Takano, K. Kubo, and H. Sakamoto, J. Phys.: Condens.\nMatter8, 6405 (1996).\n4) K. Hida and K. Takano, J. Phys. Soc. Jpn. 86, 033707 (2017).\n5) K. Hida, K. Takano, and H. Suzuki, J. Phys. Soc. Jpn. 79,\n114703 (2010).\n6) K. Hida, J. Phys. Soc. Jpn. 88, 074705 (2019).\n7) K. Hida, J. Phys. Soc. Jpn. 89, 024709 (2020).\n8) H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T.\nTonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. Ohta,\nPhys. Rev. Lett. 94, 227201 (2005).\n9) H.Kikuchi,Y.Fujii,M.Chiba,S.Mitsudo,T.Idehara,T.T one-\ngawa, K. Okamoto, T. Sakai, T. Kuwai T, K. Kindo, A. Mat-\nsuo, W.Higemoto, K.Nishiyama, M.Horovi´ c, and C.Bertheir ,\nProg. Theor. Phys. Suppl. 159, 1 (2005).\n10) E. Lieb and D. Mattis, J. Math. Phys. 3, 749 (1962).\n11) Y. Kato and A. Tanaka, J. Phys. Soc. Jpn. 63, 1277 (1994).\n12) S. Yamamoto, J. Phys. Soc. Jpn. 63, 4327 (1994); Phys. Rev.\nB51, 16128 (1995).\n13) K.Totsuka, Y.Nishiyama, N.Hatano, and M.Suzuki, J.Phy s.:\nCondens. Matter 7, 4895 (1995).\n14) A. Kitazawa and K. Nomura, J. Phys. Soc. Jpn. 66, 3944\n(1997).\n5"    },    {        "title": "0906.3534v2.Intrinsic_exchange_bias_in_Zn__x_Mn___3_x__O__4____x__leq_1___solid_solutions.pdf",        "content": "Intrinsic exchange bias in Zn xMn 3\u0000xO4(x\u00141) solid solutions\nDaniel P. Shoemaker,\u0003Efrain E. Rodriguez, and Ram Seshadri\nMaterials Department and Materials Research Laboratory\nUniversity of California, Santa Barbara, CA, 93106, USA\nIvana Sabaj Abumohor\nCentro para la Investigaci\u0013 on Interdisciplinaria Avanzada en Ciencia de los Materiales,\nDepartamento de Ingenier\u0013 \u0010a Qu\u0013 \u0010mica y Biotecnolog\u0013 \u0010a,\nUniversidad de Chile, Casilla 2777, Santiago, Chile\nThomas Pro\u000ben\nLos Alamos National Laboratory, Lujan Neutron Scattering Center,\nMS H805, Los Alamos, New Mexico 87545, USA\n(Dated: August 13, 2021)\nBulk specimens of the het\u001arolite solid solution Zn xMn 3\u0000xO4withx= 0, 0.25, 0.5, 0.75, and 1\nhave been prepared as homogeneous, phase-pure polycrystalline samples as ascertained by neutron\ndi\u000braction measurements. Samples with x= 0.25, 0.5, and 0.75 exhibit shifted magnetic hysteresis\nloops at low temperature, characteristic of exchange bias typically seen in magnetic composites.\nWe propose that the unusual magnetic behavior arises as a result of a nanoscale mixture of fer-\nrimagnetic and antiferromagnetic regions that are distinct but lack long-range order. While some\nglassy behavior is seen in AC magnetic measurements, its magnitude is not su\u000ecient to account for\nthe observed dramatic exchange bias. Furthermore, isothermal and thermoremanent magnetization\nmeasurements distinguish this material from a pure spin glass. The title system o\u000bers insights into\nthe alloying of a ferrimagnet Mn 3O4with an antiferromagnet ZnMn 2O4wherein distinct magnetic\nclusters grow and percolate to produce a smooth transition between competing orders.\nPACS numbers:\nI. INTRODUCTION\nExchange bias is a magnetic memory e\u000bect that oc-\ncurs at the interface between a ferromagnet (or ferri-\nmagnet) and an antiferromagnet.1By \feld-cooling a sys-\ntem with an ordered ferromagnet/antiferromagnet inter-\nface through the N\u0013 eel temperature TNof the antiferro-\nmagnet, exchange interactions at the interface lead to\na preferred direction of magnetization, typically along\nthe cooling \feld direction. Exchange bias has been engi-\nneered into a wide variety of materials systems and ge-\nometries: core-shell nanoparticles, granular composites,\nand thin \flm read-heads for magnetic recording media.2\nIn addition to the abrupt interfaces in thin-\flm architec-\ntures, a signi\fcant thrust has been made toward under-\nstanding the mechanisms of loop-shifting phenomena in\ndisordered and composite magnets.\nDisordered and/or dilute magnetic spins in a crystal\ncan lead to glassy behavior that gives rise to magnetic\nmemory e\u000bects as a result of slow and time-dependent\nprocesses below the spin freezing temperature Tf. Such\nglassiness can result in biased magnetization loops. Dis-\ntinctions between exchange bias and glassy magnetism\nare therefore useful. Exchange-biased systems are usu-\nally expected to have (i) two magnetic phases with a\nwell-de\fned interface, (ii) a loop shift, measured as the\nexchange \feld, HE, that goes to zero above TN, and\n(iii) zero exchange \feld (loop shift) if the cooling \feld\nis zero; exchange bias is not observed for M\u0000Hloopsacquired after zero-\feld cooling. Spin glasses, in turn,\nare associated with (i) frozen spins below Tfthat pro-\nduce a frequency-dependent peak in susceptibility, (ii) an\nabsence of long-range magnetic ordering, and (iii) some\nrelaxation on a macroscopic time scale after \feld changes\nbelowTf.3,4\nAs an illustrative example, loop shifts along the \feld\naxis were observed in the prototypical spin glass CuMn\nby Monod, et al. in 1979,5but these are not strictly con-\nsidered to be evidence for exchange bias since the mag-\nnetic phase is homogeneous and \feld-cooling is not nec-\nessary. A glassy phase can occasionally ful\fll the role of\nan antiferromagnet in a two-phase exchange biased sys-\ntem: loop shifts are commonly observed in ferromagnetic-\ncore nanoparticles with disordered surface layers, where\na spin-glass-like relaxation of the remanent magnetiza-\ntion versus time is accompanied by a loop shift.6,7,8\nGlassy spins freeze to partially align with the ferromag-\nnetic spins during \feld cooling and a preferred direction\nof magnetic orientation is therefore imparted. A detailed\nstudy of the interplay between ferromagnet/spin glass\nCo/CuMn bilayers with well-de\fned thicknesses has con-\n\frmed this behavior.9\nHere we report a detailed study of the magnetic prop-\nerties of Zn xMn 3\u0000xO4(x\u00141) solid solutions, studied in\nphase-pure polycrystalline samples. This system was re-\nported many decades ago by Jacobs and Kouvel,10who\nfound that exchange bias and \\magnetic viscosity\" ef-\nfects (meaning glassy magnetism in the current context)\nwere found to occur together in the solid solution. WearXiv:0906.3534v2  [cond-mat.mtrl-sci]  20 Jun 20092\nre-examine this system in light of the increased interest\nin nanoscale inhomogeneities in functional, and partic-\nularly correlated oxides.11,12We focus in particular on\nthe role of magnetic inhomogeneities and how they re-\nsult in competing magnetic order. We probe the question\nof whether these magnetic inhomogeneities are assocated\nwith structural inhomogeneities, in the sense of the for-\nmation of nanocomposite architectures. We also examine\nthe nature of glassy magnetism in this system and make\ndistinctions between glassiness and exchange bias.\nThe end members hausmannite Mn 3O4and het\u001aro-\nlite ZnMn 2O4are a spiral ferrimagnet and an antiferro-\nmagnet, respectively, with the former compound having\nrecently emerged as a candidate magnetoelectric mate-\nrial as a consequence of its complex magnetic ordering.13\nAt high temperatures ( >1100\u000eC) these compounds are\ncubic spinels, but they distort to the tetragonal het\u001aro-\nlite structure below 1100\u000eC as a consequence of orbital\nordering of octahedral d4Mn3+, as \frst described by\nGoodenough.14,15The octahedral site is completely oc-\ncupied by Mn3+. The tetrahedral site accommodates al-\nloying of isovalent d10Zn2+andd5Mn2+, the former\nbeing an ion that prefers tetrahedral coordination, and\nthe latter, an ion that lacks a site preference.\nWe \fnd, in agreement with, but signi\fcantly extend-\ning the original work of Jacobs and Kouvel,10that\nZnxMn 3\u0000xO4does not behave like a random solid so-\nlution in the magnetic sense, and neither does it macro-\nscopically phase-separate into ZnMn 2O4and Mn 3O4. In-\nstead, features of both are present, and the complex mag-\nnetic behavior can be explained by invoking nanoscale\nclusters of ferrimagnetic spins that gradually grow and\npercolate as xis increased. These nanoscale ferrimag-\nnetic regions always abut nanoscale antiferromagnets for\nx<1 and this results in the observed exchange bias.\nIntrinsic exchange biased systems have similarly been\nreported in perovskite manganites and cobaltites with\nmixed valent B-sites.16,17For example, the system\n(Y,Sr)MnO 3has been reported as displaying glassiness\nas well as loop shifting.18In contrast to these perovskite\nsystems, we \fnd striking magnetic complexity in the title\nsolid solution in the absence of any site disorder on the\nB-site. Additionally, the solid solution does not require\naliovalent substitution and concomitant changes in the\nvalence states of ions.\nIn general, the magnetic structure of spinel compounds\nsuch as ZnMn 2O4can be in\ruenced in two ways: through\ntuning the average size of cations in the tetrahedral\nsite, and through the addition of spins on the tetrahe-\ndral A site. Such tuning viathe A-site cation radius\nhas been studied extensively in chalcogenide spinels, but\nrarely changes the type of magnetic ordering in oxide\nspinels.19,20Tuning viathe introduction of magnetism\non the A-site has been studied in the (Zn,Co)Cr 2O4\nsystem.21In these Cr oxide spinels, like in the title Mn\nspinels, the B site is always occupied by Cr3+or Mn3+.\nIn cases where B-site Mn3+is alloyed with non-Jahn-\nTeller ions, dramatic phase separation due to dilution ofthe orbital ordering patterns is observed.22,23\nII. METHODS\nCeramic pellets of Zn xMn 3\u0000xO4were prepared by\ngrinding stoichiometric amounts of ZnO and MnO ( both\n99.9 % from Aldrich) in an agate mortar and pestle,\npressing at 100 MPa, and \fring in air at temperatures\nbetween 950\u000eC and 1200\u000efor 24 h (water quenched for x\n= 0 and 0.25) in accordance with the phase diagram of\nDriessens and Rieck.24For all calcinations, pellets were\nburied in sacri\fcial powder of the same composition in\ncovered alumina crucibles. The purity of all samples was\ncon\frmed by laboratory X-ray di\u000braction (XRD) data\nacquired on a Philips X'Pert di\u000bractometer with Cu-\nK\u000bradiation. Magnetic properties were measured using\na Quantum Design MPMS 5XL SQUID magnetometer.\nTime-of-\right (TOF) neutron powder di\u000braction on sam-\nples held in vanadium cans at the high intensity powder\ndi\u000bractometer (HIPD) at Los Alamos National Labora-\ntory. The HIPD instrument can collect high d-spacing\nmagnetic re\rections out to tens of \u0017A. However, no peaks\nwere found beyond 6 \u0017A in any of the samples studied\nhere. We limit the Rietveld re\fnement to banks 1{4,\nwith a maximum momentum transfer Qmax = 20 \u0017A\u00001\nand maximum d-spacing of 6 \u0017A. Rietveld re\fnement was\nperformed using the XND code25for X-ray data and\nGSAS26for TOF data. Crystal structures are visualized\nusing VESTA.27\nIII. RESULTS AND DISCUSSION\nTime-of-\right neutron di\u000braction is an especially use-\nful tool in examining the solid solutions studied here. In\naddition to the possibility of variable temperature stud-\nies, the availability of high resolution high momentum\ntransfer (Q) data, the ability to probe magnetic scatter-\ning, and the ability to examine Zn2+/Mn2+A-site dis-\ntribution are all advantageous. The nuclear scattering\nlengths are 5.68 fm for Zn and \u00003.73 fm for Mn, so these\nions are extremely well contrasted in the scattering.\nRoom temperature neutron TOF di\u000braction patterns\nare shown in Fig. 3, along with \fts to the pro\fles using\nthe Rietveld re\fnement method. The \fts give excellent\nmatches to the het\u001arolite structure across the solid so-\nlution. The TOF re\fnements reveal no impurities, and\nthe particles are many microns in extent as seen from\nthe narrow widths of the di\u000braction peaks. Structural\nparameters from the Rietveld re\fnement are provided in\nTable I. Trends in the relevant structural parameters as\na function of xare shown in Fig. 2.\nThe cell volume and c=aratios vary smoothly and re-\n\rect the 10 % di\u000berence in the ionic radii of tetrahedral\nMn2+(0.66 \u0017A) and tetrahedral Zn2+(0.60 \u0017A). The de-\ncrease in tetragonality as the Zn content xincreases could\nbe due to its preference for covalent bonding, and there-3\nlog counts\n1 2 3 4 5\nd-spacing (Å)Mn3O4\nZnMn2O4Zn0.75Mn2.25O4Zn0.5Mn2.5O4Zn0.25Mn2.75O4\nFIG. 1: 300 K neutron TOF di\u000braction Rietveld re\fne-\nments in the I41=amd space group con\frm the purity of all\nZnxMn 3\u0000xO4phases at 300 K. Di\u000berence pro\fles are shown\nbelow each \ft. Re\fnement results (including Rwp) are pro-\nvided in Table I.\nfore a tendency toward more regular tetrahedral coordi-\nnation. This is supported by the oxygen yandzcoordi-\nnates, which approach their least-o\u000bset values of1\n4and1\n2,\nrespectively with increasing Zn. The oxygen Uisovalues\nfor each compound are relatively close, but the small-\nest values occur for the end members, while site mixing\non the A-site leads to larger values for intermediate x.\nRandom A-site mixing of Zn2+/Mn2+is suggested by\nthe smoothly varying lattice parameters and the c=ara-\ntios versusx. This system strictly maintains a \\normal\"\ndistribution of cations: Zn2+greatly prefers tetrahedral\ncoordination by oxygen, and Mn3+is very stable in a JT\ndistorted octahedral coordination.28The A-site occupa-\ntion re\fnes to within 1 % of the nominal Zn/Mn ratio in\neach case. The JT distortion is present in all samples\nsince the B sublattice is invariant with composition x.29\nFigure 3 displays TOF di\u000braction patterns at T=\n300 K, 50 K, and 20 K over a region that contains all mag-\nnetic scattering intensity relevant to the discussion here.\nMost obvious are the numerous, intense magnetic peaks\nin the end member Mn 3O4. The top panel is on a log\nscale one order of magnitude higher than the rest. The\nonset of long-range magnetic ordering leads to a trans-\nfer of intensity from the di\u000buse scattering to the Bragg\npeaks, resulting in a much lower baseline for the 20 K\ndata than that at higher temperatures.30The magnetic\nstructure of hausmannite Mn 3O4is complex, with the\nonset of incommensurate sinusoidal magnetic ordering at\nTC= 44 K, followed by a locking in of the spin modula-\ntion to a commensurate structure below 33 K.31,32\n1.141.151.161.17c /a\n600610620630\nV (Å3)\n0 0.25 0.5 0.75 1\nx in ZnxMn3-xO40.250.2550.26zO\n0 0.25 0.5 0.75 1\nx in ZnxMn3-xO40.0040.0080.012\nO Uiso (Å2)(a) (b)\n(d) (c)FIG. 2: Structural parameters at 300 K from neutron TOF\nRietveld re\fnements show decreasing (a) c=aratios and (b)\ncell volume with Zn concentration (linear \fts, dashed), due\nto its smaller radius. The oxygen zposition in (c) decreases\ntoward the undistorted value of 0.25. In (d), chemical disorder\ncauses compounds with intermediate Zn/Mn mixing to have\nhigher thermal parameters than the end members. Error bars\nare smaller than the symbols in all panels.\n20 K\n50 K\n300 Klog Counts\n3 4 5\nd-spacing (Å)Mn3O4\nZnMn2O4Zn0.75Mn2.25O4Zn0.5Mn2.5O4Zn0.25Mn2.75O4\nFIG. 3: (Color online) Neutron TOF powder di\u000braction pat-\nterns (log scale, o\u000bset for clarity) for the Zn xMn 3\u0000xO4solid\nsolutions at 300 K, 50 K and 20 K. The Rietveld \ft to the\n300 K (non-magnetic) pro\fle is shown for all samples. Note\nthat only di\u000buse magnetic scattering is evident around d=\n5\u0017A for the sample with x= 0:5. In Mn 3O4, the baseline at\n20 K drops due to transfer of di\u000buse magnetic scattering to\nBragg peaks.4\nTABLE I: Bulk structural parameters at 300 K for Zn xMn 3\u0000xO4obtained from Rietveld re\fnement of TOF neutron di\u000braction\ndata:I41=amd (No. 141, origin choice 2); A-site Zn xMn 1\u0000xat (0,1\n4,7\n8); B-site Mn at (0,1\n2,1\n2); O at (0,y,z).\nComposition a(\u0017A)c(\u0017A)c=a y OzO OUiso(\u0017A2)Rwp(%)\nZnMn 2O4 5.71643(5) 9.2275(1) 1.1414 0.47657(8) 0.25577(5) 0.0060(2) 3.1\nZn0:75Mn 2:25O45.71955(3) 9.28628(7) 1.1481 0.47524(3) 0.25681(2) 0.00894(4) 2.8\nZn0:5Mn 2:5O45.73726(3) 9.3504(1) 1.1524 0.47499(3) 0.25751(2) 0.00702(4) 3.0\nZn0:25Mn 2:75O45.75134(4) 9.4225(1) 1.1585 0.47404(4) 0.25867(3) 0.00767(4) 3.3\nMn 3O4 5.75924(2) 9.46632(6) 1.1622 0.47273(3) 0.25913(2) 0.00534(7) 2.7\nFIG. 4: (Color online) The ZnMn 2O4het\u001arolite unit cell is\nshown in (a) with oxygen polyhedra drawn around Mn (red)\nand Zn (blue). In (b), the B-site linkages are shown. The\nB{B direct exchange net consists of a stretched pyrochlore\nlattice (four interwoven kagom\u0013 e nets) with B{B links in a\nandbdirections (dark) that are shorter than those with a c\ncomponent (light). The diamond-type A lattice is shown in\n(c).\nAt the other end of the solid solution, het\u001arolite\nZnMn 2O4has fewer and weaker magnetic peaks. While\nextensive work has been done on the magnetic order-\ning of cubic spinels where the spins are con\fned purely\non the B sublattice and are strongly geometrically frus-\ntrated,19,33,34the magnetic ordering in tetragonally dis-\ntorted hausmannite/het\u001arolite B-site compounds has re-\nceived less attention. There are three relevant tetrag-\nonal spinels to consider: ZnMn 2O4, CdMn 2O4, and\nMgMn 2O4. Zn and Cd both have a strong tendency to\noccupy tetrahedral sites, but Mg is exhibits about 10 %-\n25 % inversion on the octahedral sites.28No description\nof the magnetic structure has accompanied studies of\n(Zn,Cd)xMn 3\u0000xO4.35,36\nTo better understand the magnetic structures that are\nplausible with the data, we display various depictions\nof the het\u001arolite crystal structure in the panels of Fig. 4.\nThe B-site octahedral cation sublattice displayed in Fig. 4\ncan be described in two ways: as a pyrochlore lattice\nstretched in the cdirection, or as layers of parallel chains\nstacked at 90\u000eto each other. In cubic spinels with non-\nmagnetic A-sites, the intrachain B{B direct exchange is\nthe strongest magnetic interaction and is geometrically\nfrustrated since it occurs within ideal tetrahedra.37In\nZnMn 2O4as in Mn 3O4, the elongation along cstretches\ntwo of the pyrochlore-type B-site nets and leaves one\n(in theabplane) unchanged. This has led to the in-\n0 10 20 30 40 50\nT (K)01020304050M (emu/g)FC\nZFC\nMn3O4\nx = 0.25\nZnMn2O4x = 0.5\nx = 0.75FIG. 5: Field-cooled (FC) and zero-\feld-cooled (ZFC) mag-\nnetization curves at H= 1000 Oe for the Zn xMn 3\u0000xO4solid\nsolution show a gradual decrease in the magnetic ordering\ntemperature, as well as the magnetization from x= 0 to 1.\nThe interactions in ZnMn 2O4are antiferromagnetic changes\ncannot ben observed on this magnetization scale; these shown\nin greater detail in Fig. 6.\nterpretation of the het\u001arolite magnetic structure to con-\nsist of ferromagnetic chains of Mn3+, with antiferromag-\nnetic interchain interactions.38This simple interpreta-\ntion clearly does not capture all the details as is evident\nin the TOF neutron di\u000braction data, where the peaks in\nZnMn 2O4are di\u000buse and therefore indicate a substantial\namount of disorder over long length scales. There is a\nshift of intensity from the (101) peak at d= 4.9 \u0017A oncex\nincreases past 0.5, and the intensity of the di\u000buse peak at\nd= 5.05 \u0017A gradually increases until ZnMn 2O4is reached.\nIn the middle compound with x= 0.5, no magnetic Bragg\npeakss are present. There is only a slight increase in dif-\nfuse intensity around d= 5\u0017A, so any magnetic order at\nthis point must only be short-range in nature.\nWhile the magnetic neutron scattering data requires\na more detailed analysis that will be presented in fu-\nture work, we use the general trends to explain AC\nand DC magnetization measurements presented in this5\n2030M -1(g/emu)FC\nZFC\n0 100 200 300 400\nT (K)00.51MFC-MZFC ×103\n2 3 4 5\nd-spacing (Å)0.11Counts50 K\n100 K\n100 K fit(a)\n(b)\n(c)\nFIG. 6: (Color online) Inverse susceptibility ZFC/FC data (a)\nfor ZnMn 2O4shows Curie-Weiss behavior above room tem-\nperature with a very broad, gradual ordering of the spins that\nbegins around 260 K. Small amounts of irreversibility are seen\nin (b), which indicates a magnetic transition at T= 60 K. In\n(c), the appearance of a magnetic Bragg peak in TOF neutron\ndata between 100 and 50 K indicates the onset of long-range\nmagnetic order coinciding with the peak in (b). The antifer-\nromagnetic downturn in this sample only occurs at near 40 K.\nThe Rietveld \ft at 100 K is for structural peaks only.\nwork. DC magnetization measurements on members of\nthe ZnxMn 3\u0000xO4solid solution indicate a smooth, lin-\near decrease in both the magnetic ordering temperature\nas well as the maximum magnetization on going from\nMn 3O4(x= 0) to ZnMn 2O4(x= 1). The \feld-cooled\n(FC) and zero-\feld-cooled (ZFC) magnetization curves\nin Fig. 5 show a steady decline in the ordering temper-\nature, temperature of magnetic irreversibility (deviation\nof ZFC and FC curves), and FC moment as xgoes from\n0 to 0.75. The magnetization curves show that the neu-\ntron TOF data in Fig. 3 at 20 K is below TCfor the four\nferrimagnetic samples. The samples at x= 0.5 and 0.75\nhave signi\fcant di\u000buse intensity at 50 K, well above TC\nmeasured viaSQUID magnetization. Interestingly, the\nweak magnetic scattering intensity in x= 0.5 versus x=\n0.75 (Fig. 3) seems contradict the fact that x= 0.5 has\nthe larger magnetization and higher TC. We can there-\nfore assume that in x= 0.5 samples, ferrimagnetism is\ncaused by local regions of aligned spins which lack long-\nrange order.\nThe ZFC-FC behavior for ZnMn 2O4is much more\ncomplex than the other samples in the solid solution, and\nhas been the subject of continued investigation for many\nyears.35,38,39,40,41,42Salient features that have remained\nconsistent are Curie-Weiss paramagnetism above room\ntemperature, with a phase transition between 230 K and\n290 K that has been detected in speci\fc heat38,43and\nYoung's modulus36measurements. In our measurements\n0 0.1 0.2 0.3 0.4 0.5\nT/|Θ|-1-0.75-0.5-0.2500.250.5C/(|Θ|χ)-1ZnMn2O4\nZn0.75Mn2.25O4\nZn0.5Mn2.5O4\nZn0.25Mn2.75O4FIG. 7: (Color online) Curie-Weiss normalization of the FC\nmagnetization curves provides a view of the di\u000bering magnetic\nordering schemes in the Zn xMn 3\u0000xO4solid solution. Devia-\ntion from purely paramagnetic behavior (dashed) is ferrimag-\nnetic for samples with x < 1, withTCdecreasing with the\nnumber of A-site spins. Only ZnMn 2O4has antiferromagnetic\nordering at low temperature.\nof the ZFC/FC behavior in Fig. 6, we observe this as a\ngradual slope change in M\u00001versusT. The irreversible\nmomentMFC\u0000MZFC has a slight dip around 260 K and\na strong transition at 60 K. A new magnetic Bragg peak\natd= 5.05 \u0017A clearly arises between 100 K and 50 K and\npersists down to 20 K.\nAs Mn2+is substituted into the end member\nZnMn 2O4, ferrimagnetism is induced and can be illus-\ntrated by normalizing the FC magnetization using the re-\nsults of \ftting the high-temperature susceptibility to the\nCurie-Weiss law. The data are then displayed on a com-\nmon scale, presented in Fig. 7. The utility of such scaling\nacross solid solutions has proven crucial in previous stud-\nies of because it o\u000bers a clear view of relative strengths of\nFM/AFM interactions in similar compounds.21All sam-\nples have Curie temperatures \u0002 <0 K, indicating that\nshort-range interactions are predominantly antiferromag-\nnetic. The trend of \u0002 versus xis shown in Fig. 8(a).\nThe strength of antiferromagnetic coupling gradually in-\ncreases as Zn is added to the A-sites, possibly as a con-\nsequence of the smaller cell volume as Zn2+ substitues\nMn2+. For ZnxMn 3\u0000xO4samples with x<1, these inter-\nactions lead to ferrimagnetic order (dropping below the\ndashed line of ideal Curie-Weiss paramagnetism) with an\nordering temperature that decreases with the concentra-\ntion of tetrahedral Zn2+.\nA more curious trend develops in the paramagnetic\ne\u000bective moment \u0016effwhich is measured above 300 K.\nIn Fig. 8(b), Mn 3O4has\u0016eff= 8.04\u0016B/f.u. instead of\nthe ideal value of 9.44 for one tetrahedral Mn2+and two6\n0 0.25 0.5 0.75 1\nx in ZnxMn3-xO477.588.599.5µeff (µB/f.u.)-1000-800-600-400-2000Θ (K)(a)\n(b)\nFIG. 8: The Curie-Weiss temperature \u0002 versus composition\n(a) shows increasing dominance of short-range antiferromag-\nnetic interactions as the solid solution progresses from Mn 3O4\nto ZnMn 2O4. The dotted line is a guide to the eye. The\nparamagnetic \u0016effshown in (b) begins below the ideal L+S\ncontribution (dashed line) for Mn 3O4, but increases past the\nexpected value for ZnMn 2O4. This increase in e\u000bective mo-\nment with xis counterintuitive since Mn2+spins are being\nremoved , but could be attributed to Jahn-Teller orbital or-\ndering contributions.\noctahedral Mn3+per formula unit (including both spin\nand orbital contributions). Interestingly, the experimen-\ntal\u0016effincreases with Zn content, despite the removal\nofd5Mn2+. If the discrepancy from the ideal value were\ndue to short-range ordering in ZnMn 2O4, we would ex-\npect lowering of\u0016eff, but this is not the case.\nAll hysteresis loops measured after ZFC in this system\nare symmetric around the origin. However, FC loops\nfor 0< x < 1 measured under a cooling \feld HFC=\n50 kOe are shifted by an exchange bias \feld \u0000HE, as\nseen in Fig. 9. Such loop shifts along Hafter \feld cool-\ning are similar to what was \frst reported by Jacobs and\nKouvel.10A systematic examination of the behavior from\n0\u0014x\u00141 reveals an interesting trend. ZnMn 2O4is an-\ntiferromagnetic and displays no hysteresis. As Mn2+is\ninserted on the tetrahedral sites, ferrimagnetism arises\nwith a linearly increasing saturation magnetization. In\nthex= 0.25 and 0.5 samples, the loop shift is exactly\nequal to the coercivity{that is, HE=HCif we de\fne HC\nto be half the loop width. This implies that for a positive\nHFC, nearly allMn spins that contribute to the ferri-\nmagnetic behavior are pinned in the + Mdirection when\nHFCis \frst relieved. As the hysteresis continues to neg-\native saturation and His increased from \u000050 to 50 kOe,\nthere reaches a point where all the Mn ferrimagnetic spins\nare exactly compensating. This occurs at H= 0. The\nmagnetic saturation MSvaries smoothly from ZnMn 2O4\n0 10 20 30 40\nT (K)0100200300400\nH (Oe)HC\nHE\n-2 -1 0 1 2\nH (kOe)-1-0.500.51M (µB/f.u.)\nZn0.25Zn0.5Zn0.75(a)\n(b)\n(c)(d)\n(e)\n(f)FIG. 9: Hysteresis loops (a-c) measured at 5 K after HFC\n= +50 kOe \feld-cooling show dramatic exchange-biased loop\nshifts. The x= 0.75 and 0.5 loops are pinned so that the\ncoercive \feld HCin the +Hdirection is zero. This results in\noverlapping values of loop shift HEand half loop width HC\nversus temperature (d-f). Some shift is still evident in x=\n0.25 and disappears in Mn 3O4.\nto Mn 3O4, with a contribution of about 0.30(4) \u0016B=per\nMn2+, which has S= 5/2 and could contribute a max-\nimum of 5 \u0016B. Because the ferrimagnetic end member\nMn 3O4also obeys this relationship, we assume that the\ninserted Mn2+create nanoscale clusters of Mn 3O4that\nare the dominant source of the total magnetic moment.\nThese local FM clusters must be contained within an\nantiferromagnetic matrix because the exchange bias be-\nhavior is genuine, as indicated by the \feld-cooled loop\nshifting and centered ZFC loops.\nAs the tetrahedral Mn2+fraction increases past 50 %,\nthe loop shift changes from HE=HCtoHE= 0 for the\nend member Mn 3O4. Whenx= 0.75,HEis still present\nbut the positive HCvalue no longer resides at H= 0 as\nit does for the completely shifted x= 0.5 and 0.25 cases.\nFor a diamond-type lattice such as the A-sites in spinel\nor het\u001arolite, the site percolation threshold is 43 %.44\nAs percolation on the tetrahedral sublattice is achieved,\nloop shifting decreases while HCandMSvary gradually.\nSo only the dilute spins near edges of clusters are pinned\nduring \feld cooling, and the pinning is overcome when\nthe clusters grow large or coalesce.\nLoop shifts such as those in Fig. 9 can arise from two\nphenomena: classical exchange biasing of a ferromagnet\nand antiferromagnet, or as a consequence of spin-glass\nbehavior. In the latter case, HEcan arise from coupling\na ferromagnet to a spin glass,9glassy uncompensated\nspins at interfaces/surfaces,8or an intrinsic anisotropy7\n102103104105\nH (Oe)051015MR (emu/g)TRM\nIRM\nFIG. 10: Thermoremanent magnetization (TRM) and\nisothermal remanent magnetization (IRM) versus applied\n\feld for ax= 0.5 sample shows clear deviation up to H=\n50 kOe. Lines are guides to the eye. For a typical spin glass,\nthe two curves should join with increasing Has the \feld aligns\nthe disordered moments to saturation. In an exchange biased\nsystem, the curves remain separated as seen here.\npresent in the glass itself.3,4One method of testing for\nspin-glass behavior is the measurement of thermorema-\nnent and isothermal remanent magnetization (TRM and\nIRM, respectively) shown in Fig. 10. The TRM measure-\nment begins as a typical FC procedure: HFCis applied\nwhile cooling from above the magnetic transition, tem-\nperature is stabilized, HFCis removed, and the remanent\nmomentMRis measured. For an IRM measurement,\nthe sample is zero-\feld cooled, the temperature is stabi-\nlized,His applied for a substantial length of time (here\nwe use 30 min.), the applied \feld is removed, and MR\nis measured. In glassy systems, TRM is greater than\nIRM for low HFCbecause additional alignment is in-\nduced while cooling through the high-susceptibility glass\ntransition.45,46At highHFCthe values coincide when the\napplied \feld overcomes intrinsic anisotropy and aligns all\nspins, regardless of thermal history. In an exchange bi-\nased material, antiferromagnetic spins are notreversed\nby high \felds, so the TRM and IRM curves remain sep-\narated even at high \felds. Indeed, we can see in Fig. 10\nthat for Zn 0:5Mn 2:5O4high values of HFCproduce a\nhigher value for the exchange-biased TRM than the ZFC,\nnon-biased IRM. The TRM/IRM data disallows consid-\nering the A-site Mn2+spins to be a dilute ferromagnetic\nspin glass that are coupled to an antiferromagnetic B-\nsite sublattice. This measurement further corroborates\na two-phase interaction between ferrimagnetic Mn 3O4-\ntype clusters with ZnMn 2O4-type antiferromagnetic re-\ngions.\n0 10 20 30 40\nT (K)00.0060.0120.018χ' (emu/g/Oe)100 Oe\n300 Oe\n500 Oe\n1000 Oe\n1500 Oe\n0 10 20 30\nT (K)00.0040.0080.012\nχ' (emu/g/Oe)\n0 10 20\nT (K)00.0040.0080.012χ' (emu/g/Oe)\n0 60 120 180\nH 2/3 (Oe2/3)0204060\nTf (K)Zn0.25Mn2.75O4\nZn0.5Mn2.5O4(a) (b)\n(c)\n(d)\nHfTfFIG. 11: (Color online) Magnetic AC susceptibility for\nwith mixed tetrahedral occupancy: (a) Zn 0:25Mn 2:75O4, (b)\nZn0:5Mn 2:5O4, and (c) Zn 0:75Mn 2:25O4. The AC \feld is 1 Oe\nwith di\u000berent DC \felds shown. Local maxima in (a) and\n(b) are marked with symbols and replotted in (d) to show\nde Almeida{Thouless behavior. No such trend is present\nin (c), where maxima are present only at the ferrimagnetic\nTCaround 18 K. Spin-glass freezing temperatures Tfand\ncritical \felds Hcrcan be extracted for both curves in (d):\nfor Zn 0:25Mn 2:75O4Tf= 36.9 K and Hcr= 5320 Oe; for\nZn0:5Mn 2:5O4Tf= 20.6 K and Hcr= 2020 Oe.\nNote that these phases are not ordered on the long\nrange, as evidenced most clearly by the di\u000braction pat-\ntern for Zn 0:5Mn 2:5O4in Fig. 3(c). The magnetic Bragg\npeaks disappear when x= 0.5, even though the trends\nin SQUID magnetism continue to vary smoothly. Never-\ntheless, the ferrimagnetism and exchange bias act as di-\nrect interpolations of the x= 0.25 and 0.75 samples. In\nZnMn 2O4some magnetic ordering produces Bragg peaks,\nbut a loss of Bragg intensity with xsignals the breakdown\nof this B-site ordering from the stronger (but still antifer-\nromagnetic) A-B coupling to the inserted A-site Mn2+.\nIn the AC magnetization measurements of Fig. 11, two\nmaxima are seen in \u001f0under cooling: one at TCand an-\nother at a lower temperature, which is interpreted as a\nspin-glass freezing Tf.3,4The glassy spins may be present\nat the interfaces between the A-site-induced ferrimag-\nnetic clusters or (less likely) as isolated sites. For sam-\nples withx= 0.75 and 0.5 (Figs. 11a and 11b), Tfshifts\nto lower temperatures as the DC bias magnetic \feld is\nincreased. The TfversusH2=3dependence plotted in\nFig. 11 indicates excellent agreement with de Almeida{\nThouless (AT) behavior,47which is typical not only for\nbulk frustrated and dilute spin glasses,48but also for a\nwide variety of systems with disordered spins at surfaces\nand interfaces.7,8,49No such behavior is seen in the x=\n0.25 sample, since the Mn spins now occupy 75 % of the\nA-sites and the ferrimagnetic phase has e\u000bectively per-\ncolated the entire structure. Two key values can be ex-\ntracted from the AT lines in Fig. 11(d): the freezing tem-8\n10 100 1000\nf (Hz)17.918.018.118.2\nTf (K)\n15 20 25\nT (K)5.0×10-51.0×10-41.5×10-4χ' (emu)11 Hz\n50 Hz\n100 Hz\n499 Hz\n997 Hz\n1488 Hz\nFIG. 12: (Color online) The AC magnetic susceptibility for\nZn0:5Mn 2:5O4exhibits frequency dependence in the region as-\nsociated with spin glass freezing. The T-value of the maxi-\nmum is plotted versus fin the inset. Error bars are smaller\nthan the data points. The variation of Tfwithfagrees with\nstandard spin glass behavior. The Tgextracted from this\ndata di\u000bers from that in Fig. 11 due to the large non-glassy\nferrimagnetic contribution.\nperatureTfwhere irreversibility (hysteresis) in the spin\nglass is \frst induced, and the critical \feld Hcrwhere the\napplied \feld overcomes the internal anisotropy of the spin\nglass and saturates it. Considering Zn 0:5Mn 2:5O4,Tf=\n20.6 K, which is slightly higher than the DC deviation of\nZFC-FC data in Fig. 5, as expected since the DC data\nwas collected at H= 1000 Oe. More importantly, Hcr\n= 2020 Oe. This implies that if the MRwere solely due\nto a spin glass component the TRM-IRM curves would\ncoincide at Hcr. As they do not, the number of glassy\nspins must be very small in comparison to those in ferri-\nmagnetic clusters. Thus the irreversible magnetization in\nthe hysteresis loops of Fig. 9(b) primarily arises from fer-\nrimagnetic regions of local spin alignment and not from\nglassy clusters that obey AT behavior.\nFrequency-dependent AC magnetization measure-\nments of the Tfregion in Zn 0:5Mn 2:5O4in Fig. 12 show\na deviation after cooling below Tf, further evidence of a\nsmall amount of glassy behavior. The peak centers are\nplotted versus fin the inset. The cusp in \u001f0obeys the\nrelationship \u0001 Tf=[Tf(log!)] = 0.005, which is the same\nvalue as the canonical spin glass CuMn.50The breadthof the peak indicates that there is a distribution of freez-\ning temperatures, based on the non-uniform distribution\nof glassy spins on interfaces of the ferrimagnetic clusters.\nIV. CONCLUSIONS\nThe system Zn xMn 3\u0000xO4is a homogeneous solid so-\nlution when investigated using bulk structural probes\nsuch as TOF neutron di\u000braction. However, magnetic\nmeasurements reveal intrinsic exchange bias that we be-\nlieve results from the interaction of distinct ferrimag-\nnetic and antiferromagnetic regions. For concentrations\nof Mn-doping up to 50 %, \feld-cooled hysteresis loops\nare shifted so that HE=HC. Because magnetic scatter-\ning is di\u000buse, and the Curie-Weiss temperature \u0002 is large\nand negative, the magnetic structure of the Zn xMn 3\u0000xO4\nsolid solution must consist of ferrimagnetic Mn-rich clus-\nters that do not order on a macroscopic scale. As the\nclusters grow, their contribution to MSincreases linearly\nuntil Mn 3O4is reached, and exchange bias disappears.\nThere is a glassy component to the the magnetism in\nthese systems, as evidenced by AC magnetization mea-\nsurements. However, the contribution of glassy spins to\nthe DC magnetization is minimal, which is most visi-\nble in the well-separated TRM and IRM traces even up\nto large \felds. The presence of intrinsic exchange bias\nmerits further investigation of the nanoscale ordering of\nspins in the Zn xMn 3\u0000xO4system. Small-angle neutron\nscattering, real-space total scattering, Lorentz transmis-\nsion electron microscopy, and magnetic force microscopy\ncould each help observe the evolution of magnetic order-\ning as a function of temperature and composition in this\nsolid solution.\nV. ACKNOWLEDGMENTS\nWe thank B. C. Melot for helpful discussions. This\nwork was supported by the Institute for Multiscale Ma-\nterials Studies, the donors of the American Chemical So-\nciety Petroleum Research Fund, and the National Science\nFoundation through a Career Award (DMR 0449354) to\nRS and for the use of MRSEC facilities (DMR 0520415).\nNeutron scattering was performed on HIPD at the Lu-\njan Center at the Los Alamos Neutron Science Center,\nfunded by the DOE O\u000ece of Basic Energy Sciences. 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We have found the exact results\nfor the magnetization by using recursion relation method. T he existence of the magnetization plateau has been observed\nat one third of the saturation magnetization in the antiferr omagnetic case. Some ground-state properties of the model a re\nexamined. At low temperatures, the system has two ferrimagn etic (FRI1 and FRI2) phases and one paramagnetic (PRM)\nphase. Lyapunov exponents for the various values of the exch ange parameters and temperatures have been analyzed. It hav e\nalso been shown that the maximal Lyapunov exponent exhibits plateau. Lyapunov exponents exhibit different behavior for two\nferromagnetic phases. We have found the existence of the sup ercritical point for the multi-dimensional rational mappi ng of the\nspin-1\n2Ising-Heisenberg model on diamond chain for the first time at absence of the external magnetic field and T→0in the\nantiferromagnetic case.\n1I. INTRODUCTION\nThe investigation of physical properties of the low-dimens ional quantum spin systems with competing interactions\nin an external magnetic field has been a subject of increasing ly intense research interest in the recent decades. The\nresearch interest of these systems has attracted much atten tion due to the following reasons: first, they can be solved\nexactly by using different mathematical techniques, second , they are realized in the nature, and third, these systems\npresent rich thermodynamic behavior, such as the appearanc e of magnetization plateaus, double peaks structure of\nthe specific heat and magnetic susceptibility.\nOne of the interesting low-dimensional quantum spin system is the frustrated diamond Heisenberg spin-chain. The\nphysical properties of real materials such as copper minera lCu3(CO3)2(OH)2, known as natural azurite (Copper Car-\nbonate Hydroxide) can be well described by using the quantum antiferromagnetic Heisenberg model on a generalized\ndiamond chain. The physical properties of the Heisenberg mo del on diamond chain have been investigated using\ndifferent methods they are full numerical diagonalization a nd the Lanczos algorithm [1–3], the decoration-iteration\ntransformation [4–7], the mapping transformation techniq ue [8], the density-matrix renormalization-group (DMRG)\nand transfer-matrix renormalization-group (TMRG) techni ques [9], Gibbs-Bogoliubov approach [10], cluster approac h\n[11], the generalized gradient approximations (GGA) [12], the density functional theory and state-of-the-art numeri cal\nmany body calculations [13].\nIntriguing properties of the azurite made it a good candidat e for studying its properties on the low-dimensional\nquantum spin systems. Kikuchi and co-workers [14] have expe rimentally studied the physical properties of the com-\npoundCu3(CO3)2(OH)2. They have shown that azurite can be regarded as a model subst ance of a distorted diamond\nchain. The temperature dependence on the magnetic suscepti bility and specific heat shows double peak structure\n(around 20 and 5 K) on both magnetic susceptibility and speci fic heat results. The existence of the magnetization\nplateau at one third of the saturation magnetization has als o been experimentally observed in the magnetization\ncurve.\nThe aim of the recursion relation method is to cut lattice int o branches and express the partition function of all\nlattice through the partition function of branches. This pr ocedure will allow to derive one- or multi-dimensional\nmapping for branches of the partition function. After which the thermodynamic quantities of the physical system\nsuch as magnetization, magnetic susceptibility, specific h eat can be expressed through recursion relation. One and\nmulti dimensional mapping allows to investigate propertie s of different models for example Ising model on Husimi\nlattice [15, 16], zigzag ladder [17], triangular lattice [1 8], two-layer Bethe lattice [19, 20], mixed-spin Ising mode l on a\ndecorated Bethe lattice [21, 22], Q-state Potts model on the Bethe lattice [16], zigzag ladder [23], phase diagrams for\nboth ferromagnetic and antiferromagnetic cases, multicri tical points, for the spin-1 Ising model on the Bethe lattice\n[24–27].\nIn this paper we have investigated some properties of the spi n-1\n2Ising-Heisenberg model on diamond chain by using\ndynamical system (recursion relation) approach. Especial ly we have investigated magnetic properties of the model and\nshown the existence of the magnetization plateau at one thir d of the saturation value. The investigation of the ground-\nstate properties of the model in the ∆−hplane shows the existence of three phases in the antiferroma gnetic case\nand two phases in the ferromagnetic case. Another interesti ng property of the model has been found by investigating\nthe behavior of Lyapunov exponent. Especially we have shown the existence of the plateau in the maximal Lyapunov\nexponent curve.\nThe rest of the paper is organized as follows: In the next sect ion using the recursion relation method we derive\nthe exact two dimensional recursion relations for the parti tion function of the spin-1\n2Ising-Heisenberg model on\ndiamond chain. The exact results for the magnetization of Is ing and Heisenberg spin sublattices have been derived.\nWe describe the ground-state properties of the model in ∆−hplane. In Sec. III we have discussed the behavior\nof Lyapunov exponent. For the antiferromagnetic case the ma ximal Lyapunov exponent for the multi-dimensional\nrational mapping is considered and it is shown that near the m agnetization plateaus the maximal Lyapunov exponent\nalso exhibits plateau structure. The supercritical point a th= 0andT→0has been found. Finally, section IV\ncontains concluding remarks.\nII. RECURSION RELATION FOR THE ISING-HEISENBERG DIAMOND CH AIN\nLet us consider the spin-1\n2Ising-Heisenberg model on diamond chain with free boundary conditions in the presence\nof an external magnetic field. The Hamiltonian operator of th e model is equal to the summation of the plaquette\n2Hamiltonians and can be written as\nH=N/summationdisplay\ni=1Hi=N/summationdisplay\ni=1[J(Sx\na,iSx\nb,i+Sy\na,iSy\nb,i+∆Sz\na,iSz\nb,i)+J1/parenleftbig\nSz\na,i+Sz\nb,i/parenrightbig/parenleftbig\nµz\ni+µz\ni+1/parenrightbig\n−hH/parenleftbig\nSz\na,i+Sz\nb,i/parenrightbig\n−hI\n2/parenleftbig\nµz\ni+µz\ni+1/parenrightbig\n], (1)\nwhereHiis Hamiltonian of each plaquette, Sα\na,i,Sα\nb,i(α=x,y,z ) andµz\nirepresent relevant components of Heisenberg\nspin-1\n2and Ising spin-1\n2operators, the parameters JandJ1stand for the interaction between the nearest-neighbourin g\nHeisenberg pairs and the nearest-neighbouring Ising and He isenberg spins, respectively and ∆is the anisotropy\nparameter. Hamiltonian (1) also includes longitudinal ext ernal magnetic fields hHandhIinteracting with Heisenberg\nand Ising spins. The first summation in Eq. (1) is correspondi ng to the interstitial anisotropic Heisenberg spins\ncoupling ( Jand∆), the second summation is corresponding to the interaction between the nearest Ising and Heisenberg\nspins and the last two summations are corresponding to the fie ld interaction with Ising and Heisenberg spins. In our\nfurther calculations we will consider the case when externa l magnetic field is uniform hH=hI. It is important to\nmention the separable nature of the Ising-type exchange int eractions between neighboring Heisenberg dimers which\nare caused from the following commutation rule between diffe rent plaquette Hamiltonians: [Hi,Hj] = 0fori/ne}ationslash=j.\nThe partition function of the system with Hamiltonian (1) is\nZ=/summationdisplay\n{µi,Sa,i,Sb,i}exp{−βH}, (2)\nwhereβ= (kBT)−1,kBis Boltzmann constant (hereafter we consider kB= 1) and T is the absolute temperature.\nBy cutting diamond chain at Sa,0andSb,0points (central plaquette) into two branches (we denote the se branches\ngn(Sa,0,Sb,0)see Fig. 1) the exact recursion relation for the partition fu nction can be derived. After this procedure\nthe partition function can be written as\nZ=/summationdisplay\n{Sa,0,Sb,0}e−β[J(Sx\na,0Sx\nb,0+Sy\na,0Sy\nb,0+∆Sz\na,0Sz\nb,0)−h(Sz\na,0+Sz\nb,0)]g2\nn(Sa,0,Sb,0), (3)\nwhereg2\nn(Sa,0,Sb,0)is contribution of both left and right branches. The sum in Eq . (3) goes over all possible\ncombinations of Heisenberg spins Sa,0andSb,0. Putting into Eq. (3) eigenvalues of the operator exp{−β[J(Sx\na,0Sx\nb,0+\nSy\na,0Sy\nb,0+∆Sz\na,0Sz\nb,0)−h(Sz\na,0+Sz\nb,0)]}we can get the partition function expressed through gn(Sa,0,Sb,0)\nZ=e−J∆\n4T+h\nTg2\nn(↑↑)+e−J\n2T+J∆\n4Tg2\nn(↑↓+↓↑√\n2)+eJ\n2T+J∆\n4Tg2\nn(↑↓ − ↓↑√\n2)+e−J∆\n4T−h\nTg2\nn(↓↓), (4)\nwhere by ↑(up) and ↓(down) we denote directions of Heisenberg spins. To find recu rsion relations for the model we\nneed to find relations between gn(Sa,0,Sb,0)andgn−1(Sa,1,Sb,1).\ngn(Sa,0,Sb,0) =/summationdisplay\n{µ1,Sa,1,Sb,1}e−β[J(Sx\na,1Sx\nb,1+Sy\na,1Sy\nb,1+∆Sz\na,1Sz\nb,1)+J1(Sz\na,0+Sz\nb,0)µz\n1+J1µz\n1(Sz\na,1+Sz\nb,1)−h(Sz\na,1+Sz\nb,1+µz\n1)](5)\n∗gn−1(Sa,1,Sb,1).\nInserting into Eq. (5) eigenvalues of the operator exp{−β[J(Sx\na,1Sx\nb,1+Sy\na,1Sy\nb,1+∆Sz\na,1Sz\nb,1)+J1(Sz\na,0+Sz\nb,0)µz\n1+\nJ1µz\n1(Sz\na,1+Sz\nb,1)−h(Sz\na,1+Sz\nb,1+µz\n1)]}we can express gn(Sa,0,Sb,0)through gn−1(Sa,1,Sb,1)\n3Figure 1: The procedure for derivation of the diamond chain.\ngn(↑↑) =(e−J∆\n4T−J1\nT+3h\n2T+e−J∆\n4T+J1\nT+h\n2T)gn−1(↑↑)+(e−J\n2T+J∆\n4T−J1\n2T+h\n2T+e−J\n2T+J∆\n4T+J1\n2T−h\n2T)gn−1(↑↓+↓↑√\n2)(6)\n+(eJ\n2T+J∆\n4T−J1\n2T+h\n2T+eJ\n2T+J∆\n4T+J1\n2T−h\n2T)gn−1(↑↓ − ↓↑√\n2)+e−J∆\n4T−h\n2T+e−J∆\n4T−3h\n2T)gn−1(↓↓),\ngn(↑↓+↓↑√\n2) =(e−J∆\n4T−J1\n2T+3h\n2T+e−J∆\n4T+J1\n2T+h\n2T)gn−1(↑↑)+(e−J\n2T+J∆\n4T+h\n2T+e−J\n2T+J∆\n4T−h\n2T)gn−1(↑↓+↓↑√\n2)\n+(eJ\n2T+J∆\n4T+h\n2T+eJ\n2T+J∆\n4T−h\n2T)gn−1(↑↓ − ↓↑√\n2)+(e−J∆\n4T+J1\n2T−h\n2T+e−J∆\n4T−J1\n2T−3h\n2T)gn−1(↓↓),\ngn(↑↓ − ↓↑√\n2) =(e−J∆\n4T−J1\n2T+3h\n2T+e−J∆\n4T+J1\n2T+h\n2T)gn−1(↑↑)+(e−J\n2T+J∆\n4T+h\n2T+e−J\n2T+J∆\n4T−h\n2T)gn−1(↑↓+↓↑√\n2)\n+(eJ\n2T+J∆\n4T+h\n2T+eJ\n2T+J∆\n4T−h\n2T)gn−1(↑↓ − ↓↑√\n2)+(e−J∆\n4T+J1\n2T−h\n2T+e−J∆\n4T−J1\n2T−3h\n2T)gn−1(↓↓),\ngn(↓↓) =(e−J∆\n4T+3h\n2T+e−J∆\n4T+h\n2T)gn−1(↑↑)+(e−J\n2T+J∆\n4T+J1\n2T+h\n2T+e−J\n2T+J∆\n4T−J1\n2T−h\n2T)gn−1(↑↓+↓↑√\n2)\n+(eJ\n2T+J∆\n4T+J1\n2T+h\n2T+eJ\n2T+J∆\n4T−J1\n2T−h\n2T)gn−1(↑↓ − ↓↑√\n2)+(e−J∆\n4T+J1\nT−h\n2T+e−J∆\n4T−J1\nT−3h\n2T)gn−1(↓↓).\nAs it can be seen from relations (6) gn(↑↓+↓↑√\n2) =gn(↑↓−↓↑√\n2)hence our recursion relation will be two-dimensional\nrational mapping. By introducing the following notations\nxn=gn(↑↑)\ngn(↑↓+↓↑√\n2), (7)\nyn=gn(↓↓)\ngn(↑↓+↓↑√\n2),\nwe can get two-dimensional recursion relation for the parti tion function\nxn=[(e−J∆\n4T−J1\nT+3h\n2T+e−J∆\n4T+J1\nT+h\n2T)xn−1+e−J\n2T+J∆\n4T−J1\n2T+h\n2T+e−J\n2T+J∆\n4T+J1\n2T−h\n2T (8)\n+eJ\n2T+J∆\n4T−J1\n2T+h\n2T+eJ\n2T+J∆\n4T+J1\n2T−h\n2T+(e−J∆\n4T−h\n2T+e−J∆\n4T−3h\n2T)yn−1]\n/[(e−J∆\n4T−J1\n2T+3h\n2T+e−J∆\n4T+J1\n2T+h\n2T)xn−1+e−J\n2T+J∆\n4T+h\n2T+e−J\n2T+J∆\n4T−h\n2T\n+eJ\n2T+J∆\n4T+h\n2T+eJ\n2T+J∆\n4T−h\n2T+(e−J∆\n4T+J1\n2T−h\n2T+e−J∆\n4T−J1\n2T−3h\n2T)yn−1]\nyn=[(e−J∆\n4T+3h\n2T+e−J∆\n4T+h\n2T)xn−1+e−J\n2T+J∆\n4T+J1\n2T+h\n2T+e−J\n2T+J∆\n4T−J1\n2T−h\n2T\n+eJ\n2T+J∆\n4T+J1\n2T+h\n2T+eJ\n2T+J∆\n4T−J1\n2T−h\n2T+(e−J∆\n4T+J1\nT−h\n2T+e−J∆\n4T−J1\nT−3h\n2T)yn−1]\n/[(e−J∆\n4T−J1\n2T+3h\n2T+e−J∆\n4T+J1\n2T+h\n2T)xn−1+e−J\n2T+J∆\n4T+h\n2T+e−J\n2T+J∆\n4T−h\n2T\n+eJ\n2T+J∆\n4T+h\n2T+eJ\n2T+J∆\n4T−h\n2T+(e−J∆\n4T+J1\n2T−h\n2T+e−J∆\n4T−J1\n2T−3h\n2T)yn−1].\n4Recursion relation (8) plays a crucial role in our further in vestigation because the thermodynamic quantities like\nmagnetization can be expressed through two-dimensional ra tional mapping. Magnetization for the sublattice of\nHeisenberg spins can be found using the following formula\nmH=< Sz\na,i+Sz\nb,i>\n2=< Sz\na,0+Sz\nb,0>\n2(9)\n=/summationtext\n{Sa,0,Sb,0}(Sz\na,0+Sz\nb,0)e−β[J(Sx\na,0Sx\nb,0+Sy\na,0Sy\nb,0+∆Sz\na,0Sz\nb,0)−h(Sz\na,0+Sz\nb,0)]g2\nn(Sa,0,Sb,0)\n2Z.\nIn Eq. (9) the sum goes over all possible combinations of Sa,0andSb,0. Putting into Eq. (9) expression for\nthe partition function and taking into account the notation (7) we can express magnetization for the sublattice of\nHeisenberg spins through recursion relations which can be w ritten as\nmH=e−J∆\n4T+h\nTx2\nn−e−J∆\n4T−h\nTy2\nn\n2(e−J∆\n4T+h\nTx2n+e−J\n2T+J∆\n4T+eJ\n2T+J∆\n4T+e−J∆\n4T−h\nTy2n). (10)\nIn the same way we can find magnetization for the sublattice of Ising spins.\nmI=< µi>=< µ1>=/summationtext\n{µi,Sa,0,Sb,0}µ1e−β[J(Sx\na,0Sx\nb,0+Sy\na,0Sy\nb,0+∆Sz\na,0Sz\nb,0)−h(Sz\na,0+Sz\nb,0)]g2\nn(Sa,0,Sb,0)\nZ. (11)\nIn this expression µ1is a part of right branch of gn(Sa,0,Sb,0). So to find magnetization for the sublattice of Ising\nspins we need to express gn(Sa,0,Sb,0)through gn−1(Sa,1,Sb,1). It is important to mention that this procedure also\nshould be done for the partition function. After this proced ure the expression for the magnetization of the sublattice\nof Ising spins can be expressed through recursion relation ( 8):\nmI=[(f1(xn−1,yn−1)−f2(xn−1,yn−1))e−J∆\n4T+h\nTxn+(f3(xn−1,yn−1)−f4(xn−1,yn−1))(e−J\n2T+J∆\n4T+eJ\n2T+J∆\n4T)(12)\n+(f5(xn−1,yn−1)−f6(xn−1,yn−1))e−J∆\n4T−h\nTyn]/[(f1(xn−1,yn−1)+f2(xn−1,yn−1))e−J∆\n4T+h\nTxn\n+(f3(xn−1,yn−1)+f4(xn−1,yn−1))(e−J\n2T+J∆\n4T+eJ\n2T+J∆\n4T)+(f5(xn−1,yn−1)+f6(xn−1,yn−1))e−J∆\n4T−h\nTyn],\nwhere\nf1(x,y) =e−J∆\n4T−J1\nT+3h\n2Tx+e−J\n2T+J∆\n4T−J1\n2T+h\n2T+eJ\n2T+J∆\n4T−J1\n2T+h\n2T+e−J∆\n4T−h\n2Ty, (13)\nf2(x,y) =e−J∆\n4T+J1\nT+h\n2Tx+e−J\n2T+J∆\n4T+J1\n2T−h\n2T+eJ\n2T+J∆\n4T+J1\n2T−h\n2T+e−J∆\n4T−3h\n2Ty,\nf3(x,y) =e−J∆\n4T−J1\n2T+3h\n2Tx+e−J\n2T+J∆\n4T+h\n2T+eJ\n2T+J∆\n4T+h\n2T+e−J∆\n4T+J1\n2T−h\n2Ty,\nf4(x,y) =e−J∆\n4T+J1\n2T+h\n2Tx+e−J\n2T+J∆\n4T−h\n2T+eJ\n2T+J∆\n4T−h\n2T+e−J∆\n4T−J1\n2T−3h\n2Ty,\nf5(x,y) =e−J∆\n4T+3h\n2Tx+e−J\n2T+J∆\n4T+J1\n2T+h\n2T+eJ\n2T+J∆\n4T+J1\n2T+h\n2T+e−J∆\n4T+J1\nT−h\n2Ty,\nf6(x,y) =e−J∆\n4T+h\n2Tx+e−J\n2T+J∆\n4T−J1\n2T−h\n2T+eJ\n2T+J∆\n4T−J1\n2T−h\n2T+e−J∆\n4T−J1\nT−3h\n2Ty.\nExpressions (10) and (12) will let us calculate the total sin gle-site magnetization of the spin-1\n2Ising-Heisenberg\nmodel on diamond chain which can be written as\nm=mI+2mH\n3. (14)\nFigure 2 shows the field behavior of the total magnetization f or antiferromagnetic case at the fixed values of interaction\nconstants J= 1.5andJ1= 1, anisotropy parameter ∆ = 1 and different values of the absolute temperature ( T).\nAt high temperatures the magnetization curve has a monotone structure (Fig. 2 (a)). At lower temperatures the\nplateau of magnetization at one third is arising in magnetiz ation curve (Fig. 2 (b)). Other plots of the magnetization\ncurves for the different values of the anisotropy parameter ∆are displayed in Fig. 2 ((c), (d), (e), (f)). Figures 2 (b),\n(c) and (d) show that the larger positive values of the anisot ropy parameter correspond to the larger width of the\nmagnetization plateau for the fixed value of the absolute tem perature. While for the negative values of anisotropy\nparameter magnetization curves remain the same (Fig. 2 (e), (f)). As it can be seen from the figures recursion relation\n5(a) (b)\n/Minus3/Minus2/Minus1 0 1 2 3/Minus0.50.00.5\nhm/Slash1ms\n/Minus3/Minus2/Minus1 0 1 2 3/Minus1.0/Minus0.50.00.51.0\nhm/Slash1ms\n(c) (d)\n/Minus4 /Minus2 0 2 4/Minus1.0/Minus0.50.00.51.0\nhm/Slash1ms\n/Minus3/Minus2/Minus1 0 1 2 3/Minus1.0/Minus0.50.00.51.0\nhm/Slash1ms\n(e) (f)\n/Minus3/Minus2/Minus1 0 1 2 3/Minus1.0/Minus0.50.00.51.0\nhm/Slash1ms\n/Minus3/Minus2/Minus1 0 1 2 3/Minus1.0/Minus0.50.00.51.0\nhm/Slash1ms\nFigure 2: The field dependence of the total magnetization wit h respect to its saturation value at exchange parameters J= 1.5\nandJ1= 1: (a)T= 0.3,∆ = 1 ; (b)T= 0.1,∆ = 1 ; (c)T= 0.1,∆ = 1.5; (d)T= 0.1,∆ = 0.5; (e)T= 0.1,∆ =−1; (f)\nT= 0.1,∆ =−1.5.\nmethod results are good agrement with other methods results such as the decoration-iteration transformation method\n(for example see [4] figure 3).\nLet us research the ground state of the spin-1\n2Ising-Heisenberg model on diamond chain via ∆andhfor the\nantiferromagnetic ( J= 1.5,J1= 1) and ferromagnetic ( J=−1.5,J1=−1) models. Depending on the value of ratio\n∆\nJ1and the magnetic field measured in unites of J1, the system exhibits two ferrimagnetic (FRI1 and FRI2) and o ne\nparamagnetic (PRM) ground-state phases (Fig. 3(a)) for the antiferromagnetic case. Phases FRI1, FRI2 and PRM\n6(a) (b)\nFigure 3: Ground-state phase diagram in the ∆−hplane for (a) antiferromagnetic case J= 1.5,J1= 1(b) ferromagnetic\ncaseJ=−1.5,J1=−1.\ncorrespond to the following values of Ising and Heisenberg s pins sublattice magnetization:\nFRI1 :mI=−0.5,mH= 0.5, (15)\nFRI2 :mI= 0.5,mH= 0,\nPRM:mI= 0.5,mH= 0.5.\nAnalytically it can be shown that for the fixed values of excha nge parameters J= 1.5andJ1= 1phase transition\nfrom FRI1 to FRI2 takes place at ∆ =1\n3. Now let us compare the displayed magnetization curves (Fig . 2) with the\nground-state phase diagram shown in Fig. 3(a). As it is alrea dy mentioned in FRI2 phase the larger positive values\nof the anisotropy parameter ( ∆>1\n3) correspond to the larger width of the magnetization platea u see Fig. 3 (b), (c)\nand (d). In FRI1 phase for the fixed values of interaction cons tants and the absolute temperature the behavior of the\nmagnetization curve remains the same (Fig. 2 (e), (f)).\nFor the ferromagnetic case there are two phases in the phase d iagram; the ferrimagnetic (FRI) and paramagnetic\n(PRM) (Fig. 3(b)). Phases FRI and PRM correspond to the follo wing values of Ising and Heisenberg spins sublattice\nmagnetization:\nFRI:mI= 0.5,mH= 0, (16)\nPRM:mI= 0.5,mH= 0.5.\nIt can be analytically shown that FRI phase ends on ∆ =−1\n3at absence of an external magnetic field.\nIII. LYAPUNOV EXPONENT AND SUPERSTABLE POINT\nIn this section we will focus on the thermodynamical equilib rium description of the spin-1\n2Ising- Heisenberg model\non a diamond chain, by studying infinite-size systems. Lyapu nov exponents near the magnetization plateau of the\nantiferromagnetic model are interesting to calculate on a d iamond chain. It is shown that the behavior of the\nmaximal Lyapunov exponent via magnetic field of multi-dimen sional rational mapping has a plateau and coincides\nwith magnetization one on one-dimensional kagome chain at l ow temperatures [35]. It was obtained that the maximal\nLyapunov exponent had a negative vanishing plateau.\nThe following values of Lyapunov exponent can be observed du ring the investigation.\n1.λ <0. Negative Lyapunov exponents show that the system is dissip ative or non-conservative. The systems with\nmore negative values of Lyapunov exponent are more stabile. Ifλ=−∞means that we have superstable fixed and\nsuperstable periodic points.\n72.λ= 0corresponding to neutral fixed point. Zero values of Lyapuno v exponents are characteristic for conservative\nsystems. At this value of Lyapunov exponents the second-ord er phase transition takes place.\n3.λ >0corresponding to unstable and chaotic systems. The systems with positive Lyapunov exponents have\nchaotic behavior.\nIn general for the mapping xn=f(xn−1)Lyapunov exponent λ(x)characterizes the exponential divergence of two\nnearby points after niterations. Lyapunov exponent may be expressed as a limit of mapping stability as [28–34]\nλ(x) = lim\nn→∞1\nnln|dfn(x)\ndx|. (17)\nIn multidimensional case with dimension n, existsnof exponents for various directions in space\neλ1,eλ2,...eλn= lim\nn→∞(eigenvalues of the productn−1/productdisplay\ni=0J(− →xi))1\nn, (18)\nwhereJ(− →x) = (∂Gi\n∂xj)is the Jacobian of the mapping− →xn+1=G(− →xn). For two dimensional mapping (8) we can\nreceive the following expression of Lyapunov exponents\nλ1,λ2= lim\nn→∞1\nnln(eigenvalues of the productn−1/productdisplay\ni=0J(xi,yi)) (19)\n(a) (b)\n/Minus3/Minus2/Minus1 0 1 2 3/Minus60/Minus50/Minus40/Minus30/Minus20/Minus100\nhΛ\n/Minus3/Minus2/Minus1 0 1 2 3/Minus80/Minus60/Minus40/Minus200\nhΛ\nFigure 4: Plot of Lyapunov exponents for spin1\n2Ising-Heisenberg model on diamond chain at exchange parame ters (a) the\nantiferromagnetic case at J= 1.5,J1= 1,∆ = 1 and temperature T= 0.1(b) the ferromagnetic case at J=−1.5,J1=−1,\n∆ = 1 and temperature T= 0.1.\nwhereJ(x,y)is the Jacobian of the mapping (8). Expression (19) will let u s count up the meanings of Lyapunov\nexponents depending from an external magnetic field ( h) at fixed values of constants of interaction (J,J1), the\nanisotropy parameter ( ∆) and temperature ( T). Figure 4 shows the dependence of Lyapunov exponents on an e xternal\nmagnetic field for the antiferromagnetic and ferromagnetic cases. As it can be seen from Fig. 4 (a) both values of\nLyapunov exponent are equal to each other at absence of an ext ernal magnetic field. Another interesting property of\nLyapunov exponent is the existence of the plateau on maximal Lyapunov exponent curve for the antiferromagnetic\ncase. It is important to mention that the locations of platea us on the maximal Lyapunov exponent curve (Fig. 4 (a))\ncoincide with the locations of plateaus on the magnetizatio n curve (Fig. 2 (b)). Such an interesting phenomena of\nLyapunov exponent has also been observed by investigating t wo, three and six spin exchange interactions Heisenberg\nmodel on kagome lattice in an external magnetic field [35]. Fi gure 4 (b) shows that for the ferromagnetic case the\nmaximum value of the maximal Lyapunov exponent tends to zero .\nIn Fig. 5 we show the behavior of Lyapunov exponent for the ant iferromagnetic case at lower temperature. As it\ncan be seen from figure the absolute values of Lyapunov expone nts are increasing by decreasing the temperature.\nNext, we turn our attention to the behavior of Lyapunov expon ent curves in different ferrimagnetic (FRI1 and\nFRI2) phases. For this purpose, Lyapunov exponent curves fo r fixed values of interaction constants and the absolute\ntemperature are plotted in Fig. (6). At low temperatures, th e minimum and maximum Lyapunov exponents have\n8(a) (b)\n/Minus3/Minus2/Minus1 0 1 2 3/Minus600/Minus500/Minus400/Minus300/Minus200/Minus1000\nhΛ\n/Minus3/Minus2/Minus1 0 1 2 3/Minus6000/Minus5000/Minus4000/Minus3000/Minus2000/Minus10000\nhΛ\nFigure 5: Plot of Lyapunov exponents at exchange parameters J= 1.5,J1= 1,∆ = 1 and temperature (a) T= 0.01(b)\nT= 0.001.\n(a) (b)\nFigure 6: Plot of Lyapunov exponents for different ferrimagn etic (FRI1 and FRI2) phases at exchange parameters J= 1.5,\nJ1= 1and the temperature T= 0.01(a)∆ = 0.6(b)∆ =−0.6.\ndifferent behavior. Only at h= 0the maximum value of the minimum Lyapunov exponent equal to v alue of the\nmaximum one, when the system is in FRI2 phase (Fig. 6 (a)), and there is no intersection of Lyapunov exponents for\nh/ne}ationslash= 0. There is a super stable point ( λmax→ ∞), whenh= 0and atT→ ∞ in FRI2 phase. There are two points of\nintersections for the maximum and minimum values of Lyapuno v exponents and coincide when h= 0in FRI1 phase\n(Fig. 6 (b)). Lyapunov exponents are tending to zero in therm odynamic limit ( T→0) in FRI1 phase at absence of\nan external magnetic field.\nNow let us investigate another interesting property of the r ecursion relation (8), namely superstability. First of all\nwe will define superstability for the one dimensional recurs ion relation. Generally one dimensional recursion relatio n\nxn=f(xn−1)is said to be superstable if the following relation takes pla ce [36–39]\ndnf(x∗)\ndx= 0, (20)\nwherex∗is the fixed point of f(x). An other way superstability can be defined by using definitio n of Lyapunov\nexponent. The system is superstable when\nλ= lim\nn→∞1\nnln|dfn(x∗)\ndx|=−∞. (21)\nIn the same way we can define superstability for two dimension al recursion relations (8). As it is already mentioned\nabove for the antiferromagnetic case the absolute values of Lyapunov exponents are increasing by decreasing the\ntemperature. Putting values of the exchange parameters ( J= 1.5,J1= 1and∆ = 1 ) into equation (17) we can see\n9that at thermodynamic limit at absence of an external magnet ic field the following relation takes place for Lyapunov\nexponents\nlim\nT→0λ1=λ2=−∞, (22)\nwhich shows the existence of the super stable point.\nWe have analyzed the behavior of the magnetization for the sp in-1\n2Ising-Heisenberg model on diamond chain for\ndifferent values of anisotropy parameter ∆. For the antiferromagnetic case ( J >0,J1>0) at fixed values of exchange\nparameters J= 1.5andJ1= 1, the temperature T and for ∆<1\n3the magnetization curves have the same appearance\nas in Fig. 2 (e). The values of characteristic Lyapunov expon ent tend to zero at absence of an external magnetic field\nand atT→0, which means that there is no supercritical behavior for the antiferromagnetic case when ∆<1\n3.\nFor the antiferromagnetic case ( J >0,J1>0) at low temperatures and for values of the anisotropy parame ter∆\n(∆>1\n3) the magnetization curves have the same behavior as shown in Fig. 2 (b). The changes of the anisotropy\nparameter ∆only brings to the changes of the width of the magnetization p lateau at one third. The values of\ncharacteristic Lyapunov exponent tend to −∞at absence of an external magnetic field and at T→0, which means\nthat there is a superstable point for the antiferromagnetic case for positive values of anisotropy parameter ∆. Usually\na super stable point lies between bifurcation points [36–39 ]. In our case for the spin-1\n2Ising-Heisenberg model on a\ndiamond chain there are no bifurcation points but the maxima l Lyapunov exponent tends to minus infinity. So we\nget the phase transition in the super stable point at h= 0andT→0. For the first time we get the phase transition\npoint at the super stable one.\nIV. CONCLUSION\nBy using the recursion relation technique, we have studied m agnetic properties of the exactly solvable spin-1\n2\nIsing-Heisenberg model on diamond chain. Recursion relati on technique allowed us to construct the exact two-\ndimensional recursion relation for the partition function . The behavior of the total magnetization with respect to\nits saturation value has been investigated. The existence o f the magnetization plateau at one third of saturation\nvalue of magnetization has been observed in the antiferroma gnetic case. The ground-state phase diagrams in ∆−h\nplane show the existence of two ferrimagnetic (FRI1 and FRI2 ) phases and one paramagnetic (PRM) phase in the\nantiferromagnetic case and one ferrimagnetic (FRI) and a pa ramagnetic (PRM) phases in the ferromagnetic case.\nThe properties of Lyapunov exponents were also discussed. T he existence of the plateau of the maximal Lyapunov\nexponent curve was observed at low temperatures. It was dete cted the different behavior for Lyapunov exponent\ncurves in two ferrimagnetic phases. We have shown that for th e antiferromagnetic case in the thermodynamic limit\n(T→0) both values of Lyapunov exponent tend to −∞at absence of an external magnetic field which is correspondi ng\nto the superstable point.\nV. ACKNOWLEDGMENTS\nThis work has been supported by the French-Armenian Grant No . CNRS IE-017 and Marie Curie IRSES SPI-\nDER, and project PIRSES-GA-2011-295302 (NA). The authors a re grateful to R. Kenna and H. 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Chakhmakhchyan, JETP L. 94 (2011) 39.\n11"    },    {        "title": "1410.5745v2.Magnetic_switching_dynamics_in_a_ferrimagnetic_two_sub_lattice_model_including_ultrafast_exchange_scattering.pdf",        "content": "arXiv:1410.5745v2  [cond-mat.mtrl-sci]  25 Mar 2015Magnetic Switching Dynamics due to Ultrafast Exchange\nScattering: A Model Study\nAlexander Baral and Hans Christian Schneider∗\nPhysics Department and Research Center OPTIMAS,\nUniversity of Kaiserslautern, 67663 Kaiserslautern, Germ any\n(Dated: March 19, 2021)\nAbstract\nWe study the heat-induced magnetization dynamics in a toy mo del of a ferrimagnetic alloy, which\nincludes localized spins antiferromagnetically coupled t o an itinerant carrier system with a Stoner\ngap. We determine the one-particle spin-density matrix inc luding exchange scattering between\nlocalized and itinerant bands as well as scattering with pho nons. While a transient ferromagnetic-\nlike state can always be achieved by a sufficiently strong exci tation, this transient ferromagnetic-like\nstate only leads to magnetization switching for model param eters that also yield a compensation\npoint in the equilibrium M(T) curve.\nPACS numbers: 72.25.Rb, 75.78.-n, 75.78.Jp, 77.80.Fm\n1I. INTRODUCTION\nHeat-induced reversal of magnetization in ferrimagnetic alloys1,2and multilayers3is a\nrecent development that has generated a lot of attention becaus e it realizes an ultrafast\ndeterministic switching without magnetic fields that may pave the way to a faster m ag-\nnetic logic. After the first phenomenological analysis of this switchin g process using the\nBaryakhtar equation,4a more microscopic understanding of this effect is now being devel-\noped using models that in one way or another include classical models of localized spins on\ndifferent sub-lattices, which are coupled by an exchange interactio ns between the different\nlattices5–7The atomistic models usually involve a thermal bath averaging, which is also\nimplicit in Landau-Lifshitz-Bloch type calculations.8–10Ref. 7 does include itinerant carrier-\nphonon scattering, but this model is based on a non-standard elec tron-phonon Hamiltonian\nand separates the charge degrees of electrons from their spins s o that its two spin systems\nessentially are also two types of localized spins coupled to a bath. Alth ough existing theoret-\nical approaches have established the importance of an exchange c oupling between localized\nmagnetic moments on different sublattices for heat-induced magne tic switching and the\noccurrence of transient ferromagnetic-like states, the microsc opic picture of magnetization\nswitching isstillunclear, asthephenomenologicalmodel4impliesacounterintuitive interplay\nbetween spin-orbit (“relativistic”) and exchange scattering. Fur ther, there are differences\nbetween existing atomistic calculations, some of which stress6thattransverse magnetization\ndynamics occurs during switching, while others do not find such a tra nsverse magnetization\ncomponent after bath averaging.5\nII. MODEL AND EQUILIBRIUM CONFIGURATION\nIn this paper we put forward a different microscopic model for magn etization dynamics—\ninspired by theories11,12for magnetic semiconductors—that (i) includes an exchange inter-\naction between delocalized and localized electrons in a band picture and (ii) that is capable\nof including the coupling of the carriers to the environment, i.e., phon ons, in a microscopic\nfashion. The model used in the following contains two bands of differe nt carrier species\nwhich have different spin and are designed to resemble itinerant 3 delectrons in iron and lo-\ncalized 4felectrons in gadolinium, respectively. For simplicity, we assume a spin o fs= 1/2\n2and a parabolic dispersion for the itinerant carriers, as well as a spin S= 1 and a com-\npletely flat band for the localized states; we thus ignore complication s that arise froma finite\nwidth of the bands which is present in real materials. Denoting the loc alized spin states\nby|ν/angbracketright=±1,0 and the electron states by |/vectorkσ/angbracketright, whereσ=±, we calculate dynamically the\nspin-resolved one-particle density matrices of the itinerant electr onsρσσ′\nkand the localized\nspinsρνν′\nlocwith an equation of motion technique. Using the spin-dependent red uced density\nmatrices we do not separate the charge and the spin of the itineran t electrons, as done in\nmodels that work with three temperatures, most notably with differ ent spin and electron\ntemperatures. “Magnetic” contributions are an antiferromagne tic exchange interaction J\nbetween itinerant and flat bands and a Stoner-like on-site coupling Uamong itinerant car-\nriers. The latter is treated in mean-field approximation and favors a ferromagnetic electron\nspin polarization. Long and short range ( U) contributions to electronic Coulomb scattering\nare neglected because (i) they do not change the itinerant spin pola rization, and (ii) for the\nconditions studied here, the electronic distributions never deviate much from Fermi-Dirac\ndistributions.13By contrast, we include both mean field and scattering contribution s from\nthe exchange interaction between both localized and itinerant spins , as well as a coupling\nof the itinerant carriers to acoustic phonons at the level of Boltzm ann scattering integrals.\nThroughout, we work with single-particle states that are obtained from a diagonalization\nincluding the exchange and Stoner mean-field contributions. The re sulting mean-field ener-\ngies are denoted by Eνfor the localized spins and ǫ/vectorkσfor the spin-split electron bands. The\nrelevant equations of motion then take the form\n∂\n∂tρνν′\nloc=i\n/planckover2pi1(Eν−Eν′)ρνν′\nloc+∂\n∂tρνν′\nloc/vextendsingle/vextendsingle\nxc(1)\n∂\n∂tρσσ′\n/vectork=i\n/planckover2pi1(ǫ/vectorkσ−ǫ/vectorkσ)ρσσ′\n/vectork+∂\n∂tρσσ′\n/vectork/vextendsingle/vextendsingle\nxc+∂\n∂tρσσ′\n/vectork/vextendsingle/vextendsingle\ne-pn−ρσσ′\n/vectork−δσσ′Fσ\n/vectork\nτsf(2)\nThe first terms in Eq. (1) and (2) are coherent contributions, whic h describe precessional\nmotion of one spin around the mean-field of the spin of the other spe cies. All the other\nterms are incoherent terms. The incoherent exchange scatterin g contributions at the level\nof Boltzmann scattering integrals are given by\n3∂\n∂tρν1ν2\nloc/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nxc=i\n/planckover2pi1/summationdisplay\n/vectork/vectork′/summationdisplay\nσ1...σ4/summationdisplay\nν3ν4ν5Wν5ν1\n/vectork′σ1/vectorkσ2(Wν3ν4\n/vectork′σ3/vectorkσ4)∗\n·δν4ν2ρν5ν3\nlocρσ1σ3\n/vectork′(δσ4σ2−ρσ4σ2\n/vectork)−δν5ν3ρν4ν2\nlocρσ4σ2\n/vectork(δσ1σ3−ρσ1σ3\n/vectork′)\nǫ/vectorkσ4−ǫ/vectork′σ3+Eν4−Eν3−iγ+(ν1↔ν2)∗\n(3)\n∂\n∂tρσ1σ2\n/vectork/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nxc=i\n/planckover2pi1/summationdisplay\n/vectork′/summationdisplay\nν1...ν4/summationdisplay\nσ3σ4σ5Wν1ν2\n/vectork′σ5/vectorkσ1(Wν3ν4\n/vectork′σ3/vectorkσ4)∗\n·δν4ν2ρν1ν3\nlocρσ5σ3\n/vectork′(δσ4σ2−ρσ4σ2\n/vectork)−δν1ν3ρν4ν2\nlocρσ4σ2\n/vectork(δσ3σ5−ρσ3σ5\n/vectork′)\nǫ/vectorkσ4−ǫ/vectork′σ3+Eν4−Eν3−iγ+(σ1↔σ2)∗\n(4)\nHere,γdenotes an infinitesimal broadening and we use the abbreviation Wνν′\n/vectorkσ/vectork′σ′≡\nJ/angbracketleft/vectorkσ|ˆ/vector s|/vectork′σ′/angbracketright/angbracketleftν|ˆ/vectorS|ν′/angbracketright, whereˆ/vector sandˆ/vectorSare the spin-operators of the localized and the itinerant\nelectrons. At this point, a comment regarding the exchange param eterJmay be in order. In\nour calculation, Jis a Coulomb matrix element that occurs in the hamiltonian and directly\nenters the equation of motion for quantum-mechanical correlatio n functions; it could be\ncalculated directly from the true electronic Bloch or Wannier functio ns. In atomistic spin\nmodels the Js are the parameters of an effective classical Heisenberg model that are ex-\ntracted from ab-initio electronic structure calculations by comput ing the (exchange) energy\nchanges for a small tilting of the magnetic moments in adjacent unit c ells.14\nThe electron-phononscattering (e-pn) contribution inEq. (2) isa standardexpression, an\nexplicit derivation of which with special attention to spin splitting is give n in Ref. 15. It is\nan important property of scattering with long-wavelength longitud inal phonons that it does\nnotlead to a transfer of angular momentum from the itinerant carriers to the lattice, it only\ncools down the itinerant carrier system and increases the tempera ture of the phonon system\nin accordance with energy conservation. The transfer of angular momentum to the lattice\nis left to a relaxation-time expression in Eq. (2) because the fundam ental mechanism is not\nthe important point of this paper. Likely it is a combination of electron -phonon/electron-\nelectron scattering and spin-orbit coupling.13,16We generally assume spin-flip processes to\nbe faster than the heat transfer to the phonons, but slower tha n the exchange scattering. In\nEq. (4),Fσ\n/vectork=f(ǫ/vectorkσ−µF) denotes a Fermi-Dirac distribution with the same energy as the\nactualρσσ′\n/vectork, but equal chemical potentials µFfor both spins.\n4We determine the equilibrium configuration self-consistently, assum ing that the equilib-\nriumreduced density matrix ρσσ′\n/vectork=δσ,σ′f(ǫσ/vectork−µ0)isspindiagonalandgiven byFermi-Dirac\ndistributions with temperature T0andequal chemical potentials µ0for bothspin states. The\naverage spins /vectorSand/vector sof the two species are parallel to the mutual exchange field. Depen d-\ning on the value of the coupling constants JandUthe itinerant system is either partially\nor fully spin polarized. In the numerical calculations below, as done in R ef. 15, we employ\ntwo-dimensional /vectorkvectors because it reduces the numerical complexity of propagat ing the\nscattering calculations over long times. This simplification also change s the equilibrium spin\npolarization, exchange splitting and the Curie temperature compar ed to three-dimensional\n/vectorkvectors. In the following numerical calculations we always assume a c ommon initial tem-\nperature of all three sub-system T0= 10K, which is far lower than the Curie temperature\nTC.\nIII. RESULTS FOR MAGNETIZATION DYNAMICS\nWe model the excitation of a short linearly-polarized laser pulse as an instantaneous\nheating of the itinerant carriers at t= 0. We assume that the localized spin system is not\nexcited optically.17Immediately after the excitation the itinerant spin density-matrix ρσσ′\n/vectork\nis assumed to be spin diagonal with the spin dependent distributions d etermined by Fermi\nfunctions with the same spin-expectation value but with an elevated initial temperature T(0)\ne\nthat usually exceeds TC. Even though the spin polarization of the carriers does not change ,\nthe chemical potentials µσbecome different.\nFigure 1 presents the key dynamical quantities for an excitation ch aracterized by T(0)\ne=\n1500K as well as a short and infinitely long spin-flip time, respectively. The parameters\nJ= 100meV and U= 400meV give rise to a Curie temperature of TC= 480K. These\nparameters together with the initial carrier temperature T(0)\nelead to a demagnetization\nscenario, regardless of the spin-flip time τsf, as shown in the top panel. The components\nof both localized and itinerant spins parallel to the mutual exchange field,S/bardblands/bardbl, show\nanultrafast symmetric decrease due to exchange scattering on a timescale of s everal ten\nfemtoseconds. The exchange scattering conserves the total a ngular momentum and energy\nof the combined system of spin-split itinerant carriers and localized s pins, so that such an\nultrafast drop does not occur for a ferromagnetic exchange cou pling.4We have checked this\n50246−0.500.51\ntime (ps)S/bardbl&s/bardbl\n0246−0.500.51\ntime (ps)S/bardbl&s/bardbl\n0246050010001500\ntime (ps)T(Kelvin)\n  \n0246050010001500\ntime (ps)T(Kelvin)\n  \n0246180220260\ntime (ps)µσ(meV)\n  \n0246180220260\ntime (ps)µσ(meV)\n  \nlower band\nupper bandlower band\nupper bandlattice\nlocalized\nitinerantlattice\nlocalized\niItinerantb) a)\nFIG. 1. Computed results for demagnetization scenario with J= 100 meV, U= 400 meV, and\nT(0)\ne= 1500 K. The spin-flip times are τsf= 200 fs (a) and τsf→ ∞ (b). Top: Dynamics of the aver-\naged localized spin S/bardbl(solid curve) and itinerant spin s/bardbl(dashed curve); middle: quasi-equilibrium\ntemperatures and TC(thin line); bottom: quasi-chemical potentials for the two itinerant bands.\nalso for our model. We will analyze the ultrafast dynamics due to the e xchange scattering\nin more detail below.\nComparedtotheintrinsictimescaleoftheexchange scattering, sp in-flipscattering, which\ndissipates only carrier angular momentum, and carrier-phonon sca ttering, which only trans-\nfers heat from the itinerant carriers to the phonon system, act o n much longer time scales of\nhundreds of fs and several ps, respectively. During the compara tively slow remagnetization\nprocess, the angular momentum and energy transfer between th e localized and itinerant\nsub-systems due to the exchange scattering is limited to the times s cales of these slower\nmechanisms. Changing the spin-flip scattering time in Fig. 1(b) as com pared to (a) does\nnot change the qualitative behavior.\nIn the middle panel of Fig. 1 we show the quasi-equilibrium temperatur es of the itinerant\n60 2.5 5−0.501\ntime (ps)S/bardbl&s/bardbl\n0 2.5 5−0.501\ntime (ps)S/bardbl&s/bardbl\n0 2.5 5150200250\ntime (ps)µσ(meV)\n  \n0 2.5 5150200250\ntime (ps)µσ(meV)\n  \nlower band\nupper bandlower band\nupper banda) b)\nFIG. 2. Top: Computed dynamics of the average localized spin S/bardbl(solid curve) and the average\nitinerant spin s/bardbl(dashed curve) for a TFS scenario with T(0)\ne= 2000 K and U= 400 meV (a)\nas well as a switching scenario with T(0)\ne= 2500 K and U= 500 meV (b). Bottom: Electronic\nquasi-chemical potentials. J= 100 meV is unchanged.\ncarriers (or “electrons”) Te, the localized spins Tlocand the phonons TL, which are obtained\nfrom the computed dynamics of the spin density matrix as the tempe ratures of thermalized\ndistributions with the same energy as the non-equilibrium distribution s. Note, in particular,\nthat the excited electrons have a quasi-equilibrium temperature Te(t) anddifferent chemical\npotentials µσ(t) for each spin species, as shown in the bottom panel of Fig. 1. With a non-\nvanishing spin-flip rate τ−1\nsfas in Fig. 1(a), the temperatures of the itinerant electrons Teand\nof the localized spins Tlocessentially converge on the time scale of the spin-flip relaxation τsf.\nThe phonon temperature TLapproaches these two on the time scale of the electron-phonon\nscattering. Figure 1(b) shows that without dissipation of angular m omentum, viz. τsf→ ∞,\nTlocdoes not get close to Te. This makes clear that exchange scattering neither simply\nequalizes the temperatures TlocandTe, nor the chemical potentials µσ.\nWe next model stronger excitations by raising T(0)\nein Fig. 2. The parameter Uin Fig. 2\n(a) is the same as before so that we have TC≃480K. In addition, we relax the lattice\ntemperature toward T0with a time constant of 10ps to include the effect of heat diffusion.\nThis guarantees a final remagnetization without affecting the fast er dynamics. For Fig. 2(b)\nwe increase Uto 500meV, which leads to TC≃1200K. For the same material parameters\n700.10.20.30.4−0.500.51\ntime (ps)S/bardbl&s/bardbl\n  \n00.10.20.30.4−0.500.51\ntime (ps)S/bardbl&s/bardbl\n  a) b)\nFIG. 3. Computed dynamics of average localized spin S/bardbl(red, starting from 1) and average itinerant\nspin (blue, starting from −0.5) after excitation with T(0)\ne= 1500 K (solid lines), T(0)\ne= 2000 K\n(dashed lines) and T(0)\ne= 2500 K (dashed-dotted lines) for U= 400 meV (a), U= 500 meV (b),\nandJ= 100 meV.\nas in Fig. 1(a), Fig. 2 exhibits a TFS starting around 50fs and persist ing up to about 2ps.\nDuring the TFS, the localized spins experience a populationinversion a nd the corresponding\nspin-temperature Tlocis no longer well defined. As the coupling to the phonons cools down\nthe itinerant carriers, the system either returns, as in Fig. 2(a), into its initial state or,\nas in Fig. 2(b), remagnetizes with an inverse orientation and thus un dergoes magnetization\nswitching(SW).NotethatinFig.2(b)theitinerantcarriersbecome fullyspin-polarized, s/bardbl≃\n1/2, around 2.5ps so that the exchange scattering cannot further remagnetize the localized\nspins. We label the eigenstates according to their energy, so that this label changes if the\neffective mean-field splitting changes its sign. Comparing the bottom panels of Figs. 1 and 2\nindicates that the difference between the chemical potentials does not drive the switching\nor demagnetization dynamics.\nIn Fig. 3 we come back to the role of the exchange scattering in realiz ing the TFS. We\nplotherefordifferentinitialtemperatures T(0)\nethecomputeddynamicsobtainedbyincluding\nonly exchange scattering. While this leads to an unphysical steady s tate, it illustrates the\nway in which exchange scattering works. In Fig. 3(a), for U= 400meV, an initial itinerant\ntemperature T(0)\ne= 1500K leads to a state with reduced localized spin, i.e., demagnetizat ion\nin each spin system. Increasing T(0)\neleads to a reversal of the itinerant spin s/bardbland thus a\nTFS on the timescale of the exchange scattering. This behavior occ urs because the exchange\nscattering redistributes the deposited energy as well as angular m omentum between the\nitinerant and localized spins all the while satisfying both conservation of total energy and\n8300 400 500 6000\n1000\n2000\n3000\nU(meV)T(0)e(K)\nFIG. 4. Color-coded dynamics: demagnetization (black), tr ansient ferromagnetic-like state (gray)\nand switching (white) as a function of Stoner parameter Uand excitation temperature T(0)\ne. The\nexchange coupling constant is J= 100 meV. The arrow marks the “critical value” of U= 445 K,\nabove which a compensation point in the equilibrium M(T) relation occurs.\ntotal angular momentum. In Fig. 3(b) the Stoner parameter is incr eased to U= 500meV,\nwhich is indicative of a more rigid itinerant carrier magnetism. Corresp ondingly, for the\nsmaller initial temperatures T(0)\nethe demagnetization of each spin system due to exchange\nscattering is reduced compared to Fig. 3(a). For the highest excit ation, i.e., T(0)\ne= 2500K\nnot only a TFS occurs, but the average itinerant spin s/bardblbecomes larger than the average\nlocalized spin S/bardbl. Thus the redistribution of the deposited energy during the ultraf ast\nexchange scattering obviously is decisive for the following behavior. Note that the exchange\nscattering alone, i.e., without spin-flip scattering, yields the initial ult rafast demagnetization\nand the subsequent TFS. This seems to be a more realistic scenario t han that described in\nRef. 4 where spin-orbit (“relativistic”) relaxation dominates the su b-picosecond dynamics,\nand the exchange interaction acts on a picosecond timescale.\nIn Fig. 4 we collect the results from individual calculations as shown in F ig. 1(a) and 2\nby plotting the type of dynamics: demagnetization, transient ferr omagnetic-like state (TFS)\nand switching vs. excitation temperature T(0)\neand the strength of the itinerant ferromag-\nnetic coupling constant Uin the carrier system. We find that there is a threshold for the\ntemperature T(0)\ne, below which the system only de- and remagnetizes into its initial orien -\ntation (without entering a TFS). Above this threshold it depends on the Stoner parameter\nUwhether “only” a TFS occurs or whether heat-induced switching is a chieved. Fig. 4\nclearly shows that switching only occurs for Stoner parameters U >(450±25)meV. With\n9the help of the Stoner parameter Uwe can, in the framework of our mean-field model,\nput the criterion for switching in correspondence with the tempera ture dependence of the\nequilibrium magnetization M(T): We find that a compensation point occurs in the M(T)\nrelation only for Stoner parameters larger than U≃445meV. Within the accuracy of our\nnumerical study these values are identical, and one can speculate a bout the importance of\nthe existence of a compensation point for heat-induced magnetic s witching (assuming that\none starts below the compensation temperature). The present r esults should be taken into\naccount when analyzing optically induced magnetization dynamics in fe rrimagnetic alloys,\nwhere both heat-induced and all-optical processes may be importa nt.18,19\nIV. CONCLUSION\nWe presented a dynamical calculation of exchange scattering afte r spin-conserving in-\nstantaneous heating (designed to model ultrafast optical excita tion) in a simple quantum-\nmechanical mean-field model of a ferrimagnetic alloy. Wefound that the exchange scattering\nprovides the essential contribution to the ultrafast switching dyn amics and established the\nimportance of the Stoner parameter Ufor the occurrence of a transient-ferromagnetic state\nand/or switching. We speculated that the occurrence of ultrafas t magnetization switching\nmay be related to the existence of a compensation point in the equilibr ium magnetization.\n∗hcsch@physik.uni-kl.de\n1I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. D¨ urr, T. A. Ostler, J. Barker,\nR. F. Evans, R. W. Chantrell, et al. , Nature 472, 205 (2011).\n2T. A. Ostler, J. Barker, R. F. Evans, R. W. Chantrell, U. Atxit ia, O. Chubykalo-Fesenko,\nS. El Moussaoui, L. Le Guyader, E. Mengotti, L. J. Heyderman, et al. , Nature communications\n3, 666 (2012).\n3N. Bergeard, V. L´ opez-Flores, V. Halt´ e, M. Hehn, C. Stamm, N. Pontius, E. Beaurepaire, and\nC. Boeglin, Nature communications 5(2014).\n4J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov, A. K irilyuk, A. V. Kimel, O. Eriksson,\nM. I. Katsnelson, and T. Rasing, Physical Review Letters 108, 057202 (2012).\n105S. Wienholdt, D. Hinzke, K. Carva, P. M. Oppeneer, and U. Nowa k, Physical Review B 88,\n020406 (2013).\n6U. Atxitia, T. Ostler, J. Barker, R. F. L. Evans, R. W. Chantre ll, and O. Chubykalo-Fesenko,\nPhys. Rev. B 87, 224417 (2013).\n7A. J. Schellekens and B. Koopmans, Physical Review B 87, 020407 (2013).\n8D. A. Garanin, Physical Review B 55, 3050 (1997).\n9U. Nowak, in Handbook of Magnetism and Advanced Magnetic Materials , edited by\nH. Kronm¨ uller and S. Parkin (John Wiley & Sons, Ltd, Chiches ter, UK, 2007) p. 858.\n10N. Kazantseva, D. Hinzke, U. Nowak, R. W. Chantrell, U. Atxit ia, and O. Chubykalo-Fesenko,\nPhys. Rev. B 77, 184428 (2008).\n11T. Jungwirth, J. Sinova, J. Maˇ sek, J. Kuˇ cera, and A. MacDon ald, Reviews of Modern Physics\n78, 809 (2006).\n12/suppress L. Cywi´ nski and L. J. Sham, Physical Review B 76, 045205 (2007).\n13B. Y. Mueller, A. Baral, S. Vollmar, M. Cinchetti, M. Aeschli mann, H. C. Schneider, and\nB. Rethfeld, Physical Review Letters 111, 167204 (2013).\n14A. I. Liechtenstein, M. Katsnelson, V. Antropov, and V. Guba nov, Journal of Magnetism and\nMagnetic Materials 67, 65 (1987).\n15A. Baral, S. Vollmar, and H. C. Schneider, Phys. Rev. B 90, 014427 (2014).\n16M. Krauß, T. Roth, S. Alebrand, D. Steil, M. Cinchetti, M. Aes chlimann, and H. C. Schneider,\nPhys. Rev. B 80, 180407 (2009).\n17A. Manchon, Q. Li, L. Xu, and S. Zhang, Physical Review B 85, 064408 (2012).\n18S. Alebrand, M. Gottwald, M. Hehn, D. Steil, M. Cinchetti, D. Lacour, E. E. Fullerton,\nM. Aeschlimann, and S. Mangin, Phys. Rev. B 101(2012).\n19S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V. Uhl´ ır, L . Pang, M. Hehn, S. Alebrand,\nM. Cinchetti, Y. Malinowski, G. Fainman, M. Aeschlimann, an d E. E. Fullerton, Nature ma-\nterials13(2014).\n11"    },    {        "title": "1502.05960v1.Classical_dipoles_on_the_kagome_lattice.pdf",        "content": "Classical dipoles on the kagome lattice\nMykola Maksymenko,1, 2V. Ravi Chandra,3and Roderich Moessner1\n1Max-Planck-Institut f ur Physik komplexer Systeme, N othnitzer Stra\u0019e 38, 01187 Dresden, Germany\n2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel\u0003\n3School of Physical Sciences, National Institute of Science Education and Research,\nInstitute of Physics Campus, Bhubaneswar, 751005 India\nMotivated by recent developments in magnetic materials, frustrated nanoarrays and cold atomic\nsystems, we investigate the behaviour of dipolar spins on the frustrated two-dimensional kagome\nlattice. By combining the Luttinger-Tisza approach, numerical energy minimization, spin-wave\nanalysis and parallel tempering Monte-Carlo, we study long-range ordering and \fnite-temperature\nphase transitions for a Hamiltonian containing both dipolar and nearest-neighbor interactions. For\nboth weak and moderate dipolar interactions, the system enters a three-sublattice long-range ordered\nstate, with each triangle having vanishing dipole and quadrupole moments; while for dominating\ndipolar interactions we uncover ferrimagnetic three-sublattice order. These are also the ground\nstates for XY spins. We discuss excitations of, as well as phase transitions into, these states. We \fnd\nbehaviour consistent with Ising criticality for the 120ostate, while the ferrimagnetic state appears\nto be associated with drifting exponents. The celebrated \rat band of zero-energy excitations of the\nkagome nearest-neighbour Heisenberg model is lifted to \fnite energies but acquires only minimal\ndispersion as dipolar interactions are added.\nPACS numbers: 75.10.Hk, 75.30.Ds, 75.40.-s, 75.40.Mg\nI. INTRODUCTION\nLong ranged dipolar interactions occur in any lattice\nsystem of interacting magnetic moments. However, the\nassessment of the relevance of dipolar interactions in\ndetermining the behavior of magnetic systems has wit-\nnessed a recalibration in the recent past. This is largely\ndue to the advent of several experimental systems that\nshifted the focus away from purely exchange coupled\nmagnets where the dipolar interaction is routinely ne-\nglected.\nWe can identify at least three broad classes of systems\nwhich have led to this renewed interest in dipolar in-\nteractions. The \frst are the A2B2O7pyrochlore oxides,\nwhich most closely resemble conventionally studied mag-\nnetic systems1. For these, as a result of an interplay of\ncrystal \feld e\u000bects, geometry and the speci\fc magnetic\nions involved, the dipolar interations can be appreciable.\nA second class are nanomagnetic arrays2, collections of\nnanomagnetic islands arranged in a regular pattern using\nlithography. The magnitude of the moments as well as\nthe strength of the dipolar interactions can be tuned to a\ngreat degree by controlling the dimensions and separation\nof the magnetic islands. These systems are much more\ntunable than the thin \flm systems studied in the past\nwith a view to analysing pattern formation and order-\ning via the dipolar interaction3. Finally, the last decade\nhas seen rapid development of magnetic systems of polar\nmolecules and atomic gases with large dipole moments\ncon\fned in optical lattices4,5.\nOf particular interest is the interplay of dipolar interac-\ntions and geometrical frustration. On frustrated lattices,\nan exchange term typically gives rise to a macroscopi-\ncally degenerate yet locally strongly constrained ground\nstate manifold, usually lacking conventional magnetic or-der. This constraint can be thought of as restricting the\nspace of states, often in a topologically non-trivial way,\nwithin which dipolar interactions are to be minimised;\nor, conversely, the dipolar interactions can be thought\nof as lifting the degeneracy, akin to the usual order-by-\ndisorder physics characteristic of quantum and thermal\n\ructuations6. The combination of exchange and dipoles\ncan lead to suprising results, such as in the case of spin\nice7, where the underlying elementary excitations can\nbe seen as doubly gauge charged8(emergent) magnetic\nmonopoles9.\nTheoretical e\u000borts to study dipolar spins are several\ndecades old10{13. An early milestone is the work of Lut-\ntinger and Tisza10who established that the ground state\nfor a simple cubic lattice of dipoles is an antiferromag-\nnetic arrangement of chains of aligned dipoles. There-\nafter, Maleev13found that the long range and anisotropic\nnature of the dipolar interactions can stabilise long range\norder in two dimensional magnets - something that is\nprohibited for short ranged isotropic exchange Hamilto-\nnians because of the Mermin-Wagner theorem. Indeed,\nfor nanomagnetic arrays and cold atoms in optical lat-\ntices the study of two dimensions is particularly rele-\nvant. For dipoles on the square lattice the ground state\nlikely consists of antiferromagnetically aligned ferromag-\nnetic legs14,15and closely related degenerate states16. For\nthe triangular lattice a ferromagnetic phase has been re-\nported for purely dipolar interactions but it was argued\nthat other phases like a 120ophase and striped antifer-\nromagnetic phases appear for increasing strength of the\nexchange interaction17{20. While there is some agree-\nment about the nature of the low temperature phase for\nseveral of these systems the precise details of the transi-\ntion to those low temperature phases are frequently un-\nder debate. The principal reason lies in the subtletiesarXiv:1502.05960v1  [cond-mat.str-el]  20 Feb 20152\ninvolved in the thermodynamic limit in the presence of\nlong ranged (and anisotropic) interactions.\nFor some otherwise well-studied lattices, not even the\ndipolar ground state is known. A case in point are\nclassical dipolar spins on the kagome lattice, the focus\nof this work. The kagome is perhaps the most-studied\ntwo-dimensional highly frustrated lattice, for which even\nthe low-temperature behaviour of a 'simple' nearest-\nneighbour Heisenberg model is remarkably intricate21,22\nWe investigate in this paper using a combination of\nLuttinger-Tisza (LT) method, spinwave calculations, nu-\nmerical energy minimisation and extensive Monte Carlo\nsimulations the interplay of exchange and dipolar inter-\nactions. We \fnd two distinct low-temperature orderings.\nFor weak dipolar interactions we observe 120othree-\nsublattice order with zero net moment; while for strong\ndipolar interactions we \fnd a peculiar ferrimagnetic state\nwith continuously varying net moment. Thus we have\ntwo di\u000berent three-sublattice k0= (0;0) states at weak\nand strong dipolar interactions (Fig. 1b). While our re-\nsults for the case of strong dipolar interactions predict\na \fnite moment per unit cell as in earlier work23, our\nextensive simulations and analytic considerations do not\nsupport the existence of a disordered non-magnetic sub-\nlattice.\nThe outline of the paper is as follows. In Section II\nwe introduce model and conventions used. In Section\nIII we present the ground-state phase diagram from the\nLuttinger-Tisza approach10. This method fails in the\ncase of strong dipolar term and hence in Section IV we\nperform a numerical search for the ground state and in\nSection V we con\frm that this state is locally stable\nvia a spinwave analysis. Finally, in Section VI we an-\nalyze our model using a parallel-tempering Monte-Carlo\nmethod which con\frms our predictions for the ground\nstates and provides the details of the corresponding rich\n\fnite-temperature phase transitions. We close with a\ndiscussion section.\nII. MODEL\nThe kagome lattice given in Fig. 1 is an Archimedean\nlattice24, a triangular lattice of triangles. The positions\nof the triangular Bravais lattice points are denoted by Rl\nwhile each site in the unit-cell is labeled by ri, so that a\nsite is labeled by Rl\nj= (Rl+rj). Throughout the paper\nthe lattice constant Rnnis set to 1=2 such that the full\ntranslation of the three site unit cell is the unit of length.\nThe general Hamiltonian of the system of Nspins is\nH=X\nk;i;l;jX\n\u000b;\fJ\u000b\f\nij(Rkl\nij)S\u000b\ni(Rk)S\f\nj(Rl);(1)\nJ\u000b\f\nij(R) =1\n2\u0014\nJ\u000e\u000b\f+DR3\nnn\u0012\u000e\u000b\f\njRj3\u00003R\u000bR\f\njRj5\u0013\u0015\n:(2)\nHereRkl\nijis the vector between two interacting classical\nO(3) spinsS\u000b\ni(Rk) andS\f\nj(Rl), of unit length. kand\nA2A1012✓D=1J=10⇡2\n\u0000⇡2\u0000⇡360FIG. 1. Kagome lattice (left) and the ground-state phase di-\nagram of the model consisting of nearest-neighbour exchange\nJ= sin\u0012and dipolar interactions of strength D= cos\u0012\n(right).A1; A 2are the basis vectors and a dark green tri-\nangle denotes the unit cell of three sites. The system exhibits\nlong-range 120oorder for\u0019=2\u0015\u0012 > \u0012 1with\u00121= 10:010\nand ferrimagnetic order for \u0000\u0019=2<\u0012<\u0012 2where\u00122= 1:03o.\nThe latter has two spins inclined with respect to one of the\nunit-cell edges by angle \u0006\u001e(\u0012). The area between two phases\npossibly contains an incommensurate intermediate regime.\nlindex the unit cell, while iandjrun over the sites of\nthe basis in the unit cell and Greek \u000band\fdenote the\ncomponents of the vectors x; y andz. The \frst term\nof the interaction matrix (2) is the nearest-neighbor ex-\nchange while the second is the dipole interaction, with\nR3\nnn, the nearest-neighbor distance, included for normal-\nization. A factor1\n2has been included to avoid double\ncounting.J >0 is the energy scale of the antiferromag-\nnetic (AFM) nearest-neighbor exchange. The dipolar en-\nergy scale is\nD=\u00160\n4\u0019\u00162\nR3nn; (3)\nwhere\u0016is the magnetic moment of the ions.\nWe parametrize the relative strength of JandDvia\nan angle\u0012(Fig. 1):\nJ= sin\u0012;D= cos\u0012; (4)\nwith the unit of energy set to J2+D2= 1.\nFourier transformation of the Hamiltonian (2) yields\nH=X\nk;i;jJ\u000b\f\nij(k)S\u000b\ni(\u0000k)S\f\nj(k) (5)\nJ\u000b\f\nij(k) =X\nklJ\u000b\f\nij(Rkl\nij) exp[\u0000ikRkl\nij]: (6)\nWe generate the interaction matrix for the dipolar inter-\nactions using Ewald summation25, which we con\frm by\nthe direct lattice summation possible in two dimensions.3\n-3-2-1 0 1 2 3 4\nΓ YX Γλ(k)a)\n-3-2-1 0 1 2 3 4\nΓ YX Γλ(k)b)\nFIG. 2. Eigenvalues of the interaction matrix J\u000b\f\nij(k) along\nlines in the Brillouin zone. a) Spectra for dominant exchange\ncase,\u0012=\u0019=4. b) spectra for D= 1,\u0012= 0. The eigenvalue\nlowest in energy is generally at the \u0000 point k0.\nIII. LUTTINGER-TISZA ANALYSIS\nWe \frst determine a ground state using the Luttinger-\nTisza (LT) method10where it applies. Decomposing the\ninteraction matrix into its Fourier components, and de-\nnoting by\u0015min(k) the lowest eigenvalue(s) of the inter-\naction matrix, we use the fact that the energy of anyspin\ncon\fguration satis\fes the bound\nH\u0015N\u0015min(k0): (7)\nIf there exists a spin con\fguration which can be decom-\nposed into a linear combination of only the 'optimal'26\nLT eigenvectors corresponding to these eigenvalues, it is\na global ground state. This happens if the \\strong con-\nstraint\" of unit length for the spins\njSij2= 1 (8)\ndoes not con\rict with the optimal eigenvectors, which\nhowever in general have entries with di\u000berent amplitudes.\nIn the latter case, not unusual for non-Bravais lattices,\nnon-optimal modes have to be admixed, and the LT ap-\nproach only yields an (often rather useful) guess at possi-\nble ground states, or at least at leading instabilities from\nthe high-temperature paramagnet.\nA. Dominant nearest-neighbor exchange.\nFor pure nearest-neighbor antiferromagnetic exchange\n\u0012=\u0019=2 (D= 0), the lowest branch of the interaction\nmatrix is exactly \rat (dispersionless) re\recting the high\nground state degeneracy27{30. Decreasing \u0012we move to\nnonzeroD > 0 which immediately lifts the degeneracy,\nselecting a ground state at wavevector k0= (0;0), as\nshown in Fig. 2 a) for \u0012=\u0019=4. The optimal eigenvec-\ntor satis\fes the constraint (8) and results in a 120ostate\nwhich is doubly degenerate re\recting two possible chi-\nralities. Further increase of Dleads to level crossing at\n\u00121= 10:010. Hence the 120ostate is certainly stable up\nto this point, as we have also con\frmed in Monte-Carlo\nsimulations.\nπ/6π/3\n-π/2-π/3-π/60π/18φ(θ)\nθFIG. 3. Inclination angle \u0006\u001eof two of the spins in the unit\ncell (Fig. 1) as function of \u0012(Eq. 4) in the ferrimagnetic\nphase.\nB. Dominant dipolar exchange.\nFor\u0012<\u0012 1LT no longer yields an exact ground state31.\nInstead, we enter an intermediate regime where neither\nspin-wave nor Monte-Carlo computations (see Sections\nV and VI) allow us reliably to conclude on the ground\nstate. This regime persists up to the point \u00122= 1:030\nbeyond which the 120ostate is no longer even a stable\nlocal minimum at k0= (0;0).\nFor purely dipolar interactions \u0012= 0 the minimal\neigenvalue\u000f0=\u00002:487 is doubly degenerate and again\noccurs at k0= (0;0), Fig. 2 b. The best state we \fnd has\ntwo of the spins are inclined approximately by \u001e\u0019\u0006160\nwith respect to one of the unit-cell edges while the third\nspin remains unchanged (right panel of Fig. 1). This sit-\nuation persists, with varying \u001e(\u0012) until the ferromagnetic\npoint\u0012=\u0000\u0019=2. However, in general no combination of\nthe pair of eigenvectors satis\fes the strong constraint on\nspin length (8). To determine the true ground state for\nhard unit length spins, we thus need to allow the admix-\ning of other modes, so that we next turn to numerics.\nIV. NUMERICAL ENERGY MINIMIZATION\nFORD\u001dJ\n. Our Monte-Carlo simulations (Section VI) do unveil\nank0= (0;0) ordering at low temperatures, suggesting\nthat hard spin constraint may optimally be satis\fed by\nadmixing higher modes at k0= (0;0) only. We therefore\nconstrain our problem to a single unit cell and perform\na numerical minimization of the Hamiltonian (5). The\nminimal energy con\fguration for the single unit cell is\nindeed the state found in full lattice Monte Carlo simula-\ntions. The ground state is a ferrimagnetic con\fguration\nin which the spins Sitake the following angles with one\nof the three edges in the unit cell\n\u001e1=\u001e; \u001e 2=\u0000\u001e; \u001e 3= 0;4\n 0 2 4 6 8 10 12 14\nΓYXΓε(k)/Sa) 0.1 1\n 0.001 0.01\u00002.6(1)pDD\n 0 1 2 3 4 5 6 7 8 9\nΓ YX Γε(k)/Sb)\nFIG. 4. Spin-wave spectra for a) 1200state at\u0012=\u0019=4 and\nfor b) the ferrimagnetic ground state obtained by energy min-\nimization at \u0012= 0. In a), the lowest branch remains almost\nperfectly \rat, while acquiring a gap /p\nD(inset).\n 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4\n 0 0.5 1 1.5 2CvS/T\nT/Sa)\nθ=π/4\nθ=0\n 0 0.05 0.1 0.15 0.2\n 0 0.5 1 1.5 2∆M\nT/Sb)θ=π/4\nθ=0\nFIG. 5. a) Speci\fc heat scaled by the temperature and b)\ntemperature dependence of the reduction of sublattice mag-\nnetization due to quantum \ructuations calculated using linear\nspin wave theory, which is controlled in the limit of small T.\nwith\n\u001e\u001936:420(9)\nAs we change\u0000\u0019=2< \u0012 < \u0012 2we can obtain a minimal\nenergy ferrimagnetic con\fguration with a drifting \u001e(\u0012),\nas shown in Fig. 3.\nV. LINEAR SPIN-WAVE THEORY\nWe next study the role of quantum \ructuations around\nthe two ground states discussed above. We \fnd that both\nstates are locally stable, and exhibit a lowest band with\nlittle dispersion, in particular for the 120ostate.\nWe evaluate the spin-wave spectrum of non-collinear\nspin structures using standard methods32{34. The Hamil-\ntonian of a Bose gas of magnons reads\nH=H(0)+X\nkX\ni\u000fi(k) (10)\n+X\nkX\ni\u000fi(k)h\nay\ni(k)ai(k) +ay\ni(\u0000k)ai(\u0000k)i\n;\nwhereai(k) are boson annihilation operators with H(0)\nthe classical ground state energy. For a stable ground\nstate spin con\fguration, His Hermitian and all the spin-\nwave eigenenergies \u000fi(k) are real. This yields the spe-\nci\fc heatCv(T) and magnetization M(T), allowing in\nprinciple for comparison with experimental data at lowtemperature, e.g. below a scale set by the gap in the\nexcitation spectrum35.\nWe \frst con\frm that the 120-degree and ferrimagnetic\nstates at\u0012=\u0019=4;0, respectively, are stable to quantum\n\ructuations. While it is known from previous studies28\nthat for the 120ostate the spin-wave excitation spec-\ntrum has a fully dispersionless (\rat) band at zero energy\nas well as two-fold degenerate acoustic mode, the ad-\ndition ofDleads to a gap in the excitation spectrum\nproportional top\nDat smallD. We plot the spin-wave\nspectra for cases \u0012=\u0019=4 and\u0012= 0 in Fig. (4) where\nit is clearly seen that in both cases the leading e\u000bect of\ndipolar interactions is pushing the zero modes to \fnite\nfrequency, expected on account of the absence of a con-\ntinuous symmetry. The dispersion of the lowest branch\nof the 1200state is only weakly a\u000bected. This fact can\nmanifest itself in \fnite energy almost k-independent res-\nonance in inelastic neutron scattering33,36. The existence\nof the gap in the spectrum a\u000bects the corresponding spe-\nci\fc heat and sublattice magnetization (Fig. 5). The\ngap leads to an exponential suppression of speci\fc heat\nCv\u0018exp[\u0000\u0001=T] or reduction of staggered magnetiza-\ntion \u0001M(T)\u0018exp[\u0000\u0001=T] with the gap \u0001.\nMoreover we have checked for both \u0012= 0;\u0019=4 that\nthese are the only stable spin-con\fgurations at k0=\n(0;0). We close this section by noting that both the\n1200and ferrimagnetic states are locally stable within\nthe boundaries of intermediate phase (Fig. 1).\nVI. MONTE-CARLO SIMULATIONS\nThis section pursues two goals. Firstly, the ground\nstates are con\frmed numerically; secondly, the corre-\nsponding \fnite-temperature phase transitions are anal-\nysed in detail. This is done with computationally inten-\nsive but tractable Monte-Carlo simulations of the system\non \fnite lattices with linear dimension of L\u001424 unit\ncells orN\u00141728 sites.\nWe employ parallel tempering with 64 to 128 replicas\nin the temperature range T= 0:125\u00002:95 for the phase\ntransition analysis and in the range T= 0:00625\u00002:95\nto investigate the low energy con\fguration of the dipoles.\nOne Monte-Carlo step corresponds to a sweep over the\nlattice in which on average every spin is touched. We\nperform\u0019106Monte Carlo steps for the thermalization,\nfollowed by\u0019106steps for every measuring round.\nWe obtain thermodynamic properties of the model\n(speci\fc heat, uniform and staggered { 1200state\n{magnetization, magnetic susceptibility, fourth order\nBinder cumulant) as well as the structure of the low-\ntemperature spin con\fguration.\nFor the set of parameters leading to the ferrimagnetic\nground state we analyze the phase transition via the be-\nhavior of the magnetic order parameter\nM=1\nNX\ni(Sx\ni;Sy\ni;Sz\ni); (11)5\n 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4\n 0.2 0.4 0.6 0.8 1 1.2C/N\nTL=6\nL=9\nL=14\nL=17\n 0 0.2 0.4 0.6 0.8 1\n 0.2 0.4 0.6 0.8 1 1.2M\nTL=6\nL=9\nL=14\nL=17\n 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65\n 0.2 0.4 0.6 0.8 1 1.2UL\nTL=6\nL=9\nL=14\nL=17\n 0 0.5 1 1.5 2 2.5 3 3.5 4\n 0.2 0.4 0.6 0.8 1 1.2χ\nTL=6\nL=9\nL=14\nL=17\nFIG. 6. Monte-Carlo results for speci\fc heat C=N , Binder\ncumulantUL, magnetization Mand susceptibility \u001ffor\u0012= 0\n(D= 1 andJ= 0).\n 1 1.5 2 2.5 3\n 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2C/N\nTL=6\nL=9\nL=14\nL=17\n 0 0.2 0.4 0.6 0.8 1\n 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2Mχ\nTL=6\nL=9\nL=14\nL=17\n 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2UL\nTL=6\nL=9\nL=14\nL=17\n 0 2 4 6 8 10 12 14 16 18\n 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2χ\nTL=6\nL=9\nL=14\nL=17\nFIG. 7. Monte-Carlo results for speci\fc heat C=N , Binder\ncumulantUL, chiral order parameter M\u001fand corresponding\nsusceptibility \u001ffor\u0012= 35:60.\nwhere the sum is taken over all the sites in the lattice.\nFor the planar 1200ground state order we investigate the\norder parameter which captures the particular chiral spin\npattern,\nM\u001f=pm\u001fm\u0003\u001f; (12)\nwhere\nm\u001f=1\nNX\nR+rjS(R+rj) exp(i\u001ej); (13)\nand\u001ejare sublattice phase angles \u001e1= 0,\u001e2= 2\u0019=3,\nand\u001e3= 4\u0019=3.\nTo investigate the corresponding \fnite-temperature\nphase transition we also compute the fourth order Binder\ncumulant\nUL= 1\u00001\n3hOi4\nhO2i2; (14)\n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1\n-2-1 0 1 2Mχ Lβ/ν\n(T/Tc-1) L1/νTc=0.6925\n1/ν =1.0\nβ/ν =0.125L=8\n  10\n 12\n 14\n 16\n 17\n 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2\n-1-0.5 0 0.5 1 1.5 2 2.5 3M Lβ/ν\n(T/Tc-1) L1/ νTc=0.4396\n1/ν =1.05\nβ/ν =0.25L=10\n 11\n 12\n 13\n 14\n  15\n  17\n 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14\n-2-1 0 1 2χ L-γ/ν\n(T/Tc-1) L1/νTc=0.6925\n1/ν =1.0\nγ/ν =1.75L=8\n  10\n 12\n 14\n 16\n 17\n 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055\n-2-1 0 1 2 3 4 5 6 7χ L-γ/ν\n(T/Tc-1) L1/νTc=0.4396\n1/ν =1.05\nγ/ν =1.50L=10\n11\n 12\n 13\n 14\n  15\n  17FIG. 8. Scaling collapse for the magnetic (chiral) order pa-\nrameter and its susceptibility for the transition into the 1200\nstate (left) and the ferrimagnetic state (right).\nas well as susceptibility\n\u001f=N\nT\u0000\nhO2i\u0000hOi2\u0001\n; (15)\nand speci\fc heat per spin\nC=N =1\nN1\nT2\u0000\nhE2i\u0000hEi2\u0001\n: (16)\nTo characterize the phase transitions we employ stan-\ndard \fnite-size scaling\n~M(L1=\u0017t) =L\f=\u0017ML;\n~\u001f(L1=\u0017t) =L\u0000\r=\u0017\u001fL;\n~C(L1=\u0017t) =L\u0000\u000b=\u0017CL; (17)\nwheret= (T\u0000Tc)=Tcis the reduced temperature. To\nobtain the critical exponent 1 =\u0017and critical point Tcwe\nuse the scaling relation for the Binder cumulant\n~U(L1=\u0017t) =UL: (18)\nWe extract \u0017; \f; \u000b; \r andTcvia data collapse.\nLet us \frst look at the \fnite temperature transition to\nthe 1200state. We perform MC simulations deep in the\nordered phase for \u0012= 35:60. Collapsing the curves for\nchiral order parameter, Binder cumulant, susceptibility\nand speci\fc heat yields critical temperature Tcas well as\nthe full set of critical exponents \u0017; \f ;\r; \u000b see Fig. 8.\nThe transition occurs at Tc= 0:692(5), consistent with\nthe 2D Ising universality class with critical exponents\n\u0017= 1; \f= 1=8; \r= 7=4; \u000b= 0, re\recting the discrete\nZ2symmetry of the chiral order parameter.\nFor dominant dipolar interactions we analyze two\npoints,\u0012= 0, and for \u0012= 10. The ferrimagnetic or-\nder has a six-fold discrete symmetry and at T= 0 di\u000bers\nonly in the angle \u001eof the two inclined spins (Fig. 1\nb)). We therefore expect the corresponding transitions\nto belong to the same universality class.6\nTABLE I. Critical exponents for the continuous phase tran-\nsition analyzed with classical Monte-Carlo.\n1/\u0017\u000b=\u0017\f=\u0017\r=\u0017 U. Class\n\u0012\u0019\u0019=5 1 0 1/8 7/4 Ising\n\u0012= 0 1.05(3) 0.10(3) 0.25(2) 1.5(2) Unknown\n\u0012= 101.17(3) 0.32(3) 0.25(3) 1.5(3) Unknown\nOur Monte-Carlo data show a clear divergence of the\nferromagnetic order parameter, speci\fc heat and suscep-\ntibility as well as crossings of fourth-order Binder cu-\nmulant curves. This suggests a single second-order phase\ntransition from a high-temperature paramagnet to a low-\ntemperature ferrimagnetic phase (Fig. 6). Both for\n\u0012= 0 and\u0012= 10we can extract critical temperatures\nTc= 0:439(2) and Tc= 0:406(5) as well as the set of\nexponents which lead to the best data collapse of Binder\ncumulant, magnetization, susceptibility and speci\fc heat\n(Table I). Note that correlation length exponent \u0017and\norder parameter exponent \fincrease monotonically with\nJwhile the ratios \f=\u0017\u00190:25 and\r=\u0017\u00191:5 remain\nconstant with a two-dimensional scaling law implying\n\u0011= 2\f=\u0017\u00190:5.\nThis appears to provide an example of the so-called\n\"weak universality\" hypothesis which states that ratios\nof exponents should be independent of the details of\nsystem Hamiltonian with \u0011= 2\f=\u0017 and\r=\u0017 univer-\nsal while\u000band\fare allowed to change37. The \"weak\nuniversality\" behavior is often observed as a drift from\nBerezinskii-Kosterlitz-Thoules (BKT) exponents to dis-\ncrete (i.e. Ising, Potts) transition exponents38,39, and\nmay of course be related to the existence of a large length-\nscale. Note that in our case, we have a correspondence to\na six-state clock model arising from a Hamiltonian with\nboth nearest-neighbor and long-range interactions. Indi-\nvidually, a six state clock model with only the former ex-\nhibits two KT transitions (not observed here)40{43while\nmean-\feld studies for the case of long range dipolar in-\nteractions suggest a single second order low-temperature\nphase transition44,45.\nA low-temperature phase transition of pure dipoles on\nthe kagome lattice was recently observed in the O(N)\nMonte-Carlo studies in the Ref. 23. The nature of the\nlow-temperature spin arrangement was however not re-\nsolved due to the high computational cost of the O(N)\nMonte Carlo algorithm inversely proportional to temper-\nature. We have investigated the system at signi\fcantly\nlower temperatures, where snapshots of the spin con-\n\fgurations give clear evidence of ferrimagnetic order at\nk0= (0;0). At the same time, the temperature depen-\ndence of the static structure factor does not indicate any\nintermediate ordering between the low-temperature ferri-\nmagnetic state and the high-temperature disordered con-\n\fguration. Together with our LT and spin-wave studies\nthis rather strongly suggests that ferrimagnetic k= (0;0)\nstate is the low-temperature con\fguration of the dipoles.In the intermediate regime, due to existence of many\nmetastable energy minima, our Monte-Carlo simulations\ndo not equilibrate even for our extensive parallel setup.\nWe thus cannot provide a clear picture of physical quan-\ntities and leave a detailed investigation of this possibly\nincommensurate regime for future studies.\nVII. DISCUSSION AND CONCLUSION\nWe have determined ground states, excitations, and\nphase transitions, of classical Heisenberg spins with ex-\nchange and dipolar interactions on the frustrated kagome\nlattice.\nOur \frst central result is a determination of the ground\nstate for classical Heisenberg dipoles. This is a ferrimag-\nnetic three-sublattice one. Note that dipolar interactions\nfor Heisenberg spins lead to ground states in two dimen-\nsional systems e\u000bectively con\fned in the plane of the lat-\ntice as a result of extensive energy cost of any \fnite out\nof plane component3,13. In our studies we indeed ob-\nserve only in-plane spin-states as the ground states of the\nmodel. Therefore, the ground states we \fnd also apply\nto classical XY spins in the plane of the lattice.\nNext, we observe that switching on a weak dipolar in-\nteraction lifts the extensive ground-state degeneracy of\nthe nearest-neighbour model which exists here as it does\nin many other frustrated lattices, e.g. the Archimedean\npyrochlore lattice in three dimensions46,47. In both cases,\nthe elementary simplices { triangles for kagome, tetrahe-\ndra for pyrochlore { have vanishing total dipole moment\nin the nearest-neighbour ground state; upon adding dipo-\nlar interactions, they enter a state where the quadrupole\nmoment of each simplex also vanishes48. However, for\nstronger values of the dipolar interaction, the suppres-\nsion of the leading multipole moment no longer seems\nto be favourable. The general principles governing the\nlow-energy states on individual clusters49, and how they\ncombine to form a large lattice, is an intriguing topic for\nfuture studies.\nThe concomitant line of phase transitions into the fer-\nrimagnetic state at dominant Dappear to have expo-\nnents\u0017and\fchange monotonically with the ratios \f=\u0017\nand\r=\u0017constant. This is known in the context of the\n\"weak universality\" hypothesis and often appears in sys-\ntems with n-fold anisotropy where exponents appear to\n`drift' from KT values to those of discrete continuous\ntransitions. The presence of an enigmatic slice of the\nphase diagram where our methods fail to produce a re-\nliable answer further focuses attention on the possibility\nof the appearance of incommensurate states for delicately\nbalanced exchange and dipolar interactions.\nMoreover, it seems rather remarkable that the \rat\nband of zero-energy excitations simply moves up in en-\nergy without acquiring almost any dispersion. We do\nnote that this phenomenon is not so uncommon, after\nall, with a range of di\u000berent perturbations capable of\nproducing a similar phenomenon, a case in point being7\nmagnetoelastic interactions50. Also, a recent preprint51\nnoted the same phenomenon for a dipolar magnet on the\nGadolinium Gallium Garnet (GGG) lattice, which has\nhistorically played an immensely important role in the ex-\nperimental study of frustrated magnetic materials. This\nmay very well be one of the best experimental handles on\ndipolar interactions, leading to an almost k-independent\nresonance in inelastic neutron scattering33,36at a non-\nzero energy scaling quite sensitively with the size of the\ndipolar interaction \u0018p\nD.\nThe prospect for experimental work in this \feld is\nprobably better now than it has been for a very long\ntime. In an large number of systems the role of dipolar\ninteractions is important or even dominant9,52. There is\nsigni\fcant progress in fabrication of dipolar nano arrays\nwith a complex frustrated lattice geometry53,54as well\nas recent progress on building a dipolar systems in opti-\ncal lattices4,55. In addition recent progress in fabricating\nthin \flms of frustrated materials56,57suggests a possi-\nble route for realization of dipolar \flms with a kagome\ngeometry. Here the possible candidates for a \flm real-\nization could be fcc kagome materials RhMn 3, PtMn 3,\nIrMn 358,59where the latter one is commonly used in thin\n\flm technology60,61. RhMn 3and PtMn 3have the fcc\ncrystal structure62{64where magnetic Mn ions reside on\nthe cube faces and the nonmagnetic (Ir) ions site at the\ncube corners. The magnetic ions can thus be viewed\nas being on ABC stacked (111) kagome planes, where\neach site has eight NNs (four in-plane, two to the plane\nabove, and two to the plane below). The (111) plane\nis perpendicular to the \flm plane in thin-\flm applica-\ntions and thus one deals with a thin stack of Lkagome\nlayers. Interestingly the bulk of IrMn 3exhibits a long-\nrange magnetic order below TN\u0019960K59which is the\n3D manifestion of the 120oq= 0 spin structure65one\nof the structures found in our studies to be stabilized by\nweak dipolar interactions. Similar magnetic order is also\nfound in RhMn 3and PtMn 3.\nWe hope that our work will provide motivation for de-\ntailed characterisation of nature and collective behaviour\nof some of these experimental systems.\nACKNOWLEDGMENTS\nWe are grateful to P. Deen, P. A. McClarty, L. Seabra\nand R. Valenti for useful discussions. We thank Rechen-\nzentrum Garching (RZG) for computing time for the par-\nallel simulations. M.M. acknowledges ICMP of NAS of\nUkraine (Lviv) where part of initial computations was\nperformed.\nAppendix A: Linear spin-wave theory\nOur spin wave analysis in the non-collinear magnetic\nsystems starts with a rotation from the global z-direction\nto the local frame for each moment. Let ~Si(Rk) pointalong its local z-axis so that it is related to the spin\noperator de\fned in the crystallographic frame via the\nrotation:\nSi(Rk) =R-1\ni~Si(Rk) (A1)\nwhere Riis the corresponding rotation matrix. In the\nlocal frame the Hamiltonian reads\nH=\u00001\n2X\ni;jX\n\u000b;\fX\nk;lJ\u000b\f\nij(Rkl\nij)~S\u000b\ni(Rk)~S\fj(Rl) (A2)\nwhere the interaction matrix components transform as\nJij(Rkl\nij) =R-1\niJij(Rkl\nij)Rj: (A3)\nFourier transforming spin operators and interaction ma-\ntrix gives\n~S\u000b\ni(Rk) =1p\nNX\nk~S\u000b\ni(k) exp\u0002\nik\u0001\u0000\nRk+ri\u0001\u0003\n(A4)\nJij\nij(k) =X\nklJij\nij(Rkl\nij) exph\n\u0000ik\u0001Rkl\niji\n(A5)\nwhereNis the number of underlying Bravais lattice\npoints. Thus, the Hamiltonian in reciprocal space is\nH=\u00001\n2X\ni;jX\n\u000b;\fX\nk~S\f\ni(k)Jij\nij(k)~S\fj(\u0000k): (A6)\nThe linearized Holstein-Primako\u000b transformation then\ngives\n~Sx\ni(k) =r\nS\n2h\ncy\ni(k) +ci(\u0000k)i\n(A7)\n~Sy\ni(k) =ir\nS\n2h\ncy\ni(k)\u0000ci(\u0000k)i\n~Sz\ni(k) =p\nNS\u000e k;0exp [\u0000ik\u0001ri]\u00001p\nNX\nk0cy\ni(k0)ci(k0\u0000k);\nwith boson operatorsh\nci(k); cy\nj(k0)i\n=\u000ei;j\u000ek;k0. Keeping\nonly terms up to second order, we obtain\nH=H(0)+H(1)+H(2)(A8)\nwhere\nH(0)=\u00001\n2NS2X\ni;jJzz\nij(0)\nH(1)=\u0000Sr\nNS\n2X\ni;jh\nFij(0)cy\ni(0) + F?\nij(0)cy\ni(0)i\nH(2)=\u00001\n2SX\ni;jX\nkh\nAij(k)cy\ni(k)cj(k) + Bij(k)cy\ni(k)cy\nj(\u0000k)\n+B?\nij(k)ci(\u0000k)cj(k) + A?\nij(k)ci(\u0000k)cy\nj(\u0000k)i\n(A9)8\nand\nFij(0) =Jxz\nij(0) +iJyz\nij(0)\nAij(k) =1\n2\b\nJxx\nij(k) +Jyy\nij(k)\u0000i\u0002\nJxy\nij(k)\u0000Jyx\nij(k)\u0003\t\n\u0000X\n\rJzz\ni\r(0)\u000ei;j (A10)\nBij(k) =1\n2\b\nJxx\nij(k)\u0000Jyy\nij(k) +i\u0002\nJxy\nij(k) +Jyx\nij(k)\u0003\t\n:\nThe equilibrium condition that on every site the e\u000bective\nmagnetic \feld be parallel to the spin direction implies the\nabsence of linear terms. 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Joseph Poon1* \n1Department of Physics , University of Virginia, Charlottesville, Virginia 22904 USA  \n2Department of Electrical and Computer Engineering , University of Virginia, Charlottesville, \nVirginia 22904 USA  \n3Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030 USA  \n* sjp9x@virginia.edu  \n \nNeél skyrmions originate from interfacial Dzyaloshinskii Moriya interacti on (DMI) . Recent studies \nhave explored us ing thin -film ferromagnets and ferrimag nets to host Ne él skyrmions f or \nspintronic applications. However, it is unclear if ultrasmall (10 nm or less) skyrmions can ever be \nstabilized at room temperature for practical use in high density parallel racetrack memories. \nWhile  thicker films can improve stability, DMI decay s rapid ly away from the interface.  As such, \nspins far away from the interface would experience near -zero DMI, raising question on whether \nor not unrealistically large DMI is needed to stabilize skyrmions, and whether skyrmions will also \ncollapse  away from the in terface. To address these questions, we have employed atomistic \nstochastic Landau -Lifshitz -Gilbert simu lations to investigate skyrmion s in amorphous \nferrimagnetic GdCo . It is revealed that a significant reduction in DMI below that of Pt is sufficient \nto st abilize ultrasmall skyrmions even in films as thick as 15 nm . Moreover, skyrmions are found \nto retain a uniform columnar shape across the film thickness due to the long ferrimagnetic  \nexchange len gth despite the decaying DMI . Our r esults show that increasing thickness and \nreducing DMI in GdCo can further reduce the size of skyrmions at room temperature, which is \ncrucial to improve the density and energy efficiency in skyrmion based devices.  \nIntroduction  \nMagnetic skyrmions have  topologically protected spin textures. Their potential in advancing  \nmemo ry density and efficiency  has drawn extensive  investigation in recent years1-25. In magnetic \nmaterials, skyrmions are stabilized th rough  the Dzyaloshi nskii Moriya interaction (DMI)26-27,  \ngenerated by  either  inherent chiral  asymmetries or by interfacial symmetry breaking . Intrinsic  \nDMI arises in non-centrosymmetric crystal s such as B20 alloys , where  Bloch skyrmions have been  \nfound in MnSi and FeGe at low temperature12-13. On the other hand , interfacial DMI originates  \nfrom inversion symmetry breaking by an interfacial layer with strong spin-orbit coupling . \nMultilayer stacks, such as  Pt/Co/Os/Pt,  Ir/Fe/Co/Pt  and Pt/Co/Ta , have been  found to host  40 nm \nto over 1 μm Ne él skyrmions at room temperature14-16. Several challenges remain in developing \nskyrmion based memory  and logic  devices  - for instance , skyrmion Hall effect can pr esent  a \nsignificant challenge in  guid ing skyrmion s linearly along racetracks  19-22. More critically , aggressive  \nreduction in skyrmion sizes is needed to optimize skyrmion based devices , whereupon their room \ntemperature  stability becomes a problem . Thicker magnetic layers are required  in most  cases to \nincrease stability17-18. However , for ferromagnet  (FM) /heavy meta l multilayer stacks, increase in \nthickness of the magnetic layer can lead to a loss of interfacial anisotropy  and the reduction of \nthe strength of average DMI28-31, both of which are critical for skyrmion formation. To overcome \nthese challenges, we need to consider a suite of  materials  and understand their underlying physics , \nespecially with varying film thickness.  2 \n Amorphous rare -earth-transitional -metal ( RE-TM) ferrimagnet s (FiM) are potential candidates  to \novercome  these challenges.  Several properties of RE -TM alloys provide a favorable environment  \nto host small skyrmion s at room temperature. Their isotropic structure helps with avoiding defect \npinning18, while  their intrinsic  perpendicular magnetic anisotropy (PMA) 32-35 helps stabil ize small \nskyrmion s by allowing the use of thicker films ( > 5 nm) . However, the effectiveness of interfacial \nDMI decreases significantly away from the interface28-31, which is the focus of our present \ninvestigation.  Besides PMA, the magnetization of RE -TM alloys vanishes at the compensation \ntemperature36. With near zero magnetization  and near compensated angular momentum37, the \nskyrmion Hall effect is vastly  reduced18-20,23, and the skyrmion velocity is predicted to  be maximum \nnear the compensation point of angular moment um61. Recently,  all-optical helicity  dependent \nultrafast switching  has been demonstrated in RE-TM alloys  using a circularly polarized  laser38-45. \nThis gives an additional tool to control spins in device  structures . Indeed, RE-TM alloys have begun \nto draw  interest in the field of skyrmion  research . Large skyrmions of ~150 nm have  been \nobserved in Pt/Gd FeCo/MgO23, and skyrmion bound pairs have been  found in Gd/ Fe multilayers24. \nRecently , small skyrmions approaching  10 nm were  found in Pt/GdCo/TaO x films25. Understanding \nthese results will enable f urther tuning to reduce the size of the skyrmion s. To guide experiments , \nnumerical simulation  has served  as an important tool , especially for co mplex systems such as RE-\nTM alloys40,46-50. Several  methods, such as atomistic Landau -Lifshitz -Gilbert (LLG) algorithm40,46-49 \nand micromagnetic Landau -Lifshitz -Bloch (LLB) algorithm49 have been employed  to provide \ndeeper understanding of  the magnetic properties in RE-TM alloys . \nIn this study , an atomistic LLG algorithm40,46-48 is employed to investigate  the properties  of \nskyrmions  in Gd Co with interfacial DMI . Although the sign of DM I at FM/heavy metal interface  is \nwell studied51-57, the sign of DMI involv ing a  FiM remains complex  and is rarely discussed . Here,  \nwe consider two scenario s for the DMI between Gd and Co (dGd-Co). First,  DMI  between the \nantiferromagnetic  (AFM) pair is set to the same sign as DMI  between ferromagnetic  pair, i.e. dGd-\nCo > 0. Second , the case of d Gd-Co < 0 is considered. The latter appears to be f avored by the sign of \nAFM inter action27. Moreover, t o incorporate  DMI being an interfacial effect, an exponential ly \ndecay ing DMI is utilized. Simulation results find that with a switched DMI sign, near 10 -nm \nskyrmions remain robust in Gd Co films as thick as 15 nm at room temperature .  Through numerical \ntomography  maps , we find  that skyrmions at room temperature are distributed as a  near uniform  \ncolumn in thicker samples , despite  a spatially  decaying DMI.   \nResults and Discussion  \nWe will now begin to investigate if ultrasmall skyrmions in GdCo can survive an exponential  DMI \nreduction over thick sample sizes. To incorporate the amorphous nature of  GdCo, we employ an \namorphous structure of RE 25TM 75 from ab initio molecular dynamics calculations , as shown in Fig. \n1. As shown in Fig. 2 , at 300 K , the magnetization of amorphous Gd 25Co75 is 5 x 104 A/m , and it has \na compensation  temperature near 250 K. We begin with a n exponential  DMI  decay  away from the \ninterface, as shown in Fig. 3. The DMI value discussed herein is the interfacial DMI D 0. The decay \nlength of DMI is based on both previous simulations and experiments.  DMI calculation in Co/Pt \ninterface finds a significant decrease in DMI beyond the second layer of Co from the Pt interface30. \nExperimental  results also find similar decay in Co -Alloy/Pt interface52-54. In amorphous GdCo, we \nadapted the “second layer” as decay length for direct comparison wi th these  findings.  \nA range of interfacial DMI values, from dCo-Co = 0.1 x 10-22 J to dCo-Co = 2.0 x 10-22 J (D = 0.12 to 2. 38 \nmJ/m2), and three thicknesses, 5 nm, 10 nm, and 15 nm, are considered. Only those show \nskyrmions are shown herein. To shorten computational time, thicker samples of 10 nm and 15 nm 3 \n are simulated using  a 5 nm  thick  sample  by conserving DMI energy density  across the  film. To \nconserve the total DMI energy, a faster decay is employed in a 5 nm sample to simulate 10 nm \nand 15 nm thick samples to keep the sum of DMI energy to be the same. To check the validity of \nthis simplification, we have compared the results of 10 nm  thick samples and 5 nm thick samples  \nwith faster DMI decay to  verif y that the two sets of samples produce identical results.  First, we \nconsider two scenario s for the sign of d Gd-Co, as both + and - signs have been reported in \nantiferromagnetically coupled  systems59,60. Fig. 4 shows the color maps of equilibrium spin \nconfigurations at 300 K for both d Gd-Co > 0 and d Gd-Co < 0 . For the case of d Gd-Gd, dCo-Co > 0 and dGd-\nCo > 0, the simulation with  dCo-Co = 0.25 x 10-22 J, dGd-Gd = 2.96 x 10-22 J and dGd-Co = 0.86 x 10-22 J \ncorresponds to an average DMI of D = 0. 21 mJ/m2. The value of dGd-Gd and dGd-Co is calculated from \ndCo-Co by multiplying the ratio of Gd moment μGd over Co moment μCo. Further discussion in the \nsupplementary  material shows that for a given average DMI, the energy minimum and thus \nskrymion size  is independent of how each DMI term varies . Eq. 3 shows the formula used for \nconverting  atomistic DMI to average DMI for Gd xCo1-x. \nD=2\nπ1\n𝑛̅[(1−𝑥)(𝑛𝐶𝑜−𝐶𝑜 ̅̅̅̅̅̅̅̅̅𝑑𝐶𝑜−𝐶𝑜\n𝑟𝐶𝑜−𝐶𝑜 ̅̅̅̅̅̅̅̅2+𝑛𝐶𝑜−𝐺𝑑 ̅̅̅̅̅̅̅̅̅|𝑑𝐺𝑑−𝐶𝑜|\n𝑟𝐶𝑜−𝐺𝑑 ̅̅̅̅̅̅̅̅2)+x(𝑛𝐺𝑑−𝐺𝑑 ̅̅̅̅̅̅̅̅̅𝑑𝐺𝑑−𝐺𝑑\n𝑟𝐺𝑑−𝐺𝑑 ̅̅̅̅̅̅̅̅̅2+𝑛𝐺𝑑−𝐶𝑜 ̅̅̅̅̅̅̅̅̅|𝑑𝐺𝑑−𝐶𝑜|\n𝑟𝐺𝑑−𝐶𝑜 ̅̅̅̅̅̅̅̅2)](1) \nWhere 𝑛̅ is the average number of nearest neighbor s around all atoms, 𝑛𝐴−𝐵̅̅̅̅̅̅̅ is the average \nnumber of atom s A that are nearest neighbor s to atom B,  𝑟𝐴−𝐵̅̅̅̅̅̅ is average distance between \natom s A and nearest neighbor ing atom B. The 2\nπ factor comes from averaging of the cross product \n𝒔𝒊×𝒔𝒋 in DMI energy.  \nFor 5 nm GdCo , with d Gd-Co < 0, dCo-Co < 0.25 x 10-22 J, only ferrimagnetic states are observed. At dCo-\nCo > 1.0 x 10-22 J, skyrmions are elongated  due to boundary effect in the simulation or stripes states \nare observed. The range of DMI, where skyrmions are found, is smaller compared to calculation \nby Cort et al.21. This is due to a reduction  in anisotropy an d exchange stiffness in GdCo.  With less \nDMI energy required to create skyrmions, smaller DMI value is needed to create skyrmions and \nstripes in FiM. Furthermore, experiment results have measured DMI value  greater than 1 mJ/m2 \nonly at ordered FM/heavy metal  interface50-56. The DMI value at amorphous  FiM/heavy metal \nremains unknown . Due to disorder nature of  amorphous materials,  the DMI value in amorphous \nFiM can be much smaller than the DMI value observed in ordered FM.  \nAs shown in Fig. 4, with ferromagnetic DMI (d Gd-Gd and d Co-Co) that are  positive, two scenarios  of \nAFM  DMI (d Gd-Co) are considered.  At 300 K, in all thicknesses, larger DMI is needed to form \nskyrmions with positive  dGd-Co than with negative dGd-Co. In 5 nm sample, D = 0.5 5 mJ/m2 is needed \nto stabilize skyrmions  with dGd-Co > 0. In comparison, with  dGd-Co < 0, a smaller DMI  of D = 0.21 \nmJ/m2 is needed to stabilize skyrmions. Similar behaviors are also found in 10 nm and 15 nm \nsamples. With dGd-Co > 0, the smallest skyrmions are found at D = 1.26 m J/m2 in 10 nm sample and \nD = 2.31 m J/m2 in 15 nm sample. On the other hand, with dGd-Co < 0, the smallest skyrmions are \nfound at D = 0.84 m J/m2 in 10 nm sample and D = 1.68 m J/m2 in 15 nm sample.   \nTo understand  such  intriguing behavior in a FiM, the in-plane spin configurations and the chirality \nof the skyrmion wall are investigated. Fig. 5 summarizes the chirality of the skyrmion wall s in the \nCo sublattice . Using  dGd-Gd, dCo-Co > 0 and dGd-Co < 0, in the  Co sublattice , the spins are turning in \ncounter -clockwise direction  across the skyrmion wall . For Gd sublattice, the spins in the skyrmion \nwall are also turning counter -clockwise. This can be explained by the DMI in the system . AFM  \ncoupling s between Gd and Co align the spins of Gd and Co in nearly antiparallel direction s, except \na small canting due to the presence of DMI. With  dGd-Gd and dCo-Co > 0, turning counter -clockwise  4 \n is energetically favorable . However , with dGd-Gd, dCo-Co > 0 and dGd-Co > 0, the chirality of the \nsimulated skyrmion wall  is found to be opposite. The DMI torque between  the AFM pairs  now \noppose s the DMI torques with in each sublattice . In the presence of a strong er inter -sublattice \nDMI torque , the spins  in each sublattice  now turn clockwise across the skyrmion wall.  \nTo better illustrate  the change in ch irality, the total DMI energies  between Co -Co, Gd -Gd and Gd -\nCo are computed using the equilibrium configurations at 0 K. Table 1 summarizes the sign of the \ntotal DMI energies for different nearest neighbor pair s. With dGd-Gd, dCo-Co > 0 and dGd-Co> 0, spins \nare turning  counter -clockwise . With this configuration, the total DMI energy between Gd -Gd pair \nEDMI(Gd-Gd) and Co-Co pair E DMI(Co-Co) are negative, and the total DMI energy between Gd and \nCo pair E DMI(Gd-Co) is also negative . This means that with dGd-Gd, dCo-Co > 0, it is energetically \nfavorable for spins to turn counterclockwise.  On the other hand,  with dGd-Gd, dCo-Co > 0 and dGd-Co > \n0, spins are revea led to turn clockwise from  the simula ted configurations. As a result of the sign \nchange in chirality,  EDMI(Gd-Gd) and E DMI(Co-Co) become  positive . On the other hand, EDMI(Gd-Co) \nremains  negative , bec ause both chirality and d Gd-Co change s sign. This implies  that it is \nenergetically favorable for Gd -Co pair to turn clockwise across, but it is energetically unfavorable \nfor Gd -Gd and Co-Co pairs to do so. In other word , AFM DMI d Gd-Co is able to overcome \nferromagnetic DMI d Gd-Gd and dCo-Co, resulting in energy favorable configuration s for Gd -Co pairs.  \nTo summarize , in a FiM, if the DMI of ferromagnetic pair and AFM  pair have the same sign, a \ncancellation of DMI occurs because it is preferable for a ferromagnetic pair to turn in the opposite \ndirection of an AFM  pair. No cancellation occurs if the DMI of ferromagnetic pair and AFM  pair \nhave the opposite  sign. These  also explain the differences in size of skyrmion  between d Gd-Co < 0 \nand d Gd-Co > 0. The dGd-Co < 0 scenario has larger  skyrmions because both ferromagnetic and AFM  \npairs are contributing to the formation of a skyrmion, which means the DMI effect is stronger  \noverall.  \nTo investigate  the minimal size  of room temperature skyrmions in GdCo, D -K phase diagrams with \nexponential ly decaying DMI at 300 K are simulated for 5, 10 and 15 nm GdCo films. In this section, \nwe focus  on the d Gd-Co < 0 scenario . Since energy barrier is a function  of exchange stiffness and \nthickness18, the minimal skyrmions size found in d Gd-Co < 0 scenario can also apply to d Gd-Co > 0 \nscenario, except a larger DMI is required . For each thi ckness, anisotropy ranges from 0.05 x 105 \nJ/m3 to 4 x 105 J/m3 are investigated.  Experimentally, GdCo h as anisotropy in the order of 104 \nJ/m3.25,36 For DMI, larger  interfacial DMI is explored in thicker samples, because a s thickness \nincreases, the average  DMI decreases,  and larger interfacial DMI is needed  to stabilize skyrmions.  \nIn 5 nm samples, interfacial DMI of 0 to 2 mJ/m2, which corresponds to dCo-Co of 0 to 2.38 x 1022 J, \nare investigated . Fig. 6 (a) shows the D -K phase diagram of 5 nm GdCo at 300 K. In 5 nm GdCo, \nskyrmions range from 12 nm to 40 nm are stabilized in the simulated range of interfacial DMI and \nanisotropy. Lines of 15 to 30 nm indicate the size of skyrmions at various DMI and anisotropy. As \nDMI decreases or anisotropy increase s, skyrmions become smaller and eventually collapse into \nFiM states. At the opposite side of D -K diagram, with large DMI and  small anisotropy , skyrmions \nlarger than 40 nm becomes elongated or collapsed due to the boundary of the simula tion space \n(50.7 nm x 50.7 nm).  This elongation of skyrmions was also seen earlier in Fig. 4 at large DMI \nvalues.  Overall, for a given anisotropy, as interfacial D MI increases from 0 to 2.0 mJ/m2, the \nequilibrium configuration goes from FiM to skyrmions, then to stripes. For a fixed DMI, as \nanisotropy increases , size of skyrmions decreases, and finally, skyrmions collapse into FiM states.  \nThese  behavior of skyrmion s in FiM GdCo as a function of DMI and anisotropy  is the same as  what \nhas been observed in a ferromagnet17,18.  5 \n For 10 nm and 15 nm GdCo, DMI of 0 to 3 mJ/m2 (dCo-Co of 0 to 3.57  x 1022 J) and 0 to 4 mJ/m2 (dCo-\nCo of 0 to 4.76 x 1022 J) are explored respectively.  The overall trend  of skyrmions as a function of \nDMI and anisotropy in 10 nm and 15 nm GdCo are identical to that of 5 nm GdCo , where  increase \nin DMI leads to larger skyrmions, and increase in anisotropy  results in smaller skyrmions. However, \none difference in thicker sample s from 5 nm sample is that ultrasmall skyrmions as small as 7 nm \nare stable in room temperature. For both 10 nm and 15 nm GdCo, there is a region of DMI and \nanisotropy where ultrasmall  skyrmions are stablized . In 10 nm GdCo,  ultrasmall skyrmions are \nfound in the region of DMI ranges from 0.8 to 1.0 mJ/m2 and anisotropy ranges from (0.1 to 0.8) \nx 105 J/m3. For 15 nm GdCo, this region lays within DMI ranges from 1.5 to 1.8 mJ/m2 and \nanisotropy ranges from (0.1 to 1.0) x 105 J/m3. For both 10 nm and 15 nm GdCo, the anisotropy \nfalls within the same range as what has being measured experimentally in GdCo25,36, which is in \nthe order of 104 J/m3. However, the interfacial DMI is  less than what has been observed at a Pt \ninterface. Ab-inito calculation has found Interfacial DMI of up to 12 mJ/m2 is reported at a n ideal  \nPt/Co interface30. On the other hand,  the interfacial  DMI measured in Co/Pt and other Co -alloy/Pt \nfilms are around 1.2 to 1.5 mJ/m2. 52-54 Thus, some reduction s of DMI from that of Pt are needed \nto experimentally obtain ultrasmall  skyrmion in  5 and 10 nm  GdCo films. Reduction of DMI can be \nobtained by sandwiching GdCo between two Pt layers with one Pt layer being diluted by other \nelements. Since GdCo is amorphous, we have more flexibility of tuning the underlayer and the \ncapping layer of a multilayer sandwich. With its intrinsic anisotropy and flexibility , GdCo films are \npromising materials to obtain ultrasmall skyrmions at room temperature  through DMI tuning .  \nFor device applications, especially in thicker films, we will also need to consider the growth of \nskyrmions away from the interface. With decaying DMI away from the interface, spins at the top \nof a thicker  sample experience effectively zero DMI. Without DM I, one might expect spins near \nthe top to align parallel for FM neighbors and antiparallel for AFM neighbors, and skyrmions to \ndisappear far away from the interface. If skyrmions collapse far away from the interface, the \nreliability of such memory devices would be vastly reduced. To investigate whether skyrmions \nremain robust in thicker samples, a numerical tomography is employed to image simulated \nultrasmall  skyrmions at 300 K. Fig. 7 shows the numerical tomography plot of a ultrasmall \nskyrmion  in 10 nm Gd Co. This skyrmion corresponds to D = 0.84 mJ/m2 and K = 0.3 x 105 J/m3. The \nsame skyrmion was shown  in Fig. 4(b) and as the smallest skyrmions (Star Symbol) at K = 0.3 x 105 \nJ/m3 in Fig. 6(b). In the 3D plots at the center of Fig. 7, color s are made to be somewhat \ntransparent to reveal the skyrmions structure near the center. For Co sublattice, red to orange \ncolor shows that most of the spins are pointing down . A region of green and blue  that appears \nnear the center corresponds to the simula ted skyrmion  at 300 K. As evidenced by the columnar \ndistribution of blue color, the skyrmion retain s a uniform columnar growth from the bottom to \nthe top. Columnar distribution of skyrmion is also found in Gd sublattice , where a column of red \nis distribute d uniformly from the bottom to the top . This feature can be understood in terms of \nthe large magnetic exchange length >20 nm  due to the low magnetization  in the ferrimagnet . \nTo further demonstrate  the uniform  columnar distribution  of skyrmion, in -plane and out -of-plane \ncross sections of the skyrmion are also plotted in Fig. 7. On the left of Fig. 7, in-plane cross section \nof spin configuration within 0.5 nm of the interface  and 0.5 nm of the top are mapped. The \nskyrmion s at the interface and near the top have  identical  size an d shape.  Compare to the \nmapping of spin configuration s in Fig. 4(b), size of the skyrmion remain the same. This shows that \nthe size of skyrmions rema in the uniform throughout a sample.  On the right side of Fig. 7, out -of-\nplane cross section s are shown for Gd and Co sublattice s. The blue color in Co sublattice and the \nred color in Gd sublattice correspond  to the center of the skyrmion. For both sublattice, out -of-\nplane cross sections show a columnar distribution of skyrmion from the bottom in terface to the 6 \n top. These results provide important evidences that skyrmion remain robust through a thicker  \nsample, and further support of using thicker GdCo samples  to increase skyrmion stability at room \ntemperature . \nConclusions  \nUsing atomistic stochastic  LLG simulations, ultrasmall skyrmions  are shown  to be stable at room \ntemperature  in ferri magnetic GdCo. Despite the rapid decay of Dzyaloshinskii Moriya interaction \n(DMI)  away from the interface, a rea listic range of DMI values  is seen to stabilize skyrmions in \nGdCo films as thick as 15 nm  irrespective of the sign of DMI between antiferromagnetic coupled \nGd and Co,  Furthermore, the low DMI values needed to form ultrasmall skyrmion in GdCo  indicate \nopportunity for design ing magnet ic materials to host ultrasmall Neel skyrmions . Through \ntomography of an ultrasmall skyrmion in 10 -nm thick GdCo  film, it is discovered that the skyrmion  \nassumes a columnar configuration that extends uniformly across the film thickness  despite  having \nnear zero DMI far away from the interface. These findings argue for using thicker magnetic films \nto host  ultrasmall skyrmions, providing an  important strategy for developing high density and high \nefficiency  skyrmion based devices.    \nMethods   \nThe classical atom istic Hamiltonian H in Eq. (1) is employed to investigate magnetic textures in \namorphous FiMs.  \n𝐻=−1\n2∑ 𝐽𝑖𝑗𝒔𝒊∙𝒔𝒋\n<𝑖,𝑗>−1\n2∑ 𝐷𝑖𝑗∙(𝒔𝒊×𝒔𝒋)\n<𝑖,𝑗>−𝐾𝑖(𝒔𝒊∙𝑲𝒊̂)2 \n−𝜇0𝜇𝑖𝑯𝒆𝒙𝒕∙𝒔𝒊−𝜇0𝜇𝑖𝑯𝒅𝒆𝒎𝒂𝒈 ∙𝒔𝒊  (2) \nwhere 𝒔𝒊,𝒔𝒋 are the normalized spins and  𝜇𝑖,𝜇𝑗 are the atomic moments at sites i, j respectively. \nThe atomic moment is absorbed into the exchange constant, 𝐽𝑖𝑗=𝜇𝑖𝜇𝑗𝑗𝑖𝑗, the DMI interaction \n𝑫𝒊𝒋=𝜇𝑖𝜇𝑗𝒅𝒊𝒋, which is proportional to ri x rj, the po sitional vector between the atoms i, j and the \ninterface , and the effective anisotropy  𝐾𝑖=𝜇𝑖𝑘𝑖. 𝑯𝒆𝒙𝒕 and 𝑯𝒅𝒆𝒎𝒂𝒈  are the external field and \ndemagnetization field respectively.  \nOnly nearest neighbor interactions are considered in the exchange and DMI interactions. Periodic \nboundary conditions  are enforced in the x and y direction s.  \nTo find the ground state, the spins are evolved under the following stochastic Landau -Lifshitz -\nGilbert (LLG) equation,  \n𝑑𝑴\n𝑑𝑡=−𝛾\n1+𝛼2𝑴×(𝑯𝒆𝒇𝒇+𝝃)−𝛾𝛼\n(1+𝛼2)𝑀𝑠𝑴×[𝑴×(𝑯𝒆𝒇𝒇+𝝃)] (3) \nwhere 𝛾 is the gyromagnetic ratio, 𝛼 is the Gilbert damping constant,  𝑯𝒆𝒇𝒇 is the effective field, \n𝝃 is the Gaussian white noise term for thermal fluctuations and 𝑀𝑠 is the saturation magnetization.  \nThe parameters used in our simulation are listed in Table 2. Exchange couplings 𝐽𝑖𝑗are calibrated \nbased on Ostler et al.49 to maintain the same Curie temperature and compensation temperature \nfor a given compensation. At 300 K, the magnetization of Gd 25Co75 is 5 x 104 A/m. Anisotropy \nenergy is determined based on Hansen et al.36 7 \n To incorporate the amorphous short range order, an amorphous structure of a 1.6 nm x 1.6 nm x \n1.6 nm box containing  250 atoms is generated from ab initio  molecular dynamics calculations by \nSheng et al .58. The composition used in the simulation is Gd 25Co75. Fig. 1 shows a plot of RE and \nTM atoms in the amorphous structure. For a 4.8 nm thick sample, replicas of th is box (32 x 32 x 3) \nare placed next to each  other to expand the simulated sample  to 50.7 nm x 50.7 nm x 4.8 nm and \n768000 atoms. On average, we find that each Co atom has 6.8 Co neighbor s and 4.1 Gd neighbor s, \nwhile  each Gd atom has 11.7 Co neighbors  and 3.5 Gd neighbor s. We have also employed a FC C \nlattice to study skyrmions in GdCo. We found that with the same compensation temperature and \nmagnetization, a larger DMI is needed to stabilize skyrmion in a FCC lattice structures  than \namorphous structure. 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Measuring and tailoring the Dzyaloshinskii -Moriya interaction in \nperpendicularly magnetized thin films. Phys. Rev. B 90, 020402(R) (2014).  \n53. Stashkevich,  A.A. et al. Experimental study of spin -wave dispersion in Py/Pt film structures in  \nthe presence of an interface Dzyaloshinskii -Moriya interaction . Phys. Rev. B 91, 214409 (2015).  \n54. Ma, X. et al. Interfacial control of Dzyaloshinskii -Moriya interaction in heavy \nmetal/ferromagnetic metal thin film heterostructures . Phys. Rev. B  94, 180408( R) (2016).  \n55. Tacchi, S. et al . Interfacial Dzyaloshinskii -Moriya Interaction in Pt/CoFeB Films: Effect of the \nHeavy -Metal Thickness . Phys. Rev. Lett.  118, 147201  (2017).  \n56. Cho, J. et al. The sign of the interfacial Dzyaloshinskii –Moriyainteraction in ultrathin \namorphous and  polycrystalline magnetic films . J. Phys. D: Appl. Phys.  50, 425004  (2017).  \n57. Simon, E., Rózsa, L., Palotás, K. &  Szunyogh , L. Magnetism of a Co monolayer on Pt(111) \ncapped by overlayers of 5d elements: A spin -model study . Phys. Rev. B  97, 134405 (2018).  10 \n 58. Sheng, H.W., Luo, W.K., Alamgir, F.M., Bai, J.M., & Ma, E. Atomic packing and short -to-\nmedium -range order in metallic glasses. Nature 439, 419 -425 (2006).  \n59. Zhang, X., Zhou, Y., & Ezawa, M.  Antiferromagnetic Skyrmion: Stability, Creation  and \nManipulation . Sci. Rep.  6, 24795 (2016).  \n60. Baker, J., & Tretiakov, O. A., Static and Dynamical Properties of Antiferromagnetic Skyrmions \nin the Presence of Applied Current and Temperature . Phys. Rev. Lett.  116, 147203  (2016).  \n61. Kim, K -J. et al. Fast domain wall motion in the vicinity of the angular momentum \ncompensation temperature of ferrimagnets . Nature Materials 16, 1187 -1192 (2017).  \nAcknowledgements:  \nThis work was supported by the DARPA Topological Excitations in Electronics (TEE) program (grant \nD18AP00009).  The content of the information does not necessarily reflect the position or the \npolicy of the Government, and no official endorsement should be inferred. Approved for public \nrelease; distribution is unlimited . This work was  partially supporte d by NSF -SHF-1514219.  \nAdditional Information  \nAuthor Contributions  \nS.J.P. conceived the project and supervised the simulation, C.T.M.  performed the simulation, \nY.X. assisted in the simulation and provided comments, A .W.G. discussed skyrmion stability and \nsuggested improve ment to the manuscript , H.S. provided the amorphous structure  from ab \ninitio molecular dynamics calculations.  \nCompeting Interests  \nThe authors declare no competing interests.  \n \n \n \n \n 11 \n   \nFigure 1 Amorphous st ructure of RE 25TM 75 from ab initio molecular dynamics calculations . Red \natoms are rare -earth, and blue atoms are transition -metal.  \n \nFigure 2 Simulated saturation magnetization vs. temperature of amorphous Gd 25Co75. The \ncompensation temperature of amorphous Gd 25Co75 is near 250 K , and the magnetization is small \nat room temperature.  \n12 \n  \nFigure 3 Exponential decay DMI in 5 nm GdCo as function of distance from bottom interface (z) . \nIn this model, DMI remains constant  (D0) within 0.35 nm of the bottom int erface, as indicated by \nthe red line. Away from the interface, the strength of DMI decays exponentially as shown.  \n13 \n  \nFigure 4 Color mapping of equilibrium spin configurations  for various DMI values (exponentia lly \ndecay ing DMI) at 300K for dGd-Co > 0 and dGd-Co < 0 in (a ) 5 nm, (b) 10 nm, and (c) 15 nm GdCo . \nOut-of-plane component s of reduced magnetizations  (mz) are mapped  in the x -y plane  using  the \ncolor bar shown in (c). Ultrasmall skyrmions are revealed in 10 nm and 15 nm GdCo samples.  \n14 \n  \nFigure 5 Simulated s kyrmion configurations  of Co sublattice for dGd-Co < 0 and dGd-Co > 0 with metal \ninterface  at the bottom . (Top) A n overhead view of simulated skyrmion configurations for dGd-\nCo < 0 and dGd-Co > 0. (Bottom) For dGd-Co < 0, the skyrmion wall is turning counter -clockwise. The  \nd(FM) vector is pointing in the opposite direction of  Si x Sj. EDMI (FM)= dij · (Si x Sj ) is negative, \nwhich is favorable.  For dGd-Co > 0, the skyr mion wall is turning clockwise. The d(FM) vector  and Si x \nSj are pointing in the same direction, resulting in positive  EDMI (FM) . Identical signs of the DMI \nenergy are also found in the Gd sublattice.  \nScenario  EDMI(Gd-Gd)   EDMI(Co-Co)  EDMI(Gd-Co)  \ndGd-Gd, dCo-Co > 0, dGd-Co < 0 - - - \ndGd-Gd, dCo-Co > 0, dGd-Co > 0 + + - \nTable.  1 Sign of total DMI energy E DMI computed from equilibrium spin configurations at 0 K.   \n15 \n  \nFigure 6 D-K phase diagram of (a) 5 nm, (b) 10 nm and (c) 15 nm GdCo at 300 K with dGd-Co < 0. \nStar corresponds to smallest skyrmions simulated at K = 0 .3 x 105 J/m3. Ultrasmall skyrmions are \nrevealed in 10 nm and 15 nm GdCo. Due to limits of simulation space (50.7 nm x 50.7 nm), with \nperiodic boundary conditions in x -y direction, large skyrmions (> 40 nm) become either elongated \nor collapsed.  \n \n16 \n  \n \nFigure 7 Tomograph  of a  simulated  ultrasmall  skyrmion in 10 nm GdCo  at 300 K . It reveals \ncolumnar skyrmion distribution throughout the 10 nm GdCo sample.  The figure  shows Co -\nsublattice spins (top box), Gd -sublattice spins (bottom box), in -plane cross sections of near the \ntop and bottom interface (left), and out -of-plane cross sections (right).    \nParameter  Value  \nGyromagnetic ratio (ϒ) 2.0023193  \nGilbert Damping (α) 0.05 \nGd moment ( μGd) 7.63 μB \nCo moment ( μCo) 1.72 μB \nGd-Gd exchange constant (J Gd-Gd)  1.26 x 10-21 J \nCo-Co exchange constant (J Co-Co) 3.82 x 10-21 J \nGd-Co exchange constant (J Gd-Co) -1.09 x 10-21 J \nMagnetic Field (H)  0.01 T  \nTable. 2 Values of parameters used in the simulation.  \n \n"    },    {        "title": "1910.00124v3.Disorder_induced_ferrimagnetism_in_sputtered_Mn___x__CoGe_thin_films.pdf",        "content": "arXiv:1910.00124v3  [cond-mat.mtrl-sci]  15 Mar 2023Disorder-induced ferrimagnetism in sputtered Mn xCoGe thin films\nD. Kalliecharan,1J. S. R. McCoombs,1M. M. E. Cormier,1B. D. MacNeil,1R. L. C. Molino,1and T. L. Monchesky1,∗\n1Department of Physics and Atmospheric Science,\nDalhousie University, Halifax, Nova Scotia, Canada B3H 3J5\n(Dated: March 5, 2023)\nInvestigations into the magnetic properties of sputtered M nxCoGe films in the range 0 .8≤x≤2.5\nuncovered ferrimagnetic order, unlike the ferromagnetic o rder reported in bulk samples. These films\nformed hexagonal Ni 2In-type structures when annealed at temperatures below 600 °C. While the\nCurie temperatures of the films are comparable to those of hex agonal bulk MnCoGe, there is a re-\nduction in the magnetization of the Mn xCoGe films relative to bulk MnCoGe, and a magnetization\ncompensation point is observed in the x <1 samples. To understand the behavior, we calculated\nthe magnetic moments of Mn-antisite defects in MnCoGe with d ensity-function theory (DFT) cal-\nculations. Models constructed from the calculation sugges t that films become ferrimagnetic due\nto the presence of Mn on the Co and Ge sites. In the x <1 samples, these defects arose from\nthe disorder in the films, whereas for x >1, the excess Mn was driven onto the antisites. Mean\nfield modeling of the temperature dependence of the magnetiz ation provides additional evidence for\nferrimagnetism. Our mean field and DFT models provide a descr iption of how the variation in film\ndefects with composition will transition the magnetic beha vior from a compensated (V-type) to an\nuncompensated (Q-type) ferrimagnet.\nPACS numbers:\nI. INTRODUCTION\nThe manganese germanides comprise a rich phase dia-\ngramwithadiverserangeofmagneticstructures. Mn 3Ge\nforms one of two polytypes. The Mn 3GeD019hexago-\nnal structure is a frustrated non-collinear antiferromag-\nnet with a large topological Hall effect,1while the tetrag-\nonalD022Heusler is a high-anisotropy ferrimagnet of\ninterest for memory applications.2There have been re-\ncent proposals for tuning the magnetic properties of this\nstructure via chemical substitutions Mn 3–yXyGe.3Sub-\nstitution of Ni, for example, decreases the moment and\nincreases the coercivity.4\nA related family of compounds – the inversetetragonal\nHeuslers – is obtained by replacing Mn on one of the 4d\nWyckoff sites in the D022structure, (0, 1/2, 1/4), with\nanother element. This lowers the symmetry from D3hto\nthe non-centrosymmetric D2dpoint group and turns on\nthe Dzyaloshinskii-Moriya interaction that is responsible\nfor the non-collinear magnetic structures in Mn 2RhSn5\nand Mn 1.4PtSn.6The stability of the Mn 2XGe Heusler\ncompounds have been explored by density-functional\ntheory (DFT) calculations,7many of which are pre-\ndicted to form the inverse tetragonal structure, including\nMn2CoGe. The initial motivation forthe workin this pa-\nper was to create Mn 2CoGe Heusler alloy films by mag-\nnetron sputtering. We fabricated Mn xCoGe in the com-\npositional range 0 .8≤x≤2.5, but were unsuccessful in\nproducing Heusler alloys. The entire composition formed\neither a hexagonal structure or an orthorhomic struc-\nture related to the magnetocaloric material, MnCoGe.\nThis paper reports on alloys which formed the hexagonal\nstructure.\nAt low temperature, MnCoGe forms an orthorhombic\nC23TiNiSi-type structure (space group No62,Pnma).It is a collinear ferromagnet with a Curie temper-\nature,Tortho\nC= 355K and a magnetic moment of\nm= 3.86µB/formula unit (f.u.). At a temperature Tt,\nthe material undergoes a martensitic transformation to a\nhexagonalB82Ni2In-type structure (space group No194\nP63/mmc).8The resulting 3.9% contraction in volume\nleads to a broadening of the Mn d-bands producing a\nsmaller moment and lower TC.9In this hexagonal poly-\ntype,m= 2.78µB/f.u.10andThex\nC≈260K. The marten-\nsitic transition is very sensitive to defects. Johnson et\nal.found that Ttvaried between 398K to 453K,8while\nKanomata et al.reportedTtas high as 650K.10When\nTtlies between Thex\nCandTortho\nCthe material undergoes a\nfirst-order transition from an orthogonal ferromagnet to\na hexagonal paramagnet that gives rise to a large mag-\nnetocaloric effect.\nFIG. 1: (Color online) The primitive unit cell of the MnCoGe\nNi2In-type phase. Isometric view (top) and c-axis projection\n(bottom), showing the 2a-Mn sites (grey), the 2c-Co sites\n(red) and 2d-Ge sites (blue).2\nWhat makes MnCoGe particularly attractive is that\nits martensitic temperature can be chemically tuned in-\ndependent of TC. The transition temperature Ttis very\nsensitive to Co vacancies,11,12as well as Mn vacancies.13\nWith only a few percent vacancies on either site, Ttcan\nbe reduced to room temperature with little effect on ei-\ntherThex\nCorTortho\nC. This is potentially drivenby areduc-\ntion in the numberofvalenceelectrons, asthe sameeffect\nis also observed in Mn 1+xCo1–xGe alloys.13,14Numerous\nstudies have explored the influence of other defects and\nsubstitutions in MnCoGe; a comprehensive summary of\nsuch studies is given in the appendix of Ref. 15.\nIn the Ni 2In-phase, Mn resides on the 2a (0, 0, 0)\nWyckoff sites and forms low density (001) planes. These\nare separated by dense CoGe planes with Co on the 2c\n(1/3, 2/3, 1/4) sites and Ge on the 2d (2/3, 1/3, 1/4)\nsites (see Fig. 1).\nWe found that Mn xCoGe films prepared by DC mag-\nnetron sputtering were much more disordered than typ-\nical bulk material, which had two important conse-\nquences. Firstly, the hexagonal B8 2phase was obtained\nat room temperature after annealing at T= 500 °C\nand remained in this phase upon cycling down to low\ntemperature, consistent with other reports of sputtered\nMnCoGe films.16Secondly, the films display ferrimag-\nnetic rather than ferromagnetic order reported in other\ninvestigations of this material. We support the analysis\nof the magnetic properties with DFT calculations that\nshow the spins from Mn-antisite defects align in the op-\nposite direction to the spins on the Mn-sites.\nII. GROWTH\nFilms were deposited on thermally oxidized Si wafers,\nas SiO 2acts as a diffusion barrier for Mn, Co and Ge.17\nSi(001) wafers (manufactured by Prolog semicor Ltd. )\nwere cut into 20mm ×20mm squares and were soni-\ncated in acetoneand methanol baths for15minutes each.\nBefore removing the wafers from the methanol bath, de-\nionized nanopure water was slowly added and allowed to\noverflow in order to remove any contaminants from the\nliquidsurface. Thewaferswereheatedin adryfurnaceat\n900°C for 5 hours to create a SiO 2layer, approximately\n300nm in thickness.\nThe sonication treatment was then repeated prior to\nloadingsamplesintoa Corona Vacuum Coater V3T mag-\nnetron sputtering deposition system with a base pres-\nsure of 3.0×10−7Torr. The Ar pressure during sputter-\ning was 2.0×10−3Torr. Films were deposited at room\ntemperature, with no external heating. Sputtering rates\nwere calibrated by measuring the weights of the samples\nbefore and after growth. The Mn sputtering rate was\nfixed at 8.61nmolcm−2s−1, while the Co and Ge rates\nranged from 4 .13nmolcm−2s−1to 12.9nmolcm−2s−1,\ndepending onthe stoichiometry. Film thicknesswasmea-\nsured using a Vecco Dektak contact profilometer . All\nfilms studied in this work were between 475nm and550nm in thickness. The compositions were verified us-\ning aThermo iCAP Q laser ablation inductively coupled\nplasmamassspectrometer (LA-ICP-MS). The resultsare\nshown in Table I.\nThe as-grown films where crystallized ex-situby an-\nnealing in an Ar environment in a Modular Process\nTechnology RTP600s Rapid Thermal Annealer (RTA).\nThe RTA reached the desired temperatures within 20s\n(15°Cs−1to 35 °Cs−1), and were cooled at a rate of\napproximately 2 °Cs−1. An X-ray photoelectron spec-\ntroscopy(XPS) depthscanwasperformedonselectedan-\nnealed samples, revealing that oxide contamination only\nexists at the surface, within the top 2% of the film thick-\nness.\nIII. STRUCTURAL CHARACTERIZATION\nThe crystal structures of the films were investigated\nwith conventional X-ray diffraction (XRD) θ−2θmea-\nsurements on a Siemens D500 Diffractometer equipped\nwith a Cu source and monochrometer. To determine\nthe strain in the films, the XRD measurements were\ncompared to grazing angle X-ray diffraction (GAXRD)\nmeasurements, where the incident X-ray beam is fixed\natθi= 6°. The alignment of the diffractometer was\nchecked with a Si powder sample for both the XRD and\nthe GAXRD geometries.\nAs-deposited XRD data shows that the films are ei-\nther nanocrystalline or amorphous and discernible crys-\ntallographic phases only appeared after annealing. An-\nnealing times and temperatures were selected to pro-\nduce single phase samples. Five sets of samples –\nMn0.8CoGe, Mn 0.9Co0.8Ge, Mn 1.4CoGe, Mn 1.8Co0.8Ge\nand Mn 2.5CoGe – were annealed within the temperature\nrange of 375 °C to 600 °C for times between 2 minutes\nand40minutes, yieldingNi 2In-typepolycrystallinefilms.\nHigh temperature annealing resulted in mixed phase\nsamples: annealing at 700 °C produced a mixture of the\nhexagonal Ni 2In-type and the orthorhombic TiNiSi-type\nphases. The properties of the samples annealed at high\ntemperature are not discussed further. Figure 3shows\nfits to GAXRD measurements of the Ni 2In-type samples\nthat demonstrate the phase is stable across the entire\ncomposition range, 0 .8≤x≤2.5.\nEstimates of the grain size were calculated from the\ndiffraction peak widths (Fig. 3) by using the Scherrer\nequation:18\nτ=Kλ\nβcosθ, (1)\nwhereτis the grain size, λis the X-ray wavelength,\nandβis the full-width at half-maximum of the diffrac-\ntion peak at a Bragg angle θ. The Scherrer constant\nKis the crystallite-shape factor, chosen to be 0.9 for\nthese samples. For each stoichiometry, the XRD grain\nsize was averaged over several peaks and both Cu K α13\nand Cu Kα2contributions to the peak were considered\nin determining β. The average grain sizes measured by\nXRD are summarized Table I. Grain sizes determined by\natomic force microscopy (AFM) were largely in agree-\nment with these estimates. Figure 2show representa-\ntive micrograms of the Mn 0.8CoGe, Mn 0.9Co0.8Ge and\nMn1.4CoGe samples. The average grain diameter was\ntaken as the first minimum in the autocorrelation of\nthe height, ( h(r)−h(r0))2. The extracted diameters\nfor the Mn 0.8CoGe, Mn 0.9Co0.8Ge samples were 56nm\nand 84nm, respectively, in agreementwith the XRD esti-\nmates shownin Table I. For Mn 1.4CoGe, the autocorrela-\ntionfunction yieldedavalueof72nm, nearly3timesthat\nfrom XRD. Figure 2(c) shows the presences of smaller\nfeatures on top of the larger 70 nm diameter grains that\nare 23 nm in diameter on average, which is within error\nof the grain size extracted from XRD. This suggests that\nthe larger 70 nm features in 2(c) are in-fact composed of\nsmaller crystallites.\nThe lattice parameters extracted from the GAXRD\nfits (Table I), are comparable to the values of bulk\nMnCoGe,a= 4.087(1),c= 5.316(3)˚A.19The Rietveld\nrefinements were performed using Rietica version 4.0\n(http://rietica.org ). We note that the (101) peak in-\ntensity is much lower that expected from bulk MnCoGe\nsamples. The discrepancy could be accounted for with\n20% vacancies on the 2c-site occupied by Co. The pres-\nence of vacancies is supported by ICP-MS measurements\nthat show Mn concentrations are lower than the nominal\nvalue. The intensity of the (2 ¯10)-peaks is higher than\nexpected. As the annealing process can lead to preferred\ngrain orientation, it is not possible to separate this effect\nfrom the possibility of vacancies.\nTABLE I: The composition of the Mn xCoGe films determined\nby LA-ICP-MS, together with lattice parameters determined\nfrom GAXRD. Grain sizes determined via XRD ( cf. Eq. 1)\nare also given, which agree with those found from AFM.\nxχMn/χGeχCo/χGea(˚A)c(˚A)τXRD(nm)\n0.8 0.79 0.93 4.02 5.21 56.04\n0.9 0.89 0.79 4.03 5.23 84.03\n1.4 1.42 0.97 4.05 5.30 23.99\n1.8 1.83 0.78 4.05 5.36 42.00\n2.5 2.47 1.07 4.07 5.29 33.56\nThe GAXRD peak positions were found to be system-\natically lower than the XRD measurements. This shift\nwas not present in the control Si powder sample. A com-\nparisonbetweenGAXRDandXRDisshowninFig. 4(a).\nWhile XRD probes the lattice parameters of planes that\nare parallel to the substrate surface, GAXRD measures\ninteratomic planes whose normal is further and further\nfrom the film normal as the detector angle θincreases.\nWe defineψ=θ−θias the angle between the film’s nor-\nmal and the scattering vector. As we show, the shift in\nthe GAXRD peaks relative to those in the conventional\nXRD measurements is due to strain in the films.\n5.0 nm\n0.0 nm\n18.0 nm\n9.0 nm\n20.0 nm\n2.0 nm(a) Mn0.8CoGe\n(c) Mn1.4CoGe(b) Mn0.9Co0.8Ge\nFIG. 2: (Color online) AFM images of (a) Mn 0.8CoGe, (b)\nMn0.9Co0.8Ge and (c) Mn 1.4CoGe. The average grain sizes\ncalculated via autocorrelation for (a) and (b) were 56nm and\n84nm, respectively. The average grain size was calculated\nby inspection of the micrograph for (c) and was found to be\n23nm.\nTo determine the influence of film strain on the\nGAXRD measurements, we assume a uniform biaxial\nstrainofthepolycrystallinematerial, where ǫ⊥andǫ/bardblare\nthe out-of-plane and in-plane strains, respectively. The\nstrain for planes that are at an angle ψwith respect to\nthe film surface is given by Eq. 13 in Ref. 20 for the case4\n050(a) Mn0.8CoGe101\n002\n1022\n10\n201\n212\n202\n0100(b) Mn0.9Co0.8Ge\n025(c) Mn1.4CoGe\n050(d) Mn1.8Co0.8Ge\n020(e) Mn2.5CoGe\n20 30 40 50 60 70 80(f)\n2θ(deg)Intensity (CPS)\nFIG. 3: (Color online) XRD data (black) with Reitveld re-\nfinements (orange) and residuals (blue): (a) Mn 0.8CoGe, (b)\nMn0.9Co0.8Ge, (c) Mn 1.4CoGe, (d) Mn 1.8Co0.8Ge and (e)\nMn2.5CoGe, with Ni 2In-type peak locations (green) in (f).\nof zero shear strain, ǫ(ψ) =ǫ/bardblsin(ψ)2+ǫ⊥cos(ψ)2,\nfrom which one obtains the ratio of planes spacing mea-\nsured for the scattering vector along ψrelative to those\nalongψ= 0 :\nd(ψ)\nd(0)=/parenleftBigg\n1+ǫ/bardblsin2ψ+ǫ⊥cos2ψ\n1+ǫ⊥/parenrightBigg\n.(2)\nFor small strain, Eq. 2can be written as\n∆d(ψ)/d(0) = [d(ψ)−d(0)]/d(0)≈(ǫ/bardbl−ǫ⊥)sinψ2,\nwhich allows us to extract ǫ/bardbl−ǫ⊥= 0.012±0.01 for the\nMn1.4CoGe sample in Fig. 4(b).\nThe strain, which was observed for all Mn xCoGe films,\nis likely induced by the annealing process. The thermal\nexpansion coefficients for metals are typically about one\norderofmagnitude largerthan the Si substrate. The film\ncrystallizes at high temperature; since the film contractsFIG. 4: (Color online) (a) XRD and GAXRD measurements\nof Mn 1.4CoGe. The lower panel shows the XRD peak posi-\ntions relative to GAXRD peaks. Note that the intensity of\nthe Si(004) peak at 2 θ= 69.9°was reduced by offsetting the\nsample angle by2 °. (b)The fractional change in the measured\ninteratomic plane spacing. Data has been linearized and the\nsolid line shows the fit to the data using Eq. (2).\nmore than the substrate upon cooling, the film develops\nan in-plane tensile strain (and through the Poisson ratio,\nit develops an out-of-plane compressive strain).\nIV. MAGNETIC MEASUREMENTS\nMagneticmeasurementswereperformedusinga Quan-\ntum Design Physical Properties Measurement System\n(PPMS), equipped with a P500 AD/DC Magnetometry\nSystem(ACMS).Sampleswerecutinto5 .8mm×5.8mm\nsquares and wedged into a plastic straw that was placed\nin the PPMS. The field was applied in the plane of the\nfilm.\nMagnetization loops were recorded as the field was cy-5\ncled between µ0H=±9T. TheM−Hloops for all\nfive samples measured at T= 5K are qualitatively sim-\nilar, as shown in Fig. 5. However, hysteresis loops with\nx>1 show larger HCwith more rounding, suggestive of\na larger mean effective anisotropy with a broader distri-\nbution.\nThe remanent magnetization, MR, was measured on\nwarming from T= 5K after saturating the film in a 9T\nfield. The temperature dependence of MRis shown in\nFig.6. Some of the samples with composition x= 2.5\nhad a small remanent magnetization above T= 270K.\nAlthough no impurity phase could be detected in the X-\nray measurements, additional annealing in the RTA was\nable to remove this additional ferromagnetic contribu-\ntion. The x= 1.4 and 1.8 samples also show a small\nMRaboveT= 270K but further annealing could not\nremove the impurity phase. Unexpectedly, the compo-\nsitions with x <1 exhibited a distinctly ferrimagnetic\nbehavior: above a compensation point of approximately\nT= 230K, the MRreverses sign. Though the MR−T\ncurves forx >1 may resemble those of a ferromagnet,\nwe will argue in subsequent sections that each sample ex-\nhibits anMR(T) curve consistent with Q-type or V-type\nferrimagnetism.\nThe Curie temperature is estimated from the knee in\ntheMR−Tplot asMRapproacheszero. As shown in the\nTableII,TCis comparable to the bulk Thex\nC≈260K of\nthe hexagonal phase, and is relatively insensitive to the\ncomposition x, as observed in bulk.13However, the table\nalso shows that the total magnetic moment per primi-\ntive unit cell is significantly lower that the bulk value for\nMnCoGe, 5 .56µBper primitive unit cell.\nTABLE II: The saturation magnetization, MS, the magnetic\nmoment per primitive unit cell, m, the coercive field µ0Hext\nand Curie Temperature TCfor Mn xCoGe films.\nx M s(kA/m) m(µB)HC(mT)TC(K)\n0.8 380 2.99 26 267\n0.9 384 3.02 20 260\n1.4 497 4.19 78 277\n1.8 394 3.19 69 272\n2.5 353 2.87 89 267\nV. COMPUTED MAGNETIC MOMENTS FROM\nDENSITY-FUNCTIONAL THEORY\nTo explore the origin of the drop in magnetic mo-\nment and the appearance of ferrimagnetic behavior,\nwe considered the influence of atomic disorder in the\nNi2In structure on the individual magnetic moments. In\nthe ordered phase, nuclear magnetic resonance (NMR)\nshows that Mn on the 2a-site has a magnetic moment\nofmMn= 2.4µB, while the moment of Co on the 2c-\nsite couples ferromagnetically to the 2a-site with a mo-\nmentmCo= 0.4µB.10These values are in good agree-−1.0−0.50.00.51.0(a)\nMnxCoGe\nx=0.8\nx=0.9\n−0.4−0.20.0 0.2 0.4\nApplied Field, µ0Hext(T)−1.0−0.50.00.51.0(b)\nx=1.4\nx=1.8\nx=2.5Magnetization, M/MS\nFIG. 5: (Color online) Normalized hysteresis curves of\nMnxCoGe films for compositions (a) x <1 and (b) x >1\nmeasured at T= 5K. The saturation magnetizations are\ngiven in Table II.\nment with the measured magnetization and consistent\nwith neutron scattering experiments.21However, we note\nthat DFT overestimates the magnetic moment of Mn in\nMnCoGe,9,10,22and so has to be rescaled to compare to\nexperiment.\nPreviously published DFT calculations of Ni 2In-type\nMn2Ge predict ferrimagnetic behavior due to the anti-\nparallel coupling between Mn on 2a- and 2c-sites.23This\nis consistent with tight-binding (TB) calculations for\nMnCoGe that show a reduction in the average Mn mo-\nment when it is distributed on both of these sites. The\nTB calculations show that Co on the other hand is lit-\ntle affected by either moving it to the 2a-site, or by the\npresence of Mn-antisite defects, as supported by DFT\ncalculations.22However, there are very few studies of the\nNi2In-type structureandthe magneticbehaviorofMnon\nthe 2c- and 2d-sites (Mn 2cand Mn 2d) remains unclear.\nDFT24,25computations were performed within\nthe spin-polarized general gradient approximation\n(GGA)26using the Vienna Ab-initio Simulation Pack-6\n0200\n(a)x=0.8\nFit\n0200\n(b)x=0.9\n0250\n(c)x=1.4\n0200\n(d)x=1.8\n0 100 200 300\nTemperature, T(K)0200\n(e)x=2.5Magnetization, M(kA/m)\nFIG. 6: (Color online) Remanent magnetization vs temper-\nature after field-cooling to T= 5 K. The nominal struc-\ntures Mn 0.8CoGe and Mn 0.9Co0.8Ge show V-type ferrimag-\nnetic behavior due to Mn occupancies on 2a and 2c sites.\nMn1.4CoGe, Mn 1.8Co0.8Ge and Mn 2.5CoGe are Q-type ferri-\nmagnets. Mn 1.4CoGe and Mn 1.8Co0.8Ge show a secondary\nmagnetic phase. Fits are provided based on a two-sublattice\nferrimagnetic model, described in Eq. 4.\nage (VASP).27–30Local magnetizations are obtained\nby projecting the ground state crystal orbitals onto\natomic-like orbitals centered at each crystallographic\nsite (i.e.atom-centered). Since the magnetization can\nbe strongly dependent on the inter-atomic distances,\nfull cell relaxations were performed for all structures,\nconverging forces to better than 10meV //RingA and stresses\nto within 1MPa by enforcing a sufficiently dense k-point\nsampling of the first Brillouin zone. We used projector\naugmented wave (PAW) datasets with 7, 9, and 4\nvalence electrons for Mn, Co, and Ge, respectively. The\nground state energies were converged to better than\n1meV/f.u.using a plane-wave energy cut-off of 550eV.\nWe attempted to converge both ferromagnetic and\nferrimagnetic solutions for all structures. In some cases\nboth solutions converged, but we present here only the\nlowest energy solutions.\nOur computed magnetic moments for Ni 2In-type\nMnCoGe and Mn 2Ge agree well with previously calcu-lated values. In MnCoGe, our computations show a\nslightly smaller Mn moment, 2 .75µBcompared to the\nvalue calculated in Ref. 22 (3 .09µB), but one that is\ncloser to the experimental value. We obtain a moment of\n0.5µBon Co, and −0.1µBon Ge that are in good agree-\nment with Ref. 22, as well as experimental values. In\nMn2Ge, our computed magnetizations for Mn 2a, 2.9µB\nand Mn 2c,−2.0µB, agree exactly with previously pub-\nlished DFT results.31. Unlike what has been published\npreviously, we find that the ferrimagnetic state is not\nthe ground states of the systems: a spin-configuration\nwith ferromagnetically aligned spins on the 2a-sites in\nthe (001) plane but with antiferromagnetic alignment\nbetween neighboring (001) planes and zero moment on\nthe 2c-sites results in a lower energy state. However,\ngiven that antiferromagnetism is not observed in any of\nthe samples, the moments in the ferrimagnetic state of\nMn2Ge provides a better reference for the spins in our\nsamples and are used in the discussion below.\nxin Mn 1+xCo1−xGe\n−3−2−10123\n(a)Mn2a\nMn2cCo2c\nGe2d\n0.00 0.25 0.50 0.75 1.00\nxin Mn 1+xCoGe1−x−3−2−10123\n(b)Mn2d\nCo2cMagnetic moment, m(µB/atom)\nFIG. 7: (Color online) DFT computed moments for\nMn1+yCo1–yGe (top) and Mn 1+yCoGe 1–y(bottom). The\ndotted lines represent linear interpolations used to model the\nexperimental data.7\nTo determine the effect of Mn 2c, 2×2×2 supercells\nwere built by repeating the MnCoGe hexagonal unit cell\n(6 atoms) twice along each lattice vector resulting in\n16 Mn, 16 Co, and 16 Ge atoms. We considered the\nMn1+yCo1–yGe solid solution where the excess Mn, y,\nreplaces Co on the 2c site. For the dilute limit we placed\n1 Mn on the 2c-site per supercell ( y= 0.06); in the con-\ncentrated limit 15 of the 16 2c-sites were occupied by Mn\n(y= 0.94), We note that the case of y= 0 andy= 1 cor-\nrespond to MnCoGe and Mn 2Ge. The results are shown\nin Fig.7(a). The influence of Mn substitution onto the\n2d site with an analogous Mn 1+yCoGe1–ysolid solution\nis shown in Fig. 7(b).\nThe Mn 2chas little impact on the magnetic moments\nof either the Mn 2amoments or the Co or Ge moments.\nHowever Mn 2cdoes have a significant compositional de-\npendence and is antiferromagnetically coupled to the\nMn2amoments. In the dilute limit, the Mn 2cmoment\nof−0.6µBis opposite in sign but comparable in mag-\nnitude to the Co moment. The magnetic moment of\nMn2creached−1.86µBin the concentratedMn 2cregime,\nwhich approaches the calculated value for Mn 2Ge, as ex-\npected.\nDespite the identical symmetry of the 2c- and 2d-sites,\nMn behaves very differently when it is substituted on\nthe Ge-sites due to its magnetic Co neighbors in the\n(001) plane. In the dilute limit, its moment is slightly\nlarger than the 2a-moment giving a total moment of\n0.08µB/f.u. As the concentration yincreases the magni-\ntude of the Mn 2a- and Mn 2c-moments both decrease and\nso the heavy compensation continues for larger concen-\ntrations.\nWe also performed additional DFT calculations to ex-\namine the influence of Mn on both the 2c- and 2d-sites.\nWe replaced one Co and one Ge atom in the 2 ×2×2\nsupercells with Mn to give Mn 1.12Co0.94Ge0.94. Two dif-\nferentconfigurationsofthis stoichiometryweregenerated\n– one where the Mn on the 2c-site was nearest to the Mn\non the 2d-site, another where it was farthest. All three\nconfigurations resulted in the same magnetic moment of\n−2.9µBfor Mn on the 2d-site and an unchanged mag-\nnetic moment for Mn on both the 2a- and 2c-sites.\nVI. DISCUSSION\nTo understand whether the antiferromagnetically\naligned Mn 2cmoments can explain the observed reduc-\ntion in the magnetization, we construct a model of the\ndefect distribution in the unit cell and use DFT calcu-\nlations to estimate the magnetic moments. Although\nDFT and the measured compositions differ somewhat\nin the amounts of Co and Ge, the magnetization is\ndominated by the size of the Mn moments. Therefore\nwe use the moments calculated for Mn 1+yCo1–yGe and\nMn1+yCoGe1–ythat correspond to the same concentra-\ntion of Mn in Mn xCoGe, given by y= 2(x−1)/(x+2).\nBased on the X-ray analysis, we consider the possibil-TABLE III: The distribution of atoms for Mn xCoGe relative\ntothe (x+2)atoms in the formula unit , for the case ∆ Mn≥0.\nThe fraction of Mn that is in excess of the available 2a sites\nis ∆Mn=x/(x+2)−(1+ν)/3.νis the number of vacancies\nper (x+2) atoms.\nMn Co Ge\n2a1+ν\n3−δ(1+ν)\n30δ(1+ν)\n3\n2c∆Mn1−2ν\n2−ν1−2ν\n3−∆Mn1−2ν\n2−ν0\n2d∆Mn1+ν\n2−ν+δ(1+ν)\n31\nx+2−1−2ν\n3+∆Mn1−2ν\n2−ν1\nx+2−δ(1+ν)\n3\nTABLE IV: The distribution of atoms for Mn xCoGe relative\ntothe (x+2)atoms in the formula unit , for the case ∆ Mn<0.\nMn Co Ge\n2ax\nx+2−δ(1+ν)\n3|∆Mn|\n2|∆Mn|\n2+δ(1+ν)\n3\n2c 02\nx+2−|∆Mn|−1+ν\n30\n2dδ(1+ν)\n31+ν\n3−1\nx+2+|∆Mn|\n21\nx+2−|∆Mn|\n2−δ(1+ν)\n3\nity of vacancies on the 2c-sites, which changes the rela-\ntive number of 2c-sites relative to the 2a- and 2d-sites.\nIn a sample that contains fformula units of Mn xCoGe,\nthere aren=f(x+ 2) atoms. However, in the pres-\nence ofnνvacancies on the 2c-sites, the natoms require\na total ofn(1 +ν), crystallographic sites. When fill-\ning these sites, we need to distinguish compositions ac-\ncording to the number 2a-sites relative to the number of\nMn atoms. For the case where the difference between\nthe number of Mn atom and the number of 2a-sites,\n∆Mn=nx/(x+2)−n(1+ν)/3, is greater than zero,\nour model assumes that the excess is distributed on the\nremaining sites according to the relative number of 2c-\nand 2d-sites. Therefore we place ∆ Mn(1−2ν)/(2−ν)\nMn atoms on 2c, and ∆ Mn(1+ν)/(2−ν) on 2d, as\nshown Table IIItogether with the Co and Ge distribu-\ntions.\nIn the case where ∆ Mn<0, all of the Mn is accommo-\ndatedon the 2asites, andthe remaining2a-sitesarefilled\nby|∆Mn|/2 Co atoms and |∆Mn|/2 Ge atoms. Table IV\nshows the distribution of atoms for this case.\nWe require additional site disorder to account for the\nreduction in the magnetization observed in our films. We\nconsidered both disorder between the 2a- and 2c-sites, as\nwell as between the 2a- and 2d-sites. While both mod-\nels can explain the size of the moments in our Mn xCoGe\nsamples, the 2a-2d site disorder is required to explain the\nmean-field results described below. We therefore intro-\nduce a parameter δthat characterizes the fraction of the\nMn2athat is exchanged with Ge 2d.\nTo calculate the moments of the ∆ Mn≥0 samples, we\nuse the interpolated DFT moments shown by the dotted8\n1.0 1.5 2.0 2.5\nxin Mn xCoGe012345Magnetic moment, m(µB/cell)0.00\n0.05\n0.10\n0.15\n0.20\n0.25Disorder, δ\nFIG. 8: (Color online) The diamonds show the measured\nmagnetic moment mper primitive unit cell of Ni 2In-type\nMnxCoGe films. The color-plot shows the expected varia-\ntion in the magnetic moment due to the disorder parameter δ\nwith 20% vacancies on the 2c-sites ( ν= 0.07). The solid and\ndashed lines show the calculated moment for δ= 0.03, and\n0.17 respectively.\nlines in Fig. 7. We use the Mn 2cand Mn 2doccupancies\nobtained from Tables IIIandIVto determine the rela-\ntive weights of the two sets of moments shown in Fig. 7.\nSince DFT overestimates the Mn 2amoment by a factor\n2.75/2.4,we rescaleall the predicted Mn moments by the\ncorresponding amount. For the ∆ Mn<0 samples, DFT\nresults in Ref. 22 show that the Co 2aand Ge 2aantisite\ndefects do not significantly affect the Mn 2amoments and\ntherefore use our x= 1 calculated values.\nThe calculated magnetic moment for 20% vacancies on\nthe 2c-sites (3 ν/(1+ν) = 0.2) as a function of xandδ\nis shown by the lines and color plot in Fig. 8. The peak\nin the plot occurs for x= 1.11,δ= 0, corresponding\nto ∆Mn= 0, the maximum in the possible fraction of\nMn on the 2a-sites. Below ∆ Mn= 0, the modeled mo-\nment drops with decreasing xdue to a reduction in the\navailable Mn. Above ∆ Mn≥0, the moment drops with\nincreasingxas more Mn is forced onto the 2c-sites and\n2d-sites. The color scale reflects the decrease in magnetic\nmomentwithincreasing2a-2dsitedisorder;acomparison\nwith the data points allows an estimation of the disor-\nder,δ. The model suggest that the disorder could be as\nlarge asδ= 0.17 for ∆ Mn<0 (dashed white line), and\nthen drop below δ= 0.05 for ∆ Mn<0 (solid white line).\nWe note that the x= 2.5 sample has a moment that is\nlarger than can be explained by our model. One possible\nsource for the discrepancy may be due to the inaccura-\ncies of interpolating the DFT results. Nevertheless, themodel captures the general trend in the variation of the\nsaturation magnetization with Mn concentration. The\nmodel for the data and DFT results indicate that ferri-\nmagnetism exists for all samples, not just the ∆ Mn<0\nsamples where compensated ferrimagnetic is observed.\nFerrimagnetism for the ∆ Mn>0 samples is not imme-\ndiately obvious from the magnetometry measurements.\nHowever, a closer inspection of the shape of the M−T\nplots in Fig. 6reveals features that are observed in\nother ferrimagnets,33such as the linear MR(T) region in\nFig.6(d) between 80 K and 220 K. To explore the shape\nof the magnetization curves in more detail, we fitted the\nMR(T) curves with N´ eel’s molecular field model.32Since\nthe DFT calculations show that Mn and Co behave sim-\nilarly on the 2c-sites and the 2d-sites, we approximated\nthe system with a two-sublattice model where Arefers\nto the moments on the 2a-sites and Bcontains both the\n2c- and 2d-sites. The molecular fields experienced by\nsublatticeAandBare given by the usual mean-field\nparameters λij,\nHA(T) =λAAMA(T)+λABMB(T) (3)\nHB(T) =λABMA(T)+λBBMB(T).\nThe temperature-dependent magnetization of each sub-\nlattice is then calculated by solving the two coupled non-\nlinear equations,32\nMA(T) =MA(0)BJA/parenleftbiggµ0mAHA(T)\nkBT/parenrightbigg\n,\nMB(T) =MB(0)BJB/parenleftbiggµ0mBHB(T)\nkBT/parenrightbigg\n,(4)\nwhere B Ji(y) is the Brillouin function. The molecular\nfield coefficients are related to the exchange constant of\nthe Heisenberg model of the form\nH=−/summationdisplay\n/angbracketlefti,j/angbracketrightJijSi·Sj (5)\nthrough the relationship\nJij=µ0(gµB)2λij\n2zijVuc(6)\nwhereVuc=√\n3a2c/2 is the unit cell volume and zijis\nthe number of j-sublattice nearest neighbours to atoms\non sublattice i.\nWe use the moments obtained from our DFT-based\ndefect model, described by Table IIIandIV, as ini-\ntial guesses for the mean-field sublattice moments\nmi=gµB/radicalbig\nJi(Ji+1). The three molecular field coef-\nficientsλij, together with the two sublattice moments\nmiare treated as fitting parameters. The resulting least-\nsquares fits to the MR(T) data are shown by the black\nlines in Fig. 6, with the corresponding fitting parameters\nplotted in Fig. 9. It should be noted that attempts to fit\nthex>1 samples with Weiss’ ferromagnetic mean field9\nmodel were unsuccessful. The fitted moments have been\nscaled to the saturationmagnetizationslisted in Table II.\nThe features below 100K in the MR(T) curves of the\nMn-deficient samples cannot be captured with this two-\nsublattice model. The atypical drop in the MRbetween\n5 K and 100 K is likely due to domain relaxation as the\nanisotropy for these samples is smaller that the x >1\nsamples, as seen in Fig. 5. For these samples, no phases\nother than the Ni 2In-typeB82were observed in XRD,\nand therefore it is unlikely that a secondary magnetic\nphase is contributing to the magnetic signal. We there-\nfore limit the fit for the x= 0.8 and 0.9 samples to tem-\nperatures above T= 100 K, where the two-sublattice\nmodel is able to capture the shape of the M−Tdata.\nFigure9(a) show the fitted moments on the A- and\nB-sublattices compared to the same moments estimated\nfrom the defect model of Fig. 8. The mean-field values\nfollowthe sametrend asthe DFT-basedmodel with com-\nparable values. However, the B-sublattice moments from\nthe DFT-based model for the two x <1 samples are\nsmaller than would allow for V-type compensated ferri-\nmagnetism. For these compositions, Table IVshows that\na non-zero δis necessary to create a ferrimagnetic sam-\nple. The reason why we have added disorder between the\n2a and 2d sites is because DFT shows that Mn 2dis sub-\nstantially larger than Mn 2c, althoughδ≃0.17 obtained\nfrom a fit to msatdoes not create a 2d moment that is\nlarge enough.\nFigure9(b)showsthattheexchangeconstantsbetween\ntheAandBsublattices is small for x <1 but is anti-\nferromagnetic, consistent with the presence of Mn 2cde-\nfects dominating the inter-sublattice interaction. With\nincreasing Mn concentration, JABincreases as expected\nfrom the increasein Mn 2cdefects inferred from the DFT-\nbased model. In contrast, the intra-sublattice interac-\ntions are ferromagnetic at low compositions, but reverse\nsign above x≃1.4.\nThe evolution in exchange parameters can be mapped\nonto N´ eel’s general ferrimagnetic phase diagram after\naccounting for the difference between the A- and B-\nsublattice moments32,33. The phase diagram is repro-\nduced in Fig. 10(a) where each of the colored regions\nin theα≡ −λAA/λBBandβ≡ −λBB/λABparame-\nter space corresponds to a different shape for M(T), as\nshown in Fig. 10(b). The grey region labelled G is para-\nmagnetic for all finite temperatures. The x= 2.5 sample\nresides in the Q-region, near the P-Q phase boundary, as\nshown by the black point . As xdecreases, the reduction\nof Mn on the 2c-sites decreases λABand increases in the\nintra-site exchange coupling, which drives the material\ntowards the Q-V boundary and leads to a straighten-\ning of theM−Tcurve at intermediate temperatures in\nFig.6. This trend continues for xbelowx≃1, and\npushes the system into the V-region where compensated\nferrimagnetism is observed.−20246Moment, m(µB/unit cell)\nmtot\nmA\nmB\n1.0 1 .5 2 .0 2 .5\nMn composition, x−7.5−5.0−2.50.02.55.0Exchange constant, Jij(meV)JAB\nJAA\nJBB\nFIG. 9: (Color online) Mean-field fitting parameters obtain\nfrom the fits in Fig. 6 for sublattices A (2a-sites) and B (2c +\n2d-sites). a) The magnetic moments per unit cell are shown\nby the filled colored points. For comparison, the open grey\npoints show the moments from the the DFT-based model.\nb) The exchange constants for the inter-sublattice interac tion\nJABand the intra-sublattice interactions JAA,JBB.\nVII. CONCLUSION\nSputtered Mn xCoGe compounds formed a metastable\nNi2In-type structure over the entire compositional range\n0.8≤x≤2.5 explored in this study. The unexpected\nferrimagnetic behavior is explained by the presences of\nMn anti-site defects on the 2c/2d-sites. DFT calcula-\ntions show that these Mn defects are antiferromagnet-\nically coupled to the Mn on the 2a-sites. An atomic\nmodel of the distribution of defects in the unit cell us-\ning the DFT predicted values explains the general trend\nin the variations in the saturation magnetization with\ncomposition. We provide supporting evidence for the\nferrimagnetism with mean-field modeling that both cap-\ntures variations in the shape of the M(T) curves and the\ntrends in the size of the sublattice moments that follow\nthe DFT-based model. The analysis demonstrates that10\nG\nFIG. 10: (Color online) The partitioning of molecular field\nparameter space for a two-sublattice ferrimagnet. Boundar ies\nin (a) were calculated for values of mAandmBobtained for\nMn2.5CoGe. Region G is paramagnetic. The exchange pa-\nrameters for Mn 2.5CoGe are shown by the black dot in the\nQ-type region. While the precise boundaries in the phase di-\nagrams vary with Mn composition, they remain qualitatively\nthe same for all x. Representative MR(T) curves for each\nregion are provided in (b).by increasing the concentration of Mn anti-site defects,\nthe inter-sitebecomes increasinglyantiferromagneticand\nthe intra-site coupling changes sign, which drives the fer-\nrimagnetism from V-type to Q-type.\nThis work suggests the possibility of controlling the\nferrimagnetism through defect engineering to generate\ncompensated ferrimagnets in alloys that would otherwise\nbe ferromagnetic. Interest in ferrimagnetism has been\nrevived with the discovery of ultrafast dynamics at the\nangular momentum compensation point35–38. Such dy-\nnamicscouldbevaluableinapplicationsforspintronics34,\ncomplementaryto approachesproposed for devices based\non antiferromagnets.\nVIII. ACKNOWLEDGMENTS\nWe would like to thank Jeff Dahn for use of the sput-\ntering machine, as well as Andrew George and Michel\nJohnson for technical assistance with XRD and PPMS\nmeasurements. We also wish to thank James Brenan for\nthe use of the LA-ICP-MS and Erin Keltie for assistance\nin the collection and analysis of the data. Thank you\nto Ulrich R¨ oßler for helpful discussion about DFT and\nCameron Rudderham and Andrey Zelenskiy for insight-\nful conversations.\n∗tmonches@dal.ca\n1Ajaya K. Nayak, Julia Erika Fischer, Yan Sun, Bing-\nhai Yan, Julie Karel, Alexander C. Komarek, Chandra\nShekhar, Nitesh Kumar, Walter Schnelle, J¨ urgen K¨ ubler,\nClaudia Felser, and Stuart S. P. Parkin. Large anoma-\nlous Hall effect driven by a nonvanishing Berry curvature\nin the noncolinear antiferromagnet Mn 3Ge.Science Ad-\nvances, 2(4), 2016.\n2H. Kurt, N. Baadji, K. Rode, M. Venkatesan, P. Stamenov,\nS. Sanvito, and J. M. D. Coey. 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J.ˇSaf´ arik University, Park Angelinum 9, 041 54 Koˇ sice, Slov ak Republic\nAbstract\nWe study a bilayer Ising spin system consisting of antiferro magnetic (AF) and fer-\nromagnetic (FM) triangular planes, coupled by ferromagnet ic exchange interaction,\nby standard Monte Carlo and parallel tempering methods. The AF/FM bilayer\nis found to display the critical behavior completely differen t from both the single\nFM and AF constituents as well as the FM/FM and AF/AF bilayers . Namely, by\nfinite-size scaling (FSS) analysis we identify at the same te mperature a standard\nIsing transition from the paramagnetic to FM state in the FM p lane that induces\na ferrimagnetic state with a finite net magnetic moment in the AF plane. At lower\ntemperatures there is another phase transition, that takes place only in the AF\nplane, to different ferrimagnetic state with spins on two subl attices pointing paral-\nlel and on one sublattice antiparallel to the spins on the FM p lane. FSS indicates\nthat the corresponding critical exponents are close to the t wo-dimensional three-\nstate ferromagnetic Potts model values.\nKey words: Ising model, AF/FM bilayer, Triangular lattice, Geometric al\nfrustration, Monte Carlo simulation\n1 Introduction\nMagnetic bilayers and multilayers are of considerable theoretical int erest since they\nallow studying the cross-over phenomena between the two- and th ree-dimensional\nsystems [1,2]. On the experimental side, recent techniques facilita te fabricating a\nvariety of such layered structures in a highly controlled and tunable way [3–5], which\ncan lead to useful technological applications such as magneto-opt ical discs [6].\n∗Corresponding author.\nEmail address: milan.zukovic@upjs.sk (M.ˇZukoviˇ c).\nPreprint submitted to Physics Letters A 22 June 2021Properties of Ising bilayers consisting of two different ferromagne tic layers cou-\npled by either ferromagnetic (FM) or antiferromagnetic (AF) exch ange interactions\nof varying strengths have been investigated in a number of studies [7–16]. Regarding\ntheir critical behavior, it has been found that they belong to the sa me universality\nclass as a two-dimensional Ising model and their critical temperatu re is controlled\nby the so call shift exponents that depends on the interlayer to int ralayer interaction\nratio. In the absence of an external magnetic field due to symmetr y reasons these\nfindings apply tobothFMandAF layered systems, aslong asthelattic eis bipartite.\nApparently, thesituation iscompletely different if oneconsiders anA F bilayer on\nanonbipartite, suchastriangular,lattice.Asinglelayer, i.e.atwo-d imensional trian-\ngular lattice Ising antiferromagnet (TLIA) is exactly known to show no long-range\nordered (LRO) phase down to zero temperature due to high geome trical frustra-\ntion [17]. A recent study of cross-over phenomena in a layered syst em obtained by\nstacking of individual TLIA layers on top of each other revealed a ra ther exotic be-\nhavior termed “stiffness from disorder” leading to a low temperatur e reentrance of\ntwo Berezinskii-Kosterlitz-Thouless transitions [18]. Nevertheles s, this phenomenon\nis only observed in multilayer systems exceeding a certain critical num ber of lay-\ners but not in the bilayer. The latter shows critical properties similar to the two-\ndimensional TLIA.\nConsidering the above, we find it interesting to study an Ising bilayer system\nconsisting of one AF and one FM triangular planes. As mentioned abov e, the critical\nbehavior of the decoupled planes is very distinct. While the AF one sho ws no LRO\ndown to zero temperature due to high geometrical frustration, t he FM one displays\na single standard Ising universality class phase transition to the FM L RO phase.\nIn the present Letter we show that in the system consisting of the coupled AF/FM\nplanes the competing ordering and disordering tendencies enforce d by the respective\nlayers result in the critical behavior completely different from both t he separate FM\nand AF planes as well as the FM/FM and AF/AF bilayers.\n2 Model and Methods\nThe model Hamiltonian can be written as\nH=JA/summationdisplay\n/angbracketlefti∈A,j∈A/angbracketrightσiσj−JB/summationdisplay\n/angbracketleftk∈B,l∈B/angbracketrightσkσl−JAB/summationdisplay\n/angbracketlefti∈A,k∈B/angbracketrightσiσk, (1)\nwhereσi=±1 is an Ising spin on the ith lattice site, the first two sums run over\nnearest neighbors (NN) within AandB planes, coupled by the exchan ge interactions\nJAandJB, respectively, and the third sum runs over NN between the planes A and\nB, coupled by the exchange interaction JAB. In the following we will restrict our self\nto the fully isotropic case of JA=JB=JAB≡J >0.\nIn order to obtain temperature dependencies of various quantitie s we use stan-\ndard Monte Carlo (MC) simulations following the Metropolis dynamics. L inear lat-\n2tice sizes of the individual planes range from L= 24 up to 168 and periodic (open)\nboundary conditions are applied within (out of) planes. Simulations st art from high\ntemperatures and random initial states. Then the temperature is gradually low-\nered with the step kB∆T/J= 0.05 and simulations at the next temperature are\ninitiated using the final configuration obtained at the previous temp erature. For\nthermal averaging we use 105MC sweeps (MCS), after discarding another 2 ×104\nMC for thermalization. Error estimates are obtained from three ind ependent simu-\nlation runs.\nIn the low-temperature region, which is potentially problematic for f rustrated\nspin systems due to time-consuming tunneling through multimodal en ergy land-\nscapes resulting inextremely slow relaxation, we verify thereliability o f theobtained\nresultsbyapplying theparalleltempering (PT)orreplicaexchangem ethod[19].The\nmethodovercomesenergybarriersbyarandomwalkintemperatur espaceandallows\nexploration of complex energy landscapes of frustrated systems . We roughly tune\nthe simulation temperature set by preliminary runs monitoring replica -exchange ac-\nceptance rates. For each lattice size, replica swaps at neighboring temperatures are\nproposed after each of 106MCS.\nTo obtain critical exponent ratios, we perform finite-size scaling (F SS) analysis,\nin which case we apply the reweighing techniques [20]. The reweighing is p erformed\nat lattice-size-dependent pseudo-critical temperatures, estim ated from the standard\nMC simulations, using 107MCS for statistical averaging and errors are estimated\nby applying the more reliable and precise Γ-method [21].\nWe measure the following basic thermodynamic quantities: The intern al energy\nper spin\ne=∝angbracketleftH∝angbracketright/L2, (2)\nthe magnetizations per spin of the separate planes A and B\n(mA,mB) = (∝angbracketleftMA∝angbracketright,∝angbracketleftMB∝angbracketright)/L2=/parenleftbigg/angbracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈Aσi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketrightbigg\n,/angbracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈Bσj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketrightbigg/parenrightbigg\n/L2,(3)\nthe staggered magnetization per spin (order parameter) within th e AF plane A\nms=∝angbracketleftMs∝angbracketright/L2= 3/angbracketleftbigg\nmax\ni=1,2,3(MA,i)−min\ni=1,2,3(MA,i)/angbracketrightbigg\n/2L2, (4)\nwhereMA,i(i= 1,2,3) are the three sublattices within the AF plane A and ∝angbracketleft···∝angbracketright\ndenotes thermal average. From the above quantities we further calculate the specific\nheat per site\nc=∝angbracketleftH2∝angbracketright−∝angbracketleftH∝angbracketright2\nL2kBT2, (5)\nthe susceptibility per site χx, corresponding to the parameter Mx,x= A,B,s.\nχx=∝angbracketleftM2\nx∝angbracketright−∝angbracketleftMx∝angbracketright2\nL2kBT, (6)\n3and the derivative and logarithmic derivative of ∝angbracketleftMx∝angbracketrightwith respect to β= 1/kBT\ndmx=∂\n∂β∝angbracketleftMx∝angbracketright=∝angbracketleftMxH∝angbracketright−∝angbracketleftMx∝angbracketright∝angbracketleftH∝angbracketright, (7)\ndlmx=∂\n∂βln∝angbracketleftMx∝angbracketright=∝angbracketleftMxH∝angbracketright\n∝angbracketleftMx∝angbracketright−∝angbracketleftH∝angbracketright. (8)\nIn the FSS analysis we employ the following scaling relations:\nχx,max(L)∝Lγx/νx, (9)\ndlmx,max(L)∝L1/νx, (10)\ndmx,max(L)∝L(1−βx)/νx, (11)\ncmax(L)∝Lαx/νx,(α∝negationslash= 0) (12)\ncmax(L) =c0+c1ln(L),(α= 0) (13)\nβx,max(L) =βc+axL−1/νx, (14)\nwhereβcis the inverse transition temperature and βx,max(L) are the inverse pseudo-\ntransitiontemperatures, estimated aspositionsofthemaxima oft heabovefunctions\nfor a given L.\n3 Results and discussion\nFrom the Hamiltonian (1) it is easy to verify that the ground-state a rrangement\nof the system corresponds to the ferromagnetically ordered plan e B with mB= 1\nand ferrimagnetically ordered plane A with two spins parallel and one a ntiparallel\non each elementary triangle, which corresponds to mA= 1/3 andms= 1. Tem-\nperature dependencies of the calculated quantities, presented in Fig. 1, provide the\npicture of the thermodynamic behavior of the system at finite temp eratures as well\nas its dependence on the system size. In particular, in Fig. 1(a) one can observe\ntwo anomalies in the internal energy dependencies associated with t wo phase transi-\ntions. Fig. 1(b) shows that the high-temperature anomaly at some temperature Tc2\ncorresponds to the ferromagnetic (FM) long-range ordering (LR O) of the B plane,\nwhile the low-temperature one at Tc1< Tc2reflects the onset of the ferrimagnetic\n(FRM1) LRO within the A plane, associated with the order parameter ms. Never-\ntheless, one can notice that in the antiferromagnetic A plane there is a non-zero net\nmagnetization mAalso within Tc1< T < T c2and is virtually independent on the\nlattice size. Such a ferrimagnetic state in the A plane is induced by the FM ordering\nin the B plane at Tc2and we will refer to it as FRM2 phase.\nThe character of the respective phase transitions can be studied from the scal-\ning of the corresponding response functions near criticality. The la tter are shown in\nFigs. 2(a)-2(c) for various Lon semi-log plots. At criticality one should expect linear\ndependence of their maxima with ln( L), according to the scaling relations (9)-(13),\n40 1 2 3 4 5−2.2−2−1.8−1.6−1.4−1.2−1−0.8\nkBT/Je/J\n  \nL=24\nL=48\nL=72\nL=96\nL=120(a)\n0 1 2 3 4 500.20.40.60.81\nkBT/J(b)\nmB\nms\nmA\n0 1 2 3 4 510−1100\nkBT/Jc\n  \nL=24\nL=48\nL=72\nL=96\nL=120(c)\nFig. 1. Temperature variations of (a) the internal energy e/J, (b) the magnetizations mA,\nmBandms, and (c) the specific heat c, plotted for L= 24−120.\nwith the slopes (critical exponent ratios) providing information abo ut the transition\norder and/or the universality class. The FSS analysis is performed u sing the above-\ndefined quantities, corresponding to the parameters mx(x= A,B,s), as marked in\nFigs. 2(a)-2(c) by FSS1-3.\nHowever, the relatively large temperature step and the relatively s mall number\nof MCS used in standard MC simulations are not sufficient to locate the desired\nextremal values with high precision. Therefore, it is more convenien t to resort to the\nreweighing techniques performed on much longer time series of the d ata obtained at\nthe respective pseudo-transition points roughly located by the st andard MC simu-\nlations. As already pointed out above, particularly simulations perfo rmed at lower\ntemperatures, such as in the vicinity of the low-temperature tran sition temperature\nTc1, involve some risk of getting stuck in local minima, which can be eliminate d by\nthe PT method. In Fig. 2(d) we confront the results for the stagg ered susceptibility\npeaks obtained by the reweighing of the standard MC data and the P T methods.\nOne can notice that even for the largest lattice sizes, which are the toughest to equi-\nlibrate, a fairly goodagreement is achieved, thus boosting our confi dence in reaching\n50 1 2 3 4 510−1100101\nkBT/JχA\n  \nL=24\nL=48\nL=72\nL=96\nL=120(a)\nFSS2\n0 1 2 3 4 510−1100101102\nkBT/JχB\n  \nL=24\nL=48\nL=72\nL=96\nL=120FSS1(b)\n0 1 2 3 4 510−1100101102\nkBT/Jχs\n  \nL=24\nL=48\nL=72\nL=96\nL=120FSS3(c)\n0.94 0.96 0.98 11.02 1.04102103\nkBT/Jχs\n  \nL=168\nL=24L=48L=72L=96L=120L=144FSS3(d)\nFig. 2. Temperature variations of the susceptibilities χx, corresponding to the parameters\nmx(x= A,B,s), plotted for L= 24−120. In (d) more detailed results in the vicinity of\nthe critical point Tc1are presented for the staggered susceptibility χsdata obtained from\nthe reweighing (dashed curves) and PT (symbols) techniques , forL= 24−168.\nequilibrium solutions.\nIn Fig. 3 we present scaling results of the functions (5)-(8) at the high- and\nlow-temperature transition temperatures Tc2andTc1. We note that mBis the order\nparameter characterizing thehigh-temperatureparamagnetic- ferromagnetic (P-FM)\ntransition in the B plane and msis the order parameter of the low-temperature tran-\nsition tothe ferrimagnetic FRM1state inthe Aplane. However, as alr eadyindicated\nin Fig. 1(b), at both transitions there are abrupt changes also in th e quantity mA\nand the corresponding susceptibility (Fig. 2(a)) seems to diverge w ith the system\nsize.\nIndeed, all the log-log plots in Fig. 3 show power-law (or logarithmic in c ase of\ncmax) scaling, however, the respective phase transitions are governe d by distinctly\ndifferent critical exponent ratios. Those obtained for the P-FM tr ansition in the\nB plane are consistent with the Ising universality class values: αI= 0, 1/νI= 1,\nγI/νI= 1.75 and (1 −βI)/νI= 0.875. Even the exponent ratios of the P-FRM2\n63 3.5 4 4.5 51234567\nln(L)  \nln(χB,max)\nln(dlmB,max)\nln(dmB,max)\nc0+c1ln(L)1/νB = 1.006(3)\n(1−βB)/νB = 0.875(2)\nαB = 0\nγB/νB = 1.760(3)FSS1 (a)\n3 3.5 4 4.5 5−202468\nln(L)  \nln(χA,max)\nln(dlmA,max)\nln(dmA,max)\nc0+c1ln(L)1/νA = 1.01(2)\nαA = 0\n(1−βA)/νA = 0.88(2)\nγA/νA = 1.67(2)FSS2 (b)\n3 3.5 4 4.5 5 5.5012345678\nln(L)  \nln(χs,max)\nln(dlms,max)\nln(dms,max)\nln(cmax)\n(1−βs)/νs = 1.049(3)\nαs/νs = 0.346(2)FSS3\n1/νs = 1.180(5)γs/νs = 1.870(6)\nFig. 3. Critical exponent ratios obtained in finite-size sca ling analyses FSS1-3 from the\nscaling relations (9)-(13).\ntransition in the A plane coincide within error bars with the standard I sing values,\nexcept for the susceptibility exponent γA. One can notice that smaller lattice sizes\ndo not comply with the power-law scaling and, therefore, we had to c onsider larger\nsizes up to L= 168 and drop some smaller ones from the fitting. Nevertheless, th e\nbest fitted value γA/νA= 1.67(2) is clearly smaller than the Ising value. The re-\nspective critical temperatures can be estimated from the scaling r elation (14). As\npresented in Fig. 4, the scaling is better behaved for the P-FM tran sition in the\nplane B (Fig. 4(a)), while in the P-FRM2 transition the linear ansatz is o nly satis-\nfied forL≥120 and the estimated values of the inverse critical temperature βcfrom\ndifferent quantities are more scattered (Fig. 4(b)). Nevertheles s, the two transition\ntemperatures cannot be distinguished within our limits of accuracy.\nThe low-temperature phase transition occurs in the A plane due to s pin rear-\nrangementfromthepartiallyorderedferrimagneticstateFRM2wit h|mA|>0tothe\nferrimagnetic state FRM1 with 2/3 of the spins aligned parallel and 1/ 3 antiparallel\nto theferromagnetically ordered B-planespins, resulting inthegro und-statevalue of\n|mA|= 1/3. Here, the role of the ferromagnetically ordered plane resembles that of\n700.005 0.01 0.015 0.02 0.0250.2720.2730.2740.2750.2760.2770.2780.2790.280.2810.282\nL−1/ν\nBβB,max\n  \nβmaxχ\nB\nβmaxdlm\nB\nβmaxdm\nB\nβmaxc\nBFSS1 (a)\n00.005 0.01 0.015 0.02 0.0250.2720.2730.2740.2750.2760.2770.2780.2790.280.2810.282\nL−1/ν\nAβA,max\n  \nβmaxχ\nA\nβmaxdlm\nA\nβmaxdm\nA\nβmaxc\nA(b)FSS2\nFig. 4. FSS fits of the inverse pseudo-transition temperatur esβx,max , according to the\nscaling relation (14), for the functions of Mxand (a)x= B and (b) x= A.\nan external magnetic field trying to align the spins in the antiferroma gnetic A plane\ninto its direction. The latter is believed to belong to the same universa lity class as\nthe three-state ferromagnetic Potts model [22,23] with the crit ical exponent ratios:\nαP/νP= 6/15, 1/νP= 6/5,γP/νP= 26/15 = 1.7¯3 and (1−βP)/νP= 16/15 = 1.0¯6.\nThe present values obtained from the fits of the FSS3 analysis are r ather close to\nthe standard three-state Potts values, nevertheless, the valu esγs/νs= 1.870(6) and\nαs/νs= 0.346(2) apparently deviate beyond the error bars. However, eve n though\nthe dependencies in Fig. 3(c) look linear a closer inspection reveals slig ht down-\nward curvatures in the respective plots, indicating that the asymp totic regime may\nnot have been reached. A more careful analysis involves evaluation of the running\nexponents γs/νs(L),1/νs(L),(1−βs)/νs(L), andαs/νs(L), corresponding to local\nslopes estimated from three consecutive values of L. The results presented in Fig. 5\nindicate that for increasing Lalso the ratio γs/νs(L) seems to converge to the Potts\nvalue, nevertheless, αs/νs(L) still remains below the standard value. We suspect\nthat this deviation of αs/νs(L) might, at least partly, be caused by neglecting the\nnondivergent “background” term in Eq. (12). However, it is also po ssible that the\npresent values indeed deviate from the standard Potts ones and s how some variation\nwith the strength of the coupling to the ferromagnetic layer. Similar behavior was\nalso observed in the TLIA model the critical exponents of which disp layed some\ndependence on the field value [23,24].\n4 Conclusions\nCritical properties of an Ising bilayer spin system consisting of antif erromagnetic\n(AF) and ferromagnetic (FM) triangular planes, coupled by ferrom agnetic exchange\ninteraction, were studied by standard Monte Carlo and parallel tem pering methods.\nAt higher temperatures we identified in the FM plane a standard Ising universality\n84060801001201401600.20.40.60.811.21.41.61.82\nL1/νs(L)\n(1−βs)/νs(L)γs/νs(L)\n6/5\n6/1516/1526/15\nαs/νs(L)\nFig. 5. Running exponents γs/νs(L),1/νs(L),(1−βs)/νs(L),αs/νs(L), for the low-tem-\nperature phase transition at Tc1. The dashed lines indicate the standard Potts values of\nthe respective critical exponent ratios.\nclass phase transition from a paramagnetic to a ferromagnetic sta te, which at the\nsame critical temperature induces via the interlayer couplings a fer rimagnetic spin\narrangement with non-vanishing magnetic moment in the adjacent A F plane. At\nlower temperatures, there is another phase transition in the AF pla ne to a differ-\nent ferrimagnetic state with two sublattices aligned and one anti-alig ned with the\nspins in the ferromagnetically ordered FM plane. This state resemble s the ferrimag-\nnetic arrangement of the TLIA model in moderate external magne tic fields [25–28].\nThe latter are believed to belong to the standard two-dimensional t hree-state fer-\nromagnetic Potts universality class and the present results sugge st that also the\nlow-temperature phase transition in the present model belongs to same universality\nclass.\nNevertheless, we think that it is likely that, similar to the TLIA model in a\nfield, the critical exponents of the low-temperature phase trans ition in the present\nmodel can show some variation with the exchange interaction ratio. The effect of the\nexchange interaction anisotropy is of great interest also because it allows studying\na dimensional cross-over phenomena in the system and may lead to q ualitatively\ndifferent critical behaviors. Such a study is currently underway.\nAcknowledgments\nThis work was supported by the Scientific Grant Agency of Ministry o f Education\nof Slovak Republic (Grant No. 1/0331/15).\n9References\n[1] T.W. Capehart, M.E. Fisher, Phys. Rev. B 13 (1976) 5021.\n[2] L. de Jongh, in: L. de Jongh (Ed.), Magnetic Properties of Layered Transition Metal\nCompounds, Kluwer, Dordrecht, 1990, p. 1.\n[3] L.J. de Jongh, Physica B 82 (1976) 247.\n[4] M. Ba/suppress landa, R. Pe/suppress lka, T. Wasiuty´ nski, M. Rams, Y. Naka zawa, Y. Miyazaki, M. Sorai,\nR. Podgajny, T. Korzeniak, B. Sieklucka, Phys. Rev. B 78 (200 8) 174409.\n[5] W. Shi, R. Liang, S. Xu, Y. Wang, C. Luo, M. Darwish, S.K. Sm oukov, The Journal\nof Physical Chemistry C 119 (2015) 13215.\n[6] K. Shimazaki, S. Ohnuki, H. Fujiwara, N. Ohta, J. Magn. Ma gn. Mater., 104107 (1992)\n1017.\n[7] A.M. Ferrenberg, D.P. Landau, J. Appl. Phys. 70 (1991) 62 15.\n[8] P.L. Hansen, J. Lemmich, J.H. Ipsen, O.G. Mouritsen, J. S tat. Phys. 73 (1993) 723.\n[9] T. Horiguchi, N. Tsushima, Physica A 238 (1997) 295.\n[10] A. Lipowski, Physica A 250 (1998) 373.\n[11] Z.B. Li, Z. Shuai, Q. Wang, H.J. Luo, L. Sch¨ ulke, J. Phys . A: Math. Gen. 34 (2001)\n6069.\n[12] H. Kim, J. Korean Phys. Soc. 38 (2001) 435.\n[13] M. Ghaemi, B. Mirza, G. Parsafar, J. Theor. Comp. Chem. 3 (2004) 217.\n[14] J.L. Monroe, Physica A 335 (2004) 563.\n[15] K. Sza/suppress lowski, T. Balcerzak, Physica A 391 (2012) 2197.\n[16] K. Sza/suppress lowski, T. Balcerzak, Thin Solid Films 534 (2013 ) 546.\n[17] G.H. Wannier, Phys. Rev. 79 (1950) 357.\n[18] S.-Z. Lin,1 Y. Kamiya, G.-W. Chern, C.D. Batista, Phys. Rev. Lett. 112 (2014) 155702.\n[19] K. Hukushima, K. Nemoto, J. Phys. Soc. Jpn. 65 (1996) 160 4.\n[20] A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 61 (19 88) 2635.\n[21] U. Wolff, Computer Physics Communications 156 (2004) 143 .\n[22] S. Alexander, Phys. Lett. B54 (1975) 353.\n[23] W. Kinzel, M. Schick, Phys. Rev. B 23 (1981) 3435.\n[24] S.L.A. de Queiroz, T. Paiva, J.S. de S´ a Martins, R.R. do s Santos, Phys. Rev. B 59\n(1999) 2772.\n10[25] B.D. Metcalf, Phys. Lett. 45A (1973) 1.\n[26] M. Schick, J.S. Walker, M. Wortis, Phys. Rev. B 16 (1977) 2205.\n[27] R.R. Netz, A.N. Berker, Phys. Rev. Lett. 66 (1991) 377.\n[28] M. ˇZukoviˇ c, M. Borovsk´ y, A. Bob´ ak, Phys. Lett. A 374 (2010) 4 260.\n11"    },    {        "title": "2012.06823v1.Ultra_fast_Double_Pulse_All_Optical_Re_switching_of_a_Ferrimagnet.pdf",        "content": "Ultra-fast Double Pulse All-Optical Re-switching of a Ferrimagnet\nC. Banerjee, K. Rode, G. Atcheson, S. Lenne, P. Stamenov,  J. M. D. Coey \nand J. Besbas*\nCRANN, AMBER and School of Physics, Trinity College Dublin, Dublin 2, Ireland\n*besbasj@tcd.ie\nAbstract\nAll-optical  re-switching  has  been  investigated  in  the  half-metallic  Heusler  ferrimagnet\nMn2Ru0.9Ga, where Mn atoms occupy two inequivalent sites in the XA-type structure. The\neffect of a second 200 fs 800 nm pump pulse that follows a first pulse, when both are above\nthe threshold for switching, is studied as a function of t12, the time between them. The aims\nare to identify the physical mechanisms involved and to determine the minimum time needed\nfor re-switching. The time trajectory of the switching process on a plot of sublattice angular\nmomentum, S4a vs S4c, is in three stages; When t < 0.1 ps, the sublattice moments are rapidly\ndisordered, but not destroyed, while conserving net angular momentum via optical spin-wave\nexcitations. This leads to transient parallel alignment of the residual Mn spins in the first\nquadrant. The net angular momentum associated with the majority sublattice then flips in\nabout 2 ps, and a fully-reversed ferrimagnetic state is then established via the spin-lattice\ninteraction, which allows re-switching provided t12 > 10 ps. \nSingle-pulse all-optical switching of magnetization (SP-AOS) is of both fundamental\nand technological interest  [1–3]. Despite intense scrutiny over the last two decades, the\nmicroscopic origin of the effect is still poorly understood, but the possibility of switching the\nmagnetisation of a thin film between two stable states on a picosecond timescale without\nrecourse to an external magnetic field is intriguing and technologically relevant in the quest\nfor ever-faster and more energy-efficient information technologies [4–6]. Here we establish\nthe minimum time that must elapse between two pulses, if the second one is to re-establish\nthe  original  state.  Our  results  advance  the  fundamental  understanding  of  SP-AOS  and\nhighlight its potential for future application in technology.We  have  recently  shown  that  the  near-cubic  XA-ordered  (F-43m)  ferrimagnetic\nHeusler alloy Mn2RuxGa (MRG) exhibits  SP-AOS  [7], and that switching  is  driven by\nantiferromagnetic  exchange  between  the  crystallographically-inequivalent  4 a and  4c Mn\nsublattices  [8]. The inequivalence is the source of two key properties of MRG. First, the\nstates close to the Fermi level are associated predominantly with one of the sublattices, which\nwe identified in compounds with x ≈ 0.7 as 4c [9], resulting in half-metallic character. This\nsublattice  dominates  magneto-optic  Kerr  effect  (MOKE)  [10].  All  MOKE-based\nmeasurements shall therefore be understood as reflecting the response of the 4 c sublattice.\nSecond, due to the hierarchy of the intra- and inter-sublattice exchange constants, Jaa > − Jac>\n|Jcc| [11], the 4c sublattice exhibits the higher moment at  T = 0 K, but its magnitude falls\nfaster with temperature than that of 4 a so that magnetic compensation occurs at a temperature\nTcomp where the two sublattice magnetizations are equal but opposite in sign [9,12]. We found\nthat SP-AOS is only possible below Tcomp, when at equilibrium the absolute value of the z-\nprojection of the angular momentum of 4 c manganese exceeds that of 4 a manganese [7,8].\nThe  MRG  sample  studied  here,  Mn 2Ru0.9Ga,  was  grown  by  DC  magnetron  co-\nsputtering from Mn 2Ga and Ru targets on MgO (001) single-crystal substrates heated to\n425°C using an ultra-high vacuum DCA multi-chamber deposition and characterisation tool\n(Trifolium Dubium, National Access Facility). The film was capped by 2 nm protective layer\nof naturally  oxidized  AlO x, deposited  at  room  temperature,  followed by 8 nm of SiO 2.\nBiaxial, substrate-induced strain induces a slight tetragonal distortion of the cubic Heusler\nstructure, resulting in perpendicular magnetocrystalline anisotropy of the film  [13] and a\nroom-temperature coercivity of 450 mT. The compensation temperature was found to be 469\nK from a thermal scan of the remnant magnetization, measured by SQUID magnetometry on\nanother sample prepared in identical conditions. Further details on the structural, magnetic,\nmagneto-optic and magneto-transport properties of MRG can be found elsewhere [9,12,13].\n200 fs laser pulses (λ = 800 nm) were sourced from a mode-locked Ti:sapphire-based\nlaser system. The system was operated in single-pulse mode for ex situ imaging, whereas for\nstroboscopic time-resolved magnetisation dynamics, the pulse repetition rate was 1 kHz. A\nportion of the beam was used for second harmonic generation in a -barium borate crystal\ncreating the probe beam (λ = 400 nm). Its delay with respect to a pump beam, was adjusted\nby a mechanical translation stage. A pair of pulses with variable delay were generated from a\nsingle pulse using a Michelson interferometer on the pump beam path with one arm mounted\non a mechanical translation stage. MOKE imagery was recorded ex situ after exposure using\nan EVICO Kerr microscope with red light in zero applied magnetic field. For all stroboscopicmeasurements, an applied field of 950 mT was applied perpendicular to the sample surface\nusing an electromagnet.\nFigure 1(a) illustrates the ‘toggle’ nature of SP-AOS. After the first pump exposure,\nthe irradiated spot reverses its magnetisation, and subsequent pulses toggle the magnetisation\nback and forth. We also show MOKE micrographs for varying pump powers (Fig. 1(b)) from\nwhich the Gaussian pump beam diameter and the threshold for switching were determined\nusing the Liu method [14]. We find a threshold of 3.5 mJ cm-2 and a spot size of about 190\nμm.\nIn Figure 1(c) we plot the time evolution of the MOKE after a single pump of 8.9 mJ\ncm-2, well above threshold. The solid line is a bi-exponential fit to the data with characteristic\ntimes 100 fs and 1.9 ps. Since our probe pulse has duration ~200 fs, while the pump is\nslightly stretched to ~250 fs, due to additional optical elements in the beam path, our time\nresolution close to the pump is ~325 fs. The two characteristic times are in agreement with\nour understanding of the SP-AOS process in MRG: Immediately after the pump, the two\nsublattices demagnetise while conserving net angular momentum such that d Sz4a/dt = − dSz4c/\ndt[7,8]. This step is governed by the inter-sublattice exchange, and it leads to a state where\nthe average z-projections of the two sublattice moments are aligned parallel because | Sz4c| > |\nSz4a|. This is referred to as the transient ‘ferromagnetic-like’ state [1], and it is a necessary but\nnot  sufficient  condition  for  switching  [15].  The  associated  time  scale  is  ~150  fs  for\nGd(FeCo)3 [1] and ~50 fs for MRG on account of the ~3 times stronger intersublattice\nexchange constant in the manganese alloy [11]. We infer that for times  t ≤ 325 fs, the 4a\nsublattice has switched its orientation while 4 c has not. At longer times, t > 325 fs, a second\nprocess becomes dominant. Angular momentum is no longer conserved, which allows the 4 c\nsublattice to switch at t ~ 1 ps and a quasi-static state is reached at t ~ 10 ps, consistent with\nthe spin-lattice relaxation time in MRG [8]. On a longer timescale of ~ 300 ps, the lattice\ncools down to near-ambient temperature by heat flow into the substrate.\nFigure 2(a) illustrates the switching with two pump pulses. A first pulse at  t = 0\nreverses the magnetisation; a second pulse at  t12  = 110 ps toggles it back. To confirm that\nmagnetic switching actually occurred, we first recorded a MOKE field loop before any\nexcitation (Fig. 2(b)), then one at t = 15 ps after a first pump pulse (Fig. 2(c)), and another at\nt = 285 ps, after both (Fig. 2(d)). The sign reversal of the loops confirms the magnetic\nswitching.\nWe then determine the minimum value of  t12 that allows the second pulse to toggle the\nmagnetisation. Figure 3(a) shows MOKE micrographs after the sample has been irradiatedwith two pulses of 4.1 mJ cm-², separated by  t12 = 9, 11, 11.7 or 12 ps. The first pulse\nswitches the area where the intensity of the Gaussian beam exceeds the threshold at room\ntemperature, and also increases the lattice temperature by approximately 65 K in about 2 ps\n[7]. This increased temperature decays slowly by heat flow to the substrate. As the threshold\nfluence decreases with increasing temperature (decreasing net magnetisation), the second\npulse toggles an area that is bigger than the first. This is clearly visible in Fig. 3(a) for 12 ps\npump separation: the central bright spot was switched once by the first pump, then toggled\nback again by the second, while the dark ring surrounding it is unchanged magnetically by\nthe  first  and  switched  by  the  second.  The  threshold  fluence  at  this  transient  higher\ntemperature (365 K) is only 2.9 mJ cm-², determined from the ratio of the toggled areas. \nRe-switching does not occur at a pump separation of 9 ps, whereas at t12 = 12 ps it is\ncomplete. For pump separations of 11.0 and 11.7 ps we find a third central region where re-\nswitching was not achieved because the higher peak pump intensity requires a longer time to\nreach equilibrium, even though the relevant time constants are the same. This is illustrated in\nFig. 3(a) where we increase the fluence of the first pump to 8.2 mJ cm-² and the second to 6.1\nmJ cm-². For these fluences, the areas switched by the first pump and re-switched by the\nsecond are nearly equal, and the central non-reswitching area remains visible up to a pump\nseparation of 70 ps. The results are summarized in Fig. 3(c) where we show the re-switched\nfraction as a function of pump separation  t12 for fluences of 4.1, 4.8 and 8.2 mJ cm-². We\nhighlight two points in the data. First, the lattice temperature does not need to exceed Tcomp to\nensure switching, as is observed in Gd(FeCo) 3 − excessive heating actually prevents re-\nswitching. Second, the fundamental limit on repetition rate is not uniquely determined by the\nheat transfer to the substrate. The relevant time is the spin-lattice relaxation time, the time\nneeded for magnetic damping.\nBased on the original studies of amorphous Gd(FeCo) 3  [1,2], SP-AOS was believed\nto depend on two conditions. First, it was thought that the demagnetisation times of the two\nsublattices needed to be substantially different, so that the  z-projections of their moments\ncould  cross  zero  at  different  times.  Second,  it  was  thought  that  high  spin  polarisation\ninhibited efficient demagnetisation. These expectations were overturned by our observation\nof switching  in MRG. There, two sublattices  composed of the  same element  would be\nexpected to demagnetize at similar rates. Furthermore, although overwhelmingly one of the\nsublattices contributes the majority of the states at the Fermi level, the material nevertheless\nexhibits SP-AOS. We now discuss the situation in light of our new findings.The on-atom Coulomb interaction integrals for 3 d5 manganese (Slater F2 and F4) are\n0.6 and 0.4 Ry (8.2 and 5.4 eV) and the first thermally excited configuration is (5/7 – 25/49)\nF2 + (5/7 – 190/441) F4 higher in energy, corresponding to an energy of 3.2 eV [16,17]. For\nMRG we infer that the atomic moment and the exchange integrals remain, to a very good\napproximation, time independent. The corresponding energies are 3.5, 0.35, 0.14 and 0.07 eV\nfor Gd (4f7), Tb (4f9), Co (3d5) and Fe (3d6) respectively, suggesting that this will not be the\ncase for Tb, Co and Fe [18]. Optically-induced transitions to excited states do not change S\ndue to the magneto-optical selection rules [16]. We must therefore discuss our findings in the\nlanguage of spin waves and precession [19], noting however that the usual models for spin\nwaves assume that the x- and y-projections of the atomic moments are small (S z >> Sx, Sy), an\nassumption that is clearly invalid for SP-AOS.\nFerrimagnets exhibit at least two orthogonal spin wave modes if axial symmetry is\nunbroken. In one mode, the two sublattices precess together without changing the angle\nbetween them; in the other, they precess in antiphase. The two are frequently referred to as\nthe ‘acoustic’ or ‘ferromagnetic’ and ‘optical’ or ‘antiferromagnetic’ modes, respectively. In\namorphous Gd(FeCo) 3, axial symmetry is broken by structural inhomogeneity [19] (Gd tends\nto cluster) whereas in MRG non-collinearity of the ferrimagnetic ground state [20], four-fold\nsublattice-specific  magnetocrystalline  anisotropy  of  opposite  signs [21],  and  preferential\nabsorption by light of one sublattice play the same role. The intense electronic excitation\nprovided  by  the  pump  pulse  excites  a  multitude  of  magnons.  In  the  absence  of  axial\nsymmetry, the optical mode can be efficiently excited  [22], leading to the first-quadrant\n‘ferromagnetic’ aligned state discussed earlier. The relevant times are those associated with\nexchange energies via the uncertainty principle ~ 100 fs, which are comparable to the 200 fs\nduration of the pulse in our experiments  [23]. We note that this process is fast because it\nconserves angular momentum. It only depends on the magnetic system absorbing the energy\ndeposited by the pump. This is often called exchange scattering [24]. Thermodynamically,\nthe maximum energy that can be absorbed by the magnetic system while conserving angular\nmomentum leads to S4cz min = (naa + nac)/(naa + ncc + 2nac) (S4az0 + S4cz0), where n (nij > 0) are the\nWeiss  molecular  field  constants.  Sz4c remains  positive,  while  Sz4a has  switched  and  the\nassociated time is that needed for the first stage of switching.\nFollowing this, the magnetic system loses energy by coupling to the lattice and the 4 c\nsublattice reverses its magnetic polarity while the 4a, that has already changed polarity, is\nincreasing. When the 4 c crosses zero, the inter-sublattice exchange will align it antiparallel to\n4a. This process does not conserve net angular momentum and probably requires emission ofoptical phonons. The experimentally determined timescale is 1.9 ps which represents the time\nneeded for the second stage, as both sublattices are now antiparallel to their initial directions.\nWe believe this second stage is driven by continued demagnetisation of the 4 c sublattice. It is\ntelling that the threshold for switching decreases when the temperature increases towards\nTcomp, as the residual z-projection at t = 325 fs is reduced. We speculate that very close to Tcomp\nthe  threshold  fluence  for  SP-AOS  could  be  very  substantially  reduced,  albeit  only  for\nextremely short pump pulses [8]. \nLastly,  the  optical  magnons  scatter  into  the  long  wavelength  acoustic  modes  of\nfrequency ~ 100 GHz [23] and are damped on the spin-lattice relaxation timescale (~10 ps)\nwhen Sz4c regains a higher magnitude than S z4a, unless the lattice has already heated above\nTcomp. This is the time that finally marks the completion of the magnetic reversal, and it then\nbecomes possible to repeat the process and toggle the magnetisation with a second pulse. The\nhalf metallicity of MRG is beneficial, as it will increase the damping of the 4 c sublattice due\nto Fermi surface breathing [25] and allow it to relax faster than 4 a, decreasing the time that\nmust elapse between subsequent toggle events. The whole three-stage process is illustrated on\nFig. 4 by the track from initial to final states, where the relevant times after the pump are\nmarked on a logarithmic scale on the red trajectory. Plausible spin configurations at different\ntimes are illustrated in the inset. The four-quadrant representation of a two-sublattice magnet\nin Fig. 4 has been used by Mentink et al. [26], originally inspired by Bar’yakhtar [24].\nFinally, we comment on the energy requirements for a potential application. We have\nshown that threshold fluences, as low as 2.9 mJ cm-² or 0.3 fJ, suffice to switch a (10 nm)³\nelement, assuming that 35% of the light is absorbed by a 30 nm thick MRG thin film. This is\nan order of magnitude more than current records for transparent magnetic insulators  [3].\nHowever, the metallic nature of MRG permits integration with other spin electronic circuitry,\nthereby creating an opportunity to bring the speed of optics to magnetism and electronics.\nPossible applications include beam steering using diffraction elements, such as Fresnel zone\nplates, where MRG (or some future material) forms the ‘dark’ elements. The focal point of\nthe zone plate could thus be changed every 10 ps.\nIn conclusion, the relevant timescales for SP-AOS are the exchange time and the spin-\nlattice relaxation time, which we evaluate from our two-pulse experiments. We infer that SP-\nAOS requires axial symmetry breaking, either by structural inhomogeneity, or by competition\nbetween magnetocrystalline anisotropy and exchange. The hierarchy of exchange constants in\na  ferrimagnet  is  critical  to  promote  low-energy  SP-AOS.  Repeated  toggle  switching  is\nenvisaged at rates as high as 100 GHz, provided the lattice temperature remains below Tcomp.Reducing the spin-lattice relaxation time could increase this frequency. To our knowledge\nthis is the fastest switch from one stable magnetic state to another ever observed. \nAcknowledgements. \nThis work was supported by Science Foundation Ireland under contract 16/IA/4534 ZEMS\nand the European Union Horizon 2020 research and innovation grant agreement 737038\n‘TRANSPIRE’.  Dr. C. Banerjee is grateful to the Irish Research Council for her postdoctoral\nfellowship. The work was carried out in the CRANN Photonics Laboratory, where we are\ngrateful  to  Dr.  Jing  Jing  Wang  for  technical  support.  The  Trifolium  Dubium\ndeposition/characterisation platform was funded by Science Foundation Ireland under grant\n15/RI/3218.Figure 1: (a) All-optical toggle switching of magnetization in Mn 2Ru0.9Ga at a fluence of 8.2\nmJ cm-2. (b) Domain size as a function power. (c) Magnetization dynamics for a pump of\nfluence of 8.9 mJ cm-2. The solid line is a guide to the eye based on three exponentials with\ncharacteristic times 100 fs, 1.9 ps and 320 ps.Figure 2: (a) Transient Kerr signal in presence of two pump pulses separated by 110 ps. (b) \nHysteresis loop measured by the probe beam in absence of any pump excitation. (c) and (d) \nare field loops at delays t12 = 15 ps and 285 ps.Figure 3: (a) Kerr micrographs of the irradiated region taken after dual pump excitation for \ndifferent times t12 separating the pulses. Both pump fluences are 4.1 mJ cm-2. For t12 = 12 ps, \nthe bright center has been switched by the first pulse and toggled by the second, while the \nsurrounding dark ring has been switched by the second pulse as described in the text.  (b) \nSame as (a) but the first and second pump fluences are 8.2 mJ cm-2 and 6 mJ cm-2. 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Rode, ArXiv 1903.04432 (2019).\n[22]A. Kamra, U. Agrawal, and W. Belzig, Phys. Rev. B 96, 020411(R) (2017).\n[23]G. Bonfiglio, K. Rode, G. Y. P. Atcheson, P. Stamenov, J. M. D. Coey, A. V. Kimel, Th. Rasing, and \nA. Kirilyuk, ArXiv:2003.01420 (2020).[24]V. G. Bar’yakhtar, J. Exp. Theor. Phys. 60, 863 (1984).\n[25]V. Kamberský, Czech. J. Phys. 26, 1366 (1976).\n[26]J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov, A. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. \nKatsnelson, and Th. Rasing, Phys. Rev. Lett. 108, 057202 (2012)."    },    {        "title": "1907.07997v2.Electric_bias_controlled_switching_of_magnetization_of_ferrimagnetically_coupled_Mn_delta_layers_in_a_GaAs_AlGaAs_quantum_well.pdf",        "content": "arXiv:1907.07997v2  [cond-mat.str-el]  12 Feb 2020Highlights\nElectric bias-controlled switching of magnetization of fe rrimagnet-\nically coupled Mn delta-layers in a GaAs-AlGaAs quantum wel l\nN. V. Agrinskaya, A. M. Kalashnikova, V. I. Kozub\n•We suggest a design of an artificial ferrimagnet consisting of two Mn -\ndoped delta-layers with different concentrations placed in a GaAs-\nAlGaAs quantum well and coupled via extra holes in the well.\n•We analyze a scenario of the magnetization switching in the artificial\nferrimagnet realized via heating the holes by a picosecond electric bia s\npulse which facilitates exchange scattering of the holes on Mn ions.Electric bias-controlled switching of magnetization of\nferrimagnetically coupled Mn delta-layers in a\nGaAs-AlGaAs quantum well\nN. V. Agrinskaya, A. M. Kalashnikova, V. I. Kozub\nIoffe Institute, 194021 St. Petersburg, Russia\nAbstract\nWe suggest a model of synthetic ferrimagnetic semiconductor str ucture based\non GaAs-AlGaAs quantum well doped by two Mn delta-layers. The cou pling\nbetween the delta-layers is mediated by extra holes, and can be swit ched\nbetween ferro- and antiferromagnetic one by gating the structu re. A proper\nchoice of Mn concentrations in the delta-layers and of local degree of dis-\norder enables fabrication of a ferrimagnetic structure supportin g ultrafast\nswitching of magnetization by short pulses of electric bias without an ex-\nternal magnetic field. The switching mechanism in the structure relie s on\nkinetic spin exchange between the two delta-layers which is mediated by ex-\nchange scattering of electric-pulse heated holes by magnetic ions w ithin the\nlayers. Owing to specific interplay between characteristics of the e xchange\nscattering, spin decay times, and the heat withdraw in the suggest ed syn-\nthetic ferrimagnetic semiconductor, the necessary parameters of electric-bias\npulse are within the technologically accessible range, and do not cont radict\ntypical thermal kinetics of semiconductor structures.\nKeywords: synthetic ferrimagnet, GaAs quantum well, ultrafast heating,\nultrafast switching of magnetization\nPACS: 75.50.Gg, 75.50.Pp, 75.50.Ss, 71.70.Gm, 75.78.-n\n1. Introduction\nRecently, a great attention [1] was attracted by experiments dem onstrat-\ning extremely fast ( ∼10−12s) magnetization reversal triggered by a single\nfemtosecond laser pulse in a ferrimagnetic metallic rare-earth (RE) - tran-\nsition metal (TM) alloy GdFeCo without external magnetic field [2, 3, 4 ].\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials February 13, 2020Follow-up studies have suggested that laser-induced magnetizatio n switch-\ning can be also realized in other RE-TM alloys, as well as in a variety\nof the engineered ferrimagnetic structures, including exchange c oupled RE-\nTM multilayers and heterostructures comprised by two transition m etal lay-\ners antiferromagnetically coupled through a nonmagnetic metallic int erlayer\n[5, 6, 8]. Most importantly, experimental studies have demonstrat ed that the\nall-optical reversal of magnetization is not precessional and relies on subpi-\ncosecond quenching of the magnetizations [7] of RE and TM sublattic es [3].\nThe time-resolved X-ray [9, 10] and optical [11] experiments have u nveiled\nunconventional distinct dynamics of these sublattices leading to em ergence\nof a transient ferromagnetic-like state. Such nonequilibrium dynam ics is be-\nlieved to enable deterministic magnetization reversal, without any ne ed for\nany other stimulus defining the magnetization direction [4, 12].\nNaturally, microscopical mechanism underlying unconventional res ponse\nof magnetization of a ferrimagnetic metallic system to a femtosecon d laser\npulse is the subject of intense discussions nowadays. Several the oretical and\nmodelling approaches were employed to account for main features o f the all-\noptical switching. Atomistic and multiscale calculations based on a Lan dau-\nLifshitz-Bloch equation [13] for an ensemble of the exchange-coup led spins\nwere developed in Refs. [4, 14, 15, 16, 17] for describing the all-op tical rever-\nsal in single phase alloys, as well as in exchange-coupled multilayers [18 ], and\nto examine a feasibility of switching in ordered RE-TM alloys [17]. In [19] a\ncomprehensive phenomenological model based on the Onsager’s re lations [20]\nwas developed introducing an exchange-dominated regime of laser- induced\ndynamics in a ferrimagnet, which allows the reversal of magnetizatio n solely\ndue to the ultrafast heating. This work highlighted importance of th e an-\ngular momentum exchange between the sublattices of the ferrimag net. In\norder to gain insight into microscopical nature of the all-optical swit ching,\ndissipationless energy and angular momentum exchange between TM and\nRE sublattices mediated by 5 d-4fcoupling in RE ions has been explored in\n[21], and the exchange electron-electron scattering as the driving mechanism\nof the magnetization reversal was discussed in [22]. In [23] a genera l micro-\nscopic approach based on the rate equations was suggested for a ddressing\nthe problem of the angular momentum exchange between two noneq uivalent\nmagnetic sublattices in a metal. In the latter work, the exchange sc attering\nwas found to be the driving mechanism of the switching. Similarly, switc hing\nenabled by the exchange scattering was also considered in [24], with h owever,\nprincipally different model of a RE-TM system. Importantly, since a la ser\n2pulse serves as an ultrafast heating pulse only [4, 12], in [23] it was sug gested\nto realize electric-bias induced switching in a metallic ferrimagnetic str uc-\nture. Recently, the first report appeared on the experimental o bservation\nof the switching of the magnetization of the GdFeCo wire by a picosec ond\ncurrent pulse [25].\nAlong with unveiling a nature of unconventional dynamics of the spin s ys-\ntem, the goal of these theoretical studies is to provide recipes fo r novel sys-\ntems and alternative stimuli which enable ultrafast switching of magn etiza-\ntion under technologically accessible conditions [18, 17, 26]. Here we p ropose\na synthetic semiconducting ferrimagnet where the exchange scat tering-based\nmagnetization switching can be realized upon application of an electric -bias\npulse of a picosecond duration. The ferrimagnet is comprised by two fer-\nromagnetic Mn-doped delta-layers in a GaAs-AlGaAs quantum well (Q W).\nAntiferromagnetic coupling between the delta-layers is supported by the ex-\ntra holes in the QW, and can be controlled by proper gating the struc ture.\nFerrimagnetism of the whole structure is realized due to different Mn con-\ncentration in the ferromagnetic delta-layers. Using the general m icroscopic\napproach developed earlier in [23], we show that ferrimagnetic prope rties\nof this structure support switching of the net magnetization via tr ansient\nferromagnetic-like state in zero applied magnetic field. Occurrence of the\ntransient ferromagnetic-like state and consequent switching of t he net mag-\nnetization of the structure is enabled by different magnetizations a nd Curie\ntemperatures of the delta-layers, while the rates of exchange sc attering of\nfree carriers (holes) at the magnetic ions (Mn) are similar in both laye rs.\nThe latter is in contrast to the all-optical or electric-bias switching in RE-\nTM alloys, where different exchange scattering rates for the TM an d RE\nmetals were playing a decisive role. Further, we argue that semicond ucting\nproperties make the switching more energetically profitable than in m etals,\nand also ease constrains regarding short duration of the electric b ias pulse\nrequired for switching.\n2. A syntheticferrimagnet based ontwoMn delta-layers ina G aAs-\nAlGaAs QW\nAs a candidate structure for a synthetic ferrimagnet we consider a QW\nGaAs-GaAlAs containing two delta-layers of Mn with concentrations N1and\nN2. The delta-layers are separated by a distance D, which is comparable\nto the well width L∼15 nm (Fig.1). Recently a system comprised by a Mn\n3delta-layer in GaAs-based QW attracted attention [27, 28] as an a lternative\nto a bulk diluted ferromagnetic semiconductor GaMnAs [29]. These st udies\nimply that the most promising realization of such a ferromagnet is the one\nwith a delta-layer situated in the barrier in the vicinity of the QW. The\nferromagnetic ordering is then realized due to indirect exchange su pported\nby holes within the well. It was suggested that in such a way the holes w ithin\nthe well do not experience a disorder imposed by Mn delta-layer [27, 28].\nThis supposedly allows obtaining higher Curie temperatures TCdespite of\nthe tunneling exponent which is necessary to pay for the holes to co ntact Mn\nions.\nRecently an observation of ferromagnetism in delta-doped GaAs-A lGaAs\nQWs with Curie temperatures around 200 K at unusually small Mn dopin g\nlevels of∼5·1012cm−2was reported in [30, 31] The indirect exchange in the\ndelta-layer in this case was supported by the holes supplied by Mn ato ms\nthemselves. The lower concentration of Mn dopants lead to a decre ase of\ndisorder potential which favors higher TCdue to, in particular, a suppres-\nsion of concentration of Mn interstitials which are known to be compe nsating\ndefects. This fact was proved experimentally by demonstrating th at an in-\ncrease of Mn concentration leads to a suppression of ferromagne tism in the\nin delta-doped GaAs-AlGaAs QWs [30]. This result advocates attempt s to\nfabricate ferromagnetic structures by doping a region within a QW, which\nmagnetic properties are controlled by holes within the well. Then, one can\nalso fabricate twodelta-layers of Mn in the same QW. Importantly, in this\ncase the layers will be coupled to the same holes subsystem localized w ithin\nthe well.\nHere we consider a particular case when N2> N1, withN1being large\nenough to support ferromagnetism in the corresponding delta-lay er. The\nseparation distance Dbetween the delta-layers is much larger than the lo-\ncalization length for a Mn hole. Then the ferromagnetism of each delt a-layer\nis supported by own holes of Mn acceptors [30]. In this case the laye r with\nMn concentration N2is characterized by a lower value of coupling between\nthe ferromagnetic ions due to an increase of degree of disorder. T he rough\nestimate of the disorder effect on TC2can be given by TC2∝exp(−N−1/2\n2/l).\nHerelis the mean free path for the holes in the vicinity of the second layer,\nand it is somewhat less than the distance between the Mn atoms ∼N−1/2\n2.\nNote that the condition l < N−1/2\n2 can be also reached by any additional\nintended contamination of the corresponding interface region. In any case\n4one can control the value of Tc2for a given value of N2> N1. Thus we\nconsider the system of two ferromagnetic layers 1 and 2 with Tc1> Tc2and\nthe saturation magnetizations Ms1< Ms2(Fig. 1).\nUnder the assumptions considered above, the delta-layers are no t ex-\nchange coupled. An exchange coupling of a RudermanKittelKasuyaY osida\n(RKKY) type between the layers can be realized by additional carrie rs sup-\nplied by doping of the barriers with shallow Be acceptors. Doping both\nbarriers with nearly the same surface concentrations N3/2 allows cancelling\nthe electric fields produced by the barrier acceptor layers within th e well.\nThus, these fields do not affect the bare well potential. The RKKY ex change\ncoupling between the delta-layers could be either ferromagnetic or antifer-\nromagnetic, depending on the band holes energies. We note that ad ditional\nholes within the well would improve ferromagnetic ordering in the delta -\nlayers via RKKY exchange coupling. We further note that a slightly hig her\ndoping level in the barrier closest to the second layer can impose add itional\ndisorder potential on this layer resulting from the screening the ad ditional\nbarrier charges by the holes within the second layer.\nHere we have to address the role of the second Hubbard band for t he Mn\nacceptors, since the Hubbard energy for them is at least twice lowe r than\nionization energy [32]. Thus one expects that the barrier dopants w ill ini-\ntially lead to partial filling of the upper Hubbard bands of the Mn accep tors.\nHowever, the complete filling of the upper Hubbard band correspon ds to\nadditional hole and thus additional positive charge per any site within the\ndelta-layer. The concentration of Mn-sites allowing ferromagnetic ordering\nstarts around 1013cm−2giving the intersite distances of order of 3 nm [31],\nand the complete filling of the upper Hubbard bands leads to large incr ease\nof the on-site hole energies resulting from coupling to the holes on th e neigh-\nboring sites (at distances at least of the order of distance to the n egatively\ncharged layers within the barriers). Such holes will be inevitably push ed to\nthe band states which are delocalized within the well. Indeed, any cha rged\nneighbor at the distance 3 nm gives additional energy of the order o f 30 meV\n∼300 K while the ionization energy for Mn is 100 meV. Correspondingly,\nonly relatively small part of the upper Hubbard band can be situated lower\nthan the bottom of the band resulting from lateral quantization. A t par-\ntial occupations of the upper Hubbard bands the states of these bands are\nexpected to be delocalized along the corresponding layer giving addit ional\nsupport to the ferromagnetic ordering within the layer. For larger concentra-\ntion of the additional holes within the well, N3−(N1+N2)> βN1(β << 1),\n5the presence of the upper Hubbard band can be neglected due to a presence\nof the band holes. As a result, the whole system can be considered a s the\nferromagnetically ordered delta-layers with the delocalized band ho les.\nConcentration of these holes of the order of 1019cm−3gives the total\nsurface concentration of the holes of the order of 1013cm−2[31], that is of\nthe order of the surface concentration within the ferromagnetic layers. In this\ncase the screening length is of the order of ∼2 nm for the holes with energy\n∼30 meV ( ∼300K). Such distance is smaller or, at least, comparable to the\nintersite distances within the layers. Thus, at corresponding conc entrations\nand the free holes energies we can consider the potentials of the do ubly\noccupied sites to be screened. This gives and additional argument t o neglect\na filling of the upper Hubbard band in our model.\nWe assume N1∼5·1012cm−2,N2∼1.5·1013cm−2,N3∼1013cm−2.\nFor the width of the well L∼1.5·10−6cm and the hole mass 10−27g one\nestimates the energy of the first quantization level of /planckover2pi12π2/2mL2∼2 meV,\nthe Fermi wavelength of the order of 10−6cm and concentration of the order\nof 1012cm−2. For concentration of the order of 1013cm−2distributed over\nthe well one has the interhole spacing ∼101/2·10−7cm that is around λF∼\n3 nm. It corresponds to occupation of the 3 dlevel of lateral quantization\nwhich gives oscillating wave function providing a possibility to have RKKY\nexchange coupling of any sign between the Mn delta-layers. Here we also\nnote that the Fermi energy of the band holes at the considered co ncentrations\nbecomes to be of the order or larger than the temperature of the system (200\n- 300 K). Thus, at the temperatures close to the critical ones, wh ich are also\naround 200 - 300 K, the situation is close to degeneration which facilit ates\nthe RKKY interaction.\nWe note that several attempts to realize and study exchange inte ractions\nbetween two Mn delta-layers imbedded into GaAs matrix were report ed (see\ne.g. [33, 34]). However the main attention in these studies has been p aid to\nthe role of the carriers supplied by Mn itself, and the interlayer dista nce was\nassumed to be small in comparison to localization length of the Mn holes\nin the GaAs host. This is in strong contrast to the structure consid ered in\nthe present work, where the additional holes are supplied by dopan ts within\nthe AlGaAs barriers. As a result the distance between Mn layers can be at\nleast of the order of several localization lengths, and the waveleng th of the\ndoped holes is comparable to the interlayer distance, and the situat ion is\nclearly corresponds to the RKKY interactions. Importantly, this in terlayer\nRKKY exchange coupling can be controlled by the gate situated in any of\n6the barriers. Indeed, the gate potential allows to control the co ncentration\nof the band holes within the well, which, in turn, affects the characte r of\ninterlayer coupling.\n3. Switching the magnetization of the delta-layers by elect ric-bias\npulse\nNow we consider an evolution of the magnetizations of the two delta-\nlayers with antiferromagnetic RKKY exchange between them in resp onse to\na short electric bias pulse of a duration of a few picoseconds. The ap plication\nof the electric bias pulse rapidly increases the temperature of the h oles in the\nQW. To treat the following evolution of the spin system we use the mod el\ndeveloped in [23]. In brief, it describes evolution of the occupation nu mbers\nof twodifferent ferromagnetic sublattices coupled antiferromagnetically via\ndelocalized carriers. The spin exchange between the localized ferro magnetic\nsubsystems is mediated by delocalized carriers which temperature is rapidly\nincreased above critical temperatures of both magnetic sublattic es. Either\nfemtosecond laser excitation or applying short electric bias pulse ca n serve\nfor the carriers heating. Spin exchange results in the switching of t he net\nmagnetization of the system without any additional stimuli, such as e xternal\nmagnetic field, circular polarization of light, or spin polarization of a cu rrent.\nThe switching of the net magnetization in this model relies on a delicate bal-\nance between the exchange scattering, spin relaxation, and coolin g times.\nImportantly, this model is not restricted to the case of RE-TM alloy s or het-\nerostructures, and is also applicable for the case of the structur es composed\nby two different transition metals.\nThe important difference between the system analyzed here and th e RE-\nTM metallic systems considered in [23], is that here two ferromagnet ic sub-\nsystems, i.e. two delta-layers, are comprised by the same Mn ions. T hus,\nthe exchange scattering of the holes on Mn ions is nearly equal for t he both\nsubsystems. We can estimate the characteristic times of this scat tering in the\nfollowing way. A 3D cross-section for Coulomb scattering of the mob ile holes\nby an unscreened ion is of ( e2/κε)2∼10−12cm2. 3D geometry is consid-\nered since the holes are delocalized in the direction normal to the well plane.\nHowever, the effect of nonlinear screening by the holes, in particula r by the\nones supplied by Mn atoms itself, diminish this estimate down to 10−13cm2\nwhich is controlled by the distance between Mn atoms. The cross-se ction for\nthe exchange scattering is expected to be less by a factor γ2, whereγcharac-\n7terizes the relative strength of the exchange potential, and can b e estimated\nto be of ∼0.3. Now we take into account that the inverse mean free path\nwith respect to the exchange scattering is given by a product of th is cross-\nsection and 3D concentration of Mn ions in the well ( ∼1019cm−3). This\ngives the value of 10−5cm for the mean free path. Correspondingly, for the\nholes velocity ∼107cm/s [38] the exchange scattering time can be estimated\nto be ofτex∼1 ps.\nAnother important difference between the spin dynamics in the cons id-\nered artificial ferrimagnet and in the RE-TM metallic alloy is that the sp in\nrelaxation times in the semiconductor GaAs appear to be considerab ly longer\nthan the exchange scattering times. In [35] the spin relaxation tim e for the\nholes in GaAs structures was found to be of ∼20 ps. This value is signifi-\ncantly larger than the spin relaxation time ∼0.1−1 ps in metals. Therefore,\nat the first stage of the electric-bias driven evolution of the spin ba lance in\nthe system, the effect of spin relaxation can be neglected. We note that this\nassumptions corresponds to the model suggested in [24], where th e two fer-\nromagnetic subsystems with different magnetizations and similar exc hange\nscattering times were considered. As we discussed above, such as sumptions\nare justified in the case of Mn layers within the semiconductor QW, wh ile\ntheir applicability to the metallic RE-TM alloy is arguable.\nRight after the application of the electric bias pulse, the temperatu re of\nholes in the QW is increased to the value T∗\nh, and the evolution of the spin\nsystem at this stage is reduced to redistribution of the total angu lar momen-\ntums(N2−N1), wheresis the spin of a hole, between the two ferromagnetic\nsubsystems. When the holes temperature reached T∗\nh, the resulting momenta\nsn1andsn2of the layers are then given by\nn1=(N2−N1)\nN2+N1N1, n 2=(N2−N1)\nN2+N1N2. (1)\nAs it is seen from Eqs. (1), the sign of the magnetization for the bot h sub-\nsystems now coincide since we consider the system with N2> N1. That is\nat this stage the mutual spin orientations in the two delta-layers co rrespond\nto a ferromagnetic-like state.[9]\nEvolution of delta-layers magnetizations from their equilibrium values to\nthe ones given by Eqs. (1) occurs on the time scale given by the estim ated\nexchange scattering time τex∼1 ps. We note that, in contrast to the evolution\nof the TM and RE magnetizations in the metallic RE-Tm alloys [23], the\nparticular time when the Ms1vanishes appear not to play crucial role, since\n8the spin relaxation times are much slower that the exchange scatte ring times.\nOnly if the spin relaxation is fast or if the temperature of the system remains\nelevated (i.e. no cooling is present) then the total angular moment, and,\ncorrespondingly, the total magnetization would finally vanish.\nTo describe the evolution of the system from the ferromagnetic-lik e state\nwe take into account the cooling down process. First, we consider a case\nwhen the dominant cooling mechanism is related to optical phonons wit h\na frequency ω0. This is relevant when the temperature of the holes is high\nenough. In this case the cooling time necessary to lower the temper ature\nof the holes from T∗\nh, down to some value Th, can be roughly estimated as\nt=τh−phkB(T∗\nh−Th)/(/planckover2pi1ω0), where the characteristic hole-phonon relaxation\ntime isτh−ph∼10−12s [39, 40].\nAt temperatures lower that /planckover2pi1ω0the further cooling is assisted by acous-\ntical phonons. To estimate the corresponding relaxation time we ta ke into\naccount that the holes are coupled to phonons with momentum equa l or less\nthan the holes momentum due to a momentum conservation. The hole s ve-\nlocities are of ∼107cm/s for temperature range in a vicinity of the Curie\ntemperatures of the layers, i.e. in the range of 200 - 300 K [38]. The wavevec-\ntors of corresponding phonons are less than 107cm−1. Thus, one concludes\nthat the phonon energies /planckover2pi1ωare about 10 times lower than the holes energies\nεh. In this case the energy relaxation time is larger by a factor ∼(εh//planckover2pi1ω)2\nthan the typical hole-phonon relaxation time τh−ph∼10−12s, provided that\nthe temperature This not much below the room temperature. Thus, we\nobtain characteristic time t=τh−ph(kBTh//planckover2pi1ω)2for the further energy relax-\nation.\nIn our estimates we assume that the heat withdrawal from the qua ntum\nwell is efficient enough, which can be ensured by the purity of the AlGa As\nbarriers and small mismatch of the elastic constants between GaAs and Al-\nGaAs. Thus, the phonon transport outside of the well can be cons idered\nas a ballistic one, and the characteristic time of the phonon escape c an be\nestimated as D/w , whereDis of the order of the thickness of the QW, and\nwis the sound velocity. Than the phonon escape time is of the order of\n10−12s which is comparable to the hole-phonon relaxation time τh−ph. The\nheat capacity of phonon subsystem is much larger than that of the holes\nsubsystem. For hole concentrations around 3 ·1012cm−2the corresponding\nfactor is around 104. Thus at time scale less than 10−8s the problem of heat\nwithdrawal can be neglected.\nSince the critical temperature for the first subsystem, Tc1is larger, then\n9that for the second subsystem ( Tc2), the temperature of the holes reaches the\nvalue ofTc1first upon cooling. Once the holes temperature is below Tc1the\nmagnetization of the first subsystem starts to restore accordin g to standard\nthermodynamical law as (see e.g. [36])\n∆Ms1∼Ms1/parenleftbiggTc1−Th\nTc1/parenrightbigg1/2\n, (2)\nwhere the This controlled by a process of cooling. Thus at a moment when\nN1/parenleftbiggTc1−Th\nTc1/parenrightbigg1/2\n∼(N2−N1)\nN2+N1N2 (3)\nthe magnetization of the layer 1 starts to dominate over the magne tization of\nthe layer 2 provided that it this moment the holes temperature is still larger\nthan the critical temperature Tc2.\nAt the final stage of the process, i.e. when the holes temperature is\nlowered down to the critical temperature Tc2of the layer 2, the magnetization\nof the latter starts to restore as well. The preferential direction for the layer\n2 magnetization is now set by the antiferromagnetic RKKY coupling to the\nmagnetization of the layer 1. Correspondingly, the net magnetizat ion of the\nsystem is switched at this stage.\n4. Conclusions\nWe suggested a model of a synthetic ferrimagnetic structure bas ed on\nthe Mn-doped GaAs-AlGaAs quantum well. It contains two Mn delta-la yers\nwithin the well with different Mn concentrations and different degree s of dis-\norder. Two symmetric layers of barrier acceptors are situated on different\nsides of the well in order to provide extra holes to the QW. These ext ra holes\nare essential since they participate in the RKKY coupling between th e delta-\nlayers. The sign of the RKKY coupling can be controlled by concentra tion of\nthese holes, by distance between Mn layers, and by gating the stru cture. As\na result, a structure of two antiferromagnetically coupled Mn delta -layers can\nbe fabricated with one layer possessing larger saturation magnetiz ation and\nlower critical temperature that the second layer. Using recently s uggested\ntheory for the switching of a ferrimagnet driven by the kinetic exch ange\nscattering, we show that the magnetization of such semiconducto r-based syn-\nthetic ferrimagnet can be switched by a short pulse of electric bias w ith no\n10additional external stimuli, e.g. external magnetic field, required t o set the\nmagnetization direction.\nWe note that the suggested switching scenario of the synthetic se miconductor-\nbased ferrimagne can be also realized by using short laser pulses inst ead of\nelectric bias ones, since the former are known to trigger ultrafast demag-\nnetization in GaMnAs [37]. It may be important from a point of view of\nintegration of Mn-doped GaAs-AlGaAs QW allowing toggle switching of\nmagnetization with the ultrashort semiconducor laser sources. Ga As and\nAlGaAs are among media for developing such lasers (for a review see, e.g.,\n[41] and more publications on this subject, e.g. [42]). Thus, one could en-\nvisage designing a device contained both a ultrafast laser source an d the\nmagnetic toggle switch ”on a single chip”. For the later to be realized, the\nCurie temperatures of the Mn-doped GaAs-AlGaAs QW should be abo ve\nroom temperature. Mn delta-doped GaAs possesses one of the hig hest tem-\nperatures among the III-V semiconductors reaching 250 K [27]. O n the other\nhand, among the bulk ferromagnetic III-V semiconductors, thos e doped with\nFe ions have the Curie temperature higher than (Ga,Mn)As [43]. Ther e-\nfore, it would be important to investigate if the III-V semiconducto r-based\nstructures with Fe delta-layers can be fabricated for the ultrafa st switching.\nPresently, however, there are no reports on successful increa se of the Curie\ntemperature by Fe delta-doping of a III-V semiconductor [44].\n5. 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L. Dalal, A. B. Dreeben, and A. Triano, J. Appl. Phys. 42, 2864\n(1971).\n[39] P. Yu, M. Cardona, Fundamentals of semiconductors (4th ed., Springer,\n2010), page 213.\n[40] R. Scholz, J. Appl. Phys 77, 3219 (1995).\n[41] B. W. Tilma, M. Mangold, Ch. A. Zaugg, S. M. Link, D. Waldburger,\nA. Klenner, A. S. Mayer, E. Gini, M. Golling, and U. Keller, Light: Sci.\nAppl.4, e310 (2015).\n[42] B. Mayer, A. Regler, S. Sterzl, T. Stettner, G. Koblmuller, M. K aniber,\nB. Lingnau, K. L¨ udge, and J. J. Finley, Nature Commun. 8, 15521\n(2017).\n[43] Nguyen Thanh Tu, Pham Nam Hai, Le Duc Anh, and M. Tanaka, Phy s.\nRev. B92, 144403 (2015).\n14[44] Kento Nishijima, Nguyen Thanh Tu, Masaaki Tanaka, and Pham N am\nHai, J. Cryst. Growth 511, 127 (2019).\n15Tc1Tc2(<TC1)M1M2\nFigure 1: (a) Schematic picture of the ferrimagneticstructure co mprisedby two Mn-doped\ndelta-layersplaced within the GaAs-AlGaAs QW. Barrier is doped with s hallow Be accep-\ntor. Also shown are the Mn (blue line) and Be levels (green line), and lat erally quantized\nhole levels (black lines). (b) Schematic diagram showing the spin and he at withdraw ex-\nchange channels considered in the model: blue lines show the exchang e scattering and the\nspin relaxation channels, the black lines show the heating of the QW ho les by the electric\nfieldE(t) pulse and its cooling down via the phonon system.\n16"    },    {        "title": "2311.07350v1.Magnetoresistive_detection_of_perpendicular_switching_in_a_magnetic_insulator.pdf",        "content": " \n1 \n Magnetoresistive detection of perpendicular switching in \na magnetic insulator  \nSilvia Damerio1,*, Achintya Sunil2, M. Mehraeen2, Steven S. -L. Zhang2 and Can O . Avci1,* \n1Institut de Ciència de Materials de Barcelona, Campus de la UAB, Bellaterra, 08193, Spain  \n2Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA  \n \n \nSpintronics offers promising routes for efficient memory, logic, and computing technologies. \nThe central challenge in spintronics is electrically manipulating and detecting magnetic states \nin devices. The electrical control of magnetization via spin -orbit torques is effective in both \nconducting and insulating magnetic layers . However, the electrical readout of magnetization in \nthe latter is inherently difficult, limiting its use in  practical applications. Here, we demonstrate \nmagnetoresistive detection of perpendicular magnetization reversal in an electrically insulating \nferrimagnet, terbium iron garnet (TbIG). To do so, w e use TbIG|Cu|TbCo , where TbCo  is a \nconducting ferr imagnet and  serves as the  reference layer, and Cu is a nonmagnetic spacer . \nCurrent injection through Cu|TbCo allows us to detect the magnetization reversal o f TbIG with \na simple resistance readout during an external magnetic field sweep. By examining the effect of \nmeasurement temperature, TbCo composition, and Cu thickness on the sign and amplitude of \nthe magnetoresistance, we conclude that the spin -dependent electron scattering at the TbIG|Cu \ninterface is the underlying cause. Technologically -feasible magnetoresistive detection of  \nperpendicular switching in a ferrimagnetic garnet is a breakthrough, as it opens  broad  avenues \nfor novel insulating spintr onic devi ces and concepts . \nFerrimagnetic garnets (FMGs) are ubiquitous in solid -state physics. They host a wide variety of useful \nproperties , such as ultralow damping, high magnon density, tunable magnetic anisotropy and \nsaturation magnetization, and highly ordered  single -crystal structures with  sharp surfaces and \ninterfaces.1 Furthermore, single -crystal epitaxial FMG s can be grown from a few micrometers  down  \n2 \n to a few nanometers of thickness via physical and chemical deposition techniques on suitable \nsubstrates and, yet, retain bulk -like properties. As a result, FMGs have served as an excellent material \ntestbed for a multitude of spintronic phenomena in t he past few years.2–8 More recently, there has \nbeen  a growing effort in exploiting FMGs as active components in magnonic integrated circuits9 and \nspintronic memory devices.10 Researc h into the latter has been particularly fruitful owing to the \ndevelopment of ultrathin FMGs with perpendicular magnetic anisotropy (PMA)11,12 and the highly \nefficient control of their magnetization vector by current -induced spin -orbit torques.8,10,13 –17  \nWhile the above advances are promising for the integration of FMGs in future computing \ntechnologies, the electrical detection of magnetization  in this material family is still a critical \nchallenge. The spin Seebeck, local and non -local spin Hall magnetoresistance, and thermal spin drag \neffects have been utilized for the electrical detection of magnetization in FMGs.3–5,18 However, all of \nthese phenomena rely on voltages induced by inverse spin Hall effect in an adj acent nonmagnetic \nmetal, such as Pt, hence fundamentally limiting the output signal amplitudes and materials that can \nbe used for the purpose. Electrical detection of the perpendicular magnetization  vector  in FMGs \npossessing PMA, crucial  for domain wall an d skyrmion -based racetrack devices, and for downscaling \nthe lateral size of memory cells, is particularly strenuous. Thus far, the anomalous Hall effect (AHE)  \ndriven by interfacial spin -dependent scattering4,19–21 is used as the sole electrical method for \nmagnetization vector detection in PMA FMGs,10,22 yielding minute output signals , nonetheless in a \ntechnologically -prohibitive four-contact Hall cross device geometry .  \nMagnetization vector detection in conventional spintronic devices is realized by tunnel and giant  \nmagnetoresistances (GMR).23–26 The latter, discovered in the 1980s, has led to miniaturized GMR \nsensors, revolutionizing both the magnetic non -volatile  data stora ge and field sensing technologies.27 \nWhat lies at the core of the GMR  is the spin valve effect, arising from  the spin -dependent electron \nscattering at the bulk and interfaces of ferromagnetic materials.28 GMR spin valves typically consist \nof conducting  magnetic layers separated by nonmagnetic metal spacers. The electrical resistance of  \n3 \n such systems exhibits a large asymmetry  between the parallel and antiparallel alignment of the  \nmagnetizations of  adjacent magnetic layers , which serve as  the reference and free layers , respectively . \nGMR can  then be used to effectively probe the magnetization reversal or rotation of the free layer  \nwith respect to the reference, also in a current -in-plane (CIP) geometry, leading to simple two -\nterminal devices. To date, FMGs have been excluded from such device concepts even though  the \nGMR -based  sensing  of FMGs’ magnetization vector would be highly desirable. Few e arlier reports \nhave characterized  the spin valve effect at other magnetic insulator |metal interfaces , but these were \nlimited to in -plane magnetization systems and cryogenic temperature measurements.29–31 While the \npresence of the spin valve effect is allowed at an FMG |metal interface  in principle , its experimental \nobservation remains elusive.  \nIn this work, we show a robust spin valve effect at the interface of a typical FMG, terbium iron garnet \n(Tb 3Fe5O12, TbIG ), enabling us to detect 180º perpendicular magnetization reversal via simple \nresistance measurements at room temperature. We achieve such spin valve behavior in a \nTbIG|Cu|TbCo trilayer ( see Fig. 1a) where the TbIG is engineered to be the free layer wherea s TbCo \nacts as the reference magnetic layer. We observe abrupt longitudinal resistance changes during a \nswept magnetic field upon crossing the coercive fields of the respective layers, analogous to the CIP -\nGMR, albeit with a smaller amplitude with respect to a fully metallic stack. The effect of measurement \ntemperature, TbCo layer composition, and spacer layer thickness on the sign and amplitude of the \nmagnetoresistance collectively pinpoint the spin valve effect at the TbIG|Cu interface as the \npredominant origin of the observed phenomenon. Theoretical calculations based on layer -resolved \nBoltzmann transport equations are in excellent agreement with our experimental findings and further \nconsolidat e the presumed origin of the magnetoresistance effect. The eff icient reading of the \nmagnetiza tion vector in FMGs in a  two-terminal  CIP geometry reported here provides a new platform \nfor novel spintronic device concepts using magnetic insulators as active components.  \nLayer composition and magnetic characterization   \n4 \n We deposited TbIG(25)|Cu(1 -5)|TbxCo1-x(8)|Pt(3) (thicknesses in nm, subscript x denotes atomic %) \nstacks  and other reference layers  by means of magnetron sputtering, and patterned them into standard \nHall bar devices (see Fig. 1b and Methods). Here, TbIG and TbxCo1-x (TbCo for short, x = 0.3 unless \nspecified otherwise) are used as the soft and hard magnetic layers, respectively, due to the lower \n(higher) coercivity of the former (latter). We preferred a Tb -based ferrimagnetic metallic alloy over \nan elemental t ransition metal such as Co, since the former exhibits a large and tunable bulk PMA \nover a broad range of composition, thickness, and temperature, independently of the under/over layer, \nthus eliminating the laborious engineering of the interfacial PMA in th e latter.32–34 The thickness of \nthe Cu spacer is varied between 1 and 5 nm and Cu is chosen due to its high spin conductivity when \ninterfaced with FMGs and TbCo.35,36 Finally, Pt is used as capping to avoid oxidation of TbCo and it \nhas no active role in the spin transp ort experiments reported in this paper, therefore will not be \nmentioned in the remainder of the text.  \nThe magnetization behavior of TbIG and TbCo were examined by polar magneto -optic Kerr effect \n(MOKE, Fig. 1c) and AHE (Fig. 1d) measurements with a swept out-of-plane field ( Hz). In the former, \nwe observe d a clear hysteresis loop typical of PMA systems with 100% remanence, and large \ncoercivity Hc ~35 mT.  A closer look -up to the lower field region reveals minor signal jumps at ~±10 \nmT. The inner loop measurement focusing on this region (see inset  – dark green curve ) indicates a \nsecond magnetic signal with a much lower amplitude, superimposed to the larger hysteresis loop. The \nHc value of this inner loop corresponds to the one measured f or TbIG prior to the Cu|TbCo deposition  \n(see inset - light green curve) , hence this signal corresponds to TbIG . The similar coercivity of the \nTbIG before and after deposition of the Cu|TbCo indicates negligible exchange coupling between the \ntwo magnetic la yers (see Supplementary Information). The Hall resistance ( RH) measurement shows \nthe hysteresis loop originating from the AHE in TbCo  with Hc ~35 mT, consistent with the MOKE \nresult. However, an inner loop measurement similar to Fig. 1c - inset does not sh ow any signal related \nto the TbIG reversal. Because of the absence of current flow through TbIG, the AHE signal cannot  \n5 \n be generated in this layer  and the interfacial AHE is also inoperative due to the negligible spin Hall \neffect in Cu . These measurements clearly demonstrate that the magnetization vector of TbIG in the \nabove stack can only be probed optically but not electrically with standard Hall measurements.  \nMagnetoresistive detection of perpendicular switching  \nWe then examine the lon gitudinal resistance behavior of the above stack and compare it with \nreference material systems. The GMR in magnetic transition -metal multilayers is known to  originate \nfrom the spin -dependent scattering in the spin -split d bands,37,38 typically  leading to an increase of the \nresistance when the magnetization ( M) of consecutive layers is antiparallel compared to the parallel \ncase. Figure 2a shows the resistance behavior of a fully metallic Co|Cu|TbCo  PMA spin valve device \nduring an Hz sweep. We observe two reversal events at low and high fields, corresponding to the \nswitching of Co and TbCo, respectively, resulting in two distinct resistance levels in the parallel and \nantiparallel configurations, expectedly of the CIP -GMR. The magnitude of the GMR is 4.9%, \ncomparable to previous reports on similar systems.36,39 Next, we measure Cu/TbCo deposited on a \nnon-magnetic gadolinium gallium garnet ( Gd 3Ga5O12, GGG) substrate (Fig. 2b). Here, the spin valve \nbehavior is not observed. Instead, the resistance displays a field -dependent variation resembling a \nbowtie shape . This unconventional resistance behavior was previously reported and attributed to the \nmagnon magnetoresistance40 and/or magnetoresistance due to the sperimagnetism of TbCo .41 This \neffect requires a conducting magnetic layer and exhibits  a quasi -linear positive or negative slope \ndepending on the relative direct ion of M and Hz.  \nFigure 2c shows the magnetoresistance of the TbIG|Cu|TbCo trilayer. The resistance clearly \nundergoes four distinct switching events corresponding to the perpendicular switching of TbIG and \nTbCo , superimposed to the quasi -linear background magnetoresistance described above. Initially, at \nlarge positive fields, the magnetizations of TbIG and TbCo  are parallel and aligned with Hz. Upon \nreducing Hz, the resistance increases linearly until ~ -10 mT at  which field the magnetization of TbIG \nreverses from up to down and a sudden resistance increase occurs. As Hz is reduced further , the  \n6 \n resistance increase continues, until the coercivity of TbCo is reached at ~ -35 mT, where the resistance \ndrops sharply , followed by further gradual Hz-dependent decrease . A reciprocal sequence is observed \nwhen Hz is swept from the negative to positive direction. To better illustrate the GMR -like behavior, \nin Fig. 2d, we plot the magnetoresistance after subtraction of the back ground due to the unusual \nmagnetoresistance of TbCo not related to the GMR effect. Finally, Fig.  2e shows the minor loops, \ncorresponding to the reversal of M of TbIG, when the TbCo  layer is pre -set in the up (+z) and down \n(-z) direction. In these measureme nts, we observe the hysteretic resistance behavior in both cases with \nan opposite sign due to the reversal of the reference layer. Based on the amplitude of the jump in Fig. \n2e, we estimate the relative change of resistance R/R0 =1.4x10-5, where R0 is the  resistance at zero \napplied field.  \nThe signals reported in Fig. 2c-e are consistent with a positive GMR in TbIG|Cu|TbCo.  The sign of \nthe GMR in all -metallic systems is dictated by the spin -dependent scattering asymmetry at the bulk \nand interfaces of the m agnetic layers. In TbCo, the spin -polarized conduction is mostly dominated by \nthe s-d electrons of Co, hence the Co magnetization  dictates the bulk and interface contribution to the \nGMR as well as the sign of  the AHE. The positive AHE in Fig. 1d and the positive GMR in Fig. 2a \nestablish that the net magnetization in TbCo is parallel to the Co sublattice in the TbIG|Cu|TbCo \nstack. Likewise, independent measurements reported in our earlier study ( see Ref.[12]) and \nSupplementary Information show that the net magnetization in TbIG is Fe sublattice dominated. Since \nTbIG is an insulator, only the TbIG/Cu interface contributes to the GMR. Therefore, the positive \nGMR in TbIG|Cu| TbCo demonstrates that the sign of the spin -dependent scattering asymmetry at \nboth the TbIG|Cu and Cu|TbCo interfaces is the same and positive wit h respect to the standard \nconvention.  \nTemperature -dependence of the magnetoresistance  \nTo examine further the effect of the relative orientation of the ferrimagnetic sublattices  on the sign, \nand the influence of the measurement temperature on the amplitude of R/R0, we studied the  \n7 \n magnetoresistance in TbIG|Cu|TbCo as a function of temperature  (T), across the magnetic \ncompensation temperature ( TM) of TbIG. In Fig. 3a, we plot representative data during a Hz sweep \nmeasured at T = 225  K and 175 K, i.e., above and below the TM of TbIG, respectively. We found TM \nof TbIG ~200 K for this specific sample (see Supplementary Information). Because of the linear \nmagnetoresistance background from TbCo th e overall shape of the signal becomes  intricate  for T ≤ \n200 K .  Nevertheless, we observe that the resistance change s due to TbIG reversal (marked with black \narrows) is upward for T = 225  K and downward for T = 175 K , indicating a clear sign change of \nΔR/R 0. This is further highlighted in Fig. 3b with the minor loop measurements corresponding to the \nmagnetization reversal of TbIG when the TbCo layer is fixed along the up (+z) direction. Above TM \n(T = 225 K) we observe  a negative hysteresis loop (similar to the room temperature measurement \nreported in Fig. 2e) whereas below TM (T = 175 K) the loop becomes positive.  \nIn Fig. 3c we plot ΔR/R 0 in the entire temperature range tested in this study. The sign reversal of \nΔR/R 0 at TM of TbIG confirms that the spin -dependent reflection at the TbIG|Cu interface, similarly \nto the interfacial  AHE42, is not governed by the net magnetization, but instead , by the orientation of \nthe Fe sublattice. In  spite  of the sign change at T ~ 200 K, the absolute value of ΔR/R 0 increases \nsmoothly and monotonically (open dots in Fig.  3c) with the decreasing temperature. This is expected \nfrom the strong temperature dependence of the GMR effect due to the increased spin conductivity  of \nCu at lower temperatures.43 \nTbCo composition and Cu spacer thickness dependence  \nThe amplitude and sign of the GMR can be modulated by the TbCo  composition and Cu spacer \nthickness. To investigate the former, we measured ΔR/R 0 in TbIG|Cu(2)|Tb xCo1-x(8nm) layers with x \nranging between 0.3 and 0.65 and plotted in Fig. 4 – left. We note that for x < 0.3 the films were  \nmagnetized in -plane and for x > 0.65 they were  paramagnetic (see Supplementary Information). We \nfind that ΔR/R 0 strongly depends on the Tb content, displaying a rapid decrease going from x = 0.3  to \n0.5. When the Tb amount reaches x = 0.55 , the magnetization becomes Tb sublattice dominated and  \n8 \n ΔR/R 0 becomes negative (i.e. R is lower when the magnetization of TbIG and Tb xCo1-x are \nantiparallel). This is analogous to the sign reversal observed upon crossing  the TM of TbIG reported \nin Fig.  3c, but obtained at room temperature with a modification of the Tb xCo1-x composition instead. \nOverall, |ΔR/R 0| undergoes a ~22 -fold decrease when the Tb concentration  is increased from 0.3 to \n0.6, and finally vanishes at x = 0.65. This result indicates  the essential role played by Co in the spin -\npolarized current generation and spin -dependent scattering in the bulk and interfaces of TbCo.  \nFigure 4 – right reports the effect of the Cu spacer thickness ( t) on ΔR/R 0 in TbIG|Cu( t)|TbCo layers. \nThe largest effect is found for Cu(2 nm). Increasing t results in a decrease of ΔR/R 0, as expected due \nto the increased current shunting and reduced spin coherence.43 On the other hand, for a Cu thickness \nof 1 nm, a slightly lower ΔR/R 0 is observed with respect to Cu(2 nm), whereas the effect  is, in \nprinciple, expected to be enhanced. We associate this behavior to the lower uniformity of such thin \nCu layer, causing additional resistance, not participating in the magnetoresistance and negatively \naffecting the spin -dependent properties and interface quality with the TbCo layer. Indeed, we  observe \na correlation between the Cu thickness and the coercivity and PMA of TbCo, indicating a sharper \ninterface when the Cu thickness is increased, promoting a higher quality TbCo. This might be the \nreason why the magnetoresistance is still substantial despite larger current shunting when the Cu \nspacer is increased to 5 nm.  The strong dependence of the magnetic properties of TbCo alloys on the \nunderlayer material, thickness and interface quality is well  known .44 Additionally, discrepancies in \nthe spin -dependent properties of the TbIG|Cu interface for TbIG grown on different dates  on different \nsubstrates could influence the data presented in Fig.  4 – right. Although the essential features and \ntrends expected of the thickness dependence of ΔR/R 0 are observed, the data in Fig. 4 –  right cannot \nreflect a universal behavior. Therefore, we believe that a more systematic study of the influence of \nthe spacer layer thickness on ΔR/R 0 should be conducted by continuously varying the Cu thickness \non a single substrate to ensure the unifo rmity of the TbIG properties, which falls outside our technical \ncapabilities. Finally, to confirm the relevance of the spacer layer with low spin -orbit coupling (high  \n9 \n spin c onductivity ) on the magnetoresistance, we replaced Cu with 2 nm of Pt  which exhibit s large \nspin-orbit coupling . We did not detect any spin valve effect within our experimental detection limit. \nThis negative result is anticipated due to  the very short spin diffusion length of Pt  (typically <2 nm) , \ndephasing nearly the entire spin current generated in the TbCo and at the Pt/TbCo interface .36 \nFurthermore, the absence of spin valve signal with the Pt spacer excludes the magnetic proximity \neffect at the TbIG |metal interface as the potential origin of the magnetoresistance, since Pt is much \nmore suscept ible to proximity magnetism than Cu. We n ote that ΔR/R 0 values of all the materials \nexamined in this work can be found in Table S1 of the Supplementary Information.  \nBoltzmann Transport Calculations  \nWe calculated the magnetization -dependent resistivity of T bIG|Cu|TbCo trilayer by solving the \nfollowing linearized Boltzmann transport equation with a layer -by-layer approach :38,45 \n                                                                       𝑣𝑧𝜕𝑔\n𝜕𝑧−𝑒𝐸\n𝑚𝜕𝑓0\n𝜕𝑣𝑥=−𝑔\n𝜏                                                              (1) \nHere 𝑚 and 𝜏 are the effective mass and momentum relaxation time and mass  of conduction \nelectrons, 𝑓0 is the equilibrium electron distribution fun ction, 𝑔(𝒗,𝑧) is the deviation from f0 induced \nby the external electric field with amplitude E applied along x. Full translation symmetry is assumed \nin the x -y plane (i.e., the layer plane) so that the spatial dependence of 𝑔 only occurs in the thickne ss \ndirection parallel to the z -axis. To further simplify our calculations, we assumed the same effective \nmass, relaxation time, and Fermi energy 𝜀𝐹 for the conducting Cu and TbCo layers, which would be \nsufficient for the purpose of an order of magnitude estimation of the ΔR/R 0 ratio of the trilayer system. \nThe calculation can be extended straightforwardly to involve different values of the above -ment ioned \nmaterials parameters without essential change to the formulation.  \nIn order to capture the interfacial spin -dependent scattering, we divide 𝑔 into four additive \ncomponents depending on the orientation of an electron’s spin moment with respective to the  \n10 \n magnetization of a magnetic layer and the sign of vz (i.e., the z -component of the electronic group \nvelocity). The general solutions of Eq. (1) can be written as  \n                                              𝑔±↑(↓)(𝒗,𝑧)=𝑒𝐸𝜏\n𝑚𝜕𝑓0\n𝜕𝑣𝑥[1+𝐶±↑(↓)𝑒𝑥𝑝(∓𝑧\n𝜏|𝑣𝑧|)],                                              (2) \nwhere the “ +” and “ −” signs denote electrons with 𝑣𝑧 being positive and negative , respectively, the \narrow ↑(↓) characterizes orientation of spin parallel (antiparallel) to the local magnetization of the \nmagnetic layer in question. The general solutions take the same form for electrons in the Cu and TbCo \nlayers, except for the coefficients, 𝐶±↑(↓), which need to b e determined by boundary conditions.  \nTo calculate the ΔR/R 0 ratio, scatterings at three interfaces of the trilayer need to be taken into account \nthrough the boundary conditions, namely, 1) the spin -dependent reflection at the out surface of the \nTbCo layer , 2) the specular transmission and reflection at the interface between the metallic TbCo|Cu \ninterface, and 3) the spin -dependent specular reflection at the Cu|TbIG interface. In addition, when \nthe magnetizations of the TbCo and TbIG layers are antiparallel , we also take into account the change \nof spin quantization axis in the middle of the Cu layer. It is reasonable to assume specular reflection \nat the metallic interface is negligible and scattering at the outer surface of the TbCo layer is completely \ndiffu sive.  And near the magnetic interface between the Cu and TbIG layer, the portions of electron \nfluxes that are specularly reflected from the interface are, in principle, also spin dependent ,21 as “spin -\nup” and “spin -down” electrons see different energy barriers effectively due to the exchange coupling \n(𝐽𝑒𝑥) between them and the magnetization of the TbIG layer even though the magnetic layer is \ninsulating.  \nBy inserting the general solutions of 𝑔±↑(↓) for each layer into these boundary conditions, the \nunknowns in Eqs. (2) can be determined, allowing us to further evaluate the spatially -averaged \nlongi tudinal conductivity of the ith layer via 𝜎(𝑖)=1\n𝑑𝑖𝐸∫𝑑𝑧∫𝑑3𝒗𝑣𝑥(𝑔↑+𝑔↓), where 𝑑𝑖 is the \nthickness of the ith layer. The total resistivities for parallel and antiparallel magnetization \nconfigurations , denoted by 𝜌↑↑ and 𝜌↑↓respectively , are obtained by inverting the corresponding  \n11 \n conductivity tensors.  Finally, the ΔR/R 0 ratio is evaluated via ∆𝑅𝑅0⁄ =(𝜌↑↓ −𝜌↑↑)𝜌↑↑⁄. For the \nTbCo|Cu interface, the transmission coefficients are taken to be 𝑇↑=0.5 and 𝑇↓=0.95 (note that \nthe roughness of the interface can be characterized phenomenologically by the spin -dependent \ndiffusive scattering parameter defined as 𝐷↑(↓)=1−𝑇↑(↓).45 For 𝐽𝑒𝑥~0.01 eV and the averaged \nenergy barrie r of the insulator 𝑉𝑏~12 eV,2,21 the reflection coefficients are estimated to be 𝑅↑=\n0.4995  and 𝑅↓=0.5005 . This spin asymmetry of electron scatterings at the two magnetic interfaces \ngives rise to a ΔR/R 0 ratio of 6.2x10-5, close to the experimental value of 7.0x10-5 measured at 10 K. \nDetailed calculation s are provided in the Supplementary  Information.  \nConclusions  \nIn summary, we demonstrate a simple magnetoresistive detection of perpendicular magnetization \nreversal in an insulating ferrimagnetic material TbIG. The detection relies on current -in-plane \nmagnetoresistance measurements in a TbIG|Cu|TbCo tril ayer system where the conducting \nferrimagnet TbCo is used as a reference magnetic layer and spin polarizer. The material and \ntemperature  dependence of the magnetoresistance and theoretical calculations collectively pinpoint \nthe spin -valve effect at the TbI G|Cu interface as the underlying cause of the observed phenomenon. \nThese results will open a new chapter in the field of insulating spintronics as they will stimulate \nresearch into a wide spectrum of FMG, spacer and spin polarizer layer combinations, and e nable a \nwhole new range of device ideas and architectures based on magnetic insulators. While the effect is \nrelatively small for any microelectronic applications as of yet, it can be enhanced by orders of \nmagnitude by material s and device engineering , and support from theory .  \nOn the materials side, ferromagnetic materials with higher spin polarization such as Heusler alloys \nwith an optimized thickness could provide a pathway to increase the magnetoresistance. To obtain \neven larger gains, the spacer layer c an be chosen from those having a very long spin coherence length \nand promoting better spin mixing conductance with the insulating and conducting magnetic layers \nforming the device. Potential candidates include two -dimensional materials such as graphene,  \n12 \n transition metal dichalcogenides (TiSe 2, MoS 2, etc.), conducting oxides (SrVO 3, etc.) and other light \nmetals (Cr, Mn, etc.). Our work will also stimulate theoretical efforts , as the most suitable materials \ncould be determined by using relevant first -principl es calculations.  \nOn the device side, the magnetores istive reading reported here will  enable non -volatile binary \nmemory cells where a magnetic insulator could be used as an active component instead of \nconventional magnetic conductors. In such devices, usin g a magnetic insulator would bring about a \nseries of advantages such as higher  structural stability, broader magnetic tunability , ultrafast \nswitching times,  and low power consumption , among others. It is also possible to use the \nmagnetoresistance output to identify and characterize the skyrmions  and domain walls  in an insulating \nracetrack memory  and study their field/current -driven dynamics in real time . Insulating domain wall - \nand sky rmion -based devices enabled by magnetoresistive reading could pave the way for novel \nanalog -like memory concept s that can be used in neuromorphic computing.  \nMethods  \nSamples preparation  \n25 nm -thick TbIG thin films were deposite d on GGG(111) substrates by radio frequency (r.f.)  \nsputtering at 800 ºC from a stoichiometric target. The deposition rate was ∼0.4 nm/min at the applied \npower of 150W in 3mTorr of a mixture of Ar and O 2 with a ratio of 30:2 and the base pressure in \nthe chamber was below 7x10-8 Torr. Detailed characterization and optimization procedures of our \nTbIG films are described in Ref.[12]. To fabricate the six -terminal Hall bar devices, the continuous \nTbIG films were covered in photoresist and patterned by laser -writer  optical lithography. Finally, the \nmetallic stack described further below was deposited and lift-off was performed. The Hall bar \ndimensions are l=30 μm for the current line length, l/4 its width and l/10 the Hall branch width.  \nThe metallic stack consisted of M( t)|Tb xCo1-x(8 nm)|Pt(3  nm) multilayers deposited by d.c.  magnetron \nsputtering at room temperature in 3  mTorr Ar. The Tb xCo1-x alloys were obtained by co -sputtering  \n13 \n pure Co and Tb targets and the relative atomic concentration of the two elements was controlled by \nthe relative sputtering power. The thickness t of the metal (M) spacer layer varied between 1 and 5 \nnm. The deposition rates were as follows: 0.13514 nm/s at 100 W for Cu, 0.104 nm/s at 200 W for \nCo, 0.082 nm/s t 50 W for Tb, 0.186 nm/s at 50 W for Pt and 0.057 nm/s at 200 W for Ti. The \nfollowing referen ce samples were also prepared: Cu(2  nm)|Tb 0.3Co0.7(8 nm)|Pt(3  nm) on GGG(111) \nsubstrate,  Ti(3  nm)|Pt(3  nm)|Co(1  nm)|Cu(2  nm)|Tb 0.3Co0.7(8 nm)|Pt(3  nm) on Si, Ti(3  nm)|Pt(3  \nnm)|Co(1  nm)|Cu(2  nm)|Tb 0.65Co0.35(8 nm)|Pt(3  nm) on Si and Pt(1.5  nm)|Cu(1.5  nm)|Tb 0.3Co0.7(8 \nnm)|Pt(3  nm) on TbIG.  \nMagnetic and Electrical Characterization  \nThe magnetic hysteresis and anisotropy axis of the films were examined using a home -built magneto -\noptic Kerr effect (MOKE) setup in a polar geometry  with a 532 nm wavelength gr een laser . The  room \ntemperature  magnetoresistance measurements were performed by recording the resistance ( R) as a \nfunction of H between the extremes of the current line using a Keithley DMM6500 digital multimeter \nand DC test current of 1 mA.  For t he temp erature dependence of the magnetoresistance , selected \nsamples were inserted  inside a physical property measurement system and measured in the range of \n10–300K with an AC probing current  of 1 mA  (root mean square ) and frequency 𝜔/2𝜋 = 999Hz. \nThe Hall effect  measurements were performed at room temperature using a Zurich Instruments MFLI \ndigital lock -in amplifier. An AC probing curren t of amplitude 0.5 -1mA (root mean square)  and \nfrequency 𝜔/2𝜋 = 1092Hz was sent across the current line and the first h armonic voltage ( R𝐻) was \nmeasured across the Hall arms.  \nData availability  \nAll the data supporting the findings of this study are available upon request from the corresponding \nauthor.  \nCode availability   \n14 \n The computer code used for data analysis is available  upon request from the corresponding author.  \nReferences  \n1. Yang, Y., Liu, T., Bi, L. & Deng, L. 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Lett.  106, 242403 (2015).  \n45. Camle y, R. E. & Barna, J. Theory of giant magnetoresistance effects in magnetic layered \nstructures with antiferromagnetic coupling. Phys. Rev. Lett.  63, 664 (1989).  \n \nAcknowledgments  \nS.D. and C.O.A. acknowledge funding from the European Research Council (ERC) under the \nEuropean Union’s Horizon 2020 research and innovation program (project MAGNEPIC, grant  \n18 \n agreement No. 949052). C. O. A. acknowledges funding from the Spanish Ministry of Science and \nInnovation through grant reference  no. CNS2022 -136060.  Work by A.S., M.M. and S.S. -L.Z was \nsupported by the College of Arts and Sciences, Case Western Reserve University.  Authors  thank M. \nFettizio for the insightful discussions.  \n \nAuthor Informa tion \nContributions  \nC.O.A. conceived the idea and supervised the study. S.D. designed and prepared the samples, carried \nout the measurements and analyzed the data. A.S. , M.M.  and S. S.-L.Z. constructed the theoretical \nframework. S.D. and C.O.A. wrote the manuscript. All authors discussed the results and commented \non the manuscript.  \nCorresponding author  \nCorrespondence to Silvia Damerio ( sdamerio@icmab.es ) and Can Onur Avci (cavci@icmab.es)  \n \nCompeting Interests  \nThe au thors declare no competing interests.  \n   \n19 \n Figures and Tables  \n \nFig.1 Illustration of the spin valve effect in TbIG|Cu|TbCo, device schematics, and optical and \nelectrical characterization . a Schematic representation of the spin valve structure in the high and \nlow resistance states at room -temperature. The red and blue arrows indicate the direction of the \nmagnetization ( M) and corresponding spin s of the majority and minority carriers.  Cyan sph eres \nrepresent conduction electrons undergoing lower and higher scattering events in the parallel  and \nantiparallel magnetic configurations, respectively. b Schematic representation of the device with the \ngeometry used for the electrical measurements. c Plot of the polar MOKE signal of TbIG|Cu|TbCo. \nThe inset shows the minor loops corresponding to the switching of TbIG prior (light green) and after \n(dark green) the deposition of TbCo.  d Transverse Hall resistance ( RH) in TbIG|Cu|TbCo  as a function \n-50 0 50-1.0-0.50.00.51.0\n-15 015-0.030.000.03MOKE (a.u.)\nHZ (mT)\n-50 0 50-0.8-0.40.00.40.8RH ()\nHZ (mT)a\nbc\ndi\niTbCo\nTbIGCu\nTbCo\nTbIGCuM\nM\nM\nMParallel state -Low Resistance\nAntiparallel state -High Resistance\nz\nx y\nRz\nx y\nRHie-\ne- \n20 \n of out-of-plane field ( HZ), corresponding to the AHE in TbCo.  No fingerprints of TbIG switching are \nobserved in the inner loop (gray).  \n \nFig.2 Magnetoresistance in TbIG|Cu|TbCo and other reference layers . a Resistance ( R) as a \nfunction of the out-of-plane fie ld (HZ) in an all -metallic Co(1)|Cu(2)| TbCo (8) trilayer. b in \nGGG(subs.)|Cu(2)| TbCo (8) and c in TbIG(25)|Cu(2)| TbCo (8). The latter clearly shows a spin valve \nsignal similar to the one observed in the all -metallic trilayer.  d Plot of the resistance  from panel c \nafter removal of the unconventional magnetoresistance contribution from TbCo (see text for more \ndetails). The red and blue arrows indicate the direction of M in TbIG and TbCo , respectively. e Inner \nloops showing the switching of TbIG when the  direction of TbCo  is fixed up ( red) or down ( blue) \nwith respect to the film normal.  \n-100 0 100-0.010.000.010.020.03R-R0 ()\nHZ (mT)\nTbCo\nCo\nTbCo\nTbIG\nTbCo\nGGG\n-200 0 200360365370375380R ()\nHZ (mT)\n-50 0 50160.23160.24R ()\nHZ (mT)\n-100 0 100942.65942.70R ()\nHZ (mT)a b c\nd e\n-20 -10 0 10 20-0.010.000.010.02R-R0 ()\nHZ (mT) TbCo up           TbCo down \n21 \n  \nFig.3 Temperature dependence of the magnetoresistance in TbIG|Cu| TbCo . a Plot of the \nresistance ( R) as a function of out -of-plane field ( HZ) for 2 selected temperatures above (red ) and \nbelow (blue) the compensation temperature ( TM) of TbIG. The black arrows indicate the resistance \njump corresponding to TbIG magnetization reversal . b Plot of the inner loops showing the switching \nof TbIG when M of TbCo  is fixed out -of-plane for two selected temperatures above (red) and below \n(blue) TM. The linear background has been subtracted. c Plot of the temperature ( T) dependence of \nthe magnetoresistance (close dots) and its absolute value (open dots).  \n0.00.51.01.5\n \n0.60 0.55 0.35Cu(2)/TbxCo1-xCu(t)/Tb0.3Co0.7\n5 2 3 1\nx t (nm)R/R0 x10-5\n0.4 0.3 0.5 0.65\n \nFig.4 Amplitude of the magnetoresistance as a function of Tb xCo 1-x composition and Cu spacer  \nthickness . Left: plot of ΔR/R 0 in TbIG(25)|Cu(2)|Tb xCo1-x(8) alloy as a function of Tb content ( x) \nfrom 0.3 to 0.65. We note that ΔR/R 0 undergoes a sign change for x > 0.5 and it is below our detection \n-200 -100 0 100 200R-R0\nHZ (mT)0.01 \n-200 0 200R-R0 \nHZ (mT)0.05 175K225Ka\n0 100 200 300-8-4048\nT (K)R/R0 x10-5\n225K\n175Kb c\nTM \n22 \n limit for x = 0.65.  The dotted lines represent the absolute value of the GMR amplitude  for visual \nillustration of the monotonic decay of ΔR/R 0. Right:  plot of ΔR/R 0 in TbIG(25)/Cu(t)/Tb 0.3Co0.7(8) as \na function of  Cu spacer thickness ( t) from 1 to 5 nm.  \n   \n23 \n Supplementary Information for “Magnetoresistive \ndetection of perpendicular switching in a magnetic \ninsulator”  \nSilvia Damerio1,*, Achintya Sunil2, M. Mehraeen2, Steven S. -L. Zhang2, and Can  O. Avci1,* \n1Institut de Ciència de Materials de Barcelona, Campus de la UAB, Bellaterra, 08193, Spain  \n2Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA  \n \nSI1 – Exchange bias in TbIG|Cu|TbCo  spin valves  \nWe found that the magnetization of the magnetic layers in the spin valves of this study is \ncoupled via a small exchange bias for the Cu thickness ≤2 nm. This interaction appears as a \nlateral shift towards negative (positive) values of the hystereis loops of the “soft” TbIG when \nthe magnetization of Tb 0.3Co0.7 is fixed  in the up (down) direction (see Fig. 2e of the main \ntext). We estimated the shift ( ΔH) as a function of the thickness of the Cu spacer layer and \nplot the trend in Fig. S1. Due to the low amount of samples available we could not perform \na detailed statistical analysis of these values, but the results seem to indicate that the exchange \nbias decreases upon increasing spacer layer thickness. Because the measured exchange bias \nis much lower than the coercivity o f TbIG, we consider it negligible in this study.  \n0 2 4 6036H (mT)\nCu t (nm)\n \nFig.S1 Exchange bias in TbIG|Cu|TbCo  spin valves. a  Plot of the difference of the \ncoercive field of TbIG ( ΔH), measured from the R vs H loops, when the magnetization of \nTbCo is fixed in the up and down direction as a function of the Cu spacer layer thickness ( t). \n \nSI2 – Magnetic and magneto -transport properties of Tb xCo1-x alloys  \nFerrimagnets are a class of magnets with unbalanced antiparallel -aligned sublattice moments, \nwhich results in a finite, albeit small, magnetization. In RE -transition metal alloys and \nmultilayers, the dominant sublattice (i.e. the sublattice whose moment is  parallel to the net \nmagnetization) varies with the stoichiometry and temperature. Measurement of the \nanomalous Hall effect (AHE) provides information on the dominating sublattice, as the sign \nof the AHE coefficient is opposite for RE and transition -metals .   \n24 \n  \nBefore utilizing Tb xCo1-x in the fabrication of spin valve devices we characterized the \nmagnetic and magneto -transport properties of the alloy as a function of Tb concentration. To \nthis end, we grew 8 nm thick Tb xCo1-x films by co -sputtering on Si substrates with 3 nm Ti \nbuffer and capping layers. The samples were patterned into Hall -bar devices ( with sizes l=30 \nμm for the current line length, l/4 its width and l/10 the Hall branch width)  by standard optical \nlithography and lift -off. Figure S2a shows the plot of the Hall resistance ( RH) as a function \nof out -of-plane field ( HZ) of the Tb xCo1-x films below the compensation composition. Here, \nPMA is only achieved above 25% of Tb due to the shape anisotropy that tends to orient the \nmagnetization in -plane when the films are patterned into micron -sized devices. Here, HC \nincreases from 65 mT at x=0.25 to 750 mT for x=0.5. Comparable values are observed in the \nspin valves of the main text (see Table S1) with a small difference due to the different buffer  \nlayer (Cu instead of Ti). Between x=0.50 and x=0.55 the compensation composition of the \nTbxCo1-x alloy is reached and the sign of the AHE reverses, as shown in Fig. S2b. Upon \nfurther increasing Tb concentration, HC starts to decrease again, reaching 20 mT  at x=0.65. \nThis allows us to conclude that Tb xCo1-x films are Co -dominated up to x=0.5 and Tb -\ndominated above x=0.55.  \n \nFig.S2 Characterization of the AHE in Tb xCo1-x alloys as a function of Tb content.   Plot \nof the Hall resistance ( RH) as a function of HZ for patterned Tb xCo1-x films with a x between \n0.25 and 0.5 and b x between 0.55 and 0.65.  \n \nSI3 – Ferrimagnetic sublattices and sign of the magnetoresistance  \nThe giant magnetoresistance (GMR) provides information on the relative orientation of the \nmoment of the transition metal sublattice in two neighboring layers, as it originates from the \nscattering of d -conduction electrons with the transition metal moments.  Figure S3a shows \nthe AHE of a Co(1)|Cu(2)|Tb 0.3Co0.7(8) (thickness in nm) reference spin valve. As it can be \nseen, here the AHE coefficient is positive for both Tb 0.3Co0.7 and Co layers, indicating that \nthe magnetization ( M) of Tb 0.3Co0.7 is parallel to the Co sublattice. Consistently, the GMR \n(Fig. S3b also Fig. 2a of the main text) is positive, indicating that in both layers the Co \n-1000 -500 0 500 1000-0.4-0.20.00.20.4RH (a.u.)\nHZ (mT) x=0.25\n x=0.30\n x=0.35\n x=0.40\n x=0.50\n-500 0 500-0.4-0.20.00.20.4RH (a.u.)\nHZ (mT) x=0.55\n x=0.60\n x=0.65a b \n25 \n sublattice, which dominates the transport properties, is aligned with the magnetic field ( HZ). \nFigures S3c-d show the measurements of a reference Co(1)|Cu(2)|Tb 0.65Co0.35(8) spin valve. \nHere, due to the large amount of Tb, the alloy becomes RE -dominated and thus both the AHE \nand the GMR change sign. Figure S3e-f show the Hall resistance ( RH) as a function of HZ in \na TbIG(25)|Pt(1.5)|Cu(1.5)|Tb 0.3Co0.7(8) spin valve. In this case, the insertion of a thin (1.5 \nnm) layer of Pt in contact with the TbIG is necessary to read its magnetization direction via \nthe spin Hall magnetoresistance (SMR) effect. Here, we observe  a major hysteresis loop ( Fig. \nS3e) corresponding to the AHE of Tb 0.3Co0.7 with positive AHE coefficient, and a minor loop \n(Fig. S3f) corresponding to the SMR -AHE of TbIG. The latter is negative, indicating that \nTbIG is above its compensation temperature a nd thus its magnetization is parallel to the \nmoment of the tetragonal Fe3+ sublattice (notice that 3d metal -dominated TbIG and TbCo \nhave opposite sign of the AHE coefficient). In this type of spin -valve the GMR is positive, \nas shown in Fig.2c of the main t ext. On the other hand, Fig. S3g shows the negative AHE in \na TbIG(25)|Cu(2)|Tb 0.55Co0.45(8) spin valve, which displays the negative GMR of Fig. S3h, \nas the TbIG and TbCo are respectively 3d metal and RE dominated. Here, the TbCo coercive \nfield is significa ntly higher, being closer to the magnetic compensation composition.  These \ndata confirm that the transport properties in ferrimagnetic insulator|spacer|metal spin valves \nare dominated by the transition metal sublattice in both layers and show that the spin -\ndependent reflection coefficient at the TbIG|Cu interface has the same sign as that of the \nwell-known Co|Cu interface.  \n \nTable S1 Amplitude of the magnetoresistance in different TbIG|spacer|magnetic metal \ntrilayers at room temperature.  \n \nMetallic layer  \n(t nm) Spacer layer  \n(t nm) HC oxide  \n(mT)  HC metal  \n(mT)  R/R 0 \nTb0.25Co0.75 (8) Cu (2)  7 In-plane  < detection  \nTb0.3Co0.7 (8) Cu (2)  10 35 1.44E -5 \nTb0.35Co0.65 (8) Cu (2)  10 55 7.74E -6 \nTb0.4Co0.6 (8) Cu (2)  7 150 3.16E -6 \nTb0.5Co0.5 (8) Cu (2)  6 800 2.42E -6 \nTb0.55Co0.45 (8) Cu (2)  9 350 -2.0E-6 \nTb0.6Co0.4 (8) Cu (2)  10 65 -6.3E-7 \nTb0.65Co0.35 (8) Cu (2)  6 20 < detection  \nTb0.3Co0.7(8) Cu (1)  50 0 1.01E -5 \nTb0.3Co0.7 (8) Cu (3)  10 48 7.57E -6 \nTb0.3Co0.7 (8) Cu (5)  10 40 7.74E -6 \nTb0.3Co0.7 (8) Pt (2)  15 73 < detection  \n  \n26 \n  \nFig.S3 AHE and GMR across the ferrimagnetic compensation.  Plot of a the Hall \nresistance ( RH) and b the longitudinal resistance ( R) as a function of out -of-plane magnetic \nfield ( HZ) of a Co(1)|Cu(2)|Tb 0.3Co0.7(8) reference spin valve. Plot of c RH and d R as a \nfunction of HZ of a Co(1)|Cu(2)|Tb 0.65Co0.35(8) reference spin valve. e-f Plot of RH as a \nfunction of HZ of a TbIG(25)|Pt(1.5)|Cu(1.5)|Tb 0.3Co0.7(8) (Co-dominated) and spin valve. \nPlot of g RH and h R as a function of HZ of a TbIG(25)|Cu(2)|Tb 0.55Co0.45(8) (Tb -dominated) \nspin valve.  \n \n \n−200 0 200−0.6−0.4−0.20.00.20.4RH ()\nHZ (mT)\n-200 0 200360365370375380R ()\nHZ (mT)\n-50 0 50-0.4-0.20.00.20.4RH ()\nHZ (mT)\n-50 0 50493494R ()\nHZ (mT)\n-80 0 80-0.4-0.20.00.20.40.6RH ()\nHZ (mT)\n-10 -5 0 5 10-0.050.000.05RH (m)\nHZ (mT) inner\n-1000 -500 0 500 1000-0.6-0.4-0.20.00.20.40.6RH ()\nHZ (mT)\n−400 0 4001122.601122.611122.62R ()\nHZ (mT)a b\nc d\ne f\ng h \n27 \n SI4 – Temperature dependence  \nThe magnetic compensation temperature of TbIG can be determined by measuring the \ntemperature dependence of the Anomalous Hall effect (AHE) in TbIG|Pt heterostructure, as \nshown in Ref.[12]and Fig. S4a. Here we observe a sign reversal of the AHE at approximately \n200 K, which coincides with the maximum coercivity (Fig. S4b), due to the reduced Zeeman \nenergy on the vanishingly small net magnetization. Simil arly, from the magnetoresistance \nmeasurements of a TbIG|Cu|TbCo spin valve shown in Fig. S4c, we can determine the \nmagnetic compensation of TbIG as the point at which the sign of the GMR reverses (see Fig. \n3c of the main text) and the coercive field diverg es. Above ~200K, the resistance ( R) displays \nan upward jump when the magnetization of the TbIG layers reverses and a downwards jump \nin correspondence of the coercivity of TbCo. The amplitude of the second jump is larger than \nthe first, as it is given by th e sum of both the GMR effect and the MMR effect, which also \nleads to a decrease of R. Below the TbIG compensation ( TM ~200K) the GMR is negative, \nthus the resistance becomes lower when the magnetization of TbIG reverses. In this case, due \nto the GMR effect , we expect an increase of R at the TbCo coercivity. However, here we \nobserve a second jump of R towards lower values. This is due to the sum of the GMR and \nMMR contributions, the latter being larger than the former, and thus resulting in a second \ndecrease  of R. Figure S4d summarizes the temperature dependence of the coercivity of the \nTbCo  and TbIG layers inferred from the magnetoresistance measurement of Fig. S4c. While \nthe Hc of both layers increases gradually as the temperature is reduced, due to the stronger \nexchange interaction between the sublattices and enhanced PMA, the HC of TbIG shows an \nadditional peak at TM = 200 K. The HC vs T trend for the TbIG layer obtained from AHE (Fig. \nS4b) and magnetoresistance (Fig. S4d) measurements is essentially the same.   \n28 \n  \nFig.S4 Temperature dependence.  a Plot of the Hall resistance ( RH) and as a function of out -\nof-plane magnetic field ( HZ) of a TbIG/Pt(4) bilayer measured between 170 and  250 K . b \nPlot of the coercive filed ( HC) of TbIG as a function of T inferred from the AHE measurement \nof panel a. c Plot of the Resistance ( R) and as a function of HZ of a TbIG(25)|Cu(2)|TbCo(8) \nspin valve measured between 10 and 300 K . d Plot of HC of Tb IG (red) and TbCo (blue) as \na function of T inferred from the magnetoresistance measurement of panel c.  \n \n  \n-15000 0 1500010K100K\n50K300K\n275K\n225K250K\n150K\n125K175KR-R0 \nHZ (Oe)0.5 200K\n-800 -400 0 400 800180K\n170K190KRH-R0 \nHZ (mT) 250K\n 240K\n 230K\n 220K\n 210K\n 200K2 m\n180 200 220 2400200400600TMHC (mT)\nT (K)\n0 100 200 30005001000TMHC (mT)\nT (K) TbCo\n TbIGa b\nc d \n29 \n SI5 – Theoretical model  \nWe compute the conductivity of the trilayer by solving the following linearized Boltzmann \ntransport equation with a layer -by-layer approach:  \n                                                            𝑣𝑧𝜕𝑔\n𝜕𝑧−𝑒𝐸\n𝑚𝜕𝑓0\n𝜕𝑣𝑥=−𝑔\n𝜏                                                          (S1) \nHere 𝑚 and 𝜏 are the effective mass and momentum relaxation time of co nduction electrons, \n𝑓0 is the equilibrium electron distribution function, 𝑔(𝒗,𝑧) is the deviation from f0 induced \nby the external electric field with amplitude E applied along x. Full translation symmetry is \nassumed in the x -y plane (i.e., the layer plane) so that the spatial dependence of 𝑔 only occurs \nin the thickness direction parallel to the z -axis. To further simplify our calculations, we \nassumed the same effective mass, re laxation time, and Fermi energy 𝜀𝐹 for the conducting \nCu and TbCo layers, which would be sufficient for the purpose of an order of magnitude \nestimation of the MR ratio of the trilayer system. The calculation can be extended \nstraightforwardly to involve d ifferent values of the above -mentioned materials parameters \nwithout essential change to the formulation.  \n \nFig.S5 Coordinate system for calculating the magnetoconductivity in the \nTbCo|Cu|TbIG trilayer.  The TbCo and TbIG layers are denoted as Layer 1 and Layer 4, \nrespectively, the Cu spacer is divided into two parts with equal thickness (a) by a dotted line \nwhere change of spin quantization axis occurs when the magnetizations of the two magnetic \nlayers a re antiparallel.  \n \nIn order to capture the interfacial spin -dependent scattering, we divide 𝑔 into four additive \ncomponents depending on the orientation of an electron’s spin moment with respective to the \nmagnetization of a magnetic layer and the sign of 𝑣𝑧 (i.e., the z -component of the electronic \ngroup velocity). The general solutions of Eq. (1) for each layer (as sketched in Fig. S5) can \nbe written as  \n                               𝑔±,↑(↓)(𝑖)(𝒗,𝑧)=𝑒𝐸𝜏\n𝑚𝜕𝑓0\n𝜕𝑣𝑥[1+𝐶±,↑(↓)(𝑖)𝑒𝑥𝑝 (∓𝑧\n𝜏|𝑣𝑧|)],                                      (S2) \n \n30 \n where the “ +” and “ −” signs denote electrons with positive and negative 𝑣𝑧 components \nrespectively, the arrow ↑(↓) characterizes orientation of spin parallel (antiparallel) to the \nlocal magnetization of the magn etic layer in question, and the superscripts are the layer \nindices.  \nThe twelve integration constants, 𝐶±,↑(↓)(𝑖) (𝑖=1,2,3) in Eqs. (S2) are determined by the \nfollowing twelve boundary conditions at the interfaces between Layer 1, 2, and 3. For the \nease of notation, we will suppress the group -velocity variable of 𝑔±,↑(↓)(𝑖).  \n Reflection of electrons at the outer surface of the TbCo layer ( 𝑧=𝑏): \n                                                      𝑔−,↑(1)(𝑧=𝑏)=𝑅↑(𝑏)𝑔+,↑(1)(𝑧=𝑏)                                      (S3-1) \n                                                     𝑔−,↓(1)(𝑧=𝑏)=𝑅↓(𝑏)𝑔+,↓(1)(𝑧=𝑏)                                       (S3-2) \nwhere 𝑅↑(↓)(𝑏) is the reflection parameter for spin -up (spin -down) electrons  respectively. The \nvalue of 𝑅↑(↓)(𝑏) ranges from 0 to 1 with “0” corresponds to fully diffusive reflection and “1” \ncorresponds to specular reflection.  \n Spin-dependent electron reflection and transmission at the interface ( 𝑧=𝑎) between \nthe TbCo and Cu l ayers:  \n                                    𝑔+,↑(1)(𝑧=𝑎)=𝑇↑(𝑎)𝑔+,↑(2)(𝑧=𝑎)+𝑅↑(𝑎)𝑔−,↑(1)(𝑧=𝑎)                         (S3-3) \n                                  𝑔+,↓(1)(𝑧=𝑎)=𝑇↓(𝑎)𝑔+,↓(2)(𝑧=𝑎)+𝑅↓(𝑎)𝑔−,↓(1)(𝑧=𝑎)                           (S3-4) \n                                  𝑔−,↑(2)(𝑧=𝑎)=𝑇↑(𝑎)𝑔−,↑(1)(𝑧=𝑎)+𝑅↑(𝑎)𝑔+,↑(2)(𝑧=𝑎)                           (S3-5) \n                                 𝑔−,↓(2)(𝑧=𝑎)=𝑇↓(𝑎)𝑔−,↓(1)(𝑧=𝑎)+𝑅↓(𝑎)𝑔+,↓(2)(𝑧=𝑎)                           (S3-6) \nwhere 𝑅↑(↓)(𝑎) and 𝑇↑(↓)(𝑎) are, respectively, the reflection and transmission parameters for spin -\nup (spin -down) electrons at the interface 𝑧=𝑎.   \n Change of spin quantization axis in the middle of the Cu layer ( 𝑧=0): \n                𝑔+,↑(2)(𝑧=0)=𝑇↑↑(𝜃𝑚)𝑔+,↑(3)(𝑧=0)+𝑇↑↓(𝜃𝑚)𝑔+,↓(3)(𝑧=0)                 (S3-7) \n              𝑔+,↓(2)(𝑧=0)=𝑇↓↓(𝜃𝑚)𝑔+,↓(3)(𝑧=0)+𝑇↓↑(𝜃𝑚)𝑔+,↑(3)(𝑧=0)                  (S3-8) \n                𝑔−,↑(3)(𝑧=0)=𝑇↑↑(𝜃𝑚)𝑔−,↑(2)(𝑧=0)+𝑇↑↓(𝜃𝑚)𝑔−,↓(2)(𝑧=0)                  (S3-9) \n               𝑔−,↓(3)(𝑧=0)=𝑇↓↓(𝜃𝑚)𝑔−,↓(2)(𝑧=0)+𝑇↓↑(𝜃𝑚)𝑔−,↑(2)(𝑧=0)                 (S3-10) \nwhere 𝑇↑↑(𝜃𝑚)=𝑇↓↓(𝜃𝑚)=cos (2𝜃𝑚) and 𝑇↑↓(𝜃𝑚)=𝑇↓↑(𝜃𝑚)=sin (2𝜃𝑚) with 𝜃𝑚 the \nangle between the magnetizations of the TbCo and the TbIG layers.  \n Reflection of electrons at the interface between the Cu and TbIG layers ( 𝑧=−𝑎): \n                                       𝑔+,↑(3)(𝑧=−𝑎)=𝑅↑(−𝑎)𝑔−,↑(3)(𝑧=−𝑎)                                   (S3-11) \n                               𝑔+,↓(3)(𝑧=−𝑎)=𝑅↓(−𝑎)𝑔−,↓(3)(𝑧=−𝑎)                                    (S3-12) \nPlacing Eqs. (S2) into the above boundary conditions, one can determine the twelve unknowns, with \nwhich the spatially -averaged total conductivity of the trilayer structure can be calculated via   \n31 \n                           𝜎=−𝑒\n(2𝜋)3∑1\n𝑑𝑖∫𝑑𝑧∫𝑑3𝒌𝑣𝑥(𝑔↑(𝑖)+𝑔↓(𝑖))/𝐸3\n𝑖=1                             (S4) \nwhere 𝑔↑(↓)(𝑖)=𝑔+,↑(↓)(𝑖) for 𝑣𝑧>0 and 𝑔↑(↓)(𝑖)=𝑔−,↑(↓)(𝑖) for 𝑣𝑧<0. The longitudinal resistivity, \n𝜌, can be obtained by inverting the conductivity tensor. And finally, the magnetoresistance \nratio is obtained by  \n                                                                         𝑀𝑅 =𝜌𝐴𝑃−𝜌𝑃\n𝜌𝑃                                                         (S5) \nwhere 𝜌𝐴𝑃 and 𝜌𝑃 are the resistivities for the magnetizations of the two magnetic layers in \nthe antiparallel ( 𝜃𝑚=𝜋) and parallel ( 𝜃𝑚=0) configurations, respectively. In Fig. S6, we \nplot the magnetoresistance ratio as a function of applied magnetic field for a trilayer TbIG(25  \nnm)|Cu(2 nm)|TbCo (8 nm), with the materials parameters given in Table S2. The calculated \nmagnetoresistance ratio is 0.0062%, comparable to the value obse rved experimentally.  \n \n \nFig. S6 MR ratio as a function of the magnetic field applied perpendicular to the layer \nplane. The blue (red) curve denotes the change of the MR with field sweep from +1500 Oe \nto -1500 Oe (from -1500 Oe to +1500 Oe). The red and blue arrows indicate, respectively, \nthe magnetizations of the TbCo and TbIG layers, respectively. The coercive fields of TbIG \nand TbCo are taken to be 150 Oe and 500 Oe, respectively. And the materials used in the \ncalculation are listed in the table below.  \n \n  \n \n32 \n Table S 2 Reflection and Transmission parameters (Refs.[21],[45])  \n \nSymbols  Value  Parameter  \n𝑅↑(𝑏) 0 Reflection at x = b for spin -up electrons  \n𝑅↓(𝑏) 0 Reflection at x = b for spin -down electrons  \n𝑅↑(𝑎) 0 Reflection at x = a for spin -up electrons  \n𝑅↓(𝑎) 0 Reflection at x = a for spin -down electrons  \n𝑇↑(𝑎) 0.5 Transmission at x = a for spin -up electrons  \n𝑇↓(𝑎) 0.95 Transmission at x = a for spin -down electrons  \n𝑇↑↑(𝜃𝑚)  0 for 𝜃𝑚=𝜋,  \n 1 for 𝜃𝑚=0 Spin-conserved transmission at x = 0 for spin-up electrons  \n𝑇↓↓(𝜃𝑚) 0 for 𝜃𝑚=𝜋,  \n 1 for 𝜃𝑚=0 Spin-conserved transmission at x = 0 for spin -down electrons  \n𝑇↑↓(𝜃𝑚) 1 for 𝜃𝑚=𝜋,  \n 0 for 𝜃𝑚=0 Spin-flip transmission (from spin -up to spin -down) at x = 0  \n𝑇↓↑(𝜃𝑚) 1 for 𝜃𝑚=𝜋,  \n 0 for 𝜃𝑚=0 Spin-flip transmission (from spin -down to spin -up) at x = 0  \n𝑅↑(−𝑎) 0.4995  Reflection coefficient at x = −a for spin -up electrons  \n𝑅↓(−𝑎) 0.5005  Reflection coefficient at x = −a for spin -down electrons  \n \n \n "    },    {        "title": "1903.11422v1.Investigation_of_Room_Temperature_Ferroelectricity_and_Ferrimagnetism_in_Multiferroic_AlxFe2_xO3_Epitaxial_Thin_Films.pdf",        "content": "Investigation of Room Temperature Ferroelectricity and Ferrimag netism in Multiferroic \nAlxFe2-xO3 Epitaxial Thin Films \nBadari Narayana Rao1, Shintaro Yasui1, Tsukasa Katayama2, Ayako Taguchi3,4, Hiroki Moriwake3,4, \nYosuke Hamasaki5, Mitsuru Itoh1 \n1) Laboratory for Materials and Structures, Tokyo Institute of Tec hnology, 4259 Nagatsuta, \nMidori, Yokohama 226-8503, Japan \n2) Department of Chemistry, The University of Tokyo, Bunkyo-ku, To kyo 112-0033, Japan \n3) Nanostructures Research Laborator y, Japan Fine Ceramics Center,  Atsuta-ku, Nagoya 456-\n8587, Japan \n4) Center for Materials Researc h by Information Integration (CMI2) , Research and Services \nDivision of Materials Data and In tegrated System (MaDIS), Natio nal Institute for Materials \nScience (NIMS), 1-2-1 Senge n, Tsukuba, Ibaraki 305-0047, Japan \n5) Department of Applied Physics, National Defence Academy, Yokosu ka 239-8686, Japan \n \nAbstract: Multiferroic materials open up the possibility to design novel functionality in \nelectronic devices, with low ene rgy consumption. However, there  are very few materials that \nshow multiferroicity at room temperature, which is essential to  be practically useful.  \nAlxFe2-xO3 (x-AFO) thin films, belonging to the κ-Al 2O3 family are interesting because they \nshow room temperature ferrimagnetism and have a polar crystal s tructure. However, it is \ndifficult to realise its ferroelectric properties at room tempe rature, due to low resistivity of the \nfilms. In this work, we have deposited x-AFO (0.5  x  1) epitaxial thin films with low \nleakage, on SrTiO 3<111> substrates by Pulsed Laser Deposition. Magnetic measureme nts \nconfirmed room temperature ferri magnetism of the films, however  the Curie temperature was \nfound to be influenced by deposition conditions. First principl e calculations suggested that \nferroelectric domain switching oc curs through shearing of in-pl ane oxygen layers, and \npredicted a high polarization value of 24 μC/cm2. However, actual ferroelectric measurements \nshowed the polarization to be two order less. Presence of multi ple in-plane domains which \noppose polarization switching of adjacent domains, was found to  be the cause for the small \nobserved polarization. Comparing dielectric relaxation studies and ferroelectric \ncharacterization showed that oxygen-vacancy defects assist doma in wall motion, which in turn \nfacilitates polarization switching.  \nI. Introduction \nSingle-phase multiferroic materials have attracted considerable  attention among scientists, due to \nstrong drive in the industry towards device miniaturization and  prospect of new functionalities with \nlow energy consumption.\n1–6 However, most of the known multiferroic materials have very lo w \noperational temperatures, thereby limiting their application.7,8 Till date, only BiFeO 3 based \nmultiferroic materials have shown promising properties with acc eptable operational temperatures.9,10 \nHowever, the antiferromagnetic nature of BiFeO 3 makes it less favourable for application. In addition, \nthe high volatility of bismuth makes its fabrication difficult,  which has led to inconsistency in the \nresults obtained from different groups.11 The κ-Al 2O3-type family of oxides (e.g. GaFeO 3, ε-Fe 2O3, \nAlFeO 3), are one of the alternative systems possessing both ferrimagn etism and ferroelectricity.12–19 \nThe Ga xFe2-xO3 system was the first compound in this family to be discovered as multiferroic, as early \nas 1964.20 However, unlike GaFeO 3, other compounds in this family are metastable phases which \ncannot be synthesized easily, and hence did not garner much int erest. The recent advances in synthesis of nanoparticles and thin films have made stabilization of thes e phases possible. Multiferroic properties \nof these materials could be observed close to room temperature,  and are currently being explored for \nvarious applications.12,13,18–32 The ferrimagnetic nature of materials in this family make it a dvantageous \nover BiFeO 3, due to better magnetic properties.33 Orthorhombic Al xFe2-xO3 (x-AFO) with space group \nPna21, belongs to the same family of multiferroic oxides. This syste m is favourable, since it is made \nup of only ‘Al’ and ‘Fe’ cations, both of which are abundantly available, and are non-toxic in nature. \nRecently, thin films and nanoparticles of orthorhombic AlFeO 3 were successfully synthesized.14,15,34,35 \nHence, in the current work, we investigate the ferroelectric an d magnetic properties of x-AFO epitaxial \nthin films deposited by pulsed l aser deposition (PLD). \nThe orthorhombic structure of x-AFO is best described as that  consisting of combination of hex agonal \nand cubic close-packing of oxygen ions. It contains one corner sharing tetrahedral site (Al1), one \nregular octahedral (Al2) and two heavily distorted octahedral s ites (Fe1 and Fe2) which are edge shared \n[Fig. 1].  Though the cation sites are disordered in nature, the Al1 and A l2 sites are predominantly \noccupied by Al3+ due to their smaller size, while Fe3+ p r e f e r s  t h e  F e 1  a n d  F e 2  s i t e s .14,33–35 The \nferrimagnetism in x-AFO originates from the strong superexchange antiferromagnetic  interactions \nbetween Fe ions,14 where the Fe ion magnetic momen t of Fe1 and Al1 sites are anti parallel to those at \nFe2 and Al2 sites. In the case of ε-Fe 2O3, it has been recently suggested that the spins of Fe3+ in the \nAl1 site is non-collinear with res pect to other sites, thereby inducing a larger magnetization in the \nsystem.36 However, this effect may not be significant in the x-AFO system, since the occupancy of \nFe3+ in the Al1 site is small. A net magnetic moment in x-AFO mainly arises due to unequal distribution \nof Fe3+ in the four cation sites. \nWhile there are some research articles available on polycrystal line AlFeO 3 ceramics as well as thin \nfilms that have confirmed ferrimagnetism,34,35,37 the evidence for piezoelectricity or ferroelectricity \nhave not been very convincing.13,38 Even the ferroelectricity in similar systems like GaFeO 3 and  \nε-Fe 2O3 is puzzling, since the experime ntally observed polarization va lues were considerably less than \nthat predicted by ab-initio calculations.17,36,39 Recently, Hamasaki et al.  successfully synthesized \nepitaxial thin films of x-AFO on SrTiO 3(111) substrates.15 However, high leakage currents in the films \nmade direct ferroelectric measurements difficult, and only loca l information using piezoresponse force \nmicroscopy (PFM) could be obtained.35 Due to this problem, very limited work on the ferroelectric \nand dielectric property measurements of x-AFO is available, thereby making such a study very \ninteresting. The low resistance in  thin films is generally due to high density of defects like oxygen-ion \nvacancies. The oxygen vacancies can be minimized either by tuni ng the deposition conditions or by \nsuitable cation doping.18,40–42 In the present work, we successfully optimized the deposition conditions \nto obtain x-AFO (0.5  x  1) films with low leakage current, thereby enabling ferroelect ric and \ndielectric measurements. While first-principle calculations aid ed in understanding the mechanism of \nferroelectric switching, the dis crepancy in the polarization va lues obtained from theory and \nexperiments is attributed to c onstraints posed by domains. \nII. Experiment \nA. Optimization of thin film deposition \nx-AFO (0.5  x  1) films were deposited on STO( 111) single crystal substrates by pulsed laser \ndeposition (PLD) using fourth-harmonic wave of a Nd:YAG laser ( λ = 266 nm) with repetition rate of \n5 Hz. The films were also deposited on 0.5 wt% Nb-doped STO(111 ) (Nb:STO) conducting substrates \nfor ferroelectric and capacitance measurements. As a PLD target , we used x-AFO ceramic pellets \nprepared by solid state synthesis (sintered at 1450°C for 14 ho urs). Several x-AFO films were deposited, and ideal conditions were identified by systematic v ariation of different parameters such as \nlaser fluence (1 – 4 J/cm2), oxygen partial pressure (10 mTorr – 500 mTorr), substrate te mperature \n(650°C – 750°C) and annealing method. Single phase films were o btained in all the tested conditions, \nindicating a wide range of phase s tability for the system. It w as found that the deposition temperature \nis the critical parameter to obtain smooth films, while oxygen pressure ( PO2) during deposition and \nannealing is important for obtai ning single phase and controlli ng oxygen vacancies. A low deposition \nrate is favourable to obtain f ilms with good electrical propert ies, which is controlled by laser fluence, \nPO2 and target-substrate distance. A t higher laser fluence, it was  noted that large number of droplets \nejected from the target along with  the plume, which then deposi ted on the film as amorphous macro \nparticles, leading to poor quality films. A laser fluence of 1. 6 J/cm2 was found to be suitable for \ndeposition of films of all compositions. The chamber pressure i s an important parameter to control the \nshape of the plume, which improves the uniformity and smoothnes s of the film. An oxygen atmosphere \nhelps to maintain the stoichiometry of the oxides by reducing o xygen vacancies in the film. PO2 of 100 \n– 300 mTorr was found to be ideal to obtain good quality films with good deposition rate. The substrate \ntemperature controls the grain size and roughness of the films,43 and a substrate temperature of about \n710°C was found to be ideal for the film deposition. Annealing under high PO2 can decrease oxygen \nrelated defects in the film. How ever, optimum annealing tempera ture is important, since higher \nannealing temperature can lead to grain growth and increase in surface roughness. Annealing with PO2 \nof 100 Torr at 600°C for half an hour was found to be sufficien t to obtain good films. After careful \nconsideration, the following PLD conditions were used to deposi t the films for further characterization: \nlaser fluence of 1.6 J/cm2, PO2 of 100 mTorr during deposition, substrate temperature of 710°C , and \nannealing with PO2 of 100 Torr at 600°C for half an hour. The film thickness was a bout 20 – 25 nm for \nall the compositions. \nB. Thin film characterization \nThe crystal structure of the films was analysed by high-resolut ion X-ray diffraction of Rigaku Smartlab \nusing Cu-Kα 1 r a d i a t i o n .  T h e  f e r r o e l e c t r i c  m e a s u r e m e n t s  w e r e  c a r r i e d  o u t  u s i ng the Precision \nMultiferroic II tester (Radiant Inc.). The dielectric measureme nts were carried out using an LCR meter \n(Agilent, 4284A), while the samples were loaded inside a Physic al Property Measurement System \n(PPMS, Quantum Design Inc.). Out of plane piezoresponse force m icroscopy measurements were \ncarried out using a frequency tr acking DART mode of MFP-3D Asyl um Research microscope. For all \nelectric measurements, Pt top electrode (100 μm diameter) was d eposited on the films by electron \nbeam evaporation. While top-top el ectrode configuration was use d for ferroelectric measurements,44 \nthe rest of the electrical measurements used Pt as top electrod e, and the Nb:STO substrate as the bottom \nelectrode. The in-plane magnetizations of the films were measur ed using a superconducting quantum \ninterference device (S QUID) magnetometer (Quantum Design Co. MP MS XL). \nC. First principles calculation method \nThe Ab initio  calculations were performed by the projector-augmented wave (P AW) method within \nthe GGA+U formalism45 and the framework of dens ity functional theory (DFT)46,47, as implemented \nin the VASP code48,49. The exchange-correlation inter actions were treated by the gen eralized gradient \napproximation (GGA-PBE)50. The on-site Coulomb repulsion was treated at the GGA+ U level51. We \nadopted the Hubbard effective Ueff = 4.0 eV only for the Fe-3 d electrons. For the PAW potentials,  \nthe electronic configurations 3 d104s2 for Fe, 3 s23p1 for Al and 2 s22p6 for O were explicitly treated as \nvalence electrons. The plane wa ve expansion up to 600 eV was ad apted. A k-point mesh of 4×2×2 within the 40-atom unit cell was used for Brillouin zone sampli ng of primitive cells, which was based \non the Monkhorst-Pack scheme52. The lattice constants and internal atomic coordinates were \nconsidered fully optimized once the residual Hellmann-Feynman ( HF) forces were less than  \n1.0×10−2 eV/Å. The activation energy for this switching was determined using the nudged elastic band \n(NEB) method53. The polarization values wer e determined by Berry’s phase54 method implemented in \nthe ABINIT code55. \nIII. Results \nA. Structural characterization \nFig. 2(a) shows the out-of-plane 2 θ-θ XRD scan of 0.5-AFO film grown on STO(111) substrates, \nwhich shows that a single phase  is successfully o btained. Since  only 00 l peaks of the film are observed, \nit is clear that the film growth is c-axis-oriented. The thickness fringes observed in the 004 peak (inset \nof Fig. 2(a)) indicate smooth f ilm, and was observed for all th e compositions studied. The in-plane \ncrystal-domain orientations of th e films were evaluated using φ -scan about the x-AFO{201} and \nSTO{110} diffraction peaks [Fig. 2(b)]. The film showed six-fol d in-plane symmetry of the {201} \npeak, indicating three types of in-plane domains, where each [1 00] Film direction is parallel to the [11-\n2]STO, [1-21] STO, or [-211] STO direction, as illustrated in Fig. 2(c). These results are cons istent with \nprevious reports of AFO and GaFeO 3 based films on STO(111) substrates.15,56 Fig. 2(d) shows the \nvariation of the lattice paramete rs obtained from in-plane and out-of-plane X-ray diffraction, as a \nfunction of composition. A decrease in unit cell volume is obse rved with increasing Al-content  \n(Fig. 2(e)), which is consistent  with Vegard’s law, as the ioni c radii of Al3+ is smaller than Fe3+. \nB. Magnetic properties \nFigure 3 shows the magnetic properties of x-AFO films. Fig 3(a) shows the field cooled temperature \ndependence of magnetization ( MT) for different compositions, indicating that the Curie tempera ture \nfor all the films is above 400 K. Since the maximum operational  temperature of the SQUID was 400K, \nthe exact Curie temperature could not be determined. However, t he magnetic measurements on films \ngrown at 300 mTorr oxygen pressure showed relatively lower Curi e temperatures [Fig. S1], indicating \nthat Curie point can be tuned by varying oxygen pressure. From Fig. [S1], it is also clear that the \nmagnetic Curie temperature decreases with increasing x, similar to other reports.19,35 Figure 4(b) shows \nthe room temperature magnetizat ion vs. magnetic field hysteresi s (MH) plots for x = 0.5, 0.8 and 0.9, \nrevealing their ferrimagnetic nat ure. The inset in fig. 4(b) sh ows the zoomed plot of the MH curve, \nindicating higher coercive f ield for lower value of x. Figure 4(c) shows the actual variation of the \nmagnetic coercive field and the saturation magnetization as a f unction of composition. While the \ncoercive field continuously decreased with increasing x, the saturation magnetization was maximum \nat 0.8-AFO. Pure ε-Fe 2O3 (x = 0), has a very high coercive field due to the strong hybridi zation of  \nFe 3d5 at the Fe2 site with O 2 p orbital, resulting from large spin-orbit interaction57. The decrease in \ncoercive field with increasing x (decreasing Fe concentration) must be due to weakening of this  \nphenomenon, since the system is moving away from ε-Fe 2O3. The reason for maxima in saturation \nmagnetization at x = 0.8 can be attributed to the differential occupation of Al3+ in each of the four \ncation sites. This can be explained using figure 3(d), which sh ows an illustration of the magnetic \nmoments of the four cation sites. When x = 0 (pure ε-Fe 2O3), all the sites are completely occupied by \nFe3+ ions. In this situatio n, sites Fe1 and Al1 have their spins al igned in one direction and sites Fe2 \nand Al2 have the spins aligned in the opposite direction. The n on-collinear magnetic moment in the \ntetrahedral site results in a non-zero magnetic moment.36 As x increases, for lower values of x (x < 0.8), \nthe Al3+ preferentially occupy the Al1 sites. Since Al1 and Fe1 sites a re antiparallel to Al2 and Fe2 sites, the net magnetic moment is given by the algebraic sum of  moments from all the sites:  \nMAl2 + M Fe2 - M Al1 - M Fe1. Hence, with increasing x, the magnetic moment contribution from Al1 site \ndecreases, thereby increasing the net magnetic moment. However,  as x increases further ( x > 0.8), Al \nions begin to occupy the Al2 site s also, consequently decreasin g the net magnetic moment. \nC. Ferroelectric Properties \nWhile the x-AFO system has a non-centrosymmetric structure, its ferroelect ricity has never been \nadequately verified. The films are prone to leakage, which make s ferroelectric measurements difficult. \nThough we could observe domain switching as well as butterfly a mplitude loop in PFM measurements \n(Fig. S2), it is not sufficient to prove its ferroelectric natu re. This is because non-ferroelectric surfaces \nare also known to show contrast in PFM measurements under certa in conditions.58–61 Hence, we \nfocussed on direct ferroelectric measurements, which was possib le by obtaining films with improved \nleakage properties. Fig. 4 show s the polarization vs. electric- field ( PE) hysteresis loops for different \ncompositions of the films. It can be seen that all compositions  showed good hysteresis loops, and \ndomain switching is confirmed by peak in the current vs. electr ic-field ( IE) plot, corresponding to the \ncoercive field. However, it was observed that the hysteresis lo ops from these films do not saturate until \nthe breakdown field, as shown in Fig. 5a-b. Figure 5a shows the  plot of PE hysteresis measured with \nincreasing maximum electric field for x = 1, and fig. 5b confirms that both the remnant polarization a s \nwell as the coercive field do not saturate with the electric fi eld. This behaviour is different compared \nto conventional ferroelectrics, which show sudden anomaly in po larization above coercive field, and \nsaturates at higher fields. Absence of sudden jump in remnant p olarization of our films indicates that \nthe polarization switching mechanism may be different as compar ed to conventional ferroelectrics. A \ncontinuous increase in the polari zation with increasing electri c field indicates major contribution from \nparaelectric effect, which ideally has a linear relationship wi th electric field. The presence of the \nparaelectric component in the polarization of our film was furt her confirmed by the fact that the shape \nof the PE loops became slimmer with decreasing frequency [Fig. S3]. Howe ver, the peak in the IE \ncurves indicate polarization switching, which means that sponta neous polarization due to \nferroelectricity is also present. \nTo further investigate ferroelectricity in x-AFO films, the paraelectric and ferroelectric components of \npolarization were separated out by the Positive Up Negative Dow n (PUND) measurement technique62. \nFig. 5c and 5d shows comparison of the remnant polarization obt ained from PE hysteresis \nmeasurements and PUND measurement for 0.5-AFO film. We can see that the actual remnant \npolarization as obtained from PUND measurement is smaller than that determined by PE loops. Similar \nresults were obtained for other compositions of x-AFO as well. This proves our earlier proposition that \na large component of the polariz ation arises from the paraelect ric component. It must be noted that, \nthe PE loop and PUND measurement results  were susceptible to minor un avoidable differences in \ndeposition conditions. Even two films of same composition, whic h were deposited separately, showed \nslight variations in their polarization values. As a result, we  did not notice any significant composition \ndependence of the ferro electric properties. \nD. Dielectric Properties \nFigure 6 shows the dielectric da ta obtained for 0.5-AFO film be tween 2 to 350 K in the frequency \nrange from 100 Hz to 1 MHz. Frequency dispersion is observed ov er a wide temperature range, \nbeginning at about 150 K and c ontinuing well above room tempera ture. Such large dispersion is often \nattributed to motion of ferro electric domain boundaries.63 All compositions studied (0.5 x1) showed \nsimilar behavior, with small shifts in the temperatures corresp onding to the peaks in dielectric loss  (fig. 7 (a-c)). Figure 7d and 7e shows the temperature dependen ce of imaginary part of dielectric \nconstant as a function of frequency, for x = 0.5 and 1 respectively. It can be seen from the figure that \nthe dispersion occurs over a wide range of frequency. The origi n of the relaxation in the films were \nfurther analyzed by modeling the frequency dependence of the pe ak positions in the dielectric loss \ncurves, using an Arrehenius relation (Fig. 8(f)): 𝐹ൌ𝐹୭𝑒𝑥𝑝ቂିாೌ\nಳ்ቃ, where F is the measuring frequency, \nEa is the activation energy, and Fo is the attempt jump frequency. An activation energy of about  \n0.37 eV, and Fo of the order of 1010 Hz were obtained for all the compositions (Table 1). Since the  \nmobility of Al and Fe ions are negligible at such low temperatu res, we associate the relaxation process \nto be dominated by oxygen vacancies. These oxygen vacancies gen erally aggrega te near domain \nboundaries, and since TEM observations showed the domain size t o be very small (5-10 nm)35, the \ndefect density in the film could be very high. Thus oxygen vaca ncies can contribute  significantly to \nthe electrical properties of the film. We propose the ferroelec tric domain motion to be assisted by \nelectron hopping through the Fe2+-V•o-Fe3+ route, where V•o denotes electron-trapped oxygen vacancy \nfollowing the Kröger–Vink notation. This is very likely since o xygen has a rather small first ionization \nenergy (0.1 eV)64. Similar order of activation energy for electron hopping throu gh the oxygen vacancy \nhas also been observed by Ke et al. for (La,Mg) substituted BiFeO 365, by Ikeda et al.  for LuFe 2O463, \nand by Katayama et al.  for Ga xFe2-xO319.    \nSince the dielectric relaxation is found to be associated with domain motion, which in turn is assisted \nby Fe2+-V•o-Fe3+ hopping, we tried to correlate the results from dielectric ana lysis and ferroelectric \nmeasurements. It can be seen from fig. 8 that at any particular  temperature, a clear PE hysteresis is \nobtained only at frequencies clos e to the relaxation frequency observed in the dielectric data. At lower \ntemperatuers, the relaxation is  observed at lower frequencies, and consequently, a good PE hysteresis \nloop is also obtained at the same frequency. From the above obs ervation, it is clear that polarization \nswitching is intricately correla ted to oxygen vacancy defects, which are usually found in the vicinity \nof domain boundaries. Upon application of electric field, the l ocal electric dipole formed by these \ndefects trigger the actual ferro electric domain switching proce ss (shown in Fig. 9). Since the defects \nare most mobile at their relaxation frequency, even the polariz ation response is best observed at this \nfrequency (Fig. 8).  \nE. First-Principles Calculation \nTheoretically determined activation energies and polarization s witching mechanisms of x-AFO are \ndiscussed based on ab initio  calculations performed on κ-Al 2O3 and ε-Fe 2O3. One possible mechanism \nof polarization switching of κ-Al 2O3 type x-AFO is via an intermediate non-polar centrosymmetric \nstate17,66. Earlier, Stoeffler et al. calculated the activation ener gy and net polarization of isos tructural \nGaFeO 3 by considering a Pnna  space group as the intermediate state17. However, the reported \nactivation energy for the polar ization switching was 0.5 eV, wh ich is much larger than that seen in \nconventional ferroelectric compounds (e.g. BaTiO 3 – 0.02 eV67, PbTiO 3 – 0.03 eV68). Xu et al.  \nsuggested an alternative centrosymmetric space group Pbcn , which gave a much lower activation \nenergy for polarization in ε-Fe 2O336. Hence, we considered the Pbcn  space group as the non-polar \npolarized structure for our cal culation. Figure 9(a - e) show t he schematic of transition from a \nnegatively  polarized structure to centrosymmetric, and then to a positivel y polarized structure. The \ncalculation yielded activation en ergies for polarization switch ing of 0.088 and 0.155 eV/f.u for ε-Fe 2O3 \nand κ-Al 2O3 respectively (Fig. 10). We can e xpect the activation energies for the intermediate x-AFO \nstructures also to be of similar order. These values are fairly  small and acceptable, compared to the \nhigh value previously reported for GaFeO 317,66. During the polarization reversal process, the polarization switches from – Ps to + Ps, while smoothly passing through zero (Fig. 9(e & h). Viewing \nthe structure along the b-axis clearly explains the polar ization switching mechanism (Fi g 9(f-j)). Close-\npacked oxygen layers in corundum l ayers keep their octahedral s hape during the sw itching. However, \noxygens above and below the corundum layers shift along a-axis, in opposite directions relative to \neach other. This shearing motion of oxygen layers induces a coo rdination switching of cations Fe1 and \nAl1 sites. An originally tetrahedral(octahedral) Al1(Al2) site turns into octahedral(tetrahedral) \nAl2(Al1) site after the polari zation switching. This mechanism is quite different from conventional \nferroelectric perovskite oxides, where cations and anions move in opposite directions in a linear \nmanner (Slater mode69).  \nBy using the Berry’s phase approach54, the polarization of Al 2O3 and ε-Fe 2O3 was calculated to be \nabout 26 μC/cm2 and 21 μC/cm2 respectively. Since there is no structure change in the substit uted  \nx-AFO series, their theoretical polarization values will also li e in between 21 - 26 μC/cm2. While this \nvalue of polarization is comparable to theoretical values repor ted for other isostructural compounds \nlike GaFeO 3 and ε-Fe 2O3,17,36,39,66 it is about two orders of magnitude larger than that observed \nexperimentally. Similar ambiguity is observed in GaFeO 3 based films as well, and the exact reason for \nthis is not yet known. We speculate that the multi-domain struc ture of the thin films obstruct complete \npolarization reversal , and hence the actual polarization is con siderably lesser than that predicted. \nIV. Discussion \nAmong all the known multferroic s in the world, the BiFeO 3 system has attracted the largest attention, \ndue to its high Néel’s temperature and large ferroelectric pola rization. However, problems like the \nantiferromagnetic ordering of BiFeO 3 and volatility of Bi during fabrication makes it unattractive for \nmagnetic or magnetoelectric applications. Hence, it is necessar y to identify other potential multiferroic \nmaterials, thereby giving more flexibility for the electronic i ndustry. κ-Al 2O3 type ferrites like ε-Fe 2O3, \nGaFeO 3 and AlFeO 3 have recently been identified to be promising multiferroics.18,19,23,26 Especially, \nthe ferrimagnetic nature of these ferrites, which can be stabil ized above room temperature, is a prime \nadvantage over BiFeO 3. While the large coercive field and magnetic anisotropy of ε-Fe 2O3 has already \nmade it interesting for high-frequency millimeter wave absorpti on,23 the research on ferroelectric and \nmultiferroic properties of these ferrites is still in its nasce nt stage. Recently, Katayama et. al. showed \nthat the properties of κ-Al 2O3 type GaFeO 3 can be tuned by suitable cation substitution, to obtain \nexcellent ferroelectric and multiferroic properties at room tem perature.18 The x-AFO system is made \nof only Al and Fe ions, both of which are abundantly available and are non-toxic in nature. Hence, it \ncan be a potential gamechanger in the electronic industry, if g ood ferroelectric and magnetic properties \ncan be established in this system. The magnetic properties are easier to be tuned, and a Curie \ntemperature above room temperatu re could be established. The Cu rie temperature can be tuned either \nby cation substitution or by varying oxygen vacancy concentrati on. Jaffe et al.  suggested that n-type \ncarriers present due to oxygen vacancy in semiconducting ferrom agnets help mediate magnetic \ninteractions between spins70. A similar phenomena may be effecting the x-AFO system as well, as a \nconsequence of which the Curie  temperature is influenced by oxy gen vacancy concentration71. \nThough theoretical predictions state much larger polarization ( ~20-24 μC/cm2), the actual value is \nabout two order less. This ambiguity has been observed for othe r systems in the family (GaFeO 3,  \nε-Fe 2O3) as well.17,36 We suggest that the existence of multiple in-plane domains in the film could be \nthe reason behind reduced polarization. As we have shown using first-principle calculation that the \ndomain switching along c-axis (out-of-plane) takes place by shearing of oxygen layers a long a-axis, \npresence of multiple in-plane dom ains will make such a shearing  very difficult. The  X-ray diffraction -scans of the films in Fig. 2b clearly show presence of three t ypes of crystal domains, adhering to the \n3-fold symmetry of the STO(111) substrate surface. Any attempt of shearing of oxygen from  \ndomain 1 will be constricted by domains 2 and 3, and likewise, shearing in domains 2 and 3 will also \nbe restricted (Fig. 11). Hence, when electric field is applied,  the domains can orient only to a small \nextent, and they tend to go back to the original position upon removal of the elect ric field. This model \ncan simultaneously explain the low polarization values, the lar ge paraelectric contribution to the \npolarization, as well as the cor relation of polarization to die lectric relaxation of the films. If we can \ngrow films with single domain, then it is likely that polarizat ion values close to that of theoretical \ncalculations can be obtained. An e xtensive work on domain engin eering is required to obtain single \ndomain films of x-AFO. Preliminary work by Katayama et al.  on Ga 0.6Fe1.4O3 films showed that \nSTO(111) yields the minimum number of in-plane domains compared  to several other substrates.56 \nAlso, using STO(100) and STO(110) o riented substrates yielded s ix in-plane domains, which is double \nof that obtained in STO(111). He nce, among all the substrates s tudied so far, STO(111) yields the least \nnumber of in-plane domain types. Further research on more subst rates and deposition conditions is \nrequired to achieve single domain films. \nXu et al.  have shown by theoretical calculations that cation size is an important factor in stabilizing \nthe ferroelectric phase.36 A decrease in the cation size stabilizes the ferroelectric pha se, and since the \nradius of Al3+ (0.535 Å) is smaller than that of Fe3+ (0.55Å and 0.645Å in high and low spin state \nrespectively), increasing x should improve the ferroelectric property of the system. Howev er, in the \npresent study, we could not establish any composition dependenc e of ferroelectric properties. Since \nthe polarization revers al in these systems occurs in an indirec t manner, many intricate parameters like \noxygen vacancy concentration, occupancy of each cation site, de fect population, etc, may supersede \nthe effect due to cation size. Nevertheless, room temperature f erroelectricity has been clearly \ndemonstrated in the x-AFO films. The system also shows dielectric dispersion, with t he activation \nenergy about 0.37 eV, correspondin g to hopping between localize d charge carriers. In comparison to \nBiFeO 3, these systems have better magnetic properties, and hence prom ise to be more beneficial for \nmultiferroic applications.  \nV. Conclusion \nRoom temperature ferroelectric ity and ferrimagnetism has been e stablished in the x-AFO system. The \nferroelectric response of the films is highly frequency depende nt, and the deposition conditions have \nremarkable influence on the nature of the PE hysteresis loops. PUND measurements proved to be \nbetter at providing reliable remna nt polarization values. The l ow polarization in the films could be \nattributed to constraints posed by multiple in-plane domains. H owever, we hope tha t the current work \nwill lead to further development in fabrication of single domai n films, which can then yield \npolarization values close to the theoretical ones. The magnetic  measurements were consistent with \nother works, and the Curie temperature and coercive field were found to decrease with increasing x. \nThe room temperature magnetism and ferroelectricity of the x-AFO system, which comprises of \ninexpensive and nontoxic raw materi als, makes this system promi sing for multiferroic applications.  \nAcknowledgements \nB.N.R acknowledges fellowship support by JSPS(P17079). This wor k was partly supported by \nJSPS KAKENHI Grants-in-Aid for challenging Research (Pioneering ) (M.I., 1706420), (Exploratory) \n(Sh.Y., 18K19126), and MEXT Elements Strategy Initiative to for m Core Research Centre, \nCollaborative Research Project of Laboratory for Materials and Structures, Institute of Innovative \nResearch, Tokyo Institute of Technology. H.M acknowledges \"Mate rials research by Information Integration” Initiative (MI2I) pr oject of the Support Program f or Starting Up Innovation Hub from \nJapan Science and Technology Agency (JST). \nReferences \n1 W. Eerenstein, N.D. Mat hur, and J.F. Scott, Nature 442, 759 (2006). \n2 R. Ramesh and N.A. Spaldin, Nat. Mater. 6, 21 (2007). \n3 J.F. Scott, Nat. Mater. 6, 256 (2007). \n4 J.F. Scott, J. Mater. Chem. 22, 4567 (2012). \n5 T. Nan and N.X. Sun, in Compos. Magnetoelectrics , edited by G. Srinivasan, S. Priya, and N.X. \nSun (Woodhead Publishing, 2015), pp. 329–356. \n6 M. Fiebig, T. Lottermoser, D. Meier, and M. 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Itoh, MRS Adv. 1-6 ( 2019),  \nDOI: 10.1557/adv.2019.121. \n\nFigures \n \nFigure 1 : Crystal structure model of orthorhombic x-AFO with space group Pna21. Al1 indicates the \ntetrahedral cation site, whereas  Al2, Fe1 and Fe2 indicate the octahedral cation site. P and M indicate \nthe direction of ferroelectric  polarization and magnetization r espectively.  \n \nFigure 2 : a) out-of-plane XRD pattern o f 0.5-AFO film ( * indicates subs trate peaks). Inset shows the \nexpanded view of 004 peak. b) phi scans of AFO film about 201 r eflection and STO(111) substrate \nabout 110 reflection. c) schema tic of orientation relationship between film domains and the substrate. \n(d) Variation of lattice pa rameters as a function of x. e) Unit cell volume as a function of x.  \nFigure 3 . a) Field cooled magnetization vs temperature plot for various  compositions ( MT), measured \nwith a constant magnetic field of 500 Oe. b) Room temperature m agnetization vs magnetic field ( MH) \nfor different compositions, inset shows the zoomed version of t he same plot to highlight the differences \nin the coercive field. c) Compositional variation of saturation  magnetization ( Ms) and coercive field \n(Hc), at room temperature. d) Schem atic of the direction of magnet ic moments of Fe3+ ions at each site. \n \nFigure 4.  (a-f) Polarization vs Electric field ( PE) and current dependence of electric field ( IE) for \nvarious compositions of the film, collected at 10 kHz.  \nFigure 5.  (a) PE hysteresis loops of 1-AFO at 10 kHz, with increasing ma ximum electric fields. (b) \nVariation of remnant polarizati on and coercive field of 1-AFO a s a function of maximum electric field, \nas obtained from (a). (c) Remnant polarization of 0.5-AFO as ca lculated from PE hysteresis. (d) \nRemnant polarization of 0.5-AFO as  calculated from PUND measure ment. \n \nFigure 6.  a) Dielectric constant and diel ectric loss as a function of te mperature and frequency, for \n0.5-AFO   \nFigure 7.  (a) Variation of real part of dielectric constant with tempera ture. (b,c) Variation of loss \ntangent with temperature at 1 kHz and 10 kHz respectively. (d,e ) variation of imaginary part of \ndielectric constant with frequency for x = 0.5 and 1 respectively, clearly depicting the frequency \ndispersion. (f) Plot of ln(Peak Frequency) vs inverse temperatu re, to show that the system follows \nArrhenius relation. \n \nFigure 8.  Comparison of dielectric plot and PE-hysteresis at different tempe rature and frequencies for  \nx = 0.5.  \nFigure 9.  Illustration of struc tural changes upon polar ization reversal.  (a-e) structure  viewed along a-\naxis, (f-j) structure viewed along b-axis. Yellow, green, and blue shapes indicate tetrahedral, \npentahedra, and octahedra respectively. \n \nFigure 10.  Variation of relative energies of κ-Al 2O3 and ε-Fe 2O3 during polarization switching through \nan intermediate centrosymmetric structure Pbcn . \n \nFigure 11.  Illustration of constriction of the shearing of the oxygen lay ers, in the multi-domain \nstructure. \nTable 1. Activation energy ( Ea) and attempt jump frequency ( Fo) for various compositions of x-AFO, \nobtained by Arrhenius fitting of temperature dependent dielectr ic loss data. \nComposition Ea (eV) Fo (Hz) x 1010 \n0.5-AFO 0.363 4.51 \n0.6-AFO 0.378 7.76 \n0.7-AFO 0.353 5.52 \n0.8-AFO 0.402 11.03 \n1-AFO 0.399 8.75 \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Supporting Information  \n \n“Investigation of Room Temperature Ferroelectricity and Ferrima gnetism in Multiferroic \nAlxFe2-xO3 Epitaxial Thin Films” \nBadari Narayana Rao1, Shintaro Yasui1, Tsukasa Katayama2, Ayako Taguchi3,4, Hiroki Moriwake3,4, \nYosuke Hamasaki5, Mitsuru Itoh1 \n1) Laboratory for Materials and Structures, Tokyo Institute of Tec hnology, 4259 Nagatsuta, \nMidori, Yokohama 226-8503, Japan \n2) Department of Chemistry, The University of Tokyo, Bunkyo-ku, To kyo 112-0033, Japan \n3) Nanostructures Research Laborator y, Japan Fine Ceramics Center,  Atsuta-ku, Nagoya 456-\n8587, Japan \n4) Center for Materials Researc h by Information Integration (CMI2) , Research and Services \nDivision of Materials Data and In tegrated System (MaDIS), Natio nal Institute for Materials \nScience (NIMS), 1-2-1 Senge n, Tsukuba, Ibaraki 305-0047, Japan \n5) Department of Applied Physics, National Defence Academy, Yokosu ka 239-8686, Japan \n \n \n \n \n \nFigure S1.  Normalized Magnetization vs. Te mperature plot for films grown at 300 mTorr PO 2, \nshowing lower Curie temperature than in Fig. 3, as well as decr ease in Curie temperature with \nincreasing x. \n \nFigure S2.  The phase and amplitude curves of  0.5-AFO films, obtained from  piezoresponse force \nmicroscopy, clearly depicting domain switching. \n \n \nFigure S3:  Frequency dependence of PE loop, for x = 0.5, showing slimming of the PE loop with \ndecrease in frequency. \n"    },    {        "title": "1203.2782v2.Magnetic_ground_state_and_2D_behavior_in_pseudo_Kagome_layered_system_Cu3Bi_SeO3_2O2Br.pdf",        "content": "arXiv:1203.2782v2  [cond-mat.str-el]  15 Oct 2012Magnetic ground state and 2D behavior in pseudo-Kagom´ e lay ered system\nCu3Bi(SeO 3)2O2Br\nM. Pregelj\nJosef Stefan Institute, Ljubljana, Slovenia and\nLaboratory for Neutron Scattering, Paul Scherrer Insitute , CH-5232 Villigen, Switzerland\nO. Zaharko\nLaboratory for Neutron Scattering, Paul Scherrer Insitute , CH-5232 Villigen, Switzerland\nA. G¨ unther and A. Loidl\nExperimental Physics V, Center for Electronic Correlation s and Magnetism,\nUniversity of Augsburg, 86135 Augsburg, Germany\nV. Tsurkan\nExperimental Physics V, Center for Electronic Correlation s and Magnetism,\nUniversity of Augsburg, 86135 Augsburg, Germany and\nInstitute of Applied Physics, Academy of Sciences of Moldov a, MD-2028 Chisinau, Republic of Moldova\nS. Guerrero\nCondensed Matter Theory, Paul Scherrer Insitute, CH-5232 V illigen, Switzerland\n(Dated: November 20, 2018)\nAnisotropic magnetic properties of a layered Kagom´ e-like system Cu 3Bi(SeO 3)2O2Br have been\nstudied by bulk magnetization and magnetic susceptibility measurements as well as powder and sin-\ngle crystal neutron diffraction. At TN=27.4K the system develops an alternating antiferromagnet ic\norder of ( ab) layers, which individually exhibit canted ferrimagnetic moment arrangement, resulting\nfrom the competing ferro- and antiferro-magnetic intralay er exchange interactions. A magnetic field\nBC∼0.8T applied along the c-axis (perpendicular to the layers) triggers a metamagneti c transi-\ntion, when every second layer flips, i.e., resulting in a ferr imagnetic structure. Significantly higher\nfields are required to rotate the ferromagnetic component to wards the b-axis (∼7T) or towards the\na-axis (∼15T). The estimates of the exchange coupling constants and f eatures indicative of a XY\ncharacter of this quasi 2D system are presented.\nI. INTRODUCTION\nThe interest for novel frustrated layered compounds\nstems from their compelling magnetic properties, which\nattract applied as well as basic science oriented research.\nIn case of a strong ferromagnetic (FM) intralayer ex-\nchange and a weak antiferromagnetic (AFM) interlayer\ncoupling, the magnetic ground state of AFM arranged\nFM layers can be easily broken by external magnetic\nfields, which enable simple switching between zero and\nmaximum magnetization. The apparent metamagnetic1\nresponse – an abrupt change of the bulk magnetization –\nis thus most appreciated in high-density magnetic stor-\nage and spintronics devices.2Such systems are often de-\nscribed by models on the two-dimensional (2D) lattice\nwith spin dimensionality n=1 (Ising), n=2 (XY) or\nn=3 (Heisenberg),3which are interesting also from the\ntheoretical point of view. The 2D XY model exhibits a\nunique feature – a Kosterlitz-Thouless transition4from\na paramagnetic state to a phase with quasi long-range\nspinorderwithvortexandantivortexexitations. Oncon-\ntrary, the 2D Ising model shows conventional long-range\norder,5whereas the 2D Heisenberg model does not order\nat anyfinite temperature.6Yet, realmaterialsareusually\nmore complex. Frequently the planar rotational symme-try is imperfect, leading to a quasi XY behavior. The\ninter-plane exchange coupling is often sufficiently strong\nto induce 3D ordering, so also spatially such systems\nare only quasi 2D. Finally, layered systems, particularly\nfrustrated ones,7are highly susceptible to small external\nperturbations8allowing to sweep across spin- and space-\ndegrees of freedom.9\nA novel layered compound with seemingly frustrated\nmagnetic lattice is Cu 3Bi(SeO 3)2O2Br. This compound\nisorthorhombic(spacegroup Pmmn)with crystallattice\nparameters a=6.390˚A,b=9.694˚A andc=7.287˚A.10It\nis built of two different types of [CuO 4] square plackets,\nsharing apices to form copper-oxygen layers reminiscent\nofabuckledKagom´ elatticeofthemagneticCu2+(S=1\n2)\nions (Fig.1), with Cu1 and Cu2 positioned at (000) and\n(1\n41\n4z,z=0.791), i.e., at the 4(c) and the 2(a) sites, re-\nspectively. Hence, in case of the AFM nearest-neighbor\ninteractions, Cu2+spins should be exposed to a strong\ngeometricalfrustration. However, additional Cu-O- X-O-\nCu (X=Se, Bi) super-superexchange interactions might\nbe important,11as the [CuO 4] plackets are linked also\nby Se4+and Bi3+ions (Fig.1). In fact, Bi3+enables\nadditional Cu1-O-Bi-O-Cu1 next-nearest-neighbor inter-\naction along the baxis, whereas Se4+provides only an\nalternative to the already recognized Cu-O-Cu nearest-2\nFIG. 1. (Color online) (a) aband (b) bcprojections of the\nCu3Bi(SeO 3)2O2Brcrystal layer. Forclaritythe abprojection\nis slightly canted. Here the magnetic Cu1 and Cu2 ions are\nindicated as small blue and red spheres, respectively; O ion s\nare thesmallest violet spheresinthecorners ofCuO 4plackets,\nBi ions are intermediate light pink spheres, Se ions are larg er\nlight green spheres, and Br ions are the largest dark green\nspheres.\nneighbor interactions. Finally, since both, Se4+and\nBi3+, possess lone pair electrons and thus effectively re-\nduce the number of chemical bonds,12the only probable\ninterlayer coupling is through a long Bi3+-O bond.\nEarlier powder magnetic susceptibility,10measured at\n1T, suggests dominant FM interactions, as its high-\ntemperature behavior above 150K follows a Curie-Weiss\n(CW) law χ=C/(T−θ) with a FM Weiss temperature\nθ≈57K. At lower temperatures, a weak anomaly, asso-\nciated with tiny structural changes,10occurs at ∼120K,\nwhereas a sharp step, implying establishment of long-\nrange magnetic order, is found at TN≈24K.\nHere we present a detailed magnetic susceptibility,\nmagnetization and neutron diffraction study of the low-\ntemperature magnetic ground states and metamagnetic\ntransition in Cu 3Bi(SeO 3)2O2Br. We find that below\nTNindividual ( ab) layers with a canted ferrimagnetic\nspin arrangement order antiferromagnetically. When\nBC∼0.8T is applied perpendicular to the layers (along\nc), weak AFM interlayer interactions are suppressed and\na metamagnetic transition is triggered, as every second\nlayer flips – resulting in an overall canted ferrimagnetic\nstructure. Based on the determined spin arrangementsand critical fields of the metamagnetic transition we pro-\nvide an estimate of the characteristic exchange couplings\nofthesystem. Interestingly,thoughtheratiobetweenthe\ninterlayer and the effective intralayer couplings is signifi-\ncant,|J′/J|∼0.006, and the magnetic anisotropy repre-\nsentsonly ∼20% ofJ, the lowtemperature experimental\ndata sustain a quasi 2D XY behavior.\nII. EXPERIMENTAL DETAILS\nPolycrystalline material of Cu 3Bi(SeO 3)2O2Br was\nprepared by solid-state reactionsat 550◦C from the high-\npurity binary compounds. Single crystals were grown at\n500 - 550◦C by the chemical-transport-reaction method\nwith bromine as the transport agent. Magnetization\n(M) and dc susceptibility ( χ=M/H) measurements\nwere performed in commercial MPMS and PPMS\nmagnetometers in the temperature range 1.8K - 400K\nand applied magnetic fields up to 8 T.\nNeutron diffraction experiments were performed at\nSwiss Neutron Spallation Source SINQ, PaulScherrerIn-\nstitute, Switzerland. Powder diffraction patterns were\ncollected between 1.5K and 60K on the DMC powder\ndiffractometer with neutron wavelength λ= 2.46˚A. Sin-\ngle crystal diffraction was performed in the temperature\nrange between 6.5K and 300K on the TriCS single crys-\ntal diffractometer ( λ= 1.18˚A). The zero-field data col-\nlection wasperformed in a cooling machinemounted on a\n4-circle cradle, for the magnetic field measurement a ver-\ntical field cryomagnet (Oxford Instruments) and normal\nbeam geometry were used.\nIII. RESULTS\nA. Magnetization and magnetic susceptibility\nComparedto earlierreported powdermagnetic suscep-\ntibility data measured in a magnetic field of 1 T,10our\nsingle crystal results reveal new anisotropic properties of\nCu3Bi(SeO 3)2O2Br. In fact, we find that the magnetic\nsusceptibility is very anisotropic and highly sensitive to\nthe strength of the applied magnetic field. When an ex-\nternal magnetic field of 0.01T is applied perpendicular\nto the layers ( B||c),χc(T) can be described by a CW\nlaw with θc=80(5)K down to 200K, where the slope of\n1/χc(T) starts to decrease (inset in Fig.2a). For per-\npendicular orientations, i.e., the field applied within the\nlayers(B||a,b), the high temperature behavior of χa,b(T)\nshows similar response (Fig.2a), with θa,b=60(5)K. We\nnote that the difference in 1/ χc(T) and 1/χa,b(T) slopes\nsuggests different g-factors, as Curie constant C∝g2,13\nwhile the discrepancy between θcandθa,bimplies a siz-\nable (∼20%) exchange anisotropy.14\nA much more anisotropic response is observed below\nthe magnetic transition (for T <TN=27.4K), where a3\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s84 /s32/s40/s75/s41/s32/s32/s48/s46/s48/s49/s32/s84/s32/s32/s32 /s32/s32/s49/s32/s84\n/s32/s32/s48/s46/s53/s32/s84/s32/s32/s32/s32/s32 /s32/s32/s50/s32/s84\n/s32/s32/s48/s46/s55/s53/s32/s84/s32/s32/s32 /s32/s32/s53/s32/s84\n/s32/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41/s66/s32 /s124/s124 /s99/s40/s98/s41/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s32/s66/s32 /s124/s124 /s97\n/s32/s66/s32 /s124/s124 /s98\n/s32/s66/s32 /s124/s124 /s99/s40/s97/s41\n/s32/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s50/s48/s48 /s52/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s66/s32 /s124/s124 /s97\n/s66/s32 /s124/s124 /s98\n/s66/s32 /s124/s124 /s99\n/s32/s32/s49/s47 /s32/s40/s109/s111/s108/s47/s101/s109/s117/s41\n/s84 /s32/s40/s75/s41\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s66/s32 /s98\n/s66/s32 /s97/s66/s32 /s124/s124 /s99\n/s66/s32 /s40/s84/s41\n/s32/s32/s77 /s32/s40\n/s66/s41/s84 /s32/s61/s32/s50/s32/s75\nFIG. 2. (Color online) (a) Magnetic susceptibility at 0.01\nT along all three crystallographic axes. Inset: the inverse\nsusceptibilities (symbols) and fits (solid lines) to the Cur ie-\nWeiss laws. (b) Magnetic susceptibility measured in differe nt\nexternal magnetic fields B||cbetween 0.01 - 5 T. Inset: mag-\nnetization at 2K measured along all three crystallographic\ndirections.\ntypical AFM behavior is observed at low fields (Fig.2a).\nForB||cthe anomaly in χc(T) is very sharp i.e.,\nχc(T)/χc(TN) drops below 0.1 already at T/TN∼0.85,\nwhichimpliesthatthemagneticgroundstateisAFMand\nthat the c-axis is the orientation favored by the magnetic\nmoments. Considering now that the anomalies in χa(T)\nandχb(T) are significantly less pronounced and that\nχa(T)issmallerthan χb(T),wecandeducethatthemag-\nnetic moments are more easily bent towards bthan to-\nwardsa. Thus we can conclude that Cu 3Bi(SeO 3)2O2Br\nis an AFM with magnetic moments predominately lying\nalong the c-axis,acoincides with the magnetic hard axis\nandbis somewhere in-between.\nWhen the applied magnetic field B||cexceeds 0.5T,\nthe anomaly in χc(T) starts to broaden and shift to\nlower temperatures (Fig.2b). Eventually, at 1T, χc(T)\nexhibits saturation-likebehavior, indicatingthat thelow-\nfield (LF) AFM ground state is followed by a high-field\n(HF) ferri- or ferromagneticphase. On contrary, increase\nofB⊥cup to 5 T has almost no effect on χa,b(T<TN)\n(not shown).Finally, low-temperature (at 2K) magnetization M\nwas measured along all three crystallographic axes (in-\nset in Fig.2b). A sharp magnetic transition is found\nforBC=0.8T with B||c. The fact that above BCthe\nvalue of Mper Cu2+ion,µCu=0.9µB, is almost con-\nstant and very close to 1 µB, expected for the complete\nmagnetization per Cu2+(S=1\n2) ion, indicates that we\nwitness a metamagnetic transition from an AFM to a\nferro/ferrimagnetic state. A very weak field dependence\nofMaboveBCindicates that fully FM state is not\nreached yet. This suggests that a small AFM component\nis still present and that significantly stronger magnetic\nfields are required to overcome the responsible AFM in-\nteractions. On the other hand, when Bis applied along\ntheb-axis, the magnetization linearly increasesup to 7T,\nwhere it again reaches a saturation value. Similar re-\nsponse is observed also for B||a, where the slope of the\nmagnetization is a bit smaller, implying that saturation\nis reached at ∼15T. Relatively slow linear response for\nB⊥ccompared to B||cindicates the presence of the\nsizable magnetic anisotropy. Since the zero-field split-\nting, imposed by the crystal field, is expected to be neg-\nligible in the Cu2+(S=1/2) systems,13we suspect that\nanisotropicexchangeinteractionisresponsibleforthe ob-\nserved behavior – in agreement with the different values\nof the CW constants (inset in Fig2a).\nB. Neutron diffraction\n1. Powder diffraction\nTo determine the spin arrangement in the LF and HF\nstateswefirstperformedneutronpowderdiffractionmea-\nsurements. The diffraction pattern measured at 60K (in-\nset in Fig.3) confirms the crystal structure published by\nMillet et al.,10while the patterns collected below 30 K\nreveal the onset of long-range magnetic order (Fig.3).\nMagnetic reflections occur at positions distinct from the\nnuclearones,indicatingthatthemagneticgroundstateis\nAFM. Indeed, the main magnetic reflections (2 θ=17.5◦\nand 28.5◦) can be indexed by the magnetic wave vector\nk=(001\n2).\nIn the next step we performed the representation anal-\nysis of possible magnetic structures based on the prop-\nagation wave vector and crystal symmetry.15The little\ngroup consists of eight elements and the resulting irre-\nducible representations (IRR) Γ i, (i= 1-8) are given\nin TableI. It was, however, impossible to determine the\nunique magnetic ground state from the powder diffrac-\ntion data alone.\n2. Single crystal diffraction\nSingle crystal diffraction was performed on a crys-\ntalline platelet with size 10 ×10×1mm3. To determine4\n25 50 750300600\n16 17 18 28 29 300.120.14\n  \nT = 60 K  1.5 K\n 18 K\n 30 K\n  Intensity (arb. units)\n2/s61553 (deg)(0 1 0.5)(1 1 0.5)\nFIG. 3. (Color online) The enlarged part of the neutron pow-\nder diffraction pattern, in which the two most pronounced\nmagnetic reflections appear, measured at several tempera-\ntures. Inset: The measured (black line) and the calcu-\nlated (red line) powder neutron diffraction patterns at 60 K,\nwhereas the green line shows the difference between the two,\nand the markers show the calculated positions of the Bragg\nreflections.\nTABLE I. Irreducible representations for the Cu1 and Cu2\n4(c) and 2(a) sites of the Pmmn space group and the\nk=(001\n2) wave vector.\nsite Cu1 Γ 1Γ3Γ5 Γ7\nx,y,z u,v,w u,v,w u,v,w u,v,w\n-x+1\n2,-y+1\n2,z -u,-v,w -u,-v,w u,v,-w u,v,-w\n-x,y+1\n2,-z -u,v,-w u,-v,w -u,v,-w u,-v,w\nx+1\n2,-y,-z u,-v,-w -u,v,w -u,v,w u,-v,-w\nsite Cu2 Γ 2Γ3Γ5Γ6Γ7Γ8\nx,y,z 0,0,u 0,0,u 0,u,0 u,0,0 u,0,0 0,u,0\n-x,y+1\n2,-z 0,0,-u 0,0,u 0,u,0 -u,0,0 u,0,0 0,-u,0\nthe LF magnetic ground state, 65 magnetic and 30 nu-\nclear reflections were collected in zero-field at 6.5K, i.e.,\nwell below TN. The best fit15was obtained for the mag-\nnetic structure described by the Γ 3irreducible represen-\ntation for both Cu sites. The refined magnetic compo-\nnents are my= 0.72(2)µBandmz= 0.57(2)µBfor Cu1\nandmz= 0.90(4)µBfor Cu2. This corresponds to the\nmagnetic moment arrangementpresented in Fig.4a. The\nCu1 moments are aligned parallel to the c-axis with ad-\nditional alternating component along the b-axis and are\nthus canted ∼±50◦fromctowards b. The Cu2 mo-\nments, on the other hand, are strictly parallel to the\nc-axis. Such an arrangement suggests dominant FM in-\nteractions within the layer, as well as a sizable magnetic\nanisotropy. Yet the coupling between the layers is AFM\nas presented in Fig.4a.\nThe temperature dependence of the (221\n2) magnetic\nreflection (Fig.5a) indicates a continuous second order\nFIG. 4. (Color online) Refined magnetic structure at (a) zero\napplied field and (b) BC=1T. Magnetic moments at the Cu1\nsites are blue arrows, while Cu2 are red. For clarity only the\nmagnetic lattice is show, i.e., J1≈J3(violet) and J2(cyan)\nexchange interactions.\nphase transition from the paramagnetic to the LF AFM\nphase. Below 26.6K, i.e. 1 −T/TN>0.04, the mag-\nnetic intensity can be described by I∼(TN−T)2β\nwithβ≤0.23 (Fig.5d), as expected for 2D XY spin\nsystems,16,17in particular when additional weak in-plane\ncrystal-field anisotropy is present.18The exact TNwas\nderived by fitting the data in the vicinity of TN(Fig.5a),\nwhere the critical exponent βincreases to ∼0.30(2), sig-\nnifying a crossover from spatial 2D (n=2) to 3D (n=1)\nbehavior. Our result is in accord with observations for\nother layered systems e.g., K 2CuF4, BaNi 2(PO4)2and\nRb2CrCl4,3,19–21and implies that the 2D critical region\nextends in the magnetic long-range ordered state as well\nas that spin fluctuations are essentially two dimensional.\nWe presume that spin waves in this material will be sig-\nnificantlyrenormalizedbyvortexexcitations.22,23Atypi-\ncal feature of the 2D regime – diffuse magnetic scattering\n– has yet not been detected, presumably due to smallness\nof the single crystal used in the diffraction experiment.\n3. Single crystal diffraction in applied magnetic field B||c\nIn order to explore the HF state, i.e., above BC=0.8T\n(B∝bardblc), we measured the field dependence of the (221\n2)5\n/s48/s49/s48/s50/s48\n/s48/s49/s48/s50/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s56/s49/s50/s49/s54/s56/s49/s50/s49/s54\n/s48/s46/s48/s49 /s48/s46/s49/s49/s48/s50/s48/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41/s66 /s32/s61/s32/s48/s32/s84/s32\n/s40/s97/s41\n/s66 /s32/s61/s32/s48/s46/s54/s53/s32/s84/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41/s50/s32/s50/s32/s49/s47/s50/s40/s98/s41\n/s32/s84 /s32/s40/s75/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41/s32\n/s66 /s32/s61/s32/s49/s32/s84\n/s40/s99/s41/s50/s32/s49/s32/s48\n/s66 /s32/s61/s32/s48/s32/s84/s40/s100/s41\n/s40/s50/s32/s50/s32/s49/s47/s50/s41/s32/s61/s32/s48/s46/s50/s51\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s40/s49/s45 /s84 /s32/s47 /s84\n/s78/s41/s32/s61/s32/s48/s46/s51/s48\nFIG. 5. Temperature dependences of the intensities of the\n(221\n2) and (210) reflections at (a) 0T, (b) 0.65T and (c) 1T.\nTheblack dashedlineindicates themagnetic transition at TN,\nwhile the red one indicates the temperature, at which field\ndependences were measured. The solid red lines represents\na fit toI∼(TN−T)2β, withβ=0.30(2) and TN= 27.4K.\n(d) Intensity of the (221\n2) magnetic reflection as a function of\n(1−T/TN), where solid and dashed lines denote two distinct\nI∼(TN−T)2βregimes.\nand (210) magnetic reflections at 1.5K (Fig.6). As an-\nticipated, the (221\n2) reflection abruptly disappears at\n∼0.8T. On contrary, the (210)24reflection exhibits ex-\nactly the opposite response, which implies that it cor-\nresponds to the FM HF phase. This sharp transition\nis reminiscent of the metamagnetic behavior, suggesting\nthat the AFM coupling between the adjacent layers is\nbroken. To investigate the phase boundaries, we thus\nperformed aseriesoftemperature scansofthe (221\n2) and\n(210) reflections at 0T, 0.65T and 1T (Fig.5). The re-\nsults corroborate with the magnetic susceptibility data,\ni.e., the applied magnetic field suppresses the LF phase\nand induces the HF phase. At 0.65T (Fig.5b) the LF\nphase persists up to ∼19K, both phases coexist between/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s66\n/s67/s40/s49/s46/s53/s32/s75/s41/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66 /s32/s40/s84/s41/s32/s40/s50/s32/s50/s32/s49/s47/s50/s41/s32/s97/s116/s32/s49/s46/s53/s32/s75\n/s32/s40/s50/s32/s50/s32/s49/s47/s50/s41/s32/s97/s116/s32/s50/s48/s32/s75/s66\n/s67/s40/s50/s48/s32/s75/s41\n/s32/s40/s50/s32/s49/s32/s48/s41/s32/s97/s116/s32/s49/s46/s53/s32/s75\n/s32/s40/s50/s32/s49/s32/s48/s41/s32/s97/s116/s32/s50/s48/s32/s75\nFIG. 6. (Color online) Field dependence of the intensities o f\nthe (221\n2) and (210) reflections at 1.5K and 20K.\n19K and 22K, implying a first-ordermetamagnetic tran-\nsition, while between 22K and TNonly the HF phase is\nleft. At 1T,the LF phaseiscompletelysuppressed, while\nTN=27.4K seems to be field independent.\nTo determine the HF magnetic structure we collected\n40 reflections, from which 17 had significant intensity.\nThe best refinement was obtained for k=(000) retain-\ning the Γ 3IRR for both Cu sites. The order within\nthe layers appears to be almost unaffected, while the ar-\nrangement of the consecutive layers is now FM (Fig.4b).\nAt 1.5K and 1T for the Cu1 sites mx=−0.2(1)µB,\nmy= 0.73(2)µBandmz= 0.66(4)µB(m= 0.99(2)µB)\nand for the Cu2 sites mz= 0.99(9)µB.\nIV. DISCUSSION\nThe magnetic properties of Cu 3Bi(SeO 3)2O2Br pre-\nsented in this work reflect the low dimensional (2D) na-\nture of its lattice. This is most evident from the ex-\nperimentally determined magnetic structures for the LF\nand HF phases, which indicate that the main magnetic\n’building block’ of the system is a single layer. In or-\nder to understand the observed behavior, we thus fo-\ncus here on the individual layer. The two Cu sites have\ndifferent local symmetries of their square planar [CuO 4]\ncoordination, which might be responsible for the differ-\nent behavior of Cu1 and Cu2 magnetic moments. How-\never, calculation of the exchange charge model of the\ncrystal field around Cu2+ions in the actual surround-\ning indicates that the zero-field spitting, imposed by\nthe crystal field, is weak,25as expected for S=1/2 sys-\ntems. Anothersourceoftheobservedresponse,e.g.,mag-\nnetic frustration or exchange anisotropy, might lie within\nthe exchange network. In particular, the electron hop-\nping, governing the exchange interaction, appears to be\nvery different for inter- and intra-layer exchange path-\nways and might also differ between the three exchange\npathways within the layer (see Figs.1 and 4). Two of6\nthese are nearest-neighbor ones, involving Cu1-O1-Cu1\n(d=3.19˚A, bond angle φ=111◦, multiplicity m=4)\nand Cu1-O1-Cu2 ( d=3.27˚A,φ=113◦,m=8) superex-\nchange bridges, while the last connects next-nearest-\nneighbors via the Cu1-O1-Bi-O1-Cu1( d=4.84˚A,m=4)\nsuper-superexchange bridge. In the above order we as-\nsign them the coupling constants J1,J3, andJ2, respec-\ntively. Since the local symmetry of the cations is low\n(point groups S2andC2v) and crystalline fields aris-\ning from the surrounding oxygens deviate significantly\nfrom the usual octahedral and tetrahedral coordinations,\nreliable prediction of the sign and strength of the ex-\nchange interactions just by following the Goodenough-\nKanamori-Anderson rules26–28is rather bold. Neverthe-\nless, the arrangements of the magnetic moments in the\ndetermined magnetic structures imply that the J1andJ3\ncouplings are most probably FM, while the J2exchange\nis AFM.\nIn order to determine the above presented exchange\ninteractions, we first explore all possible combina-\ntions of their sign and strength within the model of\nisotropic Heisenberg interactions, using the ENERMAG\nprogram.29Here, we assume that J1andJ3are equal,\nwhich is justified by their similarity in bonding lengths\nand angles. As a result, we obtain a list of probable\nordering modes for a specific magnetic propagation vec-\ntorkand corresponding intervals of Ji(i=1,2). We find\nthattheonlymodes, whichcorrespondtoΓ 3IRRonboth\nCu-sites (see TableI) and thus meet the experimental ob-\nservations, are M z(++++++) and M y(+−−+00).\nFurthermore, we discover that for k=(001\n2) (LF phase)\nandk=(000) (HF phase) the M z(++++++) mode\nis favorable, when J1is FM, and J2can have any sign;\nwhereas the M y(+−−+00) mode is obtained, when\nall couplings are AFM. Apparently, these two modes re-\nquire different sign of couplings and thus implies that\nthe experimentally determined canted magnetic struc-\nture, which is a convolution of the two modes, is a result\nof competing magnetic interactions.\nTo resolve the canting of the Cu1 spins we thus min-\nimize the energy of the magnetic ground state. In line\nwith the experimental observations and to avoid unnec-\nessary complications, we restricted the orientation of the\nCu1 and Cu2 magnetic moments to the bcplane. The\nresults show that the experimentally determined cant-\ning of the Cu1 magnetic moments ( ∼50◦fromctowards\nb) can be achieved with isotropic, yet different, inter-\nactions. In fact, we find that J2should be AFM and\neven stronger than FM J1, i.e.,J2∼−1.6J1. This seems\ncontra-intuitive, as the exchange path for J2is signif-\nicantly longer than those corresponding to J1andJ3,\nand since it involves an additional Bi3+ion. However,\nstudies of other tellurides and selenides reveal that simi-\nlar super-superexchangepaths can have similar strengths\nas some significantly shorter superexchange bonds.11\nIn order to quantify the strengths of the exchange in-\nteractions we take the Curie-Weiss temperature into con-\nsideration, which is defined as the sum of the exchangeinteractions per magnetic site. Thus we can write the\nexpression:\nθCW=2\n3kBS(S+1)/summationdisplay\nn,i(Jizni). (1)\nHere, we assumed ( g−1)2≈1,n=1,2 counts different\nCu sites, i=1,2 counts different exchange interactions\n(J3=J1), andzniis half of the number of Cu neighbors\ncoupled by Jiexchange ( z11=2,z12=1,z21=2,z22=0).26\nHence, basedon the experimentallydetermined θc=80K\nforB||cand the evaluated J2∼−1.6J1ratio, we esti-\nmateJ1≈67K and J2≈−107K. Similarly, we obtain\nJ1≈50K and J2≈−80K from θa,b=60K for B||a,b.\nThe large AFM J2value is in agreement with AFM\ncomponent remaining beyond 8T, which reflects in the\nweak field dependence of magnetization above BC. On\nthe other hand, the derived difference between the ex-\nchange interactions parallel and perpendicular to the c-\naxis implies the presence of sizable ( ∼20%) exchange\nanisotropy. In fact, the difference between the main ex-\nchangeinteractions, ≈17K (i.e., ∼12.6T), is of the same\norder of magnitude as the observed and estimated satu-\nration magnetic fields of ∼7 and∼15T for B||aand\nB||b, respectively. The above agreement implies that\nthe magnetic fields, needed to bend the magnetic mo-\nments out of the preferred orientation, indeed compen-\nsate the exchange anisotropy and is thus in line with the\nfact that zero-field splitting, imposed by crystal filed, for\nS=1/2 systems is negligible and with the apparent g-\nfactor anisotropy,30reflected in the magnetic suscepti-\nbility. Still, the estimated anisotropy is relatively small\n(∼20%) compared to the magnitude of the exchange\nparameters, which implies that the dominant exchange\nis Heisenberg-like with a sizable exchange anisotropy.\nThe origin of this anisotropy might be the 2D nature\nof the magnetic lattice and the specific arrangement of\nthe Cu2+atomic orbitals combined with spin-orbit and\nCoulomb exchange interactions.31\nFinally, an estimate of the AFM interlayer exchange\ncoupling J′canbeobtained,withintheWeissfieldmodel,\nfrom the magnitude of the magnetic field BC=0.8T re-\nquired to flip the layers and thus overcome J′:26\ngµBBC= 2z|J′|S. (2)\nTakingz=2, i.e., considering two neighboring layeres,\nthe resulting J′equals 0.5K. Still, the significantly sup-\npressedTNcompared to Ji’s is not only a result of weak\ninterlayer couplings, but probably also reflects the com-\npetition between AFM and FM interactions within the\nlayer.\nV. CONCLUSIONS\nThe anisotropic magnetic properties of the layered\nKagom´ e-like system Cu 3Bi(SeO 3)2O2Br have been stud-\nied by bulk magnetic measurements and neutron diffrac-7\ntion. We have found that below TN=27.4K the mag-\nnetic ground state can be described as antiferromagnet-\nically coupled ablayers, with moments on the Cu2 sites\npointing along cand those on the Cu1 sites alternating\nbetween the ±50◦tilt from ctowards b. AtBC∼0.8T\napplied perpendicular to the layers (along c), every sec-\nond layer flips, resembling a metamagnetic transition to\nan almost FM structure.\nBased on the determined spin arrangements and crit-\nical fields we provide an estimate of the characteristic\nexchange couplings of the system. For intralayer ex-\nchange (along the caxis) we thus obtain that nearest-\nneighbor Cu-O-Cu exchange is FM ( J1≈J3≈67K),\nwhile next-nearest-neighbor Cu-O-Bi-O-Cu exchange is\neven stronger and AFM ( J2≈–107K). Obviously, the\nnearest-neighbor FM interactions remove geometrical\nfrustration, which would be a dominant feature of the\nKagom´ eantiferromagnet; the frustrationis onlypartially\nrestored by next-nearest-neighbor AFM interactions, re-\nsponsible for the canting of Cu1 moments. We note that\nconsiderably (20%) lower Jivalues were derived for the\nperpendicular orientation (along the layers), implying a\nsizable exchange anisotropy. Interestingly, in spite of the\nfact that the ratio between the interlayer and intralayercouplings is significant, |J′/J|∼0.006, and that the ex-\nchange interactions are predominantly isotropic, the in-\ntensity of the magnetic diffraction peak can be in a broad\nregion below TNdescribed as ( T−TN)2β, withβ≤0.23,\ncharacteristic for finite-sized 2D XY magnetic systems\nwith additional weak in-plane crystal-field anisotropy.\nThis may imply that interlayerinteractions are still weak\nenough that spin fluctuations below 0.96 TNare essen-\ntially 2D, and that the existing magnetic anisotropy\ncould be strong enough to impose a quasi 2D XY be-\nhavior.\nAcknowledgements\nThis work was supported by the Swiss National\nFoundation (SNF) project 200021-129899, the Deutsche\nForschungsgemeinschaft (DFG) via research unit 960\n”Quantum Phase Transitions” and the Transregional\nCollaborative Research Center TRR80 ”From Electronic\nCorrelations to Functionality” (Augsburg–Munich). We\nthank Dana Vieweg, Thomas Wiedenmann and Lukas\nKeller for experimental support. The neutron diffraction\nwork was performed at SINQ, Paul Scherrer Institute,\nVilligen, Switzerland.\n1I. S. Jacobs and P. E. Lawrence, Phys. Rev. 164, 866\n(1967).\n2S. A. Wold, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka-\nnova, D. M. Treger Science,294, 1488 (2001).\n3Magnetic Properties of Layered Transition Metal Com-\npounds, edited by L. de Jongh (Kluwer, Dordrecht, 1990).\n4J. M. Kosterlitz, D. J. Thouless J. Phys. C. 6, 1181 (1973).\n5L. Onzager, Phys. Rev. ,65, 117 (1942).\n6N. D. Mermin and H. Wagner, Phys. Rev. Lett. ,17, 1133\n(1966).\n7Introduction to Frustrated Magnetism , edited by C.\nLacroix, P. Mendels, and F. Mila (Springer-Verlag, Berlin,\n2011).\n8D. P. Landau and K. Binder, Phys Rev. B 24, 1391 (1981).\n9A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 (2002).\n10P. Millet, B. Bastide, V. Pashchenko, S. Gnatchenko, V.\nGapon, Y. Ksari, A. Stepano J. Mater. Chem. 11, 1157\n(2001).\n11J. Deisenhofer, R. M. Eremina, A. Pimenov, T. Gavrilova,\nH. Berger, M. Johnsson, P. Lemmens, H.-A. Krug von\nNidda, A. Loidl, K.-S. Lee, and M.-H. Whangbo, Phys.\nRev.B 74, 174421 (2006).\n12M. Johnsson, K. W. T¨ ornroos, P. Lemmens, and P. Millet,\nChem. Mater. 15, 68 (2003).\n13Kittel C., Introduction to Solid State Physics, 8th Edition\npp. 304 (New Caledonia, Wiley, 2005)\n14T. Han, S. Chu, Y. S. Lee, e-print arXiv:1202.4729 (un-\npublished).\n15FULLPROF suite, J. Rodr´ ıgues-Carvajal, Physica B 192,\n55 (1993).16S. T. Bramwell, P. C. W. Holdsworth J. Phys.: Condens.\nMatter5, L53 (1993).\n17S. T. Bramwell, P. C. W. Holdsworth Phys Rev. B 49,\n8811 (1994).\n18A. Taroni, S. T. Bramwell, and P. C. W. Holdsworth, J.\nPhys.: Condens. Matter 20, 275233 (2008).\n19K. Hirakawa and H. Ikeda, J. Phys. Soc. Jpn. 35, 1328\n(1973).\n20L. P. Regnault and J. Rossat-Mignod in Magnetic Proper-\nties of Layered Transition Metal Compounds , edited by L.\nde Jongh (Kluwer, Dordrecht, 1990), p. 274.\n21S. T. Bramwell, P. C. W. Holdsworth and M. T. Hutchings,\nJ. Phys. Soc. Jpn. 64, 3066 (1995).\n22J. M. Kosterlitz J. Phys. C. 7, 1046 (1974).\n23J. V. Jos´ e, L. P. Kadanoff, S. Kirkpatrick, D. R. Nelson\nPhys Rev. B 16, 1217 (1977).\n24We note that weak hk0 reflections with h+k=2n+1 are\npresent up to room temperature, indicating that the actual\ncrystal symmetry is lower than the proposed Pmmn.10\n25S. Klokishner private communication.\n26J. B. Goodenough, Magnetism and the chemical bond (New\nYork, Interscience, 1963)\n27J. B. Goodenough, Phys. Rev. 100, 564 (1955).\n28J. Kanamori, J. Phys. Chem. Solids 10, 87 (1959).\n29N. El Khayati, R. Cherkaoui El Moursli, J. Rodr´ ıgues-\nCarvajal, G. Andr´ e, N. Blanchard, F. Bour´ ee, G. Collin,\nT. Roisnel, Eur. Phys. J. B 22, 429 (2001) and references\ntherein.\n30A. Bencini and D. Gatteschi, EPR of Exchange Coupled\nSystems (Berilin Heidelberg, Springer-Verlag, 1990).\n31T. Yildirim, A. B. Harris, A. Aharony, O. Entin-Wohlman\nPhys Rev. B 52, 10239 (1995)."    },    {        "title": "1901.03072v2.Ultrafast_magnetization_dynamics_in_uniaxial_ferrimagnets_with_compensation_point__GdFeCo.pdf",        "content": "Ultrafast magnetization dynamics in uniaxial ferrimagnets with compensation point.\nGdFeCo\nM. D. Davydova,1, 2,\u0003K. A. Zvezdin,1, 2A. V. Kimel,3, 4and A. K. Zvezdin1, 2,y\n1Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia\n2Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia\n3Moscow Technological University (MIREA), 119454 Moscow, Russia\n4Institute for Molecules and Materials, Radboud University, Nijmegen 6525 AJ, The Netherlands\nWe derive an e\u000bective Lagrangian in the quasi-antiferromagnetic approximation that allows to\ndescribe the magnetization dynamics for uniaxial f-d(rare-earth - transition metal) ferrimagnet\nnear the magnetization compensation point in the presence of external magnetic \feld. We perform\ncalculations for the parameters of GdFeCo, a metallic ferrimagnet with compensation point that is\none of the most promising materials in ultrafast magnetism. Using the developed approach, we \fnd\nthe torque that acts on the magnetization due to ultrafast demagnetization pulse that can be caused\neither by ultrashort laser or electrical current pulse. We show that the torque is non-zero only in\nthe non-collinear magnetic phase that can be acquired by applying external magnetic \feld to the\nmaterial. The coherent response of magnetization dynamics amplitude and its timescale exhibits\ncritical behavior near certain values of the magnetic \feld corresponding to a spin-\rop like phase\ntransition. Understanding the underlying mechanisms for these e\u000bects opens the way to e\u000ecient\ncontrol of the amplitude and the timescales of the spin dynamics, which is one of the central problems\nin the \feld of ultrafast magnetism.\nPACS numbers: 79.20.Ds, 75.50.Gg, 75.30.Kz, 75.78.-n\nINTRODUCTION\nMost of the prominent advances in the \feld of ultrafast magnetism have been achieved by using thermal mechanism\nof magnetization control [1{7]. These studies rooted from the pioneering work by Beaurepaire et al. [8] on ultrafast\nlaser-induced demagnetization of Ni. In this experiment, partial destruction of magnetic order was found at much\nfaster rates that were believed to be possible prior to that publication. Since then, the \feld of ultrafast magnetism has\nbeen rapidly growing and the possible channels of ultrafast angular momentum transfer have been studied extensively\n[9]. Ultrafast demagnetization can be achieved by applying ultrashort laser pulses [1{7], or, alternatively, by using\nshort pulses of electric currents [10, 11].\nIn the last decades, GdFeCo and other rare-earth - transition metal compounds (RE-TM) have been in the center\nof attention in this regard [12]. For example, all-optical switching has been demonstrated for the \frst time in GdFeCo\n[13]. It was found that the switching is possible due to di\u000berent rates of sublattice demagnetization, which enables\nultrafast magnetization reversal to occur because of the angular momentum conservation [1, 2].\nIn many RE-TM compounds, GdFeCo and TbFeCo being part of them, realization of the magnetization compen-\nsation point is possible. At this point, the magnetizations of the two antiferromagnetically coupled sublattices with\ndi\u000berent dependencies on temperature become equal and the total magnetization of the material turns to zero. In\nthe presence of the external magnetic \feld, a record-breaking fast subpicosecond magnetization switching was found\nin GdFeCo across the compensation point [13]. In addition, a number of anomalies in the magnetic response was\nobserved near this point [14{16], which has never been explained theoretically. All said above illustrates the impor-\ntance of understanding the role of the compensation point in the dynamics and working out an appropriate tool for\nits description.\nE\u000ecient control of the amplitude and the timescales of the response of the magnetic system to an ultrafast de-\nmagnetizing impact on a medium is one of the most important issues in the area of ultrafast magnetism nowadays\n[9, 14, 15]. Understanding of the mechanisms and of the exhaustive description of the subsequent spin dynamics is\nalso a long-standing goal that will help to promote the achievements of this area towards practical applications in\nmagnetic recording [3, 17], magnonics [18] and spintronics [19]. In this work, we expand the understanding of response\nof magnetic system of a uniaxial f-dferrimagnet near the compensation point in the external magnetic \feld to an\nultrafast demagnetizing pulse, which can be induced either by a femtosecond laser or an electric current pulse. We\npresent a theoretical model and calculations, which allow to describe the ultrafast response of the system that resides\nin an angular phase before the impact. We show that in this magnetic phase the coherent precessional response is\npossible and the subsequent magnetization dynamics may become greatly nonlinear and is governed by large inter-\nsublattice exchange \feld [20]. We derive the e\u000bective Lagrangian that governs the dynamics of the system near thearXiv:1901.03072v2  [cond-mat.mtrl-sci]  28 Jan 20192\ncompensation point and obtain the torque acting on the magnetizations of the two sublattices due to demagnetiza-\ntion. In ref. [15], the critical response of the amplitude and the time of the signal rise have been found in GdFeCo\nin external magnetic \feld along the easy axis. At given laser pump \ruences, the response was found to be negligible\nin collinear phase, but it was dramatically large in angular one. We elaborate on this example and show that the\ncritical behavior of the response is the consequence of the second-order magnetic phase transition from collinear to an\nangular in the external magnetic \feld. We \fnd that these e\u000bects are pronounced in the vicinity of the compensation\npoint, where the phase transitions cross each other[21{23]. Thus, the proposed model explains a range of important\nexperimental observations as well as allows for developments of methods and tools of magnetization control by setting\nthe temperature near the compensation point and applying magnetic \feld. Moreover, by changing the composition\nof the alloy, the [24], the position of the magnetization compensation point can be tuned arbitrary close to the room\ntemperature. Our results might open new ways for technologies for ultrafast optical magnetic memory.\nEFFECTIVE LAGRANGIAN AND RAYLEIGH DISSPATION FUNCTION\nOur approach is based on Landau-Lifshitz-Gilbert equations for a two-sublattice (RE-TM) ferrimagnet. These\nequations are equivalent to the following e\u000bective Lagrangian and Rayleigh dissipation functions:\nL=Mf\n\r(1\u0000cos\u0012f)@'f\n@t+Md\n\r(1\u0000cos\u0012d)@'d\n@t\u0000\b(Mf;Md;H); (1)\nR=Rf+Rd;Rf;d=\u000bMf;d\n2\r\u0010\n_\u00122\nf;d+ sin2\u0012f;d_'2\nf;d\u0011\n(2)\nwhere\ris the gyromagnetic ratio, MdandMfare the magnetizations, \u0012d(TM) and\u0012f(RE) are the polar, 'dand\n'fare the azimuthal angles of d- andf- sublattices correspondingly in the spherical system of coordinates with z-axis\naligned along the external magnetic \feld H. \b(Mf;Md;H) is the thermodynamic potential for the system that we\ntake in the following form:\n\b =\u0000MdH+\u0015MdMf\u0000MfH\u0000Kf(Mfn)2\nM2\nf\u0000Kd(Mdn)2\nM2\nd; (3)\nwhere\u0015is the intersublattice exchange constant, nis the direction of the easy axis and Kf;dare the anisotropy\nconstants for f- andd- sublattices, respectively.\nNext, we transfer to description in terms of the antiferromagnetic L=MR\u0000Mdand the total magnetization\nM=MR+Mdvectors. In the vicinity of the compensation point the di\u000berence between the sublattice magnetizations\njMR\u0000Mdj\u001cLis small. The two vectors are parametrized using the sets of angles \u0012;\"and';\f, which are de\fned\nas:\n\u0012f=\u0012\u0000\"; \u0012d=\u0019\u0000\u0012\u0000\";\n'f='+\f; 'd=\u0019+'\u0000\f:(4)\nIn this case the antiferromagnetic vector is naturally de\fned as L= (Lsin\u0012cos';Lsin\u0012sin';Lcos\u0012).\nFollowing the work [25] we use the quasi-antiferromagnetic approximation to describe the dynamics near the mag-\nnetization compensation point. F In this approximation the canting angles are small \"\u001c1,\f\u001c1, and we can\nexpand the Lagrangian (1) and the corresponding thermodynamic potential up to quadratic terms in small variables:\nL=\u0000m\n\r_'cos\u0012\u0000M\n\rsin\u0012\u0010\n_'\"\u0000\f_\u0012\u0011\n\u0000\b;\n\b =\u0000K(l;n)2\u0000Hmcos\u0012\u0000\"MH sin\u0012+\u000e\n2\u0000\n\"2+ sin2\u0012\f2\u0001\n:(5)\nHerem=MR\u0000Md,M=MR+Md,K=KR+Kdis the e\u000bective uniaxial anisotropy constant, l=L=Lis\nthe unit antiferromagnetic vector \u000e= 2\u0015MdMRand we assume the anisotropy to be weak K\u001c\u0015M. For GdFeCo\nwith 24% Gd and compensation point near 283 K, we assume the following values of parameters: M\u0019800 emu/cc,\nK= 1:5\u0002105erg/cc,\u0015= 18:5 T/\u0016B,\u000e\u0019109erg/cc and mchanges in the range between 150 emu/cc and \u000050\nemu/cc at \felds H\u0019H\u0003\u00194 T. The characteristic values of small angles \u000fand\fare of the order of 10\u00002.3\nNext, we exclude the variables \"and\fby solving the Euler-Lagrange equations. Substituting them into the\nLagrangian (5), we obtain the e\u000bective Lagrangian, which describes the dynamics of a uniaxial ferrimagnet in the\nvicinity of the compensation point:\nLeff=\u001f?\n2 _\u0012\n\r!2\n+mcos\u0012\u0012\nH\u0000_'\n\r\u0013\n+\u001f?\n2sin2\u0012\u0012\nH\u0000_'\n\r\u00132\n+K(l;n)2; (6)\n\beff(H) =\u0000mHcos\u0012\u0000\u001f?\n2H2sin2\u0012\u0000K(l;n)2; (7)\nReff=\u000bM\n2\r\u0010\n_\u00122+ sin2\u0012_'2\u0011\n(8)\nwhere\u001f=2M2\n\u000e. In GdFeCo \u001f\u00191:6\u000210\u00003and\u000b\u00190:05. In the derivation above we assumed the gyrotropic factor\n\rand Gilbert damping constant \u000bto be the equal for both sublattices. Taking into account the di\u000berence between\nthese values for di\u000berent sublattices will lead to the angular momentum compensation e\u000bect at certain temperature.\nThe Lagrangian, Rayleigh function and equations of motion preserve the same form in this case if we substitute the\nparameters \rand\u000bwith temperature-dependent factors e\rande\u000bde\fned as:\n1\ne\r=1\n\u0016\r\u0012\n1 +M\nm\rf\u0000\rd\n\rf+\rd\u0013\n=Md\n\rd\u0000Mf\n\rf\n(Md\u0000Mf);1\n\u0016\r=1\n2\u00121\n\rd+1\n\rf\u0013\n;e\u000b=(\u000bd\rf+\u000bf\rd)\n(\rf+\rd)1\n1 +M\nm\rf\u0000\rd\n\rf+\rd(9)\nThis allows to reproduce the angular moment compensation phenomenon, which was studied experimentally in ref.\n[14].\nEXCITATION OF THE SPIN DYNAMICS\nThe proposed approach presents a powerful tool allowing analyzing coherent magnetization dynamics in ferrimagnets\nthat occurs under a broad range of conditions. Let us consider the following example that poses an important problem\nin the \feld of ultrafast magnetism. An femtosecond laser pulse strikes the uniaxial ferrimagnet (for instance, of\nGdFeCo, TbFeCo type) in the presence of external static magnetic \feld. The impact of the laser pulse leads to the\ndemagnetization of one or both of the sublattices. What coherent magnetization dynamics will occur as a consequence\nof this impact? The proposed model can be further developed in order to answer to this question and is applicable\nfor small values of demagnetization \u000eM.\nIn our framework the spin dynamics in ferrimagnet is described by Euler-Lagrange equations of the formd\ndt@L\n@_q\u0000@L\n@q=\n\u0000@R\n@_q, whereq=\u0012; ' are the polar and azimuthal angles describing the orientation of the antiferromagnetic vector\nL, correspondingly. Let us consider a particular case when the easy magnetization axis is aligned with the external\nmagnetic \feld, which leads to the presence of azimuthal symmetry in the system. In this case n= (0;0;1). In this\nparticular case the Euler-Lagrange equations can be rewritten as:\n\u001f?\n\r2\u0012=@Leff\n@\u0012\u0000@Reff\n@_\u0012;d\ndt@Leff\n@_'=\u0000@Reff\n@_'(10)\nThe nonlinear equations that are similar to Eqs. (10) and describe the spin dynamics of two-sublattice ferrimagnets\nwere obtained in the work [26] under the conditions H= 0 andReff= 0. Over the short time of demagnetization the\nsecond equation can be approximately treated as a conservation law and the conserving quantity (angular momentum\nof magnetization precession J) stays approximately constant as @L=@'= 0 due to the Noether theorem:\nJ=@Leff\n@_'=\u00001\n\r\u0014\nmcos\u0012+\u001f?sin2\u0012\u0012\nH\u0000_'\n\r\u0013\u0015\n=const (11)\nLet the moment of time t= 0\u0000denote the moment before the laser pulse impact and system initially is in the\nground state de\fned by the ground state angles \u0012(0\u0000) =\u00120,'(0\u0000) ='0, and their derivatives _ '(0\u0000) = 0, _\u0012(0\u0000) = 0.4\nDepending on the external parameters and preparation of the sample, the system might reside in one of the two\npossible antiferromagnetic collinear phases or in angular phase, which are separated by the magnetic phase transition\nlines [27]. If the demagnetization due to the laser pulse action is small, it produces the changes in the values of Mf,Md\nandMof the order of percent or less, whereas the change of m(which is approximately equal to total magnetization\nnear the compensation point) may be of several orders of magnitude, as its value is almost compensated. In what\nfollows, we assume that the demagnetization is associated only with change of m, namelym=m0+ \u0001m(t). As we\nwill see below, the change in this quantity already leads to several drastic e\u000bect in dynamics.\nTherefore, the conservation law (11) leads to the emergence of azimuthal dynamics _ '(t) at the demagnetization\ntimescales (\u0001 t) due to demagnetization pulse \u0001 m(t):\n_'(t) =\r\u0001m(t)\n\u001f?cos\u00120\nsin2\u00120(12)\nWe see that the torque is non-zero only in the angular phase, where 0 <\u00120<\u0019. Emergence of the azimuthal spin\nprecession as a result of demagnetization of the medium is similar to the well-known Einstein-de-Haas e\u000bect, where\nthe demagnetization leads to azimuthal precession of the body. Subsequently, this azimuthal spin dynamics leads to\nthe emergence of polar dynamics \u0012(t), which is most commonly measured in pump-probe experiments of ultrafast\nmagnetism, by acting as an e\u000bective \feld Heff=H\u0000_'\n\rin the Lagrangian (5). We can then view the Lagrangian\nas depending only on variable \u0012and the e\u000bective \feld Heff. At demagnetization \u000em\u00180:01Min GdFeCo the value\nof _'can reach up to 1 THz, and the corresponding e\u000bective magnetic \feld is of the order of 10 T. Note that initial\nstate of the system corresponds to the condition@\b\n@\u00120(Heff=H) = 0. We can rewrite the Euler-Lagrange equation\nfrom eq. (10) for polar angle as follows:\n\u001f?\n\r2\u0012+@\b(Heff)\n@\u0012=\u0000\u000bM\n\r_\u0012: (13)\nOr, alternatively:\n\u001f?\n\r2\u0012+msin\u0012Heff\u0000\u001f?sin\u0012cos\u0012\u0012\nH2\neff\u00002K\n\u001f?\u0013\n=\u0000\u000bM\n\r_\u0012 (14)\nBy integrating this equation over the short demagnetization pulse duration \u0001 twe obtain the state of system after\nthe laser pulse impact at t= 0+, which is characterized by the initial conditions\n\u0012(0+) =\u00120;_'(0+); '(0+) =Z\u0001t\n0_'(t)dt;_\u0012(0+) =Z\u0001t\n0\u0012(t)dt\nThe value \u0001 tis of the order of the optical pulse length. It may also include the time of restoration of the magnetization\nlength (or the value of m). After the moment of time 0+ free magnetization precession occurs in the model. Analysis\nof the spin dynamics under laser pump excitation will lead to emergence of critical dynamics near the second-order\nphase transitions to the collinear phases where \u0012= 0;\u0019, as is already seen from (12). We will discuss this behavior\nbelow.\nCRITICAL DYNAMICS\nIn a simple case of a quick decay of demagnetization (at the exciton relaxation timescales) with \u0001 m(t) = \u0001m,\n0<t< \u0001t; \u0001m(t) = 0,t>\u0001t, we obtain the initial condition from (14):\n_\u0012(+0)\u0019\u0014\n\u0000\u0012\n2cos2\u00120\nsin\u00120+ sin\u00120\u0013\nH+m0\n\u001f?cos\u00120\nsin\u00120+\u0001m\n\u001f?cos\u00120\nsin3\u00120\u0015\r2\n\u001f?\u0001m\u0001t=B(\u00120)\u0001m+O(\u0001m2): (15)\nThis quantity de\fnes the initial angular momentum of the polar spin precession that is induced in the system due to\nthe optical spin torque created by the femtosecond laser pulse. The amplitude of oscillations is proportional to the\ninitial condition (15). Its dependence on the external magnetic \feld is illustrated in Fig. 1 for di\u000berent temperatures\nfor magnetic parameters of GdFeCo uniaxial ferrimagnet. At low values of external magnetic \felds there is only\ncollinear ground state in the ferrimagnet and above certain \feld Hsfthe transition to an angular state occurs [21].\nThe schematic of the magnetic phase diagram for GdFeCo is shown in insertions in Fig. 1. At T= 275 K and T= 2885\nFIG. 1. The amplitude of the magnetization precessional response after the demagnetization due to the femtosecond laser\npulse action in GdFeCo ferrimagnet near the compensation point at di\u000berent temperatures. Insertions: the schematic of the\nmagnetic phase diagram. There are two antiferromagnetic collinear phases with Mddirected along(opposite) to the external\nmagnetic \feld above(below) the compensation temperature TM. They are separated by the \frst-order phase transition line\n(blue). Above them, an area where the angular phase exists, which is \flled with gray color. The black solid lines are the\nsecond-order phase transition lines. The dashed lines corresponds to the \fxed temperature in the plot. The red dot is the point\nof phase transition for this temperature.\nK the phase transitions are of the second order, which corresponds to a smooth transition from angle \u00120= 0 to\u00120>0,\nand the divergence of the response occurs at Hsf. Immediately above the compensation temperature the transition is\nof the \frst order and the behavior of the response above the is more complex; however, there is no critical divergence.\nThe critical behavior of the signal amplitude was observed experimentally for GdFeCo in ref. [15].\nAnother feature in the dynamics described by the proposed model is the critical behavior of the characteristic\ntimescales that occurs in the vicinity of the second-order phase transitions. To demonstrate this e\u000bect analytically,\nwe assume small deviations of \u0012during oscillations: \u0012(t) =\u00120+\u000e\u0012(t). We obtain:\n\u000e\u0012+!2\nr(\u00120)\u000e\u0012=\u0000\u000b!ex\u000e_\u0012; (16)\nwhere!2\nr(\u00120) =\r2h\nm\n\u001f?Hcos\u00120+\u0010\n2K\n\u001f?\u0000H2\u0011\ncos 2\u00120i\n,!ex=\rM\n\u001f?. The initial conditions are \u0012(0) =\u00120and eq.\n(15). In the limit of small oscillations and !r<\u000b!ex=2 (is ful\flled near the second-order transition) the solution has\nthe form\u000e\u0012(t) =Ae\u0000\ftsinh!t, where\f=\u000b!ex=2,!2=\f2\u0000!2\nr,A=B(\u00120)=!. The rise time can be estimated from\nthe condition _\u0012(\u001crise) = 0:\n\u001crise\u0019atanh!\n\f\n!=atanhp\n\f2\u0000!2r\n\fp\n\f2\u0000!2r(17)\nThe time of the oscillations decay (relaxation time) is proportional to the imaginary part of eigenfrequency and can\nbe estimated by the following expression:\n\u001crelax\u00194\u0019\f\n!2r: (18)\nNear second-order phase transition the mode softening occurs and the eigenfrequency turns to zero: !r!0,\nand we observe growth of the both timescales. The critical behavior of the rise time has been observed in GdFeCo\nexperimentally [15] and the typical values of \u001crisewere of the order of 10 ps.6\nCONCLUSIONS\nTo sum up, the developed theoretical model based on quasi-antiferromagnetic Lagrangian formalism proved to\nbe suitable for description of the coherent ultrafast response of RE-TM ferrimagnets near the compensation point\ndue to an ultrashort pulse of demagnetization in the presence of external magnetic \feld. We have found that the\ntorque acting on magnetizations is non-zero in the noncollinear phase only. We have explained the experimentally\nobserved critical behavior of the response amplitude and characteristic timescales as the consequence of the second-\norder magnetic phase transition from collinear to an angular in the external magnetic \feld and the mode softening\nnear it. These e\u000bects are vivid in the vicinity of the compensation point in external magnetic \feld. Understanding\nthe ultrafast response to demagnetizing optical or electrical pulses and subsequent spin dynamics can facilitate future\ndevelopments in the \felds of ultrafast energy-e\u000ecient magnetic recording, magnonics and spintronics.\nACKNOWLEDGMENTS\nThis research has been supported by RSF grant No. 17-12-01333.\n\u0003davydova@phystech.edu\nyzvezdin@gmail.com\n[1] T. Ostler, J. Barker, R. Evans, R. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader, E. Men-\ngotti, L. 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Phys. 82, 2731 (2010).\n[10] R. B. Wilson, J. Gorchon, Y. Yang, C.-H. Lambert, S. Salahuddin, and J. Bokor, Phys. Rev. B 95, 180409 (2017).\n[11] Y. Yang, R. B. Wilson, J. Gorchon, C.-H. Lambert, S. Salahuddin, and J. Bokor, Science Advances 3(2017), 10.1126/sci-\nadv.1603117, http://advances.sciencemag.org/content/3/11/e1603117.full.pdf.\n[12] R. Chimata, L. Isaeva, K. K\u0013 adas, A. Bergman, B. Sanyal, J. H. Mentink, M. I. Katsnelson, T. Rasing, A. Kirilyuk,\nA. Kimel, O. Eriksson, and M. Pereiro, Phys. Rev. B 92, 094411 (2015).\n[13] C. D. Stanciu, A. Tsukamoto, A. V. Kimel, F. Hansteen, A. Kirilyuk, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 217204\n(2007).\n[14] C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, Phys. Rev. B 73, 220402\n(2006).\n[15] J. Becker, A. Tsukamoto, A. Kirilyuk, J. C. Maan, T. Rasing, P. C. M. Christianen, and A. V. Kimel, Phys. Rev. Lett.\n118, 117203 (2017).\n[16] Z. Chen, R. Gao, Z. Wang, C. Xu, D. Chen, and T. Lai, Journal of Applied Physics 108, 023902 (2010),\nhttps://doi.org/10.1063/1.3462429.\n[17] A. Moser, K. Takano, D. T. Margulies, M. Albrecht, Y. Sonobe, Y. Ikeda, S. Sun, and E. E. Fullerton, Journal of Physics\nD: Applied Physics 35, R157 (2002).\n[18] B. Lenk, H. Ulrichs, F. Garbs, and M. Mnzenberg, Physics Reports 507, 107 (2011).\n[19] J. Walowski and M. Mnzenberg, Journal of Applied Physics 120, 140901 (2016), https://doi.org/10.1063/1.4958846.\n[20] M. Liebs, K. Hummler, and M. F ahnle, Phys. Rev. B 46, 11201 (1992).\n[21] B. Goransky and A. Zvezdin, JETP Lett. 10, 196 (1969).\n[22] A. Zvezdin and V. Matveev, JETP 35, 140 (1972).\n[23] C. K. Sabdenov, M. Davydova, K. Zvezdin, D. Gorbunov, I. Tereshina, A. Andreev, and A. Zvezdin, J. Low. Temp. Phys\n43, 551 (2017).\n[24] M. Ding and S. J. Poon, Journal of Magnetism and Magnetic Materials 339, 51 (2013).\n[25] A. Zvezdin, JETP Lett. 28, 553 (1979); arXiv preprint arXiv:1703.01502 (2017).\n[26] B. Ivanov and A. Sukstanskii, Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 84, 370 (1983).7\n[27] A. Zvezdin, Handbook of Magnetic Materials 9, 405 (1995)."    },    {        "title": "1004.4631v1.Andreev_reflection_in_ferrimagnetic_CoFe2O4_SrRuO3_spin_filters.pdf",        "content": "Andreev reflection in ferrimagnetic CoFe 2O4/SrRuO 3 spin filters \n \nFranco Rigato1, Samanta Piano2,3, Michael Foerster1, Filippo Giubileo3, Anna Maria Cucolo3, and  \nJosep Fontcuberta1 \n \n1Institut de Ciència de Materials de Barcel ona, CSIC, Campus UAB, Bellaterra 08193, Spain \n \n2 School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom \n \n3Physics Department and INFM-CNR SUPERMAT Laboratory, University of Salerno, Via S. Allende, 84081 Baronissi (SA), Italy \n \n \nWe have performed point contact spectroscopy  measurements on a sample constituted by a \nmetallic ferromagnetic oxide (SrRuO 3) bottom electrode and a tunnel ferrimagnetic (CoFe 2O4) \nbarrier. Andreev reflection is observed across the tunnel barrier. From the comparison of Andreev \nreflection in SrRuO 3 and across the CoFe 2O4 barrier we infer that th e ferrimagnetic barrier has a \nspin filter efficiency not larger than +13%. The observation of a moderate and positive spin filtering \nis discussed in the context of the microstructure of the barriers and symmetry-related spin filtering \neffects. I. INTRODUCTION \n \nThe spin polarization is a key quantity for the spin tronics. In the last few years, spin filters, \nconstituted by a ferro(ferri)magne tic tunnel juncti on, have emerged as promising alternative to \ncreate artificial spin polarized current sources. Ferromagnetic spin filters rely on the spin-dependent \ntransmittance of ferromagnetic tunnel barriers due to the existence of an exchange-split band gap. \nFollowing the pioneering work of Moodera et al.1 who showed spin-filtering at low temperature \nusing EuS ferromagnetic tunnel barriers, spin-filte ring has been demonstrated using perovskite \noxides2 (BiMnO 3) and more recently, spinel oxides: NiFe 2O4 (NFO)3,4 and CoFe 2O4 (CFO)5  \nbarriers. As the Curie temperature of ferrites is  well above room temperature, efficient room-\ntemperature spin-polarized sources could be obta ined using these oxides. The most simple spin-\nfilter structure is formed by two metallic electrodes: a non spin-polarized current source (M) and a ferromagnetic layer (FM), acting as a spin analyzer , with a ferro(ferri)ma gnetic tunnel barrier (FI) \nbetween them. Determination of the spin-filtering efficiency (P\nFI) of the FI barrier has been \nachieved by measuring the magnetoresistance of the tunnel junction and using the Jullière model to \nderive P FI. Using this approach low-temperature values  of +22% and -25% have been reported for \nNFO3,4 and CFO5 respectively. However, by using the Meservey-Tedrow technique, positive P CFO \nvalues (ranging from +6% to 26% depending on film growth condi tions) have been obtained in \nCFO-based spin filters6. Theoretical calculations of spin-depe ndent electronic structure of spinels \nindicate that the lowest ener gy conduction band is spin-down7, thus predicting a P FI < 0. It thus \nfollows that, in spite of its relevance for further progress in spintronics, no definitive determination \nof P FI is yet available for spinel-based spin filters.  \nThe Point Contact Andreev Reflection (PCAR) t echnique has been introduced as a tool to \nmeasure the spin polarization of carriers in ferromagnetic materials8. In the case of a normal \nmetal/superconductor junction, an incoming electron from the normal metal with energy smaller \nthan the superconducting gap cannot enter into the superconducting (S C) electrode and is reflected \nas a hole in the normal metal, simultaneously adding two electrons (a Cooper pair) to the condensate in the superconductor. This process,  known as Andreev reflection (AR), causes an \nincrease of the conductance around zero bias G(V≈0) compared to the conductance at voltages \nG(V) well above the superconducting gap ( ∆/e) by a factor of two. Since the reflected hole is \ncreated in the density of states with opposite sp in than the incoming electron, the AR process is \npartially suppressed when the respective densities of  states are not equal, as in the case of a ferromagnetic metal/superconductor (FM/ SC) interface. In particular, in  fully spin-polarized metals, \nall carriers have the same spin  orientation and the AR should be  totally suppressed because a hole \ncannot be created in the opposite de nsity of states. Thus, the absolute value ( but not the sign) of \ntransport spin polarization │P│ of a ferromagnetic material can be inferred from the grade by \nwhich AR is suppressed in a measured conductance spectra8,9. It has been recently predicted10 that \nthe insertion of a spin-filtering barrier to form a SC/FI/M structure should lead to the subsequent \nmodification of the AR process by spin-selective tunnelling across the ferromagnetic insulator.  \nIn this paper we report PCAR spectrosc opy experiments realized by pressing a \nsuperconducting Nb tip on a spin filter constituted by a CoFe 2O4/SrRuO 3 (CFO/SRO) thin film \nbilayer. We show that AR occurs across the ferrimagnetic CFO tunnel barrier of the \nNb/CoFe 2O4/SrRuO 3 (SC/FI/FM) structure demonstrating spin-preserved tunnelling through the \nCFO barrier. Analysis of the experimental co nductance data by means of the modified BTK \nmodel11 allows determining the polarization P of th e CFO/SRO bilayer. PCAR experiments have \nalso been conducted on the bare SRO layers, and we obtained P ≈ 42(1)%. These results indicate a \nvery modest filtering efficiency of the CFO barriers (< +13%). Implications of these findings are discussed. \n \n  \nII. SAMPLE PREPARATION AND PRE-CHARACTERIZATION \n \nCFO/SRO bilayers have been deposited by RF magnetron sputtering from stoichiometric \ntargets, on a single crystalline (111)SrTiO 3 substrate. The bottom electrode is a 25 nm thick \nferromagnetic (T Curie ≈ 120 K) and metallic ( ρ ≈ 200 μΩ •cm at 10K) SRO epitaxial film, deposited \nat ≈0.5 nm/min in a mixed atmosphere of argon and oxygen (ratio 3:2), with a total pressure of 100  \nmTorr; the substrate temperature was 725 ºC. The top layer is a 3 nm thick epitaxial CFO film, \ngrown at ≈ 0.1nm/min with an Ar/O 2 ratio of 10:1, and a total pre ssure of 250 mTorr at 500 °C. The \nroughness of the SRO and the CFO layer are lower than 0.3 nm. Th e saturation magnetization of a \nbare 3 nm thick CFO film12, measured by using a Quantum Design’s SQUID, was ≈524 emu/cm3. \nThis value is somewhat larger than the bulk valu e. Similar enhancement ha d been reported earlier \nfor ultrathin films of other spinel oxides13 and attributed to partial ca tion inversion of the spinel \nstructure and subsequent electronic and magnetic ordering modifications.  The bilayer structure was characterized by Conductive-Atomic Force Microscope (C-AFM), \nwith a Nanotec Cervantes AFM, using Nanosensors  tips (CDT-NCHR). Electrical measurements \nwere performed in 2-point electr ical configuration: the C-AFM probe was grounded, while the SRO \nbottom electrode was positively biased. C-AFM m easurements were conducted in dry nitrogen \natmosphere, using a feedback-force of ≈850 nN and an applied voltage of 800 mV. \nIn Fig. 1(a) we show a resistance map measured  by C-AFM: at each point of the surface the \nsystem records the current between tip and sample under constant bias voltage. This image indicates \na very high electrical hom ogeneity in a large area (3 μm x 3 μ m). The simultaneously recorded \ntopographic maps (not shown) confirmed the absence of particular defects and an extremely smooth \nsurface (rms < 0.3 nm). The histogram of the resist ance values, shown the Fig. 1(a)(inset) indicates \na narrow distribution of log R (half-width at half-maximum ≈ 3.5%), centred around ≈8.25 \n(≈178 MΩ). As the tunnelling current depends exponentia lly on the barrier thic kness, the data in \nFig. 1(a) signals also an extremely sm all variation of the barrier thickness14. Subsequent \nmeasurements were repeated in  the same area; neither the re sistance map nor the corresponding \ntopographic image evidenced significant changes in the surface properties other than a minor \nreduction of the overall resistance likely associat ed to residual surface contamination removal. No \ntraces of indentation or  scratching could be detected. Tests re peated at different locations on the \nsample surface yield very similar results thus co nfirming the homogeneity of the surface properties. \nThe Current-Voltage ( I-V) curves measured at different points on the surface (one example \nis shown in Fig. 1b) show clear characteristics of tunnelling behaviour. The asymmetry visible in \nI-V curves is due to the asymmetric contact conf iguration for forward and reverse biasing of the \ntunnel structure and it is of no re levance for PCAR experiments, performed in the subgap region, to \nbe described in the following. \n \n \nIII. POINT CONTACT SPECTROSCOPY \n \nPoint contact junctions were formed by pushing a soft Niobium tip, obtained by mechanical \ncutting and chemical etching a Nb wire (diameter ≈ 0.2 mm), onto the CFO surface of the \nCFO/SRO bilayer. The experimental setup utilized for the meas urements has been described \nelsewhere15. Due to the softness of Nb and to the hard ness of the CFO surface, we do not expect \nsignificant scratching or penetration of the CFO film by the tip. The e ffective electric contact radius d can be estimated by using the approximation16 R = 4ρ l / 3πd2 + ρ / 2d and employing the values of \nthe resistance R measured at high bias ( ≈200-300 Ohm), the resistivity of equivalent  SRO films \n(ρ ≈ 200  μΩcm) at 4 K and a mean free path l of about 15 Å, estimated from ρ17. It turns out that \nthe estimated contact size d is of about 45-60 Å, implying that although the contacts are larger than \nthe mean free path they are stil l in an intermediate regime d ≈ l. \nWe have recorded conductance curves at low temperature (T = 4.2 K) at different positions \non the film and with different pressures between tip  and sample. In Fig. 2 we show representative \nconductance spectra of two different types of junc tions we measured: Junctions 1 and 2. Data have \nbeen normalized by using the background conductance estimated at large voltage (V >> ∆/e) \nregions, where Δ is the superconducting gap of Nb ( Δ ≈ 1.5 meV). Both spectra display a clear \nincrease of the conductance around zero bias suggesting that the tr ansport is mainly due to the \nAndreev reflection process. More over, Andreev reflection, presumab ly spin-filter weighted, takes \nplace across the ferrimagnetic barrier as predicted in Re f. [10]. From data in Fig. 2 it is clear that the \nnormalized conductance G(≈0 V) < 2 thus indicating some suppr ession of AR as expected for a \nspin-polarized electron tunnelling.  \nIt is noteworthy in Fig. 2 that the features in the conductanc e spectrum appear at energies \nthat are sensibly higher than what is expected for Nb. This observation is  commonly attributed to \nthe presence of a spread resistance R S arising from the resistance of the sample between the junction \nand one of the measuring contacts18. The effect of R S is to shift the coherence peaks from V ≈ Δ to \nlarger voltages and subsequently changing the G (0) values. Following Woods et al.18 the spread \nresistance is included in our modelling of the PCAR curves by considering two contributions to the \nmeasured voltage19: \n () ()()I V I V I VS PC + =  (1)\nand so the measured conductance G(V)  will be fitted by using  \n \n()1−\n⎟\n⎠⎞⎜\n⎝⎛+ = =dIdV\ndIdV\ndVdIV GS PC (2)\n V PC and VS are the voltage drops at the FM/FI/SC junction and at R S, respectively. To \ncalculate VPC(I) we used the modified BTK model11 considering an effective P spin-polarized \ncurrent coming out from the spin-filter. For simplicity, a ballistic regime ( l >> d ) of transport across \nthe junction will be assumed. Its application to the present case, where l ≈ d, at first sight may seem \nproblematic. However, as shown in Ref. [18], th e potential errors introdu ced by applying ballistic formulas to diffusive contacts, as far as Z is not too small, have a negligible impact on the extracted \nP values and the uncertainty is translated into the barrier transparency Z. The other relevant fitting \nparameters are: the superconducting gap Δ and a smearing coefficient ( Γ) that allows modelling of \nthe broadening of the gap edges and in elastic scattering at the interfaces20. The local temperature \nhas been assumed to be the same like measured at the sample holder, which is immersed in a liquid \nHe bath. \nThe spread resistance R S was determined beforehand by setting a gap value Δ equal to the \nBCS value for Nb, thus eliminating a possible degeneracy in Δ and R S. Consequently, Δ cannot be \nconsidered a completely free parameter, but it is set close to the BCS value. In the subsequent \nfitting step, R S was kept fixed to the determined value and P, Z, Δ and Γ  were allowed to vary. The \nsolid line in Fig. 2a is the result of the optimal fit of G(V)  using P  ≈ 39%, Z ≈  0.13, Γ  ≈ 0.22 meV, \nΔ ≈ 1.50 meV and R S/RPC ≈ 0.5. The low interface barrier transparency Z used to fit our \nexperimental data indicates that our measurem ents are not significantly affected by a possible \ndependence of the values of P on Z21-23. The quality of the fit and its robustness on variations of P \ncan be better appreciated in Fig. 2c where we s how the low-voltage zoom of  data in Fig. 2a. In \nFig. 2c we also include fits obtained by fixing Δ (1.5 meV) and P  to be larger (45%) or smaller \n(35%) than P from optimal fit of Fig. 2a and allowing Z, Γ to vary. We notice in Fig. 2c that fixing \nlarger (smaller) P values lead to smaller (larger) Γ values as predicted in Ref. [22], just illustrating \nthat moderate inelasti c interface scattering ( Γ) decreases the AR probability. Thus the Γ and P \nparameters in the AR spectra mix together a nd distinction among both e ffects is challenging20. In \nspite of this, it is clear from these data that P J1 should be in the 45-35 % range for Junction 1. \nTo treat a possible degeneracy of fits  to PCAR spectra, it has been proposed25 to perform fits \nof (Δ, Z, Γ ) for different fixed P trial, and check the resulting sum of the squared deviations χ2 of the \nfits as function of P trial. In Fig. 3, results of such analysis are shown for the data  of Junction 1. A \nclearly defined minimum of χ2 at about 39(1)% is found, which falls well within the previously \ndetermined range of values, thus we conclude that P J1 ≈ 39(1)%. However, for the data of Junction \n2 (Fig. 2b, which will be discussed below), such pr ocedure proved to be ill-d efined when using the \nproximity reduced gap, due to the larger number of fit parameters. \nIt has been demonstrated25 that an error in the normalization of the conductance spectra can \nlead to false minima in χ2 vs. P trial. Therefore the χ2 analysis was repeated using a deliberately \nwrong normalization of the data in Fig 2a (divid ed by an additional factor n = 1.001). Still, χ2 \nreproduces the minimum but values of χ2 are considerably enlarged. For normalization just slightly smaller than 1 (factor 0.999) reas onable fits to the data are no longer possible and values of χ2 \nincrease drastically, demonstrating the validity of the original normalization. \nIn Fig. 2b we show an example of a G(V) curve measured in some different contacts \n(Junction 2). Data in Fig. 2b displa y characteristic features of AR reflection but also the occurrence \nof proximity effects and subsequent smaller gap formation ( ΔP)11,24. As CFO is a ferromagnetic \ninsulator it may be supposed that a gap reduction may occur in some part of th e tip, likely due to the \nstray field of the ferrimagnetic barrier. The mo dified BTK model has b een worked out including \nexplicitly two gaps ( Δ and ΔP)11. We used the corresponding expressi ons to fit the da ta in Fig. 2b. \nThe solid line is the result of the optimal fit of G(V) using, Δ ≈ 1.48 meV, ΔP ≈ 0.99 meV, \nΓ ≈ 0.0 meV and R S/RPC ≈ 0.36. The values Z ( ≈0.20) and P  (31%) determined for this contact are \nquite similar to those extracted from data of Junc tion 1 (Fig. 2a). In the zoom of the low-voltage \nregion (Fig. 2d) we include the f its obtained by fixing P to larger (35%) or smaller (25%) values \nwhile keeping Δ and Δ P constant. The small Γ values obtained from the best fits ( Γ ≈ 0.1), most \nprobably result from the fact that the broadening of the spectrum is captured during the fit by the \npresence of the two gaps required to fit the di ps. Although the available data do not allow to \ndisentangle both contributions, it is  clear from these data that P ≈ 31(3)% is a robust result for \nJunction 2. At this point it is wort h to recall that dips in PCAR c onductance spectra, as they appear \nin the data of Junction 2 have also been explai ned as arising from the superconductor reaching its \ncritical current and thus adding some normal conducting state finite resistance26. Although the \norigin of the dips in  the conductance spectr a cannot be decided with certainty, a smaller \nsuperconducting energy gap is in princi ple, consistent with the observation of critical current effects \nin Junction 2.  \nTherefore, it follows that Andreev reflecti on has been observed across a ferrimagnetic tunnel \nbarrier (Fig. 4). From the analysis of the PCAR we infer that representative values of the effective \npolarization of our CFO/SRO spin f ilter (3 nm thick) are of about P ≈ 31(3)% and 39(1)% as \ndetermined for Junctions 1 and 2 respectively. To  progress further and to extract the spin-filter \nefficiency of the CFO barrier re quires the knowledge of the spin-p olarization of electrons emitted \nfrom the SrRuO 3 electrode (P SRO).  \nSignificantly different P SRO results have been reported: Worledge and Geballe27 reported \nMeservey-Tedrow type measurements of tunnel j unctions having SRO as electrode, and determined \nPSRO = -9%; Nadgorny et al.23, using PCAR with Sn tips, inferred a much larger value \n(│PSRO│≈ 53%). Although this strong discrepancy is not fully understood23,27, consensus exist that PSRO < 0. Negative spin-polarization arises from the difference of Fermi velocities for spin-up and \nspin-down electrons emerging from SRO rather than the density of states at the Fermi level (which \nis practically identical)27. \nTo obtain a more reliable basis for assessing the PCAR results from the CFO/SRO structure, \nadditional experiments were perfor med on a SRO film equivalent to  the bottom electrodes used in \nthe bilayers. A representative PCAR spectrum of the SRO film is shown in Fig. 5. The polarization \nof SRO was again determined by fitting (solid lin e) to the modified BTK model and resulted in \nPSRO = 42%, somewhat lower but still in agreement w ith previously reported PCAR results within \ntheir variations23. In the inset of Fig. 5, the resulting χ2 for fits with various P trial to the SRO data \nwith a clear minimum at P SRO = 41.5(1)% is shown. \n \n \nIV. DISCUSSION \n \nWe are now in position to compare the effective polarization measured in  the spin filter and \nthe bare SRO electrode. One first notes that the values of P extracted from both junctions are \nsmaller than that of the bare SRO electrode. Due to the fact that P SRO is recognized to be negative \n(PSRO < 0), this observation implies that the spin pol arization of the CFO barrier must be positive \n(PCFO > 0). Notice that although PCAR experiments of a single ferromagnetic layer do not allow to \nextract the sign of the spin polar ization, this is possible in th e present structure, where two \nferromagnetic layers are involved, if the sign of the sp in polarizations of one of the layers (SRO in \nthis case) is known.  \nIt has been theoretically predic ted that the lowest energy barrie r in the exchange-split gap of \nCFO is the spin-down7; in such circumstances it could be expected that the spin-down channel is \ndominating (P FI < 0) the AR (Fig. 4), and thus one could anticipate an effective P in CFO/SRO \nbilayers larger than in the PCAR of the bare SRO electrode. Experimentally this is not the case as \nPJ1 and P J2 are found to be smaller than P SRO. We will discuss below on this discrepancy. \n We next consider the values of the spin filtering efficiency of the CFO layers. The two \nextracted values of P J1, J2 (≈ 39(1)% and ≈  31(3)%) represent two distinct situations. Whereas the \nsecond value (Junction 2) would indicat e a substantial spin filtering of CFO, this is not so for data \nfrom Junction 1, where a rather small different  with the bare SRO electrode is observed.  \nWe can define the spin-filter efficiency of the CFO barrier P CFO by28:  \nCFO SROCFO SRO\nP P 1P PP++=  (3)\nwhere P is the effective spin polarization measured in the PCAR experiment. Using P SRO = -\n42%, the two solutions  of Eq. (3) are P CFO ≈ +4% and +73% for P J1 and P CFO ≈ +13% and +67% for \nPJ2. Among these two sets of possible values, which phys ically arise due to th e fact that PCAR can \nnot determine the sign of the meas ured spin polarization, the larges t pair of values (73% and 67%) \nare most likely inappropriate as they could lead to  spin filtering efficiencies for CFO much larger \nthan reported values5,6. On the other hand, we notice that P CFO ≈ +13%, as determined for Junction \n2, is within the range of Meservey-Tedrow results (+6%  ≤  PCFO ≤ +26%, depending on the growth \nconditions)6 obtained on CFO barriers of nominally equal thickness (d ≈ 3 nm). Notably, the sign of \nthe effect agrees with the Meservey-Tedrow experi ments, which is, however, different from the one \ndetermined in tunnel magneto resistance measurements using Al 2O3 tunnel barriers. On the other \nhand data for Junction 1 appears to  indicate a very marginal spin filtering effect. This could be \nrelated to the fact that the CFO barrier is locally suppressed either by a mechanical effect associated \nto the Nb-tip pressure (although, as mentioned a bove, scratching effects have not been observed \nwhen using the harder C-AFM tip) or a locally poorer homogeneity of the insulating barrier. \nAlthough the CFO layers appears homogenous and r obust in CAFM measurem ents, the possibility \nthat the Nb tip penetrates the CFO layer either through pinholes or completely when crushed onto \nthe sample cannot be excluded. Under such circumstances, it turns out that P J2 and thus P CFO ≈ \n+13% constitutes the most represen tative value of the spin filtering efficiency of the present CFO \nbarriers. It could be argued that a similar result would be obtained if the CFO barrier does not spin-\nfilter at all but only contributes to depolarize the electron current from SRO. However, the experimental observation of spin-filt ering in CFO in tunnel structures\n5,6 does not support this view.  \nBefore concluding we would like to comment on the positive sign observed for P CFO. As \nmentioned, this observation is oppos ite to theoretical predictions7, which are based on the electronic \nconfiguration of an ideally inverse spinel structure of CFO where all Co2+ occupy the octahedral \nsites of the unit cell and the Fe3+ equally populate the octahedral and tetrahedral sites. However, \nmagnetization data of nanometric thin films13,29 of various spinels, including CoFe 2O4 and NiFe 2O4 \nhave provided conclusive evidence that the ca tionic distribution in nanometric thin films may \nlargely differ from their bulk c ounterparts. As shown by the calc ulations of Szotek et al.7, the \ninsulating gap of CFO closes by some 75% when th e cationic distribution is not that of the ideal \ninverse spinel structure but a nor mal one. It thus follows that th e tunnel transport may be overcome by other non-spin preserving transp ort channels and therefore a redu ced spin filtering efficiency \ncould be anticipated for cationically -disordered films. To what extent  the change of sign of the spin \nfiltering efficiency is related to the same effect or  not is not definitely settled. We also notice that \nthe spin filtering efficiency should not be simply related to the ex change splitting of the insulator \nbut the symmetry of the relevant wave  functions may also play a role. \n \n \nIV. SUMMARY \n \nIn summary we have shown evidence that Andreev reflection occurs at ferro(ferri)magnetic \ntunnel barriers. Data collected us ing point contact spect roscopy allowed to estimate the effective \nspin-polarization of a current th rough the interface of a spin-fil ter and a superconducting tip. It \nturned out that a possible spin filtering effect of the spinel oxide CoFe 2O4 tunnel barrier is limited to \nabout +13% with the accuracy of  these measurements. Observati on of a positive spin filtering \nefficiency is unexpected and may suggest the rele vance of spin-dependent orbital symmetry effects \non the tunnel probability in spin filters. To th e best of our knowledge this issue has not been \ntheoretically addressed yet.  \n \n \nACKNOWLEDGEMENTS \n \nFinancial support from the Ministerio de Ci encia e Innovación of th e Spanish Government \nProjects (MAT2008-06761-C03 and NANOSELECT CSD2007-00041) and from the European \nUnion [ProjectMaCoMuFi (FP6-03321) and FEDER  and Marie Curie IEF Project SemiSpinNano] \nis acknowledged. S.P. thanks B.L. Gallagher and C. J. Mellor for hosting the final stage of this research. \n REFERENCES \n1 J. S. Moodera, X. Hao, G. A. Gibson, and R. Meservey, Phys. Rev. Lett. 61, 637 (1988). \n2 M. Gajek, M. Bibes, A. Barthélémy, K. 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B 67, 174423 (2003 ). \n18 G. T. Woods, R. J. Soulen, Jr., I. Mazin, B. Nadgorny, M. S. Osofsky, J. Sanders, H. Srikanth, \nW. F. Egelhoff, and R. Datla, Phys. Rev. B 70, 054416 (2004)  \n19 S. Piano, F. Bobba, F. Giubileo, A. M. Cucolo, M. Gombos, and A. Vecchione, Phys. Rev. B 73, 064514 (2006). \n20 P. Chalsani, S. K. Upadhyay, O. Ozatay, and R. A. Buhrmann, Phys. Rev. B 75, 094417 (2007). \n21 V. Baltz, A. D. Naylor, K. M. Seemann, W. Elder,  S. Sheen, K. Westholt, H. Zabel, G. Burnell, \nC. H. Marrows, and B. J. Hickey, J. Phys. Condens. Matter 21, 095701 (2009). \n22 N. Auth, G. Jakob, T. Block, and C. Felser, Phys. Rev. B 68, 024403 (2003). \n23 B. Nadgorny, M. S. Osofsky, D. J. Singh, G. T. Woods , R. J. Soulen Jr., M. K. Lee, S. D. Bu \nand C. B. Eom, Appl. Phys. Lett. 82, 427 (2003). \n24 R. P. Panguluri, K. C. Ku, T. Wojtowicz, X. Liu, J. K. Furdyna, Y. B. Lyanda-Geller, N. \nSamarth, and B. Nadgorny, Phys. Rew. B 72, 054510 (2005). \n25 Y. Bugoslavsky, Y. Miyoshi, S. K.  Clowes, W. R. Branford, M. Lake, I. Brown, A. D. Caplin, \nand L. F. Cohen, Phys. Rev. B 71, 104523 (2005). \n26 G. Sheet, S. Mukhopadhyay, and P. Raychaudhuri, Phys. Rev. B 69, 134507 (2004). \n27 D. C. Worledge and T. H. Geballe, Phys. Rev. Lett. 85, 5182 (2000). 28 This equation is derived by considering diffe rent transmission coefficient for the two \nindependent spin channels across the spin-filte ring CFO barrier and a spin polarized current \ncoming from the SRO. \n29 F. Rigato, M. Foerster, J. Fontcuberta, unpublished Figure Captions \n \nFig. 1: (Color online) C-AFM analysis of a CF O (3 nm)/ SRO (25 nm) bilayer on (111)STO: (a) \nresistance map in logarithmic color scale; inset: hi stogram of the resistance values over the surface. \n(b) I-V characteristic, with a schematic of the measurement circuit (inset). \n \nFig. 2: (Color online) Measured conductance spectra: (a) Junction 1; (b) Junction 2; the \ncontinuous line represents the best fits. (c), (d) Zoom around low bias; for Junction 1 and 2 \nrespectively: Comparison of the best fits with simulations achieved forcing P to slightly different \nvalues, demonstrating the accuracy  of the obtained polarizations. \n \nFig. 3 Characteristics of fits obt ained for data from Junction 1. χ2 as function of P trial for \nstandard normalization (n = 1) and varied normalization (n = 1.001).   \n \nFig. 4: (Color online) Schematic  of the density of states vs. energy of a SC-FI-FM (Nb-CFO-\nSRO) structure. The Andreev refl ection (AR) process is illustrate d (solid arrow). An incoming \nelectron from the SRO side is reflected as a hole in the spin-reversed density of states, while a \nCooper pair is added to the supe rconducting condensate. Parallel to AR, electron tunnelling into \nthermally excited states may occur (dashed arrow). For simplicity, only spin-down electrons are \ndepicted and low-lying spin-dow n barrier has been assumed.  \n \nFig. 5 (Color online) Measured co nductance spectra of a SRO thin film, equivalent to the base \nelectrodes used in CFO/SRO bilayers; the con tinuous line represents th e best fit. Inset: χ2 as \nfunction of P trial for fits to the data in the main panel. \n \n \n \n \n \n \n Figure 1 \n(b)\n-1,0 -0,5 0,0 0,5 1,0-1001020\nSTO (111)SROCFO\nSTO (111)SROCFO\nSTO (111)SROCFO\n Current (nA)\nVoltage (V)\n(a) 10 \n7 Figure 2 \n \n(c) (a) \n(d) (b) Junction 1 Junction 2 \n-20 -10 0 10 200,951,001,051,101,15\nΔNb=1.48 meV\nΔproximity =0.99 meV\nT=4.2 K\nRseries /RPC=0.36\n   Z=0.21\nP=0.305\nΓ=0  meV\nV(mV)dI/dVnormalized\n- 6 - 4 - 2 02460,951,001,051,101,151,20dI/dVnormalized\n   \n P=0.305\nΓ=0.0 meV\nZ=0.21\nP=0.35\nΓ=0.1 meV\nZ=0.1 exp. data\n P=0.25\n P=0.305\n P=0.35\nV(mV)P=0.25\nΓ=0.18 meV\nZ=0.16-20 -10 0 10 200,951,001,051,101,15\nΔNb=1.48 meV\nΔproximity =0.99 meV\nT=4.2 K\nRseries /RPC=0.36\n   Z=0.21\nP=0.305\nΓ=0  meV\nV(mV)dI/dVnormalized\n-20 -10 0 10 200,951,001,051,101,15\nΔNb=1.5 meV            \nT=4.2 K\nRseries /RPC=0.5dI/dVnormalized\n   Z=0.135P=0.39\nΓ=0.22 meV\nV(mV)\n- 6 - 4 - 2 02460,951,001,051,101,151,20dI/dVnormalized\n   P=0.305\nΓ=0.0 meV\nZ=0.21\nP=0.35\nΓ=0.1 meV\nZ=0.1 exp. data\n P=0.25\n P=0.305\n P=0.35\nV(mV)P=0.25\nΓ=0.18 meV\nZ=0.16\n-10 -5 0 5 101,001,051,101,15\nP=0.39\nΓ=0.22 meV\nZ=0.135dI/dVnormalized\n   P=0.45\nΓ=0.0 meV\nZ=0.0P=0.35\nΓ=0.52 meV\nZ=0.19 exp. data\n P=0.35\n P=0.39\n P=0.45\nV(mV)Figure 3 \n0,25 0,30 0,35 0,400,01,0x10-52,0x10-53,0x10-5\n  \n n=1\n n=1.001χ2\nPtrial\n \n Figure 4 \n \n \n \n \n Figure 5 \n-60 -40 -20 0 20 40 600,981,001,021,041,061,08\n0,36 0,40 0,449,0x10-61,0x10-51,1x10-51,2x10-5\n dI/dVnormalized\nV(mV)Δ = 1.45 meV\nZ = 0.185\nΓ = 0.27 meV\nP = 0.415\nRS/RPC = 1\nχ2\nPtrial\n \n "    },    {        "title": "2111.06045v3.Kondo_effect_in_Lieb_s_ferrimagnetic_system_on_the_T_shaped_bipartite_lattice.pdf",        "content": "arXiv:2111.06045v3  [cond-mat.mes-hall]  14 May 2022Kondo effect in Lieb’s ferrimagnetic system on the T-shaped b ipartite lattice\nMasashi Tokuda1,∗and Yunori Nishikawa2,3,†\n1Dept. of Physics, Graduate School of Science,Osaka Univers ity, Toyonaka, Osaka 560-0043 Japan\n2Dept. of Physics, Graduate School of Science, Osaka City Uni versity, Sumiyoshi-ku, Osaka 558-8585 Japan\n3Nambu Yoichiro Institute of Theoretical and Experimental P hysics,\nOsaka City University, Sumiyoshi-ku, Osaka 558-8585 Japan\n(Dated: May 17, 2022)\nThe minimal ferrimagnetism by Lieb’s theorem emerges on the T-shaped bipartite lattice com-\nposed of four sites, which can be realized experimentally, j ust as Nagaoka ferromagnetism has been\ndemonstrated experimentally using a quartet quantum-dot ( J.P.Dehollain et al., Nature 579, 528\n(2020).). In this paper, the Kondo effect on this ferrimagnet ism is theoretically studied. The mag-\nnetic moment S= 1 is screened in two steps by the Kondo effect and the series co nductance gsis\nstrongly suppressed to gs≃0, while the parallel conductance gphas the maximum value gp≃4e2/h.\nThe robustness of these properties against a parameter chan ge toward reducing the Lieb’s ferrimag-\nnetism is also discussed, showing the scenarios for entangl ement of the degrees of freedom toward\nthe ground state.\nPACS numbers: 72.10.F,72.10.A,73.61,11.10.G\nI. INTRODUCTION\nItinerant magnetism has been one of the intriguing\nandchallengingtopicsin condensedmatterphysics, espe-\ncially in strongly correlated electron systems. For exam-\nple, a reliable general theory predicting magnetic transi-\ntion temperature for various magnetic materials has yet\nto be realized. Alternately, there are some exact theo-\nries predicting the existence of magnetism in the Hub-\nbard models (see, e.g., a text book on this topic1, and\nthe references therein). The Lieb’s theorem for ferrimag-\nnetism in the half-filled and repulsive Hubbard model on\nbipartite lattice is one such theory2,3and has been ex-\namined in many different material systems; for example,\nhoneycomb lattice structures such as graphene4–10. By\ndefinition, a bipartite lattice is connected and composed\nof two sublattices AandB, where any bond connect-\ning sites is in different a sublattice. The Lieb’s theorem\nstates that the ground state of the half-filled and repul-\nsive Hubbard model on bipartite lattice has the mag-\nnetic moment S=|NA−NB|/2 and is unique up to the\nspin degeneracy. Here NAandNBare the number of\nlattice points in sublattice AandB, respectively. Ac-\ncording to this theorem, the minimal ( and nontrivial11\n) Lieb’s ferrimagnetism emerges on the T-shaped bipar-\ntite lattice composed of four lattice points decomposed\nintoNA= 3 andNB= 1 and has the magnetic moment\nS= 1, which we focus on in this paper. Such a small\nlattice can be experimentally realized as a quantum-\ndot array using recent nanotechnology. Actually, a con-\ntrollable quartet quantum-dot plaquette has been fabri-\ncated and the Nagaoka ferromagnetism in the Hubbard\nmodel on the plaquette lattice has been demonstrated\nexperimentally12. Similarly, an experimental realization\nof the minimal Lieb’s ferrimagnetism mentioned above\nwould be possible in the near future. Incidentally, the\nKondo effect, a screening of a magnetic moment in an\nitinerant electron system by the many-body effect, hasbeen investigated using a quantum-dot array connected\nto leads as itinerant electron reservoirs13–21. When the\nminimal Lieb’s ferrimagnetismis realized experimentally,\nit would be interesting and challenging to investigate the\nKondo effect on such an intriguing magnetism. In this\npaper, we theoretically investigate the Kondo effect on\nthe minimal Lieb’s ferrimagnetism on the T-shaped lat-\ntice connected to reservoirs. Using the numerical renor-\nmalization group (NRG) calculation and the local Fermi\nliquid theory, we predict the two-step Kondo screening of\nthe ferrimagnetic moment, and the strongly suppressed\nand perfect conductivity through the T-shaped lattice\nunder the Kondo screening, respectively, for two kinds of\nconfiguration. The robustness of these properties against\na parameter perturbation toward reducing the Lieb’s fer-\nrimagnetism is also predicted.\nThis paper is organized as follows. In Sec. II, the\nmodel and the formulation we use in this paper are pre-\nsented. We show our results in Sec. III. First of all, we\ninvestigate the isolated Hubbard model on the T-shaped\nlattice in Sec. IIIA. After showing the reliability of our\nmethod in Sec. IIIB, we present our main results in Sec.\nIIIC. The robustness of our findings against parameter\nperturbations is discussed in Sec. IIID. Section IV is\ndevoted to the Conclusion.\nII. MODEL AND FORMULATION\nThe model we consider is a Hubbard model on the T-\nshaped bipartite lattice decomposed into the sublattice\nA={1,3,4}andB={2}, which connects two reservoirs\nat the left (L) and right (R) by the symmetrical tunnel-\ning matrix elements v, as illustrated in Fig. 1 (a). The2\nHamiltonian His given by H=HT+Hres+Hhybwith\nHT=/summationdisplay\nσ,i∈A,j∈Btijd†\niσdjσ+/summationdisplay\ni,σ(εd,iniσ+Uini↑ni↓),(1)\nHres=/summationdisplay\nν,k,σενkc†\nνkσcνkσ, (2)\nHhyb=v/parenleftBig\nd†\n1σψLσ+h.c./parenrightBig\n+v/parenleftBig\nd†\n4σψRσ+h.c./parenrightBig\n,(3)\nwherediσannihilates an electron with spin σat the site-\niin the T-shaped lattice, characterized by the intersite\nhopping matrix elements tijbetween site i∈Aand\nj∈B, the onsite energy εd,iand the intrasite repulsion\nUi. Hereniσ≡d†\niσdiσis the number operator of the elec-\ntronwith spin σatthesite-i. Thenecessaryconditionfor\nthe emergence of the Lieb’s ferrimagnetic state is a half-\nfilled repulsive Hubbard model on a bipartite lattice, so\nthat the symmetry of the lattice is not a main factor for\nemerging the Lieb’s ferrimagnetic state. Therefore, for\nsimplicity we assume εd≡εd,1=εd,2=εd,4,ε3≡εd,3\nt≡t12=t24,t3≡t23andU≡U1=U2=U4through-\nout this paperand setmainly εd=ε3, t=t3andU=U3\nunless otherwise stated. In the reservoir at ν(=R, L),\nc†\nνkσcreatesan electronwith energy ενkcorrespondingto\nanone-particlestate φνk(r) andψνσ=/summationtext\nkφνk(rν)cνkσis\nthe field operator of the conduction electron in the reser-\nvoir atrνwhere the conduction electrons in the reservoir\nmix with the electrons in the site labeled by i= 1 (for\nν=L) ori= 4 (forν=R). We assume that the hy-\nbridization strength Γ ≡πv2/summationtext\nk|φνk(rν)|2δ(ω−ενk) is a\nconstant independent of the frequency ωandν, and take\nthe Fermi energy µto beµ= 0. Hence, assuming that\nthe conduction electron in the reservoir has a flat band\nstructurewith halfbandwidth D, wehaveΓ = πv2/(2D).\nOur system has inversion symmetry, so that the even\nand odd parities are good quantum numbers. There-\nfore, it is convenient to introduce the even-parity orbitals\na1σ, a2σ, a3σand the odd-parity orbital b1σas follows;\na1σ=d1σ+d4σ√\n2,a2σ=d2σ,a3σ=d3σ,(4)\nb1σ=d1σ−d4σ√\n2. (5)\nThe retarded Green’s functions for a1σandb1σplay an\nimportant role for calculating the conductance through\nthe T-shaped lattice and the averaged electron number\nin the lattice because the orbitals d1σandd4σconnect\nto the reservoirs. Due to the inversion symmetry, at the\nzero temperature and Fermi energy, each of these two\nretarded Green’s functions is determined by a single real\nparameter, κeorκo. The parameter κp(p=e,o) is\ndefined by,\nκp=detKp\nΓdetKp,11. (6)\nHere,Kp≡ −(h(0)\np+ReΣ+\np(0)), where h(0)\npis the matrix\ncomposed of the hopping integrals among the p-parityorbitals, Σ+\np(ω) is the self-energy with the p-parity and\nKp,11is the matrix obtained by deleting the first row and\ncolumn corresponding to the orbital a1σorb1σfrom the\nmatrixKp. These real parameters determine the phase\nshiftsδeandδocorresponding to the angles of these two\nGreen’s functions in the complex plane as follows; δe=\narctan(−1/κe),δo= arctan( −1/κo). These two phase\nshiftsδeandδoof the quasi-particleswith even- and odd-\nparity characterize a local Fermi-liquid behavior of the\nwhole system described by H. The conductance gsin the\ntwo-terminal series configuration illustrated in Fig.1(a)\nand the averaged electron number n≡ /an}bracketle{tG|/summationtext\ni,σniσ|G/an}bracketri}ht\nin all sites of the ground state |G/an}bracketri}htare represented22,23as\nfollows;\ngs=2e2\nhsin2(δe−δo), (7)\nn=2\nπ(δe+δo). (8)\nFrom the same phase shifts, we can calculate the con-\nductancegpin the four-terminal parallel configuration\nillustrated in Fig.1 (b) as follows;\ngp=2e2\nh/parenleftbig\nsin2δe+sin2δo/parenrightbig\n. (9)\nWe perform NRG calculation to determine δeandδo.\nIn the NRG approach, asequenceofthe Hamiltonian HN\nis introduced, by carrying out the logarithmic discretiza-\ntion with the controlparameterΛfor the continuouscon-\nduction bands of the electron reservoirs, and trasforming\nthe discretized electron reservoirs as,\nHN= Λ(N−1)/2/parenleftBig\nHT+HNRG:hyb+H(N)\nNRG:res/parenrightBig\n,(10)\nHNRG:hyb=v/summationdisplay\nσ(d1σf†\n0,Lσ+d4σf†\n0,Rσ+h.c.),(11)\nH(N)\nNRG:res=D1+1/Λ\n2/summationdisplay\nν=R,L/summationdisplay\nσN−1/summationdisplay\nn=0ξnΛ−n/2\n×(fn,νσf†\nn+1,νσ+h.c.), (12)\nwherefn,νσannihilates an electron with spin σat\nsitenin theν-discretized electron reservoir, v=/radicalbig\n2DΓAΛ/π,AΛ=1\n2(1+1/Λ)/(1−1/Λ)logΛ, and\nξn=1−1/Λn+1\n/radicalbig\n1−1/Λ2n+1/radicalbig\n1−1/Λ2n+3.(13)\nWe keep the lowest 3600 eigen states during the NRG\niteration process and set Λ = 6 in our NRG calculations.\nWe can deduce δeandδoviaκeandκofrom the fixed-\npoint eigen energies of the NRG calculation.22,23as fol-\nlows;\nκp=v2\nΓDlim\nN→∞DΛN−1\n2gN(ǫ∗\np). (14)\nHere,ǫ∗\npis the quasiparticle energy with the p-parity ob-\ntained from the NRG fixed-point eigen energies , and gN3\nis the Green’s function for one of the isolated discretized\nelectron reservoirs.\nWe have confirmed that the numerical results for the\nfixed-point eigenvalues can be mapped onto the energy\nspectrum of the free quasiparticles in all parameter sets\nwe have examined, which justifies the assumption of the\nlocal Fermi liquid we have made in our formulation.\nUsing the NRG flow, we can calculate the impurity (T-\nshaped lattice) entropy as a function of the discretized\ntemperature TNnrg≡τΛ−(Nnrg−1)/2Dcorresponding to\nthe number Nnrgof NRG iterations24. (Hereτ=O(1) is\na fitting constant .)\n(a) (b)\n12\n34\nv vt t\nt3\n2v2v\n2v2v\nCurrent flow\nCurrent flowRight \nreservoirLeft \nreservoir\nFIG. 1. Schematic picture of (a) series and (b) parallel con-\nfigurations\nIII. RESULTS\nA. Results for the isolated Hubbard model on the\nT-shaped lattice\nFirstly, we investigate the isolated Hubbard model on\nthe T-shaped lattice before connecting the electron reser-\nvoirs, clarifying the spin state Sand the electron oc-\ncupation number Nin the model parameter space. In\nFig.2(a), we show the phase diagram of SandNon the\nmodel parameter plane spanned by (2 εd+U)/tandU/t.\nAtthehalf-filledstate N= 4, weconfirmthattheground\nstate withS= 1 is realized for any positive value of U\nin our system25, as is predicted by Lieb’s theorem. The\nregion ofN= 4 andS= 1 becomes wider as Uin-\ncreases. From the phase diagram of N, we easily realize\nthat the phase diagram of N−4, the electron occupation\nnumber from the half-filled state, is antisymmetric with\nrespect to the line 2 εd+U= 0 (the white dashed line) in\nthe phase diagram. This is because our system has the\nelectron-hole symmetry. As a result, the phase diagram\nofSis symmetric with respect to the line. Along the\nlineU= 0 in the phase diagram, the value of Nchanges\nby increments of two, while S= 0 because two electrons\nwith up and down spin occupy the energy level crossing\nthe Fermi energy at the same time.\nNext, we examine the minimal Lieb’s ferrimagnetic\nstate in more detail. The basis vectors that span the\nN= 4 andS= 1 states including the minimal Lieb’s fer-\nrimagnetic state can be classified into two types, namely,\nS-state and D-state. A basis vector that consists of only-6-4-2 0 2 4 6N=0, S=0N=4, S=1\nN=2, S=0 N=6, S=0 N=8, S=0  N=1,\nS=1/2N=7,\nS=1/2N=3,\nS=1/2N=5,\nS=1/2\n(2εd + U) / t   0 0.5 1 1.5 2 2.5 3 3.5 4 U / t \n 5PS\nPDS-state\nD-state. . . \n. . . \n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(a) (b)\nU / t  0  10  15  20  25  302εd + U = 0 \nFIG. 2. (Color online) (a) The spin quantum number and\nthe electron occupation number of the ground state of the\nisolated Hubbard model on the T-shaped lattice as functions\nof (2εd+U)/tandU/t. (b) The probabilities PSandPD\nplotted as functions of U/t. Examples of S- and D-state are\npresented in this figure.\nsingle occupied sites belongs to the S-state and a basis\nvector that has a doubly occupied site belongs to the D-\nstate. Some examples ofthe S- and D-state are presented\ninside Fig.2(b). A so-calledferrimagnetic state where the\nspins on each sublattice are in ferromagnetic order and\ntwo spins on the different sublattices are anti-parallel, is\nasuperposition state ofthe basisvectorsbelongingto the\nS-state. The ground state |Giso/an}bracketri}htof the isolated Hubbard\nmodel with N= 4,S= 1 (andSz= 1) is a superpo-\nsition state of the basis vectors belonging to the S- and\nD-state. To estimate how close the ground state is to\nthe so-called ferrimagnetic state, we calculate two prob-\nabilitiesPS≡ /an}bracketle{tGiso|ˆPS|Giso/an}bracketri}htandPD≡ /an}bracketle{tGiso|ˆPD|Giso/an}bracketri}ht,\nwhereˆPS(ˆPD) is the projection operatorto the subspace\nspanned by the basis vectors belonging to the S-state (D-\nstate). The results are shown in Fig.2(b). The value of\nPDdecreases as Uincreases because the doubly occu-\npied state is unfavorable due to the intrasite repulsion\nU. As a result, PSincreases as Uincreases because of\nthe constraint condition PS+PD= 1.\nLastly, we investigate how the spins align in the\nground state |Giso/an}bracketri}htby calculating the spin correlations\n/an}bracketle{tGiso|Si·Sj|Giso/an}bracketri}htbetween site- iand site-jas functions\nofU. The results are shown in Fig.3, where the blue cir-\ncles and the red inverted triangles represent the results\nfor the spin correlation between two sites in sublattice A\n( the intra-sublattice spin correlation ), and the spin cor-\nrelations between site- iin sublattice Aand site-j(= 2)\nin sublattice B( the inter-sublattice spin correlation ),\nrespectively. These results show that the spins in sub-\nlatticeAferromagnetically align and two spins on the\ndifferent sublattices are in antiferromagnetic order. As\nUis increased, the values of the intra- and inter- sublat-\ntice spin correlations saturate to1\n4and−5\n12=−0.4166..,\nrespectively. These saturation values can be explained as\nfollows. In the limit of large U, the ground state |GU=∞\niso/an}bracketri}ht\nis a superposition state of the basis vectors belonging to\nthe S-state only, as shown in Fig.2(b) and is expressed\nby4\n|GU=∞\niso/an}bracketri}ht=/radicalbigg\n9\n12d†\n1↑d†\n3↑d†\n4↑d†\n2↓|0/an}bracketri}ht−/radicalbigg\n1\n12(d†\n1↓d†\n3↑d†\n4↑d†\n2↑|0/an}bracketri}ht\n+d†\n1↑d†\n3↓d†\n4↑d†\n2↑|0/an}bracketri}ht+d†\n1↑d†\n3↑d†\n4↓d†\n2↑|0/an}bracketri}ht) (15)\n, where |0/an}bracketri}htis the vacuum state. Using this expres-\nsion, we obtain the two saturation values, /an}bracketle{tGU=∞\niso|S1·\nS3|GU=∞\niso/an}bracketri}ht=1\n4and/an}bracketle{tGU=∞\niso|S1·S2|GU=∞\niso/an}bracketri}ht=−5\n12\nFrom these calculations, it is found that a sizable value\nofU(≫t) is required to regard the ground state as a so-\ncalled ferrimagnetic state (we mainly set U/t= 4).\nFIG. 3. (Color online) The spin correlations between two\nsites in sublattice A(blue circle) and between site- iin sublat-\nticeAand site- j(= 2) in sublattice B(red inverted triangle)\nplotted as functions of U/t.\nB. Results for noninteracting system\nHenceforward, we investigate the Hubbard model on\nthe T-shaped lattice connected to the electron reser-\nvoirs. ForU= 0, changing the value of εd, we calculate\ngs, gp, δeandδoby using our method and compare the\ncalculated results with the exact results in Fig.4. The ex-\nact expressionsof κeandκoforU= 0 areeasily obtained\nfrom Eq.(6) because Σ+\ne(0) = Σ+\no(0) = 0 and\nKe=\nεd√\n2t0√\n2t εdt\n0t εd\n, Ko=/parenleftbigεd/parenrightbig\n.(16)\nThen we obtain the exact expressions of δeandδofor\nU= 0 as follows;\nδe= arctan( −Γ(ε2\nd−t2)/(εd(ε2\nd−3t2))),(17)\nδo= arctan( −Γ/εd). (18)\nIt is found that the agreement between the results by our\nmethod and the exact results are remarkably consistenteven for a rather large value of the discretization param-\neter Λ = 6. Therefore, the effect of the NRG discretiza-\ntion on these quantities is negligible and our method for\ncalculating these quantities is reliable.\nTwoconductances gsandgphavelargevalueswhenthe\nelectron occupation number n= 2(δe+δo)/πchanges by\ntwo because a pair of electrons with up and down spins\nfromthe reservoiroccupythe energylevelofthe impurity\nresonating with the Fermi level.\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\n-2-1.5-1-0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6\nεd /t gp\ngs : exact results\n2δ e /π\n2δ o /π\nPhase Shift  ( π/2 )Conductance  ( 2e2/h )\n : our method\nFIG. 4. (Color online) Two kinds of conductance gs,gp\nand the phase shifts δe,δoobtained using our method are\ncompared with the exact results for U= 0. Here Γ /t= 0.1.\nFor NRG, we use Λ = 6 and t/D= 0.1.\nC. Kondo effect\nWe next investigate how the Lieb’s ferrimagnetic mo-\nmentS= 1 at the half-filled state ( εd/t=−U/(2t))\nis screened by the Kondo effect. We show the impurity\nentropy as a function of the temperature, varying the\nvalues ofUin Fig.5. In the high temperature region,\nthe values of the impurity entropy for all the values of\nUshown in the figure are log(256) because all degrees\nof freedom 256 = 28of the four sites appear in this re-\ngion. Decreasing the value of TNnrgfrom the high tem-\nperature region, we observe the log(3)-plateau due to the\nLieb’s ferrimagnetism S= 1 on the T-shaped lattice.\nThis degree of freedom is screened by the Kondo effect\nin the low temperature region via the log(2)-plateau and\nthe Kondo temperature, which is the screening tempera-\nture required to reach the singlet states, decreases as the\nvalue ofUincreases. Therefore, the magnetic moment in\nthe Lieb’s ferrimagnetism on the T-shaped lattice is not\nscreenedin one step by the conduction electrons from the\ntwo symmetrically connected reservoirsbut is completely\nscreened in two steps with different energy scales. In the\nfirst Kondo screening (log(3) →log(2)), the reduced de-\ngrees of freedom f1isf1≃1(= 3−2). Therefore, the\npartialmagneticmomentoftheLieb’sferrimagneticstate\nwithS= 1 screened in the first Kondo screening can not5\ncorrespond to any positive integer or half-integer mag-\nnetic moment s1because 2s1+ 1 =f1⇐⇒s1= 0.\nTo give some insights into the screening mechanism,\nwe consider the distribution of the S= 1 moment on\nthe T-shaped lattice. The two-step Kondo screening\nbecomes more significant for large Uand small Γ as\nshown in Fig.5 and Fig.6. Therefore, it is reasonable\nto consider the momentum distribution in large Uand\nsmall Γ limit. In the limit, the ground state of the\nisolated Hubbard model on the T-shaped lattice with\nS= 1, Sz= 1 is given by Eq.(15). Using these as-\nsumptions, we can calculate the momentum distribution\nmi≡ /an}bracketle{tGU=∞\niso|1\n2(d†\ni↑di↑−d†\ni↓di↓)|GU=∞\niso/an}bracketri}htfor each site- iof\nthe T-shaped lattice as follows; m1=m3=m4=5\n12\nandm2=−3\n12. The results show that the momentum\nis mainly distributed among sites-1,3,and 4. When the\nreservoirsconnectto theT-shapedlattice, site-1andsite-\n4 directly couple to the reservoirs. Therefore, the partial\nmoments on these two sites can be screened directly by\nthe conduction electrons at the first step. In contrast,\nsite-3 does not connect to the reservoirsdirectly and thus\nthe partialmoment onsite-3hastobe screenedindirectly\nby the conduction electron through site-2, which corre-\nsponds to the second step of the Kondo screening. To\nconfirm the discussion mentioned above, we increase the\nvalue ofU3, which is the intrasite repulsion of the site-3\n(the most internalsite from the reservoirs),from U3=U,\nkeeping the relation 2 ε3+U3= 0 for the half-filled state.\nThe results are shown in the inset of Fig.5. The shape of\nthe curve of the impurity entropy from the high temper-\nature region to the beginning of the log(2)-plateau via\nthe log(3)-plateau is almost insensitive to U3/U, while\nthe length of the log(2)-plateau becomes longer as the\nvalue ofU3/Uincreases. Then the Kondo temperature\ndecreases as the value of U3/Uincreases. From these\nfacts, we can confirm that the second screening process\ncorresponds to the screening of the partial magnetic mo-\nment distributed on the most internal site-3.\nWe consider the dependence of the two-step Kondo\nscreening on the hybridization Γ, for U/t= 4. The im-\npurity entropy for the intermediate coupling Γ = 0 .2 re-\ntainsthestructureobservedintheweakcouplingΓ = 0 .1.\nFor the large hybridization Γ = 0 .3, the charge transfer\nbetween the T-shaped lattice and the reservoir brings\nthe system into a mixed-valence regime and the typical\nstructures are smeared out. The Kondo temperature is\nsensitive to the value of Γ and it increases with Γ. This\nis because the large hybridization makes the resonance\npeaks broad and it reduces effectively the correlation ef-\nfects.\nTo investigate behaviors of two conductances gsand\ngpunder the Kondo screening of the Lieb’s ferrimag-\nnetism emerging at the half-filled state shown above,\nwe calculate gs, gpandδe, δoas functions of εdand\nshow the results in Fig.7. Around the half-filled state\nεd/t=−2(=−U/(2t)), the value of gsis strongly sup-\npressedgs≃0 in spite of the existence of the Kondo\nscreening, while the value of gpreaches its maximum 0 1 2 3 4 5 6\nlog(2)log(3)log(256)\n10-3010-2510-2010-1510-1010-5100\nTN nrg / DU/t = 4U/t = 3U/t = 2 U/t = 1 0 0.5 1 1.5 2\nlog(2)log(3)\n10-50\nTN nrg / DU3 / U= 5\n10-3010-4010-2010-10100U3 / U= 4U3 / U= 3U3 / U= 2\nU3 / U= 1U / t = 4Entropy ( kB)\nEntropy ( kB)\nFIG. 5. (Color online) Temperature dependencies of the im-\npurity entropy for several values of Ucalculated using NRG\nenergy spectrum. (Inset) The U3/Udependencies of the im-\npurity entropy.\n 0 0.5 1 1.5 2\nlog(2)log(3)\n10-30\nTN nrg / DU/t = 4Entropy ( kB)\n10-2510-2010-1510-1010-5Γ/t = 0.1\nΓ/t = 0.2Γ/t = 0.3\n100\nFIG. 6. (Color online)Γ-dependence of the impurity entropy\nforU/t= 4\nvaluegp≃4e2/hbecauseδeandδorespectively have\n3π/2-andπ/2- plateaus around the half-filled state. One\npossiblereasonforthis isasfollows. Thereisapossibility\nof a residual anti-ferromagnetic correlation between the\nsite-2 and 3 resulting from the Lieb’s ferrimagnetic state,\nwhich prevents conductivity in the series configuration,\nbut causes perfect conductivity in the parallel configu-\nration. This is because the anti-ferromagnetic coupling\nbetween site-2 and 3 blocks the branch path for conduc-\ntivity in the parallel configuration. Increasing the value\nofεdfrom the half-filled state, we can see the region\nwhere the value of ntransitions from 4 to 2 via 3. In\nthis region, the graph of gshas a double-peak structure\nbecause of the dip structure in the behavior of δo. This\ninteresting behavior will be investigated elsewhere be-\ncause, in this paper, we focus on the Kondo effect on the\nminimal Lieb’s ferrimagnetism emerging at the half-filled6\nstate. Around the region where n≃1, we can see the\ntypical Kondo plateau of both conductances gsandgp,\nwhich is principally caused by the even-parity states.\n 0 0.5 1 1.5 2\n 0 0.5 1 1.5 2 2.5 3 3.5 4\n-2 -1  0  1  2  3εd / tgp\ngs\n2δ e/π\n2δ o/πn ( = 2(δ e + δ o)/π )Conductance  ( 2e2/h )Averaged occupation number                       and              Phase Shift  ( π/2 )\nFIG. 7. (Color online) Two kinds of conductance gs,gp, the\nelectron occupation number nin theT-shapedlattice, andthe\nphase shifts δeandδoplotted as functions of εd/tforU/t= 4.\nD. Robustness against parameter perturbation\ntoward reducing the ferrimagnetism\nFinally, we study in more detail the behaviors of both\nconductance gsandgpat the half-filled state mentioned\nabove, by setting t/ne}ationslash=t3and reducing the value of t3/t.\nAtt3/t= 0, our system is decoupled into the half-filled\n3-site Hubbard chain connected to two electron reser-\nvoirs and the isolated site-3 occupied by one electron. In\nthis case, the spin S= 1/2 on the half-filled 3-site Hub-\nbard chain is screened by the Kondo effect in the ground\nstate, which gives conductivities in both configurations\n(gs≃2e2/h, gp≃2e2/h)22and we have the residual\nimpurity entropy log(2) by the degrees of the freedom of\nthe spinS= 1/2 on the isolated site-3. The question\narises as to the t3/t-dependence of two conductances. To\nanswer this question, we calculate gsandgpas functions\noft3/tand show the results in Fig.8. We find that the\nvalues ofgsandgpfort3/t>0 keep the constant values\natt3/t= 1 and change discontinuously only at t3/t= 0.\nTherefore, the behaviors of gsandgpat the half-filledstate are robust against the parameter perturbation to-\nward reducing the Lieb’s ferrimagnetism. We investigate\n 0 0.5 1 1.5 2\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1gp\ngs\nt 3 / t Conductance  ( 2e2/h )\nEntropy ( kB)\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4\nlog(2)log(3)log(4)\n10-80\nTN nrg / Dt3 / t = 0.01\n10-7010-6010-5010-4010-3010-2010-10100t3 / t = 0.1\nt3 / t = 1\n12\n34 12\n34\nFIG. 8. (Color online) Two kinds of conductance gsandgp\nplotted as functions of t3/twith discontinuities at t3/t= 0.\n(Inset) Temperature dependencies of the impurity entropy f or\nt3/t= 0.01 and 0 .1 compared with the result for t3/t= 1.\nthis robustness from t3/t-dependence of the impurity en-\ntropy as a function of the temperature. For t3/t= 0.1\nand 0.01, the impurity entropies are plotted as functions\nof the temperature and are compared with the result for\nt3/t= 1 in the inset of Fig.8. The value of Uis fixed at\nU/t= 4. We find that the Kondo temperature decreases\nas the value of t3/tdecreases. For 0 .1< t3/t <1, we\nconfirmed that as the temperature decreased, the value\nofthe impurity entropydecreasesfrom the value log(256)\nand directly reaches the log(3)-plateau corresponding to\nthe Lieb’s ferrimagnetic state (an entanglement state of\nfour sites), and converges to zero via the log(2)-plateau\nby forming the Kondo singlet state (an entanglement\nstate of four sites and two reservoirs). Therefore, the\nscenario for the entanglement of the degrees of freedom\n(log(3)→log(2)→0) from the high temperature region\ntoward the ground state for 0 .1<t3/t<1 is the same as\nfort3/t= 1. Fort3/t= 0.1, we can see the short log(4)-\nplateau in the high temperature region which stems from\nthe degrees of freedom of the spin S= 1/2 on the half-\nfilled 3-site Hubbard chain and the spin S= 1/2 on the\nsite-3 regarded as isolated from the Hubbard chain in\nthis temperature region (a separable state of the com-\nposite system of the 3-site Hubbard model and site-3).\nBut as the temperature slightly decreases from this re-\ngion, we observe the log(3)-plateau due to the Lieb’s fer-\nrimagnetic state, which is finally screened in two steps by\nthe Kondo effect in the same manner as the previous re-\nsults. We confirmed that this scenario for entanglement\n(log(4)→log(3)→log(2)→0) from the high temper-\nature region toward the ground state for t3/t= 0.1 is\nthe same as for 0 .01< t3/t≤0.1. Alternatively, for\nt3/t= 0.01, the value of the impurity entropy directly7\ndecreases from the log(4)-plateau to the log(2)-plateau\nwithout via the log(3)-plateau as the temperature de-\ncreases from the high temperature region. This is be-\ncause the spin S= 1/2 on the half-filled 3-site Hubbard\nchainisatfirstscreenedbytheKondoeffect(anentangle-\nment state of the composite system of the 3-site Hubbard\nchain and two reservoirs) and the spin S= 1/2 on site-3\nremains and generates the log(2)-plateau. Therefore, it\nis suggested that we have gs≃2e2/h, gp≃2e2/hin the\ntemperature region with the log(2)-plateau. However, as\nthe temperature decreases further, the spin S= 1/2 on\nthe site-3 is screened by electrons in the rest of the sys-\ntem coming through the renormalized hopping matrix el-\nement˜t3. The impurity entropy finally convergesto zero,\nreaching the non-magnetic ground state. This scenario\nfor entanglement (log(4) →log(2)→0) from the high\ntemperature region toward the ground state is expected\nfor 0<t3/t≤0.01.\nThe two-step Kondo screening for 0 <t3/t<0.01 re-\nlates to that seen in side-coupled double quantum dot\nsystems26–31in the following aspects. The side-coupled\ndouble quantum dot system consists of quantum dot A\nconnected to two reservoirs and quantum dot B con-\nnecting or interacting only with quantum dot A. The\n1/2-spin on dot A and the 1/2-spin on dot B behave\nindependently in the high temperature region. As the\ntemperature is decreased, screening of the 1/2-spin on\ndot A is first observed, followed by a subsequent screen-\ning of the 1/2-spin on dot B. Meanwhile, the two-step\nKondo screening for t3/t >0.01 is intrinsically different\nfrom the Kondo screening mentioned above. This is be-\ncause Lieb’s ferrimagnetic state, which is an entangled\nspin-triplet state between site-3 and sites-1, 2, and 4, is\nquenched for t3/t>0.01.\nFrom these facts, we find that the t3/t-independent\nground state for t3/t >0, resulting from the t3/t-\ndependent scenarios for the screening of the degrees of\nfreedom in high temperature regions, causes the robust-\nness of the behaviors of gsandgpat the half-filled state.\nIV. CONCLUSION\nIn summary, we investigated the Kondo effect on\nthe minimal Lieb’s ferrimagnetism on the T-shaped lat-tice connected to electron reservoirs by using a reliable\nmethod. We found that the Lieb’s ferrimagnetic mo-\nmentS= 1 is screened in two steps by the Kondo effect.\nHere we estimate one of ourKondo temperatures in units\nof Kelvin using recent experimental values. In our cal-\nculations, Kondo temperatures are scaled by Dand we\nsett/D= 0.1. The value of tcan be controlled from\ntheµeV to sub meV in recent experiments12,32. This\nrange corresponds to Dbeing on the order of meV ∼10\nK. Therefore, even in the first step of the Kondo screen-\ning, the Kondo temperature estimated from Fig. 5 for\nU/t= 2 is to the order of 1 µK. Therefore, the Kondo\ntemperatures are very low in the present case because\nwe chose a relatively small Γ /tand a larger U/t. How-\never, the Kondo temperature rises as Γ /tincreases and\nU/tdecreases, which would make the value of the Kondo\ntemperaturean accessiblevalue in experiments, asshown\nin Fig.5 and Fig.6. In spite of the existence of the Kondo\nscreening, we found that the conductance gsis strongly\nsuppressed gs≃0whiletheconductance gphasthemaxi-\nmum value gp≃4e2/h. For thesebehaviors, weproposed\none possible reasonwhich should be confirmed by further\ncalculations, where the spin correlations among the sites\nof the T-shaped lattice connected to the reservoirs would\nbe clarified. We also discussed the robustness of these\nbehaviors of the conductance against the perturbation\ntoward reducing the Lieb’s ferrimagnetism. This robust-\nness is caused by the perturbation-strength-independent\nground state resulting from three perturbation-strength-\ndependent scenarios for entanglements of the degrees of\nfreedom in high temperature regions. It would be in-\nteresting that experimental investigations of the above\nmentioned properties of the Kondo effect on the minimal\nLieb’s ferrimagnetism will be carried out in the future.\nACKNOWLEDGMENTS\nThe authors acknowledge the fruitful discus-\nsions with Dr.A.C.Hewson, M.Sc.N.Shimada and\nM.Sc.M.Watanabe. One of us (M.T.) acknowledges the\nsupport by JSPS KAKENHI Grant No.JP20J20229.\nNumericalcomputationwaspartlycarriedout in Yukawa\nInstitute Computer Facility.\n∗tokuda@meso.phys.sci.osaka-u.ac.jp\n†nishikaway@osaka-cu.ac.jp, nisikawa@sci.osaka-cu.ac. jp\n1H. 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Van-\ndersypen, Nature 548, 70 (2017)."    },    {        "title": "1805.10607v1.Prediction_of_new_multiferroic_and_magnetoelectric_material_Fe3Se4.pdf",        "content": "Prediction of new multiferroic and magnetoelectric material Fe 3Se4 \nDeobrat Singh1, Sanjeev K. Gupta2,*, Haiying He3 and Yogesh Sonvane1,* \n1Advance Materials Lab, D epartment of Applied Physics, S. V. National Institute of Technology,  \nSurat 395007, India.  \n2Computational Materials and Nanoscience Group, D epartment of Physics and Electronics,  \nSt. Xavier’s College, Ahmedabad 380009, India.  \n3Department of Physics and Astronomy, Valparaiso University, Valparaiso, Indiana 46383, USA . \n \nAbstract  \nNowdays , multiferroic materials with magnetoelectric coupling have many  real-world  applications in \nthe fields of novel m emory devices. It is challenging is to create multiferroic materials with strongly  \ncoupled ferroelectric and ferrimagnetic orderings at room te mperature . The single crystal of ferric \nselenide ( Fe3Se4) shows  type-II multiferroic due to the coexistence of ferroelectric as well as magnetic \nordering at room temperature.  We have  investigated  the lattice instability , electronic structure , \nferroelectric , ferrimagnet ic ordering and transport  properties  of ferroelectric metal  Fe3Se4. The density \nof states shows considerable hybridization of Fe -3d and Se -4p states near the Fermi level confirming its \nmetallic behavior. The magnetic moments of Fe cations follow  a type -II ferr imagnetic and ferroelectric \nordering with a calculated total magnetic moment of 4.25\nB  per unit cell (Fe 6Se8). The strong \ncovalent bonding nature of Fe -Se leads to its ferroelectric properties. In addition, the symmetry analysis \nsuggests that tilting of Fe sub -lattice with 3d -t2g orbital ordering is due to the Jahn -Teller (JT) \ndistortion. This study provides further insight in the development of spintronics related technology \nusing  multiferroic materials.  \n \n*Corre sponding author s: sanjeev.gupta@sxca.edu.in   (Dr. Sanjeev Gupta)  \nyas@phy.svnit.ac.in  (Dr. Yogesh Sonvane)  \n \n  Introduction  \nRecent year, m ultiferroic and magnetoelectric materials are becoming more and more indispensable  for \nmany forms of current multi-functional technology, such as highly sensitive magnetic field sensor, \nfilter  transducers, filters, phase shifters, memory devices and oscillator s1-4. For practical applications, \nwe needs to discover multiferroic materials at room temperature which is strongly coupled ferroelectric \nand ferrimagnetism ordering. In addition to their important polarization properties, ferroelectrics are \nalso pyroel ectric5, where i t develops  a voltage across the material upon heating, while  in the case of \npiezoelectric  a voltage is developed in response to strain across the material. These properties allow \nferroelectric  materials  to be utilized in many device applications, including non -volatile memory6, \nthermal detectors7, piezoelectric applications8, and energy harvesters9. The coexistence of magnetism \nand ferroelectricity in a material is called multiferroicity, which is  of even  great er technological and \nfundamental importance. This has added o ther potential applications . For instance , it can be utilized in \ndata storage system s, which count  both magnetic and electr onic state s of the compound to store \ninformation, and in the magneto -elect ronic device s using multiferroic  thin films10. \n Some transition metal compounds, such as BiMnO 3 and BiFeO 3 with magnetic Mn3+ and Fe3+ ions, \nare ferroelectric11, owing  to the intricate interplay between  spin, charge, orbital and lattice degree of \nfreedom  in th ese material s12. The mechanisms behind ferroelectricity , however, are  not fully \nunderstood because of the complicated structures of early ferroelectrics ( i.e. BaTiO 313). The compound \nCaMnO 3 is also an interesting counter example which shows ferroelectricity and magnetic ordering due \nto the  d3 configuration of Mn atom s14. It is worthy to be  noted that in general the ferroelectricity \nbehavior require s nearly  unoccupied  subshell  orbitals of the transition metal cations , while partial ly \nfilling d orbitals are required to have magnetic moment s. This dilemma largely hinders the coexistence \nof ferroelectricity and magnetism in a material and explains the scarcity  of multiferroic  materials15. \n Recently,  Bishwas  et al.,16 studied the  synthesized manganese doped iron selenide  nanostructures \nin an attempt t o increase the energy product . Another research group Shao -jie Li et al.,17 obtained  high \nCurie temperature and coercivity performance of Fe 3-xCrxSe4 nanostructures which can be utilized in \nthe alternative low -cost hard -magnetic materials . Gen Long et al .,18 demonstrated that  at low \ntemperatures, Fe 3Se4 nanostructures exhibit giant coercivity. It was proposed that t his unusual ly large \ncoercivity originates from the large magneto -crystalline anisotropy of the monoclinic structure of \nFe3Se4 with ordered Fe vacancies . The ferromagnetic material  BaFe 2Se3 is of particular interest because \nit breaks the parity symmetry of the crystal  structure and displ ays exchange striction effects19. Indeed, \nthe iron displacements in the crystal structure are prominent, as reve aled by neutron studies20, 21. Dong \net al.22 show ed that the first nearest -neighbor distances between Fe (↑) and Fe (↑) [or Fe (↓) and Fe (↓)] at 0 K become 3.14 Å, much larger  than the Fe (↑) and Fe (↓) distance 2.88 Å. On the other hand, this \nexchange striction is not sufficient to induce Ferroelectric (FE) and Polarization (P) since it breaks \nparity but not space -inversion s ymmetry.  Most ferroelectric materials are transition metal oxides having \nvacant d subshells. In this respect the  two configurations are not so different , but study of the difference \nin filling of the d -orbitals as theoretically expected for ferroelectricity and magnetism makes these two \nordered states mutually exclusive.  According to Giovannetti et. al. [24], the ferroelectric and metallic \nstate are found in LiOsO 3 material.  Interestingly, however, i t has been  suggested that several other \nmeta llic transi tions could be \"ferroelectric\"23 such as the transition in LiOsO 324 due to the appearance of \na polar axis.  \nIn this work, we focus on the  Fe3Se4  material derived from the monoclinic  phase. We have study \nthe lattice instabilities including ferroelectric distortions in monoclinic Fe 3Se4 materials. We have \nperformed first principles calculations within the spin -polarized density functional theory (DFT) \nframework to obtain the  electronic band structure, the role of electron correlations in the metallic state ,  \nmagnetic properties  of the Fe 3Se4 material.   \nMethodology  \nIn th is work,  all calculations are based on the  density functional theory (DFT)25 as implemented in \nthe Quantum  Espresso (QE) package26. The Kohn -Sham equation s were solved using  the Perdew -\nBurke -Ernzerhof (PBE) e xchange -correlation functionals27 formulated within the  generalized grad ient \napproximation (GGA) scheme28. We have included sixteen valence electrons for Fe (3s2, 3p6\n, 3d6\n, 4s2) \nand six valence electron s for Se (4s2\n, 4p4) in our calculation s. The kinetic energy cut-off is set to  60 Ry \nwhich yielded good convergence in results of energies and ground state structural parameters. For \nstructural optimization we have used t he conjugate gradient algorithm29. The lattice parameters and the \ninternal coordinates of atoms were optimized with in the space group of I12/m1 (monoclinic ) with the \ncriteria  for force and pressure below  10−3 a.u. A Monkhorst -Pack k -point mesh30 of 9×9×11 was used \nfor electronic band structure and density of state calculations , while  the spontaneous polarization w as \ncalculated using  the Berry phase method31 with a k-point mesh of 5×5×7 . All the calculations were \nspin-polarized . Scalar relativistic Troullier -Martins ultra-soft pseudopotential s32 were employed  with \nnon-linear core correc tions33. \n \nResult s and Discussion  \nStructural and electronic properties  \n \nAs a benchmark test for the approach used in this work, we have investigate d the structural , electronic and magnetic properties of the monoclinic Fe3Se4 phase . Fe3Se4 adopts a normal spinel \nstructure  as shown in Fig. 1  and the calculated lattice parameters are listed in Table 1. The Fe3+ ions are \nlocated in the octahedral sites of the monoclinic l attice formed by Se2- anions forming magnetic \nordering . The calculated structural parameter s are in very good agreement with previous reported \nvalues (Table 1).  \n \nTable  1. Optimized  structural lattice constants of monoclinic Fe3Se4. \nReferenc es a (Å)  b (Å)  c (Å) β (º) \nPresent  6.100  3.520  11.030  91.15  \nExp34,35,16 6.208, 6.167, 6.159  3.525, 3.537, 3.493  11.2832, 11.170, 12.730  92, 92 , - \n \n \nFigure 1. (A) Relaxed  structure of monoclinic Fe3Se4 (Fe in red and Se in yellow) ; (B) distribution of \nelectric dipole moments ( black arrows) of Fe atoms due to partial ionic displacement along the a \ndirection).  \n  \n \nFigure 2. The electronic band structure of monoclinic Fe3Se4 with up spin (left side) and down spin \n(right side) shown separately.  \n \nThe electronic band structure of monoclinic Fe 3Se4 is plotted  in Figure 2 . The Fermi level EF is set \nto zero. It is clearly shown that there are bands crossing  at EF, thereby  the band gap is zero and Fe 3Se4 \nis metallic in nature.  This is further confirmed from the density of states ( DOS ) plots (see  Figure S1 in \nESI), which demonstrate  asymmetric -spin  and  -spin DOS. A careful look of the projected density \nof states (PDOS) reveals ferrimagnetic spin order ing between the iron and selenium atom s. The \nselenium atoms carried a small magnetic moment (0.14 \nB unit cell ) due to charge transfer and \ngeometric distortion . Therefore , monoclinic Fe3Se4 shows metallic behavior in the ferrimagnetic (FM) \nstate.  \nIn order to check whether the metallic behavior is an artifact of neglecting the spin -orbit coupling  \n(SOC) , we have further calculated electronic band structure of Fe 3Se4 with the inclusion of SOC \ninteraction  (shown in Figure S2 in ESI) . The spin -orbit energy splitting  is large r at points of high -\nsymmetry . For instance, t he spin -orbit splitting at gamma point  is about  14 meV and 30 m eV in valence \nband maximum (VBM) and conduction band minimum (CBM) , respectively.  Since  the monoclinic \nFe3Se4 lattice has inversion symmetry, due to the Kramers degeneracy36, each energy band line is at \nleast doubl y degenerate for both spin state s. Our results show that  the inclusion of the SOC interaction \ninfluences only three p -energy bands of Se atom s and five d -energy bands of Fe  atom s and the metallic \nbehavior remains unchanged . \n \n \nFigure 3. PDOS of Fe ions in relaxed Fe 3Se4. The numbering of Fe is the same as in Figure 1(B).  Fe-1. \nFe-2 and Fe -5 forms one layer , while  Fe-3, Fe -4 and Fe -6 forms another . \n \n The origin of the metallic behavior of the monoclinic Fe 3Se4 can also be understood from the \nPDOS plots (Figure S1 in ESI). It is clear that the spin -up and spin -down states determining the FM \nmetallicity mainly originate from the edge Fe -3d and edge Se -4p orbitals.  Fe atoms contribute more to \nDOS than the Se atoms at Fermi level and the majority of the density of states near the Fermi level for \nFe3Se4 is attributed to the Fe -3d states. The Se -3p bands overlap wit h the Fe -3d bands in the -7 eV to 3 \neV energy range, representing a hybridization of the Se -3p and Fe -3d states to form the covalent \nbonding while Se -3p and Se -3s orbitals also have small contribution to the magnetic property of Fe 3Se4. \nThe difference of t he spin -up band and spin -down band of Fe -3d orbitals show that they carry very \nlarge magnetic  moment in Fe 3Se4. Further, the spin polarization is negative near Fermi because of the \nelectronic DOS of spin -down electrons is larger than that of spin -up electr ons.  \n Furthermore, we have identified the orbital ordering (OO) state in Fe 3Se4 as shown in projected \nDOS in Figure 3 with five electrons in 3d states of Fe ions in the 3D -coordinates ( xyz) with the x and y \naxes pointing to the crystal [\n101 ] directions and z axis directed to the crystal c-axis. In the minority \nspin channel, four Fe ions (Fe -1, Fe -2, Fe -3 and Fe -4) show t 2g bands right below the Fermi level ( EF) \ndown to1.6 eV with reduced DOS at the EF. All Fe ions in the unit cell of bulk Fe 3Se4 can be \ncategorized  into two groups according to the orbital characters : spin-up/down e g bands fully \nunoccupied and spin -down t 2g bands partially occupied, respectively. Bands of Fe -1, Fe -2, Fe -3 and Fe -\n4 [Figure 3(A, B, C and D)] are of predominate d x2\n-y2and d z2 characters, while bands of Fe -5 and Fe -6 \n[Figure 3(E) and 3(F)] are of mainly d xy, dz2 and d x2\n-y2 orbitals.   \nThese result s demonstrate that the configuration  of -spin 3dt 2g OO state s in the Fe sub -lattice , \nwhere a  -spin electron of each Fe has occup ied a mixed t 2g state. The mixing of orbitals is a \ncombination of two of the t 2g states  with the third empty t 2g states37. In addition, it is  found that the -\nspin electrons of Fe from 3dt 2g OO state occupying, respectively, the canted d xy and d yz orbitals of Fe -1 \nto Fe -4 and d yz of Fe-5 and Fe -6 sublattice , as they  contribute more at the Fermi level . The OO pattern \nis clearly seen in the distribution of charge density on each Fe3+ with the distribution on Fe -1, Fe -2, Fe -\n3 and Fe -4 belonging  to one state, while with Fe -5 and Fe -6 belonging to another state.  \n \nSymmetry analysis  \nAccording to  the ionic model37, we have acquired  a metallic ground state in consistence . It is also \nshown that the tilting 3dt 2g OO states  on the Fe sub -lattice is  strongly  related to the John -Teller (JT) \ndistortions37. The main contributions in electronic structure were utilized for Fe ions to investig ate the correlation effects in 3d -electrons.  The OO states are generally found in 3de g manganite frameworks \nwhere the helpful Jahn -Teller (JT) mutilations are huge because of the solid hybridization between the \neg and O -2p electrons38. The t2g OO states with higher degeneracy and moderately weaker JT distortion  \nis additionally found in confined 3d frameworks magnetite39. In the ionic model, the five 3d electrons \nof the Fe3+ ion possess the t 2g triplet degenerate  and leave the higher e g doublet degenerate  vacant, \nunder the octahedral crystal  field (see Figure S4 in ESI) . As per Hund's rule , Fe3+ is in the high spin  \nstates with the spin  arrangement of (\n3\n2gt\n2\n2gt  ), giving rise to a magnetic moment of 4.25\nB unit \ncell: and it indicates metallic ground  state with the majority -spin at the Fermi level accountable for \nthe conductivity . Assess ing three 3d electrons, the spin  up t 2g states  is moderately confined with a \nsuppressed bandwidt h and energy of lower band, while the -spin t2g band is pushed upwards \nmarginally [se e ESI in Figure S 1]. Besides, the -spin t2g band is thus completely occupied and at E F \nthere is no band gap , giving rise to a metallic ground state in consistence with the valence setup (\n3\n2gt\n2\n2gt\n ) of Fe from ion ic model as display ed in Figure S 3 in ESI . \nThe tilting of FeSe 6 octahedral site (Fig. 1  (A)) is to accommodate  two sorts of distortion of the Se \nlattice : the two Se ions at the tip of the octahedron deform  along the z-axis (Fe-Se top bond length = \n2.44 Å), while two ions of the coplanar Se move upwards and two  ions move  downwards along the z-\naxis. The JT distortions  with Fe-Se bond length (2.46 Å , between Fe -1 and Se ) along the y-axis and \nextended bond length (2.63 Å) along the x-axis (Figure  1) split the triply degenerate  t2g orbitals into the \nlower d xz and higher d yz and d xy orbitals . The JT distortions  could  bring down the Coulomb interaction \nbetween the ions of Se  and the Fe d xz states , which is a combination  of d xy and d yz states  as appeared in \nFigure S 4 in ESI . Moreover , the d xz states  with intermediate  directions between nearest Se anions would \nadditionally stabilize the lattice distortion.  \n It is well known tha t, if we include the spin -orbit coupling (SOC) interaction then the observed \norbital -ordering in  SrRuO 3 would be destroyed ; otherwise if we do not include SOC interaction then \nthe polyhedral crystal field makes the cubic harmonics (break ing the inversion symmetry)40 a nature \nbasis set resulting in lower t 2g and higher e g bands37. The Se octahedral site also break parity in each \nsite as Figure 1  shows that Se -5 is above the ladder’s plane, but the next Se -7 is below, and the \ndistances of Se -5 and Se -7 to the iron ladder plane should be the same in magnitude and opposite in \nsign (“antisym metric”). However, the OO  introduces a fundamental modification in the symmetry.  \nNow the blocks m ade of four Fe (↑) [or four Fe (↓)] are no longer identical to pair of two Fe (↑) and \ntwo Fe (↓). Then, the heights of  Se-5 and Se-7 do not require  to be antisymmetric anymore; their \ndistances to the  xz plane can become different. A  similar mechanism works for the edge Se's, e.g., Se -1 and Se -5. As a result, the atomic positions of Se break the space inversion symmetry, creating a local \nFerroelectric (FE) Polarization (P) pointing perpendicular to the iron yz plane (almost along the x- axis).  \nTo clarify more of the features of  OO states,  we have calculated charge density distribution (0.66e/ \nÅ3) and  depicted in Figure S5 in ESI  which corresponds  to the -spin t2g bands below the Fermi level ( -\n1.80-0.0 eV) [see  ESI in Figure S1]. It has been observed that electron de nsities corresponding t o -\nspin electrons are totally different for each atom  present in the zigzag whereas when we compare both \nthe spins,  electron densities  are partially  different from  each other. This suggests that in case of Fe 3Se4, \nspin down electrons present at unsaturated edge Fe -atoms are responsible for conduction as also \nconfirmed through corresponding band structure and DOS.  The charge contour  of real space \ndistribution of spin  dependent  electron densities suggests that  the possibility  of metallic spin  \npolar ization in Fe 3Se4 structures is totally attributed  to the localization  of unpaired electrons at \nunsaturated Fe atoms  which are situated at face s of the unit cell in Figure S 5(A) and S5(B) in ESI .  \n \n The yellow  spheres show  that the Se anions attract more electrons while, the Fe cations lose more \n3d electrons in Fe 3Se4, shown by bright magenta lobes pointing along the Fe directions. By contrast, the  \ndensity difference is weak but also exists in the Se ladder plane. There is a dark yellow sphere \nintroduce d as a Se atom, with negative value: this suggests that the outmost electrons of Fe are more \nextended ( delocalized), also supporting the covalent scenario for Fe 3Se4. \n \nMagnetic and ferroelectric properties  \n \nThe calculated total magnetic moment per unit cell o f monoclinic Fe3Se4 with 14 atoms per unit cell \n(6-Fe atom and 8-Se atom) is 4.25\nB . This value is consistent  with the previous DFT calculations \n(4.34\nB /unit cell)35, but significantly higher than  the reported  experimental value  for \nFe3Se4 nanostructures  (2.2\nB /unit cell )35. This was attributed to the  spin fluctuation and the long -range \nordering as measured by experiments, which normally would be smaller than the calculated value at 0 \nK.  \nWe have calculated spontaneous polarization (Ps)  along  each of the directions  x, y and z, for the \nferroelectric phase apply ing the Berry phase approach for bulk Fe3Se4. The direction of Ps in Fe 3Se4 \nsingle crystal lies in the monoclinic ac plane and the magnitude of the segment of spontaneous \npolarization along the a-axis of the monoclinic un it cell. The spontaneous polarization (Ps) in Fe 3Se4 \nhas a value of ~44.60 μC/cm2 and the direction of Ps vector makes an angle 91.15° with the major \nsurface (normal to c-direction). The calculated values of three component s of the spontaneous polarization  vectors are P x= 23.14 μC/cm2, Py= 6.20 μC/cm2and P z=37.62 μC/cm2, respectively.  The \nmagnitude of spontaneous polarization of Fe 3Se4 is in very good agreement with previously reported  \nexperimental value41. Roy et al .,42 provided the spontaneous polarization of ferroelectric bismuth \ntitanate  with a value of 42.83  μC/cm2 which  is fairly comparable  to our calculated value for \npolarization . Another study by Ravindran  et al.,43 gave the spontaneous polarization of BiFeO 3 which is \nalso within the reported agreement between theoretical and experimental values. Therefore, it is clear \nthat the calculated values of spontaneous polarization using GGA fall in the range of 43 -68 μC/cm2. \nDielectric  and transport properties  \n Furthe rmore, we have calculated the frequency dependent optical properties including dielectric \nfunction, absorption  coefficient  using DFT within the random phase approximation (RPA)44. The \nfrequency dependent dielectric function can be written as  \n12 ( ) ( ) ( ) i       . Where\n1() and \n2()\nare the real and imaginary parts of the complex dielectric function, respectively. \n2()  is \ndetermined by summation over electronic states and \n1()  is obtained using the Kramers –Kronig (KK) \nrelationship45. \n The real and imaginary parts of the complex dielectric function \n  versus frequency for bulk \nFe3Se4 are presented in Figure (4A and 4B). The real (\n1 ) and imaginary (\n2 ) part follow the general \ntendency of a metal which can be well explained by the classical Drude theory. In Figure 4A and 4B, \nwith increasing in frequency, the\n1 decreases while \n2 initially increases and decreases above the \n1.5x1014 Hz up to 2.0x1014 Hz region, as expected in a metal46. The variation of dielectric constant \n  \nand loss tangent is shown in Figure 4C and 4D. With increasing frequency, the \n  decreases and \ntan\ninitially increases at certain frequencies and then it is decreases in all the polarization direction (in -\nplane and out -of-plane). The loss tangent measures the loss -rate of power in an oscillatory dissipative \nsystem47. It is clearly seen that the variation \ntan executes  in the same trend as \n2 . Since \n2  is lower \nthan \n1 , then the energy loss  of the materials  is relatively low. This suggests that the material  possesses \ngood optical qualities due to lower energy losses and lower  scattering of the incident radiation47. Our \ntheoretical results of the dielectric function of bulk Fe 3Se4 are in excellent agreement with previously \nreported experimental results48 except \n2 . But according to other theoretical investigation our results is \nexpected in a metal46. \n  \nFigure 4. (A) Real and  (B) imaginary  part of the complex dielectric constant, (C) magnitude of the \ncomplex dielectric function  and (D) loss tangent  verses  frequency at 1014 Hz at room temperature (300 \nK) of bulk Fe 3Se4. (E) negative value of real part of complex dielectric function is a function of photon \nenergy.  \n The real part of complex dielectric function \n1  is shown in Figure 4(E). In Fig ure 4(E) , we plot \nthe real parts of dielectric functions  in all the polarisation direction  for the materials. Character of this \nmaterial exhibit metallic behavior, i.e. Drude peaks at low energy due to intraband contribution and the \nreal part of ε(ω) crossing from negative value to positive value with increasin g frequency  and it’s \nshows o scillatory behavior upto 15 eV . The strong anisotropy in the real part of complex dielectric \nfunction is also quite obvious. Remarkably, because of this anisotropy, the real part of complex \ndielectric function, \n1  in x, y and z -direction , changes sign at a frequency which is different from \neach other , leading to an extended frequency window in which the each components have different \nsigns shown in shaded region with gray color in Figure 4(E) . Such type of sign difference is the \ncharacteristic feature in real part of dielectric function is known as indefinite media49. Generally, \nindefinite media are mostly in artificially assembled structures  which require complicated fabrication \nprocess and usua lly have high dissipation. This theoretical results suggest that crystalline solid Fe 3Se4 \nin the bulk form  would just be indefinite materials for a frequency range spanning the near infrared.  \n \nFigure 5. The variation of (A) thermal conductivity\n , (B) heat capacity \npC  as a function of \ntemperature.  \n The thermal conductivity ( ) of a material  originates from two fundamental sources: (i) electrons \nand hole s transporting heat (κ e) and (ii) phonons trave lling through the lattice (κ l). Most of the \nelectronic term (κ e) is straight  forwardly associated with the electrical conductivity which can be well \nexplained by  Wiedemann –Franz hypothesis50. From Figure 5(A), the thermal conductivity  of Fe3Se4 is \nalmost linearly increased from 50 K to 400 K , and at room temperature the thermal conductivity is 4.56  \nW/ (m K). The thermal conductivity is  dominated  by the lattice contribution  as it is about 1.0 W/m K at \nT=50 K and it increases to 5.57  W/mK at T=400 K which is  higher  than the values observed in  most  \nthermoelectric materials50, 51. Since  the heat flow in a material is directly proportional to its thermal \nconductivity , the heat flow in bulk Fe 3Se4 will be  higher  as well .  \n       Specific heat estimation  is one of  the most reliable techniques for exploring temperature reliance of \nmaterials.  The heat capacity \npC  of bulk Fe 3Se4 increas es with temperature as shown in Figure 5(B ),  \ndemonstrat ing a similar behavior  as the thermal conductivity . In addition, the variation of \npC  with \ntemperature follow s the well-known  Debye form and shows excellent fit at lower  temperatures.  Our \ntheoretical results are in  good agr eement with experimental results48. Furthermore, the estimated \nelectric conductivity as a function of temperature of Fe 3Se4 reveals nearly a linear behavior  with a \npositive slope  as plotted  in Figure 6. This could be ascribed to the way that the electrical conductivity \nwas evaluated based on  DFT by assuming a constant scattering time  \n14( 10 ) s  . In general,  in ca se of \nmetallic system s, the electrical conductivity decreases with respect to temperature owing to the \noccurance of more  collisions  among electrons and between electrons  and phonons  which shorten the \nmean free path of charge carriers. I n our case , similar behavior are found in  its metallic system , the \nelectrical conductivity of Fe 3Se4 continuously decreases  with increasing tem perature. This is likely due \nto the enhanced delocalization of hybrid orbitals.   \n \nFigure 6. The variation of electronic conductivity\n  as a function of temperature.   \n \n \n \n \n \n \n \n \n \n \n \nFigure 7. The absorption coefficient \n  in the unit of 105/cm of bulk Fe 3Se4 for all the polarization  \ndirection in the electric field. (A) The absorption coefficient up to 15 eV and (B) absorption coefficient \nup to 40 eV beyond 28 eV the absorption will be almost zero . \nThe absorption coefficient \n  in all the polarization directions of the electric field (depicted in \nFigure 7) show  two main peaks in all polarizations. The first main peak oc curs at energy around 5  eV \nthat is related to π electron plasmon ic peak s. And other  peak occurs around 15  eV that is associated  to \nπ+σ electron plasmon ic peak s. Moreov er, at the energy range of 10 -20 eV, the value of absorption for \nall cases is very high. \nConclusion s \nIn conclusion s, we investigated the  novel multiferroic material shows  ferroelectric, ferrima gnetism \nand electronic structure in monoclinic Fe3Se4 material.   The nature of orbital ordering , and its  close \nconnection to the JT distortion is unwound, which is responsible for ferroelectric like instability . Fe3Se4 \ntakes  a ferromagnetic ordering with a  total magnetic moment of 4.25\nB per unit cell (6 Fe atom and 8 \nSe atom) . The spontaneous polarization (44.60 μC/cm2) provides evidence of its ferroelectric behavior. \nOur study suggests that the monoclinic phase of Fe 3Se4 material possesses the unusual dual behavior of \nmetallicity and ferroelectricity . The stabilization of the ferroelectric structure in Fe 3Se4 coexisting with \nmetallic conductance is the consequence of a decoupling between the metallic electrons in the t 2g \nelectrons from the soft phonons which break the inversion symmetry . The development of multiferroic \nmaterial are usefull applications in spintronics -related technologies for ultrahigh -density memory and \nquantum -computer devices is underway.  \n \nAcknowledgem ents \nHelpful discussion with Drs. Pankaj Poddar and Mousumi Sen is acknowledged . S. K. G. \nacknowledges the use of high performance computing clusters at IUAC, New Delhi and YUVA, \nPARAM  II, Pune to obtain the partial results presented in this paper. S . K. G and Y. A. S  also thank the \nScience and Engineering Research Board (SERB), India for the financial support (grant nu mbers.: \nYSS/2015/001269  and EEQ/2016/000217, respectively ). D. 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Belitz2;3\n1Institute for Physical Science and Technology,\nand Department of Physics, University of Maryland,\nCollege Park, MD 20742, USA\n2Department of Physics and Institute of Theoretical Science,\nUniversity of Oregon, Eugene, OR 97403, USA\n3Materials Science Institute,\nUniversity of Oregon, Eugene, OR 97403, USA\n(Dated: November 30, 2018)\nAn earlier theory of the quantum phase transition in metallic ferromagnets is revisited and gener-\nalized in three ways. It is shown that the mechanism that leads to a \ructuation-induced \frst-order\ntransition in metallic ferromagnets with a low Curie temperature is valid, (1) irrespective of whether\nthe magnetic moments are supplied by the conduction electrons or by electrons in another band, (2)\nfor ferromagnets in the XY and Ising universality classes as well as for Heisenberg ferromagnets, and\n(3) for ferrimagnets as well as for ferromagnets. This vastly expands the class of materials for which\na \frst-order transition at low temperatures is expected, and it explains why strongly anisotropic\nferromagnets, such as UGe 2, display a \frst-order transition as well as Heisenberg magnets.\nPACS numbers: 64.70.Tg; 05.30.Rt; 75.50.Cc; 75.50.Gg\nI. INTRODUCTION, AND RESULTS\nQuantum phase transitions are a subject of great\ninterest.1,2In contrast to classical or thermal phase tran-\nsitions, which occur at a nonzero temperature Tc>0 and\nare driven by thermal \ructuations, quantum phase tran-\nsitions occur at zero temperature, T= 0, as a function\nof some non-thermal control parameter and are driven\nby quantum \ructuations. In this paper we will focus\non quantum phase transitions in metallic systems. For\nreasons discussed below, these transitions are especially\ninteresting.\nA prototypical quantum phase transition is the one\nfrom a paramagnetic metal to a ferromagnetic metal.\nIndeed, the earliest theory of a quantum phase transi-\ntion was the Stoner theory of ferromagnetism.3Stoner\nassumed that the conduction electrons are responsible\nfor the ferromagnetism, and developed a mean-\feld the-\nory that describes both the classical and the quantum\nferromagnetic transition. In an important paper, Hertz\nlater derived a Landau-Ginzburg-Wilson (LGW) func-\ntional for this transition by considering a simple model\nof itinerant electrons that interact only via a contact po-\ntential in the particle-hole spin-triplet channel.1Hertz\nanalyzed this (dynamical) LGW functional by means of\nrenormalization-group (RG) methods. He concluded that\nthe critical behavior in the physical dimensions d= 2 and\nd= 3 is mean-\feld-like. That is, as far as the static crit-\nical exponents of the transition at T= 0 are concerned,\nhe concluded that Stoner theory is exact in d= 2 and\nd= 3.\nIn the mid 1990s it was realized that the above con-\nclusion is not correct. The problem is that in metals\natT= 0 there are gapless particle-hole excitations that\ncouple to the magnetic order-parameter \ructuations andin\ruence the quantum critical behavior for all dimensions\nd\u00143. In Hertz's theory this coupling is taken into ac-\ncount only in an approximation that does not su\u000ece for\nyielding the leading critical behavior. Technically, Hertz\ntheory treats the fermionic soft modes in a tree approx-\nimation, whereas describing their in\ruence on the criti-\ncal behavior requires taking into account fermionic loops.\nPhysically, a correct description of any phase transition\nmust treat the order parameter \ructuations and all soft\nmodes that couple to them on equal footing.\nA theory that takes into account these e\u000bects was de-\nveloped by the present authors and T. Vojta. In Ref. 4\nit was shown that the quantum phase transition from a\nmetallic paramagnet to an itinerant ferromagnet in the\nabsence of quenched disorder in d= 2 andd= 3 is\ngenerically discontinuous, or of \frst order, in contrast to\nthe second-order transition with mean-\feld critical be-\nhavior predicted by Hertz theory.5The mechanism be-\nhind this phenomenon is analogous to what is known\nas a \ructuation-induced \frst-order transition in super-\nconductors and liquid crystals.6There, soft \ructuations\nof the electromagnetic vector potential (in superconduc-\ntors) or the nematic order parameter (in liquid crystals)\ncouple to the order parameter and e\u000bectively change the\nsign of the cubic term in the equation of state, leading to\na \frst-order transition. In the quantum magnetic case,\nthe role of the additional soft modes is played by the\nfermionic particle-hole excitations mentioned above that\nare massless at T= 0. Since these modes acquire a mass\natT > 0, the tendency towards a \frst-order transition\ndiminishes with increasing temperature. This leads to a\ntricritical point at a temperature Ttc>0 that separates\na line of continuous transitions at T > T tcfrom a line\nof \frst-order transitions at T < T tc. In a later paper\nwith Rollb uhler, the e\u000bects of a magnetic \feld Hwere\ninvestigated.7It was found that in the space spanned byarXiv:1203.3826v1  [cond-mat.str-el]  16 Mar 20122\nFIG. 1: Generic phase diagram of a metallic magnet in the\nspace spanned by temperature ( T), magnetic \feld ( H), and\nthe control parameter ( t). Shown are the long-range or-\ndered magnetic (LRO) and paramagnetic (PM) phases, lines\nof second-order transitions, surfaces of \frst-order transitions\n(\\tricritical wings\"), the tricritical point (TCP), and the two\nquantum critical points (QCP). The long-range order can be\nof ferromagnetic or ferrimagnetic type, and the electrons caus-\ning the long-range order can be in the same band as the con-\nduction electrons, or in a di\u000berent band. See the text for\nfurther explanation.\nT,H, and the control parameter, tricritical wings, or\nsurfaces of \frst-order transitions, emanate from the tri-\ncritical point and terminate in a pair of quantum critical\npoints in the T= 0 plane. The wing boundaries at T >0\nare given by lines of critical points that are reminiscent\nof a conventional liquid-gas critical point and connect\nthe tricritical point with the quantum critical points at\nT= 0. The resulting generic phase diagram is shown in\nFig. 1. This general picture is in good agreement with\nexperimental results for low-Curie-temperature metallic\nferromagnets, including ZrZn 2,8UGe 2,9URhGe,10and\nMnSi.11,12\nIn this paper we generalize our previous theory in three\nimportant ways. First, we show that our previous results,\nwhich had been derived under the same assumption made\nby Stoner and by Hertz, namely, that the magnetism is\ncaused only by itinerant electrons, remain valid in metal-\nlic systems where the magnetism is caused by electrons\nin a di\u000berent band than the conduction electrons.\nSecond, we show that the results are notrestricted\nto Heisenberg ferromagnets, contrary to what was im-\nplied in Refs. 4 and 13. Rather, they apply equally\nwell to metallic XY or Ising magnets, since the mag-\nnetic moments couple to conduction electrons whose\nspins have three degrees of freedom. This is an im-\nportant point, since some of the relevant materials are\nstrongly anisotropic magnets, including UGe 2(easy axis)\nand URhGe (easy plane).\nThird, we show that the phase diagram shown in Fig.1 also applies to generic metallic ferrimagnets. Ferrimag-\nnets are materials that spontaneously develop both a ho-\nmogeneous and a staggered magnetization at the same\ncritical value of either the temperature (for a classical\ntransition) or a non-thermal control parameter (for a\nquantum transition). Physically, this can happen when\nmagnetic moments of unequal magnitude on a bipartite\nlattice align in opposite directions.14\nThe unifying principle behind these generalizations is\nthe realization that coupling a homogeneous magneti-\nzation to conduction electrons will produce the same\nresults irrespective of the microscopic origin of the\nmagetization.15As a result, the phase diagram depicted\nschematically in Fig. 1 is valid for generic metallic ferro-\nmagnets in addition to itinerant ones, for ferromagnets of\nXY or Ising type in addition to Heisenberg magnets, and\nfor ferrimagnets as well as for ferromagnets. In all cases\nwe also consider the e\u000bects of nonmagnetic quenched dis-\norder. In Ref. 4 it was shown that this type of disor-\nder leads to an interesting phase diagram with a num-\nber of multi-critical points, and that su\u000eciently strong\nquenched disorder causes the \frst-order paramagnetic-\nto-ferromagnetic transition in metals to become second\norder. We will see that the same result holds for metallic\nferrimagnets. Experimentally, the e\u000bects of disorder on\neither one of these transitions have not yet been studied\nsystematically.\nII. THEORY\nWe now derive the results listed in Sec. I. To this end\nwe are interested in a theory that describes the magne-\ntization or order-parameter (OP) \feld M, the fermionic\ndegrees of freedom described by Grassmann-valued \felds\n\u0016 and , and the coupling between them. Accordingly,\nthe action will have three parts:\nA[M;\u0016 ; ] =AOP[M] +~AF[\u0016 ; ] +~Ac[M;\u0016 ; ];\n(2.1a)\nand the partition function is given by\nZ=Z\nD[M]D[\u0016 ; ]e\u0000A[M;\u0016 ; ]: (2.1b)\nWe are, however, not interested in a complete descrip-\ntion of the fermionic degrees of freedom; rather, we want\nto restrict ourselves to the fermionic soft modes and in-\ntegrate out the massive modes in the simplest approxi-\nmation that respects the symmetries of the problem to\narrive at an e\u000bective Landau-Ginzburg-Wilson (LGW)\ntheory in terms of soft modes only. If we denote the soft\nfermionic degrees of freedom collectively by q, and the\nmassive ones by P, we formally have\nZ=Z\nD[M;q]e\u0000ALGW[M;q]; (2.2a)3\nwhere\nALGW[M;q] =AOP[M]\u0000lnZ\nD[P]e\u0000~AF[q;P]\n\u0002e\u0000~Ac[M;q;P]\n\u0011 A OP[M] +AF[q] +Ac[M;q]:(2.2b)\nAs we will see later, the qare matrices formed by bilinear\nproducts of the fermion \felds, qnm(x;y) =\u0016 n(x) m(y)\nwith (n+ 1=2)(m+ 1=2)<0, and the Pare given by\nthe same products with ( n+ 1=2)(m+ 1=2)>0. Here\n n(x)\u0011 (x;!n) is the temporal Fourier transform of\nthe Grassmann \feld  (x), wherex\u0011(x;\u001c) comprises\nthe real-space position xand the imaginary-time variable\n\u001cin a Matsubara formalism, and !n= 2\u0019T(n+ 1=2)\nis a fermionic Matsubara frequency. \u0016 n(x) is de\fned\nanalogously.\nThis separation of soft and massive fermionic modes\nqandP, respectively, integrating out Pin a suitable\napproximation, and determining the consequences of the\ncoupling between qandM, is the central objective of\nthis paper. For the separation we will make use of the\ngeneral theory developed in Ref. 16.\nA. Order parameter, and coupling to fermions\nWe are interested in magnetic order, and hence the\nappropriate order-parameter \feld is the magnetization\nM(x). We write the magnetization as a part m(x) whose\naverage is the homogeneous magnetization, and a part\nn(x) whose average is a staggered magnetization,\nM(x) =m(x) +n(x)NX\nj=1cos(kj\u0001x): (2.3)\nHere thekjareNwave vectors that characterize the\nstaggered magnetic order, and both m(x) andn(x) are\nslowly varying in space and time. In particular, their\nFourier expansions contain only wave numbers that are\nsmall compared to the norms of the kj.\nIn a paramagnetic state the expectation values of m\nandnare both zero. At a transition to a ferromagnetic\nstate the expectation value of mbecomes nonzero while\nthat ofnremains zero; at a transition to an antifer-\nromagnetic state the converse is true. A ferrimagnetictransition is characterized by both mandnacquiring a\nnonzero expectation value at the same point in parame-\nter space. In this sense there is only one order parameter\n\feld for a ferrimagnetic transition; this fact will be impor-\ntan later. For the purposes of the present paper, a crucial\nquestion is the coupling of the order-parameter \ructua-\ntions to the soft fermionic degrees of freedom. Since the\nsoft parts of the latter are soft at zero wave number,\nthe leading coupling is to m. The fermions also couple\nton, but this leads to subleading e\u000bects since the stag-\ngered magnetization is soft at a nonzero wave number.\nWe will neglect this coupling in what follows. Physically,\nthe near-homogeneous magnetization \rutuations act as\na magnetic \feld proportional to mthat couples to the\nelectronic spin density\nns(x) =X\na;b\u0016 a(x)\u001bab b(x): (2.4a)\nHere\u001b= (\u001bx;\u001by;\u001bz)\u0011(\u001b1;\u001b2;\u001b3) denotes the Pauli\nmatrices, and a;b= (\";#)\u0011(+1;\u00001) are spin indices.\nThe coupling takes the form of a Zeeman term\n~Ac[M;\u0016 ; ] =cZ\ndxm(x)\u0001ns(x); (2.4b)\nwithca coupling constant. As we will see, the spin den-\nsity contains both massive and massless modes, so only\npart of Eq. (2.4b) contributes to Ac[M;q] in Eq. (2.2b).\nWe will discuss this separation next.\nB. Fermionic soft modes\nIn this subsection we separate the massless fermionic\nmodes from the massive ones by means of the technical\napparatus developed in Ref. 16. Here we will quote only\nas much of this formalism as is necessary for the further\ndevelopment, see Ref. 16 for additional details.\nThe soft fermion excitations are all two-particle ex-\ncitations; the related correlation functions are those of\nbilinear products of fermion \felds. The latter commute\nwith each other, and with individual fermion \felds, and\nhence are isomorphic to classical \felds. Denoting these\nclassical \felds by Q, we de\fne a classical matrix \feld\nQnm(x;y)\u0018=i\n20\nBB@\u0000 n\"(x)\u0016 m\"(y)\u0000 n\"(x)\u0016 m#(y)\u0000 n\"(x) m#(y) n\"(x) m\"(y)\n\u0000 n#(x)\u0016 m\"(y)\u0000 n#(x)\u0016 m#(y)\u0000 n#(x) m#(y) n#(x) m\"(y)\n\u0016 n#(x)\u0016 m\"(y)\u0016 n#(x)\u0016 m#(y)\u0016 n#(x) m#(y)\u0000\u0016 n#(x) m\"(y)\n\u0000\u0016 n\"(x)\u0016 m\"(y)\u0000\u0016 n\"(x)\u0016 m#(y)\u0000\u0016 n\"(x) m#(y)\u0016 n\"(x) m\"(y)1\nCCA: (2.5)\nHere \\\u0018=\" means \\isomorphic to\"; technically, the isomor- phism is implemented by means of a Lagrange multiplier4\n\feld, see below. We also de\fne the Fourier transform of\nQ,\nQnm(k;p) =1\nVZ\ndxdye\u0000ik\u0001x+ip\u0001yQnm(x;y):(2.6a)\nIt is further useful to de\fne\nQnm(k;q) =Qnm(k+q=2;k\u0000q=2) (2.6b)\nand\nQnm(x) =Qnm(x;x) =1\nVX\nqeiq\u0001xX\nkQnm(k;q):\n(2.6c)\nThe 4\u00024 matrixQnmcan be expanded in a spin-\nquaternion basis\nQnm(x;y) =3X\nr;i=0(\u001cr\nsi)i\nrQnm(x;y); (2.7)\nwhere\u001c0=s0=112is the unit 2\u00022 matrix, and\n\u001c1;2;3=\u0000s1;2;3=\u0000i\u001b1;2;3. An explicit inspection of\nthe 16 matrix elements shows that r= 0;3 represents\nthe particle-hole channel, i.e., products of the form \u0016  ,\nwhereasr= 1;2 represents the particle-particle channel,\ni.e., products of the form \u0016 \u0016 or  . For our purposes\nwe will need only the particle-hole degrees of freedom.\nIt was shown in Ref. 16 (see also Ref. 17) that a crucial\ncriterion for separating the fermionic degrees of freedom\ninto soft and massive modes is given by the relative signs\nof the frequency arguments of the matrix elements Qnm.\nAccordingly, we write\ni\nrQnm(x) =i\nrqnm(x) \u0002(\u0000!n!m) +i\nrPnm(x) \u0002(!n!m)\n(i= 1;2;3) (2.8)\nHere \u0002 is the step function, and we use the fact that in\nthe spin-triplet channel ( i= 1;2;3) the expectation value\nof theQ-matrix vanishes (this is since the fermionic de-\ngrees of freedom described by Qdo not by themselves\nhave long-ranged magnetic order; see the discussion at\nthe end of the current subsection), so that qandPrep-\nresent \ructuations. In what follows we will absorb the\nstep functions into the matrix \felds qandP, i.e., writ-\ningqnmimpliesn\u00150 andm< 0 andPnmimplies either\nn\u00150 andm\u00150 orn <0 andm < 0. Thei\nrqare the\nspin-quaternion elements of a matrix\nqnm(x) =X\nr;i(\u001cr\nsi)i\nrqnm(x): (2.9a)\nIt is also useful to de\fne an adjoint matrix\nq+\nnm(x) =X\ni;r(\u001c+\nr\ns+\ni)i\nrqmn(x); (2.9b)\nwhere\u001c+\nrands+\niare the hermitian conjugates of \u001crand\nsi, respectively. In addition, the theory contains a \feldq =nm(x) that has the same properties as qnm(x) except\nfor di\u000berent propagators, see below. The origin of q =is\nthe Lagrange multiplier \feld \u0015that constrains the bilin-\near products of fermion \felds to the q. In various places\nin the theory q\u0000\u0015\u0011q =appears, and the \u0015-propagator\nequals minus the q-propagator for noninteracting elec-\ntrons, whereas cross-correlations between qand\u0015vanish.\nThe net e\u000bect of \u0015is therefore to subtract the noninter-\nacting part of the q-propagator wherever the combination\nq\u0000\u0015occurs.\nTheqcorrelation functions are the basic soft modes in\nthe theory, see below. However, due to nonlinear cou-\nplings thePcouple to the qand thus have a soft compo-\nnent. This e\u000bect can be expressed by expanding Pin a\npower series in q. To quadratic order in qand to lowest\norder in the fermion interaction one \fnds\nP12(k)\u0019 \u0000 2iX\n3X\np'(3)\n132(p;k\u0000p)'\u00001\n13(p)'\u00001\n32(k\u0000p)\n\u0002\u0002\nq =13(p)q =+\n32(k\u0000p) +q =+\n13(p)q =32(k\u0000p)\u0003\n:(2.10)\nHere and it what follows we use a simpli\fed notation for\nfrequency indices, 1 \u0011n1, etc. We have dropped con-\ntributions to Pof higher order in q, and a contribution\nthat is linear in the interaction and linear in q, see Ref.\n16; neither will be needed for our purposes. We also have\nomitted a term quadratic in qand quadratic in the inter-\naction, which leads to less singular contributions to the\nfree energy than the one we keep. Note the frequency re-\nstrictions inherent in Eq. (2.10): sgn ( !n1) = sgn (!n2) =\n\u0000sgn (!n3). Here\n'12(k) =1\nVX\npG1(p)G2(p\u0000k) (2.11)\nwith!n1!n2<0 implied, and\n'(3)\n132(k1;k2) =1\nVX\npG1(p)G3(p\u0000k1)G2(p\u0000k1\u0000k2)\n(2.12)\nwhereG1(p)\u0011G(p;i!n1) is the single-particle Green\nfunction.'12has a scaling form\n'12(k) =NF2\u0019G\nk'd(Gi\n1\u00002=k)\n\u0011'(k;\n1\u00002): (2.13)\nwhereGis a coupling constant whose bare value is the\ninverse Fermi velocity, G= 1=vF,NFis the density of\nstates per spin at the Fermi level, and \n 1\u00002=!n1\u0000!n2.\nInd= 2;3, and for free electrons, we \fnd explicitly\n'd=2(z) = sgn (Im z)=p\n1\u0000z2; (2.14a)\n'd=3(z) =\u0000i\n2ln\u00121\u0000z\n\u00001\u0000z\u0013\n; (2.14b)\nwhich we recognize as the hydrodynamic part of the Lind-\nhard function. Equations (2.13) and (2.14) re\rect the5\nsoft particle-hole excitations with a linear momentum-\nfrequency relation in a metallic electron system. In par-\nticular,'(k;\nn= 0)/1=jkj, and'(k= 0;\nn)/\n1=\nn.18For later reference we also note the following\nidentities that hold for a special form of '(3):\n'(3)\n121(k;\u0000k) =\u0000'(3)\n212(k;\u0000k) =\u0000@\n@i!n1'12(k)\n\u0011'(3)(k;\n1\u00002): (2.15)\nThe fermionic action can be expressed in terms of q\nandP, and by using Eq. (2.10) and its generalizations\nto higher order one obtains a fermionic soft-mode action\nentire in terms of q. For our purposes we need only theGaussian part of this action, which reads\nAF[q] =\u00008X\nkX\n1;2\n3;4X\nr=0;33X\ni=0i\nrq12(k) \u0000i\n12;34(k)i\nrq34(\u0000k):\n(2.16a)\nHere 1\u0011n1etc., and the Gaussian vertex is given by\n\u0000i\n12;34(k) ='\u00001\n12(k) +\u000e1\u00002;3\u000042T\ri(2.16b)\nwith\ri=0=\u0000\rsand\ri=1;2;3=\rt;i, where\rs>0 and\n\rt;i>0 are the spin-singlet and spin-triplet interaction\namplitudes. The fermionic Gaussian propagator is given\nby the inverse of the vertex. One \fnds\nhi\nrq12(k)j\nsq34(\u0000k)i=1\n16\u000ers\u000eij\"\n\u000e13\u000e24'12(k)\u00002\riT\u000e1\u00002;3\u00004'12(k)'34(k)\n1\u00002\ri\u001f(0)\n1\u00002(k)#\n; (2.17a)\nwhere\n\u001f(0)\n1\u00002(k)\u0011\u001f(0)(k;\n1\u00002) =\u0000TX\n34\u000e1\u00002;3\u00004'34(k): (2.17b)\nWe see that the q-propagator is given in terms of ', and hence is soft. The \felds q =that enterP, Eq. (2.10), are\ncharacterized by Gaussian propagators\nhi\nrq =12(k)j\nsq34(\u0000k)i=hi\nrq12(k)j\nsq =34(\u0000k)i=hi\nrq12(k)j\nsq34(\u0000k)i (2.17c)\nand\nhi\nrq =12(k)j\nsq =34(\u0000k)i=\u00001\n8\riT\u000e1\u00002;3\u00004'12(k)'34(k)\n1\u00002\ri\u001f(0)\n1\u00002(k): (2.17d)\nThe last expression is just the interacting part of the\nq-propagator, Eq. (2.17a), as was mentioned after Eq.\n(2.9b).\nThe interaction amplitudes in the Gaussian fermionic\nvertex, Eq. (2.16b), warrant some comments. First, we\nnote that the three spin-triplet amplitudes \r1;2;3\nt are in\ngeneral not identical in a cyrstalline solid, and they do\nnot need to be for what follows. Second, we comment\non the two cases that result from the magnetism being\ncaused by the conduction electrons, or by electrons in\na band di\u000berent from the conduction band, respectively.\nLet us \frst assume the latter case, which is the concep-\ntually more straightforward one. Then AF[q], which de-\nscribes the conduction electrons, is independent of the\nmagnetism and contains interactions in both the spin-\nsinglet and spin-triplet channels. The only restriction is\nthat the latter are weak enough to not lead to magnetism\nby themselves. The conduction electrons are a\u000bected by\nthe magnetization, which acts as an e\u000bective magnetic\n\feld, and this is described by the Zeeman coupling term,Eq. (2.4b). The other possibility, which is conceptually\nmore complex, is that the magnetism is caused by the\nconduction electrons themselves. In this case the mag-\nnetic order parameter and the soft modes qdescribe de-\ngrees of freedom for electrons in the same band. The\nmagnetic order parameter then should be thought of as\nderiving from the spin-triplet interaction between the\nconduction electrons, e.g., via a Hubbard-Stratonovich\ndecoupling of the latter. This leaves the bare action AF\nwith a spin-singlet interaction only. However, as long as\nthe latter is present, a spin-triplet interaction will always\nbe generated under renormalization. The action AFwill\ntherefore again contain a spin-triplet interaction ampli-\ntude, albeit one that is much weaker than the one in the\nunderlying action that describes the system before the\nseparation of magnetic and fermionic degrees of freedom.\nThis is the case that was discussed, for ferromagnetism,\nin Ref. 13, which used phenomenological and symmetry\narguments to construct the fermionic part of the action.\nFinally, we mention that we assume the conduction elec-6\ntrons, in the absence of a nonzero magnetization (i.e.,\nwith the coupling constant cin Eq. (2.4b) put equal to\nzero), to indeed have three soft spin-triplet excitations at\nT= 0, which are given by Eqs. (2.17) with i= 1;2;3.\nThis is not necessarily the case. For instance, an external\nmagnetic \feld gives two of these three channels (the ones\ntransverse to the \feld) a mass, and a small concentration\nof magnetic impurities will make all three channels mas-\nsive without having signi\fcant other e\u000bects. However,\nin general the energy scales associated with these e\u000bects\nwill be small, and they will lead to a small reduction, but\nnot a complete suppression, of the tricritical temperature\nin Fig. 1. We will discuss this point in more detail in Sec.\nIII.\nC. Coupling between the order parameter and the\nfermionic soft modes\nWe are now in a position to separate the Zeeman term,\nEq. (2.4b), into parts where the order parameter couples\nto soft and massive fermionic modes, respectively. If we\nde\fne a temporal Fourier transform of the magnetization\n\feldmby\nmn(x) =p\nTZ1=T\n0d\u001c ei\nn\u001cm(x;\u001c); (2.18)\nwith \nn= 2\u0019Tn a bosonic Matsubara frequency, then\nwe can write Eq. (2.4b) in the form\n~Ac[M;Q] = 2cp\nTZ\ndxX\nn3X\ni=1mi\nn(x)\n\u0002X\nr=0;3(\u00001)r=2X\nmtr [(\u001cr\nsi)Qm;m+n(x)]:(2.19)\nBy expressing Qin terms of qandPby means of Eq.\n(2.8), andPin terms of q =by means of Eq. (2.10), we\nobtain the desired coupling Ac[M;q] between the order-\nparameter \ructuations and the fermionic soft modes q.D. Generalized Mean-Field Theory\nAn e\u000bective action, Ae\u000b[M] in terms of the order pa-\nrameter alone can be obtained by integrating out the\n\feldsq,\nAe\u000b[M] = lnZ\nD[q]eALGW[M;q]: (2.20)\nIn general the evaluation of this expression is very di\u000e-\ncult. However, it can be evaluated exactly within a gener-\nalized mean-\feld approximation that was \frst employed\nin the context of liquid crystals and superconductors6\nand is de\fned as follows. First, we ignore temporal and\nspatial variations of the order parameter, i.e. we treat\nthe \feldsm(x) andn(x) in Eq. (2.3) as numbers. If we\nassume ordering in the 3-direction, we have\nMi(x)\u0019\u000ei32\n4m+nNX\nj=1cos(kj\u0001x)3\n5; (2.21a)\nwhich implies\nmi\nn(x)\u0019\u000ei3\u000en0m=p\nT : (2.21b)\nThis mean-\feld approximation for the order parameter\nmeans that only the part of Qthat is diagonal in fre-\nquency space, i.e., Pmm, contributes to Eq. (2.19). This\nin turn means that the contribution to Pthat is linear\ninq, which we had dropped from Eq. (2.10), does not\ncontribute. Second, we restrict ourselves to quadratic\norder inq. That is, we treat the fermionic soft modes in\na Gaussian approximation with a \fxed magnetic order\nparameter. The validity of these approximations will be\ndiscussed in Sec. III B.\nWith these approximations the action Acthat couples\nqand the order parameter is quadratic in qand can be\nwritten\nAc[m;q] = 8X\nr;s=0;3X\ni;ji\nrq12(k)ij\nrs\u0000c\n12;34(k)j\nsq34(\u0000k): (2.22a)\nHere\nij\nrs\u0000c\n12;34(k) =\u000e13\u000e244cm\u0012\n0 1\n\u00001 0\u0013\nrs0\nB@0 0 0 0\n0 0 1 0\n0\u00001 0 0\n0 0 0 01\nCA\nij'(3)\n121(k;\u0000k)'\u00002\n12(k); (2.22b)\nand we have used Eq. (2.15). The matrices give the values ofij\nrs\u0000cfor the 4 possible values of ( r;s) and the 16 possible\nvalues of (i;j).\nThe integral over qin Eq. (2.20) can now easily be carried out. For the free-energy density f=\u0000TAe\u000b=Vwe\nobtain\nf=f0(m;n) + \u0001f(m): (2.23a)7\nHeref0=\u0000TAOP=Vis the mean-\feld free energy in the absence of a coupling to the fermionic soft modes. For\n\u0001f(m), which is the contribution to the free energy due to this coupling, one \fnds\n\u0001f(m) =2\nVX\nk0\nTX\nnlnN(k;\nn;m); (2.23b)\nwhereP0\nkdenotes a wave vector sum such that jkj<\u0003 with \u0003 an ultraviolet cuto\u000b, and\nN(k;\nn;m) =\u000016c2\rt;1\rt;2m2\n2\nn\u0010\n'(3)(k;\nn)\u00112\n'\u00004(k;\nn) +'\u00004(k;\nn)Y\ni=1;2h\n1\u00002\rt;i\u001f(0)(k;\nn)i\n:(2.23c)\nThe equation of state is obtained by minimizing the\nfree energy density. In the absence of a coupling be-\ntween the order parameter and the fermionic soft modes\nthis amounts to minimizing f0, which yields the ordinary\nmean-\feld equation of state. For a ferromagnet, the lat-\nter has the usual Landau form. For a ferrimagnet, the\nequation of state depends on details of the magnetic or-\nder. It can be complicated and describe several di\u000berent\nphases, see, e.g., Ref. 19. However, generically the \frst\nphase encountered as one approaches from the paramag-\nnetic state is entered via a second-order transition. After\nminimizing f0and expressing nin terms of mone thus\nhas again an ordinary mean-\feld equation of state given\nby\nh=rm+um3+O(m5); (2.24)\nwherehis an external magnetic \feld in the 3-direction,\nu>0, and the transition occurs at r= 0.20In Appendix\nA we recall a very simple model that leads to this result.\nThe second term on the right-hand side of Eq. (2.23a)\ngives an additional contribution to the equation of state,\nwhich then reads\nh=rm+um3\u000064mc2\rt;1\rt;2\n\u00021\nVX\nk0\nT1X\nn=1\n2\nn\u0000\n'(3)(k;\nn)\u00012'\u00004(k;\nn)\nN(k;\nn;m):\n(2.25)\nThis is the desired generalized mean-\feld equation of\nstate which takes into account the coupling of the order\nparameter to the fermionic soft modes.\nE. Discussion of the Generalized Mean-Field\nEquation of State\nWith some e\u000bort the integrals in Eqs. (2.23b) and\n(2.25) can be explicitly performed. However, the salient\npoints can be seen by simple scaling considerations and\ndimensional analysis. Equations (2.11) and (2.13) im-\nply that the frequency \n nscales as the wavenumber\nk, \nn\u0018k, and that '(k;\nn)\u00181=k\u00181=\nn, which\nalso can be seen explicitly from Eqs. (2.14). Equation(2.15) implies that '(3)(k;\nn)\u00181=k2\u00181=\n2\nn. Equa-\ntion (2.23c) then shows that there is a length scale Lm,\nor a corresponding frequency scale !m, that scales as\nLm\u00181=!m\u00181=m. If one attempts to expand \u0001 f(m),\nEq. (2.23b), in powers of matT= 0, then nonanalytici-\nties will occur at next-to-leading order for all d\u00143.\nAn alternative way to describe this mechanism is to\nsay that of the three soft fermionic spin-triplet excita-\ntions, Eq. (2.17a) with r=s= 0;3 andi=j= 1;2;3,\ntwo (namely, the ones transverse to the order parameter\ndirection) acquire a mass due to the coupling between\nthe fermions and the order parameter m, as can be seen\nexplicitly from Eq. (2.22b). This acquisition of a mass\nby a generic soft mode due the spontaneous breaking of\na continuous symmetry is an example of the Anderson-\nHiggs mechanism,22{24even though the broken symmetry\nin this case is not a gauge symmetry, see the discussion\nin Sec. III A. It implies in turn that the free energy is a\nnonanalytic function of m.\nAt nonzero temperatures the singularities are cut o\u000b\nbyTaccording to m\u0018T. That is, a crossover occurs\nfromm-scaling to T-scaling when the Zeeman splitting\nis comparable to the temperature, or the thermal length\nscaleLT/1=Tis comparable to the magnetic length\nscaleLmmentioned above. Taking into account the sign\nofN, Eq. (2.23c), one \fnds schematically, for 1 <d< 3,\n\u0001f(m) =\u0000vm2(m2+T2)(d\u00001)=2; (2.26a)\nand ford= 3\n\u0001f(m) =v\n8m4ln(m2+T2); (2.26b)\nwherev>0 is a positive constant.\nThe most important aspects of this result, as far as\nthe order of the transition is concerned, are the sign of\nvand the power of matT= 0. For all d\u00143 there\nis a negative term in the free energy that dominates the\nm4in the Landau free energy and hence necessarily leads\nto a \frst-order transition. Another way to see this is by\nexpanding \u0001 f(m), Eq. (2.26a), in powers of mforT >0.\nThe leading term is proportional to \u0000m4=T3\u0000d. That is,\nthere is a negativem4term whose prefactor diverges as\nT!0 for alld\u00143, which implies that there will be a\ntricritical point at some temperature. The free energy for8\nFIG. 2: Schematic sketch of the free energy for three values\nof the parameter r. The \frst-order transition occurs at r=\nr1>0. It pre-empts the second-order transition of Landau\ntheory which would occur at r= 0.\nthree di\u000berent values of ris plotted schematically in Fig.\n2. For this schematic free energy, the equation of state\nin the case d= 3, for which many experimental results\nexist, takes the form\nh=rm+v\n2m3ln(m2+T2)\n+m3\u0012\nu+v\n4m2\nm2+T2\u0013\n:(d= 3) (2.27)\nAlso of interest is the other physical dimensionality, d=\n2, where the equation of state reads\nh=rm\u00002vm(m2+T2)1=2\n+m3\u0012\nu\u0000v\n(m2+T2)1=2\u0013\n:(d= 2) (2.28)\nHere the analyticity is stronger than in the 3- dcase, with\na negative m2-term in the equation of state at T= 0.\nThis is particularly interesting in the case of Ising mag-\nnets, which display long-range order in d= 2 even at\nT > 0. The case of Heisenberg and XY magnets, which\ndo not show true long-range order in d= 2 except at\nT= 0, is more complicated.\nThese are the same results that were obtained using\na more phenomenological theory of the fermionic soft\nmodes in Ref. 13. They were discussed extensively in that\nreference, as well as in Refs. 4 and 7. There is no need to\nrepeat this discussion here, and the salient features are\nsummarized by the schematic phase diagram shown in\nFig. 1. The important conclusion of the current paper is\nthat the validity of these results, in addition to itinerant\nHeisenberg ferromagnets, extends to metallic ferromag-\nnets where the magnetism is not due to the conduction\nelectrons, to metallic ferromagnets in the XY or Ising\nuniversality class, and also to metallic ferrimagnets. The\nonly condition is that the conduction electrons are not\nsubject to strong spin-symmetry breaking e\u000bects such asmagnetic impurities. We note in passing that an inter-\nesting system is provided by the easy-plane ferromagnet\nURhGe, where an in-plane magnetic \feld transverse to\nthe magnetization has been used to tune the transition,\naccess the tricritical point, and map out the tricritical\nwings.10This situation requires a re\fnement of the the-\nory presented above, which will be reported elsewhere.21\nIII. DISCUSSION, AND CONCLUSION\nWe now discuss our results, before concluding with a\nsummary.\nA. The mechanism behind the \frst-order transition\nThe mechanism that leads to the \frst-order tran-\nsition discussed in Sec. II E is precisely analogous to\nthe \ructuation-induced \frst-order transition discussed in\nRef. 6 for the BCS-superconductor transition and the\nnematic-to-smectic-A transition in liquid crystals. An\nimportant physical ingredient is an underlying \\generic\"\nsoft mode, i.e., one that is not related to the phase tran-\nsition in question, but couples to the order parameter.\nIn the case of liquid crystals this soft mode is the ne-\nmatic Goldstone mode, in the case of superconductors,\nthe vector potential, in the present case, the spin-triplet\nparticle-hole excitation. At the transition of interest, this\nsoft mode acquires a mass that is given in terms of the\nnonzero expectation value of the order parameter. This\ngeneral mass-generating mechanism was \frst pointed out\nby Anderson, and is now known as the Anderson-Higgs\nmechanism.22{24This coupling of the order parameter to\nunderlying soft modes leads to a non-analytic term in the\nLandau free energy that is dominant over the usual quar-\ntic term and has a negative sign, leading to a \frst-order\ntransition. It should be stressed that this is only one way\nto realize a \ructuation-induced \frst-order transition; an-\nother one, for instance, is realized by a \u001e4-theory with a\ncubic anisotropy.25The current realization is analogous\nto the case of scalar electrodynamics studied by Cole-\nman and Weinberg in a particle-physics context.26It is\nalso worthwhile noting that the analogy between super-\nconductors on one hand, and liquid crystals and quantum\nmagnets on the other, breaks down in the ordered phase.\nIn the former case, the Goldstone mode gets absorbed\ninto the longitudinal component of the vector potential,\nwhich is massive, and there is no soft mode in the ordered\nphase. In the latter, there are Goldstone modes in the\nordered phases, namely, a \\smecton\" with an anisotropic\ndispersion relation in the smectic-A phase (Ref. 27, see\nalso Ref. 28) and magnons in the magnetic phase.9\nB. Universality of the \frst-order transition, and\nthe validity of the generalized mean-\feld theory\nExperimentally, all examples of clean low-T cferromag-\nnets (for disordered systems, see below; ferrimagnets so\nfar have not been systematically studied from this point\nof view) show a \frst-order transition if the Curie temper-\nature is suppressed far enough. There is not a single ex-\nample of a quantum critical point in zero magnetic \feld.\nWhile this is consistent with the generalized mean-\feld\ntheory theory presented in Sec. II, it is somewhat surpris-\ning when compared with the case of liquid crystals, where\nan analogous theory also predicts a \frst-order transition.\nIn this case, in stark contrast to that of quantum mag-\nnets, the observed transition is usually of second order,\nand only recently have examples of a (weakly) \frst-order\ntransition been found.29These observations beg the ques-\ntion whether in the case of quantum magnets the gener-\nalized mean-\feld approximation is more generally valid\nthan in classical systems.\nTo discuss this point, we \frst observe that we have\nmade three approximations to treat the action given by\nEq. (2.1a). First, we have integrated out the fermionic\nmassive modes in a saddle-point approximation that re-\nspects the Ward identity that governs the soft-mode\nstructure of the system.16,30Second, we have kept the\nsoft fermionic degrees of freedom only to Gaussian or-\nder in the soft modes q. Third, we have treated the\norder parameter in a mean-\feld approximation. These\napproximations are not independent of one another, and\nthe \frst two simpli\fcations do not constitute any addi-\ntional approximation over and above the last one. This\ncan be seen as follows.\nThe mean-\feld approximation for the order parame-\nter means that the fermionic degrees of freedom describe\nan interacting electron system that is spin-polarized by\nthe coupling to the homogeneous magnetization, which\nacts as an e\u000bective external magnetic \feld. The state\nof the fermionic subsystem is thus described by a stable\nFermi-liquid \fxed point. Corrections to the fermionic\nsoft-mode action due to massive degrees of freedom are\nirrelevant with respect to this \fxed point by at least one-\nhalf power of frequency or wavenumber in all dimensions,\nand thus cannot change the properties of system.17Sim-\nilarly, only the terms quadratic in qcontribute to the\n\fxed-point action; all higher-order terms are irrelevant\nby power counting. Keeping terms of higher order in q\nwill therefore renormalize the parameters of the theory,\nbut it cannot change its structure. In particular, it can-\nnot change the sign of the term in the equation of state,\nEqs. (2.27, 2.28), that is due to the soft fermionic \ructu-\nations and leads to the \frst-order transition.\nThis leaves the mean-\feld approximation for the order\nparameter to be discussed. If the \frst-order transition\natr=r1occurs far from the second-order transition at\nr= 0 that is pre-empted by it (see Fig. 2), then order-\nparameter \ructuations are negligible and the results of\nthe generalized mean-\feld theory are qualitatively cor-rect. If, however, the \frst-order transition occurs close\nto the putative second-order one, i.e., if the minimum\nin the free energy in Fig. 2 is very shallow, then it is\nless clear whether order-parameter \ructuations can be\nneglected. One key di\u000berence between classical liquid\ncrystals and quantum magnets is that in the former case,\nthe system is below the upper critical dimension d+\nc= 4\nfor the (unrealized) phase transition that would occur\nin the absence of any coupling between the smectic or-\nder parameter and the nematic soft modes. In contrast,\nthe quantum magnetic systems are above the correspond-\ning upper critical dimension d+\nc= 1 that follows from\nHertz theory, and even with that coupling taken into ac-\ncount, ordinary mean-\feld theory becomes exact, as far\nas the description of the phase transition is concerned,\nford>3.31This strongly suggests that order-parameter\n\ructuations are of much less importance in the case of\nquantum magnets, and it provides a possible explanation\nof the fact that the observed transition is universally of\n\frst order.\nIrrespective of these observations, the role of order-\nparameter \ructuations in quantum magnets is a topic\nthat warrants additional work. For the case where the\nmagnetism is not produced by the conduction electrons,\nthis will require an action that properly describes lo-\ncalized magnetic moments and their \ructuations, e.g.,\nthe one given in Ref. 32. For itinerant magnets, i.e., if\nthe magnetism is due to the conduction electrons them-\nselves, the theory developed in Sec. II will apply, but\nthe order-parameter \ructuations and the fermionic ex-\ncitations both need to be kept, along the lines of the\nphenomenological theory of Ref. 13. The latter reference\ngave a scenario that can lead to a second-order transi-\ntion in the magnetic case. It would also be interesting to\nexperimentally study quantum ferromagnets or ferrimag-\nnets ind= 2, where order-parameter \ructuations will be\nstronger than in d= 3.\nC. The e\u000bects of quenched disorder\nSo far we have discussed the case of clean or pure mag-\nnets. Impurities, modeled by quenched disorder, have\nimportant e\u000bects that are both needed to understand ex-\nperimental observations in certain systems, and to pre-\ndict e\u000bects that can serve to ascertain that the \frst-order\ntransition in pure samples is indeed due to the posited\nmechanism.\nQuenched disorder changes the soft-mode spectrum of\nthe fermions. It gives the ballistic soft modes that are\nrepresented by Eqs. (2.17) as mass, and leads to new soft\nmodes that are di\u000busive. In the context of the current\ntheory, this change has two principal e\u000bects. First, it\ncuts o\u000b the nonanalyticity in the clean equation of state,\nEqs. (2.27, 2.28). Second, it leads to a new nonanalytic\nterm in the equation of state that has the opposite sign\nand whose prefactor vanishes in the clean limit.31The\nresulting schematic generalized Landau theory has been10\ndiscussed in Ref. 4. A more detailed model discussion\nthat allows for semi-quantitative predictions of the e\u000bects\nof disorder will be presented elsewhere;21here we just\npresent the most pertinent aspects of such a model cal-\nculation. A good representation of the mean-\feld equa-\ntion of state for realistic values of the magnetization, the\ntemperature, and the disorder, is\nh=rm+v1=4\n4(kF`)3=2m3\nm3=2+ (bT)3=2\n+v\n2m3ln\u0002\ncm2+ (1=kF`+bT)2\u0003\n+um3;(3.1)\nwhich generalizes Eq. (2.27) in the presence of quenched\ndisorder. Here the magnetic \feld hand the temperature\nTare measured in units of the Fermi energy \u000fFand the\nFermi temperature TF, respectively, and the magnetiza-\ntionmis measured in units of the conduction electron\ndensity (we put \u0016B= 1). The dimensionless coupling\nconstantvis proportional to the fourth power of the ef-\nfective spin-triplet interaction amplitude of the conduc-\ntion electrons. It is a measure of how strongly corre-\nlated the conduction electrons are, and it is bounded\nabove by a stability criterion that requires v.0:5.\nkFis the Fermi wave number of the conduction elec-\ntrons, and `is the elastic mean-free path. Within a\nDrude model, and for good metals, one has approxi-\nmatelykF`\u00191;000=(\u001a0=\u0016\ncm), with\u001a0the residual\nelectrical resistivity. candbare dimensionless constants\nthat are equal to c= 1=45 andb= 3\u0019in a model\ncalculation.21The second factor in the second term on\nthe right-hand side is a reasonable representation, for re-\nalistic parameter values, of a more complicated scaling\nfunction\nm3=2g(kF`m;bT=m )\u0019m3\nm3=2+ (bT)3=2(3.2)\nthat depends on the disorder in addition to the temper-\nature, and we have dropped the last term in Eq. (2.27)\nfrom Eq. (3.1) since one generically expects v\u001cu.\nAtT= 0, and in a clean system, Eq. (3.1) yields a\n\frst-order transition at r1=vm2\n1=4, where the magneti-\nzation discontinuously jumps from m= 0 tom=m1=\ne\u0000(1+2u=v)=2. Withu\u00190:14 andv\u00190:02 this yields\nm1\u00194\u000210\u00003, which is reasonable for a weak ferromag-\nnet. Similarly, there is a tricritical temperature given\nbyTtc=TF= (1=b) exp(\u0000u=v); with the same parame-\nter values this yields Ttc=TF\u001910\u00004, orTtc\u001910 K for\nTF= 100;000 K, which is also reasonable. This tricritical\npoint gets destroyed by quenched disorder on the order of\nkF`\u0019bTtc=TF\u00191;000, or a residual resisitivity on the\norder of\u001a0\u00191\u0016\ncm. At this point the second term on\nthe right-hand side of Eq. (3.1) is still very small, and the\ncritical behavior at the resulting quantum critical point\nis given by ordinary mean-\feld exponents except extrely\nclose to the transition, where it crosses over to the crit-\nical behavior derived in Ref. 17. For instance, in this\nasymptotic region the critical exponents \fand\u000e, de\fnedbym(h= 0)/jrj\fandm(r= 0)/h1=\u000e, respectively,\nare given by \f= 1=2 and\u000e= 3=2, as opposed to the\nmean-\feld values \f= 1=2 and\u000e= 3. Only for sub-\nstantially larger values of the disorder, \u001a0\u0019100\u0016\ncm\nwith the above parameters, does the asymptotic critical\nbehavior extend over a sizeable range of rvalues (up to\njrj\u00190:01). This observation explains why an experi-\nment on Ni xPd1\u0000x, which shows a ferromagnetic transi-\ntion at a very small value of x(x\u00190:025) corresponding\nto weak disorder, found mean-\feld exponents consistent\nwith Hertz theory,33whereas Bauer et al.34found non-\nmean-\feld exponents, at least some of which were con-\nsistent with Ref. 17, in URu 2\u0000xRexSi2, where the ferro-\nmagnetic transition occurs at x\u00190:15 with the residual\nresistivity on the order of \u001a0\u0019100\u0016\ncm.35\nD. Conclusion\nIn conclusion, we have extended a previous theory\nof quantum ferromagnets in several important ways.\nWe have shown that the mechanism that leads to the\nparamagnet-to-ferromagnet transition at low tempera-\nture ind= 3 andd= 2 to be generically of \frst order,\nwhich was \frst reported in Ref. 4, is valid in anisotropic\nferromagnets, in ferrimagnets, and in metallic ferromag-\nnets where the conduction electrons are not the source of\nthe magnetization, in addition to the case of isotropic\nitinerant ferromagnets originally considered. This ex-\nplains why the low-temperature transition is observed\nto be of \frst order in highly anisotropic ferromagnets,\nand it much expands the class of materials for which this\nphenomenon is predicted. For clean magnets, an e\u000bec-\ntive theory of soft fermionic modes recently developed\nin Ref. 16 has provided a technical basis that improves\non the phenomenological theory of Ref. 13. In the pres-\nence of quenched disorder, the theory allows for a semi-\nquantitative description of the suppression and ultimate\ndestruction of the tricritical point. A sizeable range of\ndisorder exists where the observable critical behavior is\npredicted to be mean-\feld like, whereas for very large\ndisorder the asymptotic critical region, which is char-\nacterized by non-mean-\feld Gaussian critical exponents,\nexpands and eventually eliminates the mean-\feld region.\nAppendix A: A simple mean-\feld model of a\nferrimagnet\nHere we recall a very simple mean-\feld model of the\ntransition from a paramagnet to long-range ferrimagnetic\norder.14Consider a one-dimensional chain of alternating\nmagnetic moments \u0016a,\u0016bthat are antiferromagnetically\ncoupled. Weiss theory assumes that the a-moments and\nb-moments are subject to e\u000bective magnetic \felds\nBa=\u0000\u0015Mb (A1a)\nBb=\u0000\u0015Ma; (A1b)11\nrespectively, where \u0015>0. The magnetizations Ma;bare\ngiven by the Brillouin expressions\nMa=\u0017\u0016atanh(\u0016aH=T +\u0016aBa=T);(A2a)\nMb=\u0017\u0016btanh(\u0016bH=T +\u0016bBb=T):(A2b)\nHereHis an external magnetic \feld, Tis the tem-\nperature, and \u0017is the number of magnetic moments\nof each species. If one de\fnes reduced magnetic \felds\nha;b=H=\u0017\u0016a;b\u0015, a reduced temperature t=T=\u0017\u0016a\u0016b\u0015,\nand reduced moments ma;b=Ma;b=\u0017\u0016a;b, then one sees\nthat the Weiss mean-\feld equations (A1, A2) have a so-\nlutionma=\u0000mb= ~m, where ~mis the solution of the\nusual mean-\feld equation of state\nh=r~m+ ~m3=3 +O( ~m5); (A3)wherer=t\u00001. This simple model thus describes a\ntransition at t= 1 to ferrimagnetic order where the\nhomogeneous magnetization is given by m=Ma+\nMb=\u0017(\u0016a\u0000\u0016b) ~mand the staggered magnetization\nn=Ma\u0000Mb=\u0017(\u0016a+\u0016b) ~mis proportional to m.\nAcknowledgments\nWe gratefully acknowledge discussions and correspon-\ndence with Greg Stewart, Je\u000b Lynn, and Nick Butch.\nThis work was supported by the National Science Foun-\ndation under Grant Nos. DMR-09-29966, and DMR-09-\n01907.\n1J. Hertz, Phys. Rev. B 14, 1165 (1976).\n2S. Sachdev, Quantum Phase Transitions (Cambridge Uni-\nversity Press, Cambridge, 1999).\n3E. C. Stoner, Proc. Roy. Soc. London A 165, 372 (1938).\n4D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev.\nLett. 82, 4707 (1999).\n5In these and related statements we denote by dthe spatial\ndimensionality of the system; spin space is always consid-\nered to be three-dimensional.\n6B. I. Halperin, T. C. Lubensky, and S.-K. Ma, Phys. 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However,\nthis will not be the case generically, whereas the general-\nized mean-\feld theory predicts a generic \frst-order transi-\ntion.\n21Yan Sang, D. Belitz, and T.R. Kirkpatrick, unpublished\nresults.\n22P. W. Anderson, Phys. Rev. 130, 439 (1963).\n23P. W. Higgs, Phys. Lett. 12, 132 (1964).\n24P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).\n25D. J. Wallace, J. Phys. C 6, 1390 (1973).\n26S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).\n27P. G. DeGennes and J. Prost, The Physics of Liquid Crys-\ntals(Clarendon, Oxford, 1993).\n28T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 80, 075121\n(2009).\n29A. Yethiraj, R. Mukhopadhyay, and J. Bechhoefer, Phys.\nRev. E 65, 021702 (2002).\n30T. R. Kirkpatrick and D. Belitz, Phys. Rev. Lett. 108,\n086404 (2012).\n31D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod.\nPhys. 77, 579 (2005).\n32N. Read and S. Sachev, Phys. Rev. Lett. 75, 3509 (1995).\n33M. Nicklas, M. Brando, G. Knebel, F. Mayr, W. Trinkl,\nand A. Loidl, Phys. Rev. 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By studying the e\u000bects of the magnon-magnon interac-\ntion on the stability of the magnon BEC in a ferrimagnet and two ferromagnets, we show that the\nmagnon BEC remains stable even in the presence of the magnon-magnon interaction in the ferrimag-\nnet and ferromagnet with a sublattice structure, whereas it becomes unstable in the ferromagnet\nwithout a sublattice structure. This indicates that the existence of a sublattice structure is the\nkey to stabilizing the BEC of interacting magnons, and the di\u000berence between the spin alignments\nof a ferrimagnet and a ferromagnet is irrelevant. Our result can resolve a contradiction between\nexperiment and theory in the magnon BEC of yttrium iron garnet. Our theoretical framework may\nprovide a starting point for understanding the physics of the magnon BEC including the interaction\ne\u000bects.\nBose-Einstein condensation (BEC) has been exten-\nsively studied in various \felds of physics. The BEC is\na macroscopic occupation of the lowest-energy state for\nbosons [1]. This phenomenon was theoretically predicted\nin a gas of noninteracting bosons [2], and then it was ex-\nperimentally observed in dilute atomic gases [3{5]. This\nobservation opened up research of the BEC in atomic\nphysics [1]. Since the concept of the BEC is applicable\nto quasiparticles that obey Bose statistics, research of the\nBEC has been expanded, and it covers condensed-matter\nphysics, nuclear physics, and optical physics.\nThere is a critical problem with the magnon BEC.\nThe magnon BEC was experimentally observed in yt-\ntrium iron garnet (YIG), a three-dimensional ferrimag-\nnet [6{9]. However, a theory [10] showed that if low-\nenergy magnons of YIG are approximated by magnons of\na ferromagnet without a sublattice structure, the magnon\nBEC is unstable due to the attractive interaction between\nmagnons. Note \frst, that YIG is often treated as the\nferromagnet for simplicity of analyses [11, 12], second, in\ngeneral, the attractive interaction between bosons desta-\nbilizes the BEC [13, 14]. Thus the stabilizing mechanism\nfor the BEC of interacting magnons in a ferrimagnet re-\nmains unclear. To clarify it, we should understand the\ninteraction e\u000bects in a ferrimagnet. In addition, we need\nto understand the essential e\u000bects of the di\u000berences be-\ntween a ferrimagnet and the ferromagnet in order to un-\nderstand the reason for the contradiction between exper-\niment [6{9] and theory [10].\nIn this Letter, we study the interaction e\u000bects on the\nmagnon BEC in three magnets and propose a stabiliz-\ning mechanism. We use the Heisenberg Hamiltonian and\nconsider a ferrimagnet and two ferromagnets. By using\nthe Holstein-Primako\u000b transformation [15{17], we derive\nthe kinetic energy and interaction for magnons. Then,\nwe construct an e\u000bective theory to study the interaction\ne\u000bects on the magnon BEC in a similar way to the Bo-\ngoliubov theory [14, 18] for Bose particles. By combiningthe results for the three magnets, we show that the exis-\ntence of a sublattice structure, not the di\u000berence in the\nspin alignment, is the key to the stabilizing mechanism\nfor the BEC of interacting magnons. We also discuss the\ncorrespondence between our model and a more realistic\nmodel of YIG and several implications.\nWe use the Heisenberg Hamiltonian as a minimal\nmodel for ferrimagnets and ferromagnets. It is given by\nH= 2X\nhi;jiJijSi\u0001Sj; (1)\nwhereJijdenotes the Heisenberg exchange energy be-\ntween spins at nearest-neighbor sites, and Sidenotes the\nspin operator at site i.\nWe consider three cases. In the \frst case, we put\nJij=J,hSii=SAfori2A, andhSii=\u0000SBfori2B,\nwhereAandBdenoteAandBsublattices, respectively;\neach sublattice consists of N=2 sites. This case corre-\nsponds to a ferrimagnet with a two-sublattice structure\n[Fig. 1(a)]. In the second case, we put Jij=\u0000Jand\nhSii=Sfor alli's. In the third case, we put Jij=\u0000J,\nhSii=SAfori2A, andhSii=SBfori2B. The sec-\nond and third cases correspond to ferromagnets without\nsublattice and with a two-sublattice structure, respec-\ntively [Figs. 1(b) and 1(c)]. As we will show below, by\nstudying the BEC of interacting magnons in these three\ncases, we can clarify the stabilizing mechanism in a fer-\nrimagnet and the key to resolving the contradiction in\nthe magnon BEC of YIG. (We will focus mainly on the\nsign of the e\u000bective interaction between magnons and its\ne\u000bect on the stability of the magnon BEC.)\nWe begin with the \frst case of our model. We \frst\nderive the magnon Hamiltonian by using the Holstein-\nPrimako\u000b transformation [15{17]. After remarking on\nseveral properties in the BEC of noninteracting magnons,\nwe construct the e\u000bective theory for the BEC of interact-\ning magnons. By using this theory, we study the inter-\naction e\u000bects in the ferrimagnet.arXiv:1810.00584v2  [cond-mat.mes-hall]  2 Nov 20182\n\tB\n \tC\n \tD\nFIG. 1. Spin alignments on a plane of the cubic lattice in the three cases of our model; panels (a), (b), and (c) correspond\nto the \frst, second, and third cases, respectively. The direction and length of an arrow represent the direction and size of an\nordered spin. The ordered spins are ferrimagnetic in panel (a) and ferromagnetic in panels (b) and (c); sublattice degrees of\nfreedom are present in panels (a) and (c) and absent in panel (b).\nThe magnon Hamiltonian is obtained by applying the\nHolstein-Primako\u000b transformation to the spin Hamilto-\nnian. In general, low-energy excitations in a magnet can\nbe described well by magnons, bosonic quasiparticles [15{\n17, 19{23]. The magnon operators and the spin opera-\ntors are connected by the Holstein-Primako\u000b transforma-\ntion [15{17]. This transformation for our ferrimagnet is\nexpressed as follows:\nSz\ni=SA\u0000ay\niai; S\u0000\ni=p\n2SAay\nir\n1\u0000ay\niai\n2SA;(2)\nSz\nj=\u0000SB+by\njbj;S+\nj=p\n2SBby\njr\n1\u0000by\njbj\n2SB;(3)\nwherei2A,j2B,S\u0000\ni=Sx\ni\u0000iSy\ni= (S+\ni)y, and\nS+\nj=Sx\nj+iSy\nj= (S\u0000\nj)y;aianday\niare the operators of\nmagnons for the Asublattice, and bjandby\njare those for\ntheBsublattice. A substitution of Eqs. (2) and (3) into\nEq. (1) gives the magnon Hamiltonian.\nIn the magnon Hamiltonian, we consider the kinetic\nenergy terms and the dominant terms of the magnon-\nmagnon interaction. This is because our aim is to clarify\nhow the magnon-magnon interaction a\u000bects the magnon\nBEC, which is stabilized by the kinetic energy terms.\nSince the kinetic energy terms come from the quadratic\nterms of magnon operators and the dominant terms of the\ninteraction come from part of the quartic terms [16, 17],\nour magnon Hamiltonian is given by Hmag=Hnon+\nHint[24], where\nHnon= 2X\nqJ(0)(SBay\nqaq+SAby\nqbq)\n+2X\nqJ(q)p\nSASB(aqbq+ay\nqby\nq); (4)\nand\nHint=\u00002\nNP\nq;q0[J(0)ay\nqaqby\nq0bq0+J(q\u0000q0)ay\nqaq0by\nqbq0\n+J(q)pSASB(SAaqby\nq0bqbq0+SBbqay\nq0aq0aq)] + (H.c.):(5)\nWe have used ai=q\n2\nNP\nqeiq\u0001iaq,by\nj=q\n2\nNP\nqeiq\u0001jby\nq, andJ(q) =P\n\u000eJeiq\u0001\u000ewith\u000e, a\nvector to nearest neighbors.\nBefore formulating the e\u000bective theory for the BEC of\ninteracting magnons, we remark on several properties in\nthe BEC of noninteracting magnons in our ferrimagnet.\nTo see the properties, we diagonalize Hnonby using\n\u0012aq\nby\nq\u0013\n=\u0012cq\u0000sq\n\u0000sqcq\u0013\u0012\u000bq\n\fy\nq\u0013\n; (6)\nwherecq\u0011cosh\u0012qandsq\u0011sinh\u0012qsatisfy tanh 2 \u0012q=\n2pSASBJ(q)\n(SA+SB)J(0). After some algebra, we obtain\nHnon=X\nq\u000f\u000b(q)\u000by\nq\u000bq+X\nq\u000f\f(q)\fy\nq\fq; (7)\nwhere\u000f\u000b(q) = (SB\u0000SA)J(0) + \u0001\u000f(q) and\n\u000f\f(q) = (SA\u0000SB)J(0) + \u0001\u000f(q) with \u0001\u000f(q) =p\n(SA+SB)2J(0)2\u00004SASBJ(q)2; in Eq. (7) we have\nneglected the constant terms. Hereafter, we assume\nSA> SB; this does not lose generality. For SA> SB\n\u000f\u000b(0) = 0 is the lowest energy. Thus many magnons\noccupy theq= 0 state of the \u000bband in the BEC of non-\ninteracting magnons in the ferrimagnet for SA>SB. In\naddition, the low-energy excitations from the condensed\nstate are described by the \u000b-band magnons near q=0.\nWe now construct the e\u000bective theory for the BEC of\ninteracting magnons. To construct it as simple as possi-\nble, we utilize the properties in the BEC of noninteract-\ning magnons. As described above, in the ferrimagnet for\nSA> SBthe condensed state is the q=0state of the\n\u000bband and the low-energy noncondensed states are the\nsmall-qstates of the \u000bband. Thus we can reduce Hmag\nto an e\u000bective Hamiltonian He\u000b, which consists of the ki-\nnetic energy term of the \u000bband and the intraband terms\nof the magnon-magnon interaction for the \u000bband;He\u000b\nis given by He\u000b=H0+H0, whereH0is the \frst term of\nEq. (7), and H0is obtained by substituting Eq. (6) into\nEq. (5) and retaining the intraband terms. This He\u000bis\nsu\u000ecient for studying properties of the BEC of interact-\ning magnons at temperatures lower than a Curie tem-\nperature, because the dominant excitations come from3\nthe small-qmagnons in the \u000bband and the interband\nterms may be negligible in comparison with the intra-\nband terms. Then we can further simplify H0. Since its\nmain e\u000bects can be taken into account in the mean-\feld\napproximation, the leading term of H0is given by [24]\nH0=\u00004\nNX\nq;q0\u0000\u000b\u000b(q;q0)nq0\u000b\u000by\nq\u000bq; (8)\nwhere \u0000\u000b\u000b(q;q0) =J(0)(c2\nqs2\nq0+c2\nq0s2\nq) + 2J(q\u0000\nq0)cqsqcq0sq0\u0000J(q)pSASBcqsq(SAs2\nq0+SBc2\nq0)\u0000\nJ(q0)pSASBcq0sq0(SAs2\nq+SBc2\nq), andnq0\u000b=h\u000by\nq0\u000bq0i=\nn[\u000f\u000b(q0)] with the Bose distribution function n(\u000f). By\ncombining Eq. (8) with H0=P\nq\u000f\u000b(q)\u000by\nq\u000bq, we obtain\nHe\u000b=X\nq\u000f\u0003\n\u000b(q)\u000by\nq\u000bq; (9)\nwith\u000f\u0003\n\u000b(q) =\u000f\u000b(q)\u00004\nNP\nq0\u0000\u000b\u000b(q;q0)nq0\u000b.\nBy using the theory described by He\u000b, we study\nthe interaction e\u000bects on the stability of the magnon\nBEC. Since the magnon energy should be nonnegative,\nthe magnon BEC remains stable even for interacting\nmagnons as long as \u000f\u0003\n\u000b(0) is the lowest energy. This\nis realized if H0is the repulsive interaction. If H0is\nthe attractive interaction, the magnon BEC becomes un-\nstable. Thus we need to analyze the sign of \u0000 \u000b\u000b(q;q0)\nin Eq. (8). Since the dominant low-energy excitations\nare described by the \u000b-band magnons near q=0, we\nestimate \u0000 \u000b\u000b(q;q0) in Eq. (8) in the long-wavelength\nlimitsjqj;jq0j! 0. For a concrete simple example we\nperform this estimation in a three-dimensional case on\nthe cubic lattice. By expressing J(q) in a Taylor series\naroundjqj= 0 and retaining the leading correction, we\ngetJ(q)\u0019J(0)[1\u0000q2\n6]. Then, by using this expression\nand performing some calculations [24], we obtain the ex-\npression of \u0000 \u000b\u000b(q;q0) including the leading correction in\nthe long-wavelength limits. The derived expression is\n\u0000\u000b\u000b(q;q0)\u0019\u00002\n9J(0)q2q02(SASB)2\n(SA\u0000SB)4: (10)\nThe combination of Eqs. (10) and (8) shows that the\nleading term of the magnon-magnon interaction is re-\npulsive. Thus the magnon BEC remains stable in the\nferrimagnet even with the magnon-magnon interaction.\nThe above result di\u000bers from the stability of the\nmagnon BEC in the ferromagnet without a sublattice\nstructure. This can be seen by applying a similar\ntheory to the second case of our model and compar-\ning the result with the above result. The Holstein-\nPrimako\u000b transformation in the ferromagnet without\na sublattice structure is expressed as Sz\ni=S\u0000cy\nici,\nS\u0000\ni=cy\niq\n2S\u0000cy\nici, andS+\ni= (S\u0000\ni)yfor alli's;\nciandcy\niare the magnon operators. By using thistransformation and the Fourier transformations of the\nmagnon operators, such as ci=1p\nNP\nqeiq\u0001icq, we ob-\ntain the magnon Hamiltonian Hmag =Hnon+Hint,\nwhereHnon=P\nq\u000f(q)cy\nqcqwith\u000f(q) = 2S[J(0)\u0000\nJ(q)] andHint=\u00001\n2NP\nq;q0[J(0)cy\nqcqcy\nq0cq0+J(q\u0000\nq0)cy\nqcq0cy\nq0cq\u00002J(q)cy\nq0cqcy\nq0cq] + (H.c.). Then, by ap-\nplying the mean-\feld approximation to Hint, the lead-\ning term of the magnon-magnon interaction is reduced\ntoH0=\u00002\nNP\nq;q0\u0000(q;q0)nq0cy\nqcq, where \u0000(q;q0) =\nJ(0) +J(q\u0000q0)\u0000J(q)\u0000J(q0) andnq0\u0011n[\u000f(q0)]. Since\n\u0000(q;q0)\u00150, the magnon-magnon interaction becomes\nattractive. Thus the BEC of interacting magnons be-\ncomes unstable in the ferromagnet without a sublattice\nstructure.\nIn order to understand the key to causing the above\ndi\u000berence, we study the stability of the BEC of interact-\ning magnons in the third case of our model. As we can\nsee from Fig. 1, the di\u000berence between the third and \frst\ncases is about the spin alignment, and the di\u000berence be-\ntween the third and second cases is about the sublattice\nstructure. Thus, by comparing the result in the third case\nwith the result in the \frst or second case, we can deduce\nwhich of the two, the di\u000berences in the spin alignment\nand in the sublattice structure, causes the di\u000berence in\nthe stability of the BEC of interacting magnons.\nThe stability in the third case can be studied in a sim-\nilar way to that in the \frst case. In the third case, the\nHolstein-Primako\u000b transformation of Sifori2Ais the\nsame as Eq. (2), whereas that of Sjforj2Bis given\nbySz\nj=SB\u0000by\njbj,S\u0000\nj=p2SBby\njq\n1\u0000(by\njbj=2SB),\nandS+\nj= (S\u0000\nj)y; this di\u000berence arises from the di\u000ber-\nent alignment of the spins belonging to the Bsublattice.\nIn a similar way to the \frst case, we obtain the magnon\nHamiltonian Hmag=Hnon+Hint, whereHnonandHint\nare given by\nHnon= 2X\nqJ(0)(SBay\nqaq+SAby\nqbq)\n\u00002X\nqJ(q)p\nSASB(aqby\nq+ay\nqbq); (11)\nand\nHint=\u00002\nNX\nq;q0[J(0)ay\nqaqby\nq0bq0+J(q\u0000q0)ay\nqaq0by\nq0bq\n\u0000J(q)pSASB(SAay\nqby\nq0bq0bq+SBbqay\nqay\nq0aq0)] + (H.c.);(12)\nrespectively, with ai=q\n2\nNP\nqeiq\u0001iaqandbj=q\n2\nNP\nqeiq\u0001jbq. In addition, Hnoncan be diagonal-\nized by using aq=cq\u000bq\u0000sq\fqandbq=\u0000sq\u000bq+\ncq\fq, wherecq\u0011cosh\u0012qandsq\u0011sinh\u0012qsatisfy\ntanh 2\u0012q=\u00002pSASBJ(q)\n(SA+SB)J(0). The diagonalized Hnonis\nHnon=P\nq[\u000f\u000b(q)\u000by\nq\u000bq+\u000f\f(q)\fy\nq\fq] with\u000f\u000b(q) and\n\u000f\f(q), which are the same as those in the \frst case. Thus,4\nthe ferromagnet and ferrimagnet with the two-sublattice\nstructure have the same properties of the BEC of nonin-\nteracting magnons. Then we can construct the e\u000bective\ntheory for the BEC of interacting magnons in the third\ncase in a similar way. For SA> SB, in the third case,\nthe BEC of interacting magnons can be e\u000bectively de-\nscribed by He\u000b=P\nq\u000f\u0003\n\u000b(q)\u000by\nq\u000bqwith\u000f\u0003\n\u000b(q) =\u000f\u000b(q)\u0000\n4\nNP\nq0~\u0000\u000b\u000b(q;q0)nq0\u000b, where ~\u0000\u000b\u000b(q;q0) =J(0)(c2\nqs2\nq0+\nc2\nq0s2\nq) + 2J(q\u0000q0)cqsqcq0sq0+J(q)pSASBcqsq(SAs2\nq0+\nSBc2\nq0) +J(q0)pSASBcq0sq0(SAs2\nq+SBc2\nq). By estimating\n~\u0000\u000b\u000b(q;q0) in the long-wavelength limits in a similar way,\nwe obtain ~\u0000\u000b\u000b(q;q0)\u0019\u00002\n9J(0)q2q02(SASB)2\n(SA\u0000SB)4. Thus the\nBEC of interacting magnons is stable in the ferromagnet\nwith the two-sublattice structure.\nCombining the results in the three cases, we \fnd that\nthe di\u000berence between the interaction e\u000bects in the fer-\nrimagnet and in the ferromagnet without a sublattice\nstructure arises not from the di\u000berence in the spin align-\nment, but from the di\u000berence in the sublattice struc-\nture. This can resolve the contradiction between exper-\niment [6{9] and theory [10] because that theory uses a\nferromagnet without a sublattice structure. This also\nsuggests that the existence of a sublattice structure is\nthe key to stabilizing the BEC of interacting magnons in\nferrimagnets and ferromagnets. One possible experiment\nto test our mechanism is to measure the stability of the\nmagnon BEC in ferromagnets without and with a sublat-\ntice structure; a sublatttice structure, such as that shown\nin Fig. 1(c), can be realized, for example, by using two\ndi\u000berent magnetic ions.\nWe remark on the role of sublattice degrees of free-\ndom. As shown above, the magnon BEC remains stable\neven in the presence of the magnon-magnon interaction\nas long as a magnet has the sublattice degrees of free-\ndom. This remarkable property can hardly be expected\nfrom the properties of noninteracting magnons because in\nall the three cases, the low-energy properties can be de-\nscribed by a single magnon band. The magnon-magnon\ninteraction becomes repulsive only in the presence of the\nsublattice degrees of freedom because the magnons in dif-\nferent sublattices give the di\u000berent contributions to the\nintraband interaction for a single magnon band; the dif-\nferent contributions arise from the di\u000berent coe\u000ecients\nin the Bogoliubov transformation [e.g., see Eq. (6)].\nNext we discuss the correspondence between our model\nand a model derived in the \frst-principles study in\nYIG [25]. The latter is more complicated than our\nmodel because the magnetic primitive cell of YIG has\n20 Fe moments [26] and its spin Hamiltonian consists of\nthe Heisenberg exchange interactions for three nearest-\nneighbor pairs and six next-nearest-neighbor pairs [25].\nNote \frst, that all of the Fe ions are categorized into FeO\nand FeTions, Fe ions surrounded by an octahedron and a\ntetrahedron of O ions, respectively, and second, that YIGis a ferrimagnet due to the antiparallel spin alignments\nof the FeOand FeTions and the 2 : 3 ratio of the FeO\nand FeTions in the unit cell [27]. Although our model\ndoes not take into account all of the complex properties\nof YIG, our model can be regarded as a minimal model to\nstudy the stability of the BEC of interacting magnons in\nYIG. This is because of the following three facts: First,\nthe largest term in the spin Hamiltonian of YIG is the\nantiferromagnetic nearest-neighbor Heisenberg exchange\ninteraction between the FeOand FeTions and the others\nare at least an order of magnitude smaller. Second, the\nlow-energy magnons of YIG can be described by a single\nmagnon band around q=0. Third, the main e\u000bect of\nthe terms neglected in our theory is to modify the value\nof \u0000\u000b\u000b(q;q0) in Eq. (8). Since this modi\fcation may\nbe quantitative, our mechanism can qualitatively explain\nwhy the magnon BEC is stabilized in YIG.\nOur work has several implications. First, our results\nsuggest that a ferromagnet without a sublattice structure\nis inappropriate for describing the properties of interact-\ning magnons in ferrimagnets, such as YIG. This sugges-\ntion will be useful for future studies towards a compre-\nhensive understanding of magnon physics and spintron-\nics using magnons in YIG. Furthermore, it may be nec-\nessary to reconsider some results of YIG if the results\nare deduced by using a ferromagnet without a sublat-\ntice structure, in particular, the results depend on the\nsign of the magnon-magnon interaction. Our theoretical\nframework can then be used to study the BEC of interact-\ning magnons in other magnets as long as the low-energy\nmagnons can be described by a single magnon band. For\nthe magnets whose low-energy magnons have degeneracy,\nan extension of this framework enables us to study the\nBEC of interacting magnons. Thus our theory may pro-\nvide a starting point for understanding properties of the\nBEC of interacting magnons in various magnets.\nIn summary, we have studied the stability of the BEC\nof interacting magnons in a ferrimagnet and ferromag-\nnets, and we proposed the stabilizing mechanism. By\nadopting the Holstein-Primako\u000b transformation to the\nHeisenberg Hamiltonian, we have derived the magnon\nHamiltonian, which consists of the kinetic energy terms\nand the dominant terms of the magnon-magnon inter-\naction. We then construct the e\u000bective theory for the\nBEC of interacting magnons by utilizing the properties\nfor noninteracting magnons and the mean-\feld approx-\nimation. From the analyses using this theory, we have\ndeduced that in the ferrimagnet and ferromagnet with\nthe sublattice structure the magnon BEC remains stable\neven in the presence of the magnon-magnon interaction,\nwhereas it becomes unstable in the ferromagnet without\na sublattice. This result shows that the existence of a\nsublattice structure is the key to stabilizing the BEC of\ninteraction magnons, whereas the di\u000berence in the spin\nalignments is irrelevant. 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Rev. 117, 117 (1960).\n[17] T. Nakamura and M. Bloch, Phys. Rev. 132, 2528 (1963).\n[18] N. N. Bogoliubov, Izv. Akad. Nauk SSSR, Ser. Fiz. 11,\n77 (1947).\n[19] F. Bloch, Z. Phys. 61, 206 (1930).\n[20] F. Dyson, Phys. Rev. 102, 1217 (1956).\n[21] R. Kubo, Phys. Rev. 87, 568 (1952).\n[22] A. B. Harris, D. Kumar, B. I. Halperin, and P. C. Ho-\nhenberg, Phys. Rev. B 3, 961 (1971).\n[23] E. Manousakis, Rev. Mod. Phys. 63, 1 (1991).\n[24] See Supplemental Material, which includes Refs. 16 and\n17, for the details of the derivations of Eqs. (4) and (5),\nEq. (8), and Eq. (10).\n[25] L.-S. Xie, G.-X. Jin, L. He, G. E. W. Bauer, J. Barker,\nand K. Xia, Phys. Rev. B 95, 014423 (2017).\n[26] A. Harris, Phys. Rev. 132, 2398 (1963).\n[27] F. Bertaut, F. Forrat, A. Herpin, and P. M\u0013 eriel, Comptes\nRendus Acad. Sci. 243, 898 (1956)."    },    {        "title": "2101.03449v1.Tuning_Dzyaloshinskii_Moriya_Interaction_in_Ferrimagnetic_GdCo__A_First_Principles_Approach.pdf",        "content": "Tuning Dzyaloshinskii-Moriya Interaction in Ferrimagnetic GdCo: A First Principles\nApproach\nMd Golam Morshed,1,\u0003Khoong Hong Khoo,2Yassine Quessab,3Jun-Wen Xu,3Robert\nLaskowski,2Prasanna V. Balachandran,4, 5Andrew D. Kent,3and Avik W. Ghosh1, 6\n1Department of Electrical and Computer Engineering,\nUniversity of Virginia, Charlottesville, VA 22904 USA\n2Institute of High Performance Computing, Agency for Science,\nTechnology and Research, 1 Fusionopolis Way, Connexis, Singapore 138632, Singapore\n3Center for Quantum Phenomena, Department of Physics,\nNew York University, New York, NY 10003 USA\n4Department of Materials Science and Engineering,\nUniversity of Virginia, Charlottesville, Virginia 22904 USA\n5Department of Mechanical and Aerospace Engineering,\nUniversity of Virginia, Charlottesville, Virginia 22904 USA\n6Department of Physics, University of Virginia, Charlottesville, VA, 22904 USA\n(Dated: January 12, 2021)\nWe present a systematic analysis of our ability to tune chiral Dzyaloshinskii-Moriya Interactions\n(DMI) in compensated ferrimagnetic Pt/GdCo/Pt 1\u0000xWxtrilayers by cap layer composition. Using\n\frst principles calculations, we show that the DMI increases rapidly for only \u001810% W and saturates\nthereafter, in agreement with experiments. The calculated DMI shows a spread in values around the\nexperimental mean, depending on the atomic con\fguration of the cap layer interface. The saturation\nis attributed to the vanishing of spin orbit coupling energy at the cap layer and the simultaneous\nconstancy at the bottom interface. Additionally, we predict the DMI in Pt/GdCo/X (X = Ta ;W;Ir)\nand \fnd that W in the cap layer favors a higher DMI than Ta and Ir that can be attributed to the\ndi\u000berence in d-band alignment around the Fermi level. Our results open up exciting combinatorial\npossibilities for controlling the DMI in ferrimagnets towards nucleating and manipulating ultrasmall\nhigh-speed skyrmions.\nIntroduction. Magnetic skyrmions are topologically\nprotected spin textures and are attractive for next-\ngeneration spintronic applications, such as racetrack\nmemory and logic devices [1{7]. The interfacial\nDzyaloshinskii-Moriya Interaction (DMI), an antisym-\nmetric exchange originating from the strong spin-orbit\ncoupling (SOC) in systems with broken inversion symme-\ntry [8, 9], is one of the key ingredients in the formation of\nskyrmions in magnetic multilayers [10{12]. Controlling\nthe DMI o\u000bers the possibility to manipulate skyrmion\nproperties, i.e., size and stability [13, 14].\nOver the past few years, the underlying DMI physics\nand overall skyrmion dynamics have been studied ex-\ntensively for ferromagnetic (FM) systems [12, 15{20].\nAlthough both heavy metal (HM)/FM bilayers and\nHM/FM/HM sandwiched structures have been explored,\nmost of the reported results are based on ideal inter-\nfaces. Indeed, very few studies focus on the role of dis-\norder on DMI [21]. Furthermore, ferrimagnetic materials\nhave drawn attention due to their low saturation magne-\ntization, low stray \felds, reduced sensitivity to external\nmagnetic \felds, and fast spin dynamics, all of which fa-\nvor ultra-fast and ultra-small skyrmions [22{26]. Very\nrecently, Quessab et al. have experimentally studied\nthe interfacial DMI in amorphous Pt/GdCo thin \flms,\nand shown a strong tunability of the DMI by varying\n\u0003mm8by@virginia.eduthe thickness of the GdCo alloy and cap layer compo-\nsition [27]. However, a detailed understanding of DMI,\nincluding the impact of two-sublattice ferrimagnetism, as\nwell as the role of an experimentally realistic, chemically\ndisordered interface are both missing.\nIn this paper, we present a systematic theoretical anal-\nysis of the DMI in a compensated ferrimagnetic alloy us-\ning \frst principles calculations. In particular, we explore\nthe variation of the DMI in Pt =GdCo =Pt1\u0000xWx(Fig. 1)\nand \fnd a strong tunability from 0 to 4 :42 mJ/m2with\nvariation in the W composition (Fig. 2). We studied the\nin\ruence of atom placement and observed that the DMI\nis sensitive to structural variations such as the GdCo con-\n\fguration in the thin magnetic \flm, and the PtW con-\n\fguration at the interface. This is important to consider\nbecause, in reality, we have an amorphous alloy and the\ninterfaces in deposited \flms are not perfect. We \fnd a\nspectrum of DMI values that show an overall saturating\ntrend, as seen in the experimental data [27]. We argue\nthat the change in SOC energy in the interfacial HM lay-\ners, especially the constancy of the SOC energy at the\nbottom layer and reduction of it in the cap layer, gen-\nerates the observed saturating trend in the DMI with\npercentage of W incorporated (Fig. 3). Additionally, we\ntheoretically predict the variation of the DMI depending\non the cap layer material, speci\fcally for Pt/GdCo/X,\nwhere X = Ta ;W;Ir (Fig. 4). We \fnd that the DMI\nis highest for W in the cap layer and lowest for Ir, a\ntrend that correlates with 3 d-5dCo-X band alignment atarXiv:2101.03449v1  [cond-mat.mtrl-sci]  10 Jan 20212\nthe cap layer interface (Fig. 5). Our results identify the\nchemical and geometric factors responsible for interfacial\nDMI, and provide a potential path forward towards the\nengineering of material properties towards next genera-\ntion skyrmion based spintronic applications.\nMethod. We use the technique of constraining the mag-\nnetic moments in a supercell to calculate the DMI within\nthe Density Functional Theory (DFT) framework [15].\nThe Vienna ab initio simulation package (VASP) is used\nfor the DFT calculations [28]. We use the projec-\ntor augmented wave (PAW) potential to describe the\ncore-electron interaction [29, 30]. The Perdew-Burke-\nErnzerhof (PBE) functional form of the generalized gra-\ndient approximation (GGA) is used for the exchange-\ncorrelation functional [31]. In order to treat the on-\nsite Coulomb interaction of Gd 4 f-electrons, we use the\nGGA+ Umethod [32] with an e\u000bective value of U= 6 eV\nfor Gd, as reported in previous studies for both bulk and\nslab calculations [33{35]. We also validate the e\u000bective\nUfor our GdCo alloy by taking a range of Uvalues from\n1\u00007 eV, and con\frming a stable ferrimagnetic ground\nstate con\fguration of GdCo at U= 6 eV. A 4\u00021\u00021 su-\npercell of Pt(2) =GdCo(2) =Pt1\u0000xWx(2) (numbers in the\nparenthesis represent the number of monolayers) is used\nin all our calculations. While creating the GdCo alloy\nby replacing Gd atoms in the hcp Co(0001) slab, a 25%\nGd composition is maintained, which is the closest to the\nexperimental proportion (22% Gd [27]) achievable within\nour structural arrangement. The trilayers are formed by\naligning fcc(111) and hcp(0001) planes. The in-plane lat-\ntice constant of the slab structure is set to 2 :81/RingA, equal\nto the calculated nearest neighbor distance of bulk Pt,\nand the supercells are separated by a vacuum layer of\n10/RingA in the [001] direction. The cuto\u000b energy is set to\n500 eV, and a 4\u000216\u00021 Monkhorst-pack k-grid is used\nfor all the calculations. We verify the convergence of our\ncalculations with cuto\u000b energy, number of k-points, and\nthe thickness of the vacuum layer.\nThe three step DMI calculation procedure starts with\nionic relaxation along the atomic z-coordinate to mimic\na thin \flm, until the forces become less than 0 :01 eV/ /RingA\nand, the energy di\u000berence between two ionic relaxation\nsteps becomes smaller than 10\u00006eV. Next, in the ab-\nsence of SOC, the non-spin polarized Kohn-Sham equa-\ntions are solved to \fnd an initial charge density. Finally,\nSOC is included, and the total energy of the system is\ncalculated self-consistently for clockwise (CW) and an-\nticlockwise (ACW) spin con\fgurations (Fig. 1) until the\nenergy di\u000berence between two consecutive steps becomes\nsmaller than 10\u00006eV.\nResults. The DMI energy ( EDMI) can be de\fned as\nEDMI =X\nhi;jidij\u0001(Si\u0002Sj) (1)\nwhere Si,Sjare the nearest neighboring normalized\natomic spins and dijis the corresponding DMI vector.\nThe total DMI strength, dtot, de\fned by the summation\nFigure 1. Schematic of Pt(2) =GdCo(2) =Pt1\u0000xWx(2) struc-\nture (number in parentheses denoting the number of mono-\nlayers) corresponding to x= 12:5% for (a) CW, and (b) ACW\nspin con\fgurations. The red arrows in the \fgure show the\nspin orientations. L1 ; :::;L6 denote the layer number while\nnumbers in circles label atomic positions.\nof the DMI coe\u000ecient of each layer, to a \frst approxima-\ntion, is calculated by the energy di\u000berence between the\nCW and ACW spin con\fgurations [15], and expressed as\ndtot= (ECW\u0000EACW )=12. The micromagnetic DMI, D\nis given by D= 3p\n2dtot=NFa2[15], where NFanda\nrepresent the number of magnetic layers and the fcc lat-\ntice constant respectively.\nBefore presenting the numerical results, it is worth\nmentioning that we can only investigate a limited sub-\nset of the structures for our calculations, as exploring all\ncombinatorial possibilities is not feasible in terms of time\nand computational resources. We consider two separate\nalloy con\fgurations: (i) Gd alloying in the magnetic lay-\ners, and (ii) W alloying in the cap layers.\nIn case (i), we \frst \fx the position of the Gd atoms\nin the GdCo alloy. We maintain 25% Gd composition\nseparately in each magnetic layer, arguing that steric re-\npulsion implies two Gd atoms are energetically unlikely\nto sit in the same layer, as assumed in previous stud-\nies [36]. The Gd atoms can thus arrange themselves in\u00004\n1\u0001\n\u0002\u00004\n1\u0001\n= 16 ways. These sixteen combinations can\nbe grouped into just four distinct sets because of their\ntranslational symmetry. In Fig. 1, looking at positions\n(1\u00008) in magnetic layers (L3 & L4), it can be seen\nthat Gd in (1 ;7);(2;8);(3;5), and (4 ;6) positions repre-\nsent equivalent structures once the unit cell is period-\nically extended. Similarly, the other three groups are\n[(1;8);(2;5);(3;6);(4;7)], [(1 ;6);(2;7);(3;8);(4;5)], and\n[(1;5);(2;6);(3;7);(4;8)]. We con\frmed this equivalence\nby calculating the energy of the Pt/GdCo stack by vary-\ning all the Gd positions and indeed \fnd equal energy\nfor the four structures within the same group. For case\n(ii), we choose one representative from each of the above\nfour groups and proceed with W positional variations in\nthe cap layer. While exploring W alloy con\fgurations,\nfor lower composition (12 :5%\u000050%), W is only incor-3\n0.00.20.40.60.81.0\nW Composition (x)0.01.02.03.04.05.0DMI, D [mJ/m2]\nW(5')\nW(6')\nW(7')\nW(8')W(5',6')\nW(5',7')W(5',8')\nW(7',8')W(6',8')\nW(6',7')W(1',3',5'-8')\nW(2',3',5'-8')W(1',4',5'-8')\nW(2',4',5'-8')\nW(3',4',5'-8')W(1',2',5'-8')W(5'-8')W(1'-8')\n(a)Gd(1,7)\n0.00.20.40.60.81.0\nW Composition (x)-1.00.01.02.03.04.05.0DMI, D [mJ/m2]\n(b)Gd(1,7)\nGd(1,8)\nGd(2,7)\nGd(3,7)\n0.00.20.40.60.81.0\nW Composition (x)-0.50.00.51.01.52.02.5Surface DMI [pJ/m]\n(c)Gd(1,7)\nGd(1,8)\nGd(2,7)\nGd(3,7)\nExp.\nFigure 2. The DMI as a function of W composition ( x) in Pt =GdCo =Pt1\u0000xWx. (a) DMI variation with respect to W positions\nwhile the Gd atoms are \fxed at (1 ;7) positions. (b) Total spectrum of the DMI as both Gd and W positions are varied. For\na speci\fc W composition, each of the di\u000berent colors represents the variation of Gd atomic positions, and the scattered points\nwithin the same color represent di\u000berent W positions for that particular Gd arrangement in the structure (Fig. 1). (c) Surface\nDMI in comparison with experimentally observed DMI [27]. The numbers followed by the symbols Gd and W represent the\npositions of the respective atoms in the structure shown in Fig. 1.\nporated in layer L6. Finally, we vary all the possible W\npositions and calculate the DMI for a total of 76 struc-\ntures.\nFigure 2(a) shows the calculated DMI, D for\nPt=GdCo =Pt1\u0000xWx, as a function of W composition. At\nx= 0%, the DMI vanishes as expected because, for a\nperfectly symmetric trilayer structure, the contributions\nfrom the bottom and top interfaces are equal and oppo-\nsite. As the W composition increases from 0% to 12 :5%,\nwe \fnd a maximum DMI of 2 :93 mJ/m2. The underly-\ning mechanism behind this non-zero DMI is the inversion\nsymmetry breaking of the Pt =GdCo =Pt structure by the\ninsertion of W atoms in the cap layer. We \fnd that a\nsmall amount of W (12 :5%) gives a large DMI change,\nand subsequent to that initial rise, with increasing W\ncontent, the DMI saturates. As the composition of W\nincreases, we \fnd a maximum DMI of 4 :42 mJ/m2cor-\nresponding to 75% W composition.\nWe \fnd that the DMI is very sensitive to the struc-\ntural details, speci\fcally the positions of the Gd and W\natoms. Figure 2(a) shows the variation of D as the posi-\ntion of the W atoms changes. In Fig. 2(a), for all cases,\nGd atoms are \fxed at the (1 ;7) positions. We show the\nvariation of W positions for the structures with 12 :5%,\n25%, and 75% compositions because for the other three\ncases there is only one combination possible in terms of\nW positions. Figure 2(b) shows the total spectrum of\nthe DMI variation while varying both the Gd and W po-\nsitions in the structure. Interestingly, for all the cases,\nthe increasing trend of the DMI is very similar. We con-\njecture that changing the position of the atoms within\nthe small unit cell will change the nature of the interface\nthat gives variations in the DMI. For example, in the\ncase of Pt/GdCo/W, when Gd atoms placed at position\n(1,7), the SOC energy change in the interfacial Pt layer\nis higher than that of position (3,7), which translates to\nthe corresponding DMI as well.\nTo validate our results against the recent experi-ment [27], we calculate the surface DMI (in units of\npJ/m) by multiplying the calculated DMI, D with the\nthickness of the magnetic layers. In our calculations,\nwe use the thickness as NFa=p\n3 = 4 :6\u0017A for the mag-\nnetic layers, while the experimental thickness is 5 nm.\nFigure 2(c) shows the surface DMI from both the DFT\ncalculation and the experiment, scaled by their respec-\ntive thickness. In the experiment, a non-zero DMI of\n0:56 pJ/m (solid black line) is found for the Pt/GdCo/Pt\nstructure because of the asymmetry in the bottom and\nthe top interfaces due to the di\u000berence of interface rough-\nness and intermixing [27]. On the contrary in our DFT\nmodel, we use a perfect crystal structure that gives a\nnear zero DMI for the symmetric cases (a small non-\nzero DMI might arise from intrinsic asymmetry within\na thin crystalline GdCo \flm modeled here). We \fnd\nan overall matching trend between the DFT and exper-\nimental data for the rest of the compositions. An exact\nquantitative agreement between the DFT results and the\nexperiment is di\u000ecult to achieve because we use a crys-\ntal structure for our model, whereas, in the experiment,\namorphous or polycrystalline materials are used. Addi-\ntionally, the magnetization also di\u000bers between our model\nand the experiment as the thickness and the dimensions\nof the structure are di\u000berent. However, we argue that\nthe structural imperfections in the experiment amount\nto an ensemble averaging over the various con\fgurations\nwe theoretically explore, so that the experimental data\nfalls in the middle of the spectrum (gray shaded area) of\nour DFT data.\nIn Fig. 2, the DMI increases non-monotonically as a\nfunction of W composition as opposed to a linear in-\ncrease one may expect. This non-monotonic trend can\nbe explained by the change of spin-orbit coupling energy,\n\u0001ESOC, between CW and ACW spin con\fgurations, in\nthe HM layers adjacent to the magnetic layers in Fig. 1.\nIn Fig. 3(a), we show the \u0001 ESOC in L2 (adjacent to the\nbottom magnetic layer) and L5 (adjacent to the top mag-4\n(a)\nL2 L5-6.0-4.0-2.00.02.04.06.0\"ESOC [meV]W(0%)\nW(12.5%)\nW(25%)\nW(50%)\nW(75%)\nW(100%)\n(b)\nL1L2L3L4L5L6-4.0-2.00.02.04.0DMI, D [mJ/m2]W(0%)\nW(12.5%)\nW(25%)\nW(50%)\nFigure 3. (a) Change in SOC energy at the interfacial HM\nlayers (L2 & L5) as a result of changing spin chirality of the\nmagnetic layers (L3 & L4) from CW to ACW. All the color\nbars on the left (right) side represents the SOC energy change\nat L2 (L5) for di\u000berent W compositions. (b) Layer resolved\nDMI for structures having W composition, x= 0%\u000050%.\nnetic layer) for all W compositions (0% \u0000100%). We \fnd\nthat \u0001 ESOC in L5 changes drastically as W composition\nchanges from 0% to 12 :5%, slowing down thereafter. On\nthe other hand, distributions of \u0001 ESOC in L2 are not\nvery sensitive to the W composition. Although we \fnd\na relatively lower \u0001 ESOC at L2 for 75% and 100% W\ncompositions, the corresponding \u0001 ESOCs at L5 are pos-\nitive. In trilayer structures, the DMIs of the bottom and\ntop interface are additive [12, 37], so that the sum arising\nfrom L2, and L5 accounts for the observed non-monotonic\nchange of DMI in Fig. 2. From our \fndings, we conjec-\nture that the inversion symmetry breaking plays a vital\nrole on the DMI while the e\u000bect of W composition is\nnot that prominent, in agreement with the recent exper-\niment [27].\nTo corroborate our analysis, we calculate the layer re-\nsolved DMI. Figure 3(b) shows the layer resolved contri-\nbution of the DMI for the structures with 0% \u000050% W\ncomposition. The results show that the DMI comes only\nfrom the interfacial magnetic layers. We can see that\nthe change in the DMI contribution from the top inter-\nfacial layer (L4) with increasing W is small, generating\n(a)\nTa W Ir0.01.02.03.04.05.0DMI, D [mJ/m2]Pt/GdCo/Ta\nPt/GdCo/W\nPt/GdCo/Ir\n(b)\nL1L2L3L4L5L6-1.00.01.02.03.04.05.0DMI, D [mJ/m2]Pt/GdCo/Ta\nPt/GdCo/W\nPt/GdCo/IrFigure 4. (a) Calculated DMI in Pt/GdCo/X, where X=Ta,\nW, Ir. (b) Layer resolved DMI.\na similar trend as \u0001 ESOC shown in Fig. 3(a). Addition-\nally, the contribution from the bottom interfacial layer\n(L3) remains almost the same throughout the range of\nW compositions. The addition of the DMI from the bot-\ntom and the top interfaces produces a saturation in the\noverall DMI curve.\nFinally, our theoretical model allows us to explore\nthe tuning of DMI in ferrimagnetic systems with dif-\nferent cap layer compositions, which could be critical\nin designing suitable materials for hosting ultrasmall\nhigh-speed skyrmions. Furthermore, for applications,\nskyrmions can be driven by current-induced spin-orbit\ntorques (SOT) [38]. Changing the cap layer HM o\u000bers\nthe ability to tune the SOT e\u000eciency and DMI simul-\ntaneously. We report the DMI of Pt/GdCo/X where\nX = Ta ;W;Ir, to demonstrate the e\u000bect of cap layer\n5dtransition HM on the DMI in Fig. 4(a). W and Ta\nare known for their giant spin-Hall angle [39, 40], and\nprevious studies have shown an additive DMI for a fer-\nromagnet sandwiched between Pt and Ir [37, 41], which\nguides us to explore these structures and see which one\nof them has the largest DMI. We \fnd that W in the\ncap layer favors higher DMI than Ta and Ir. To explain\nthe DMI trend, we calculate the layer resolved DMI con-\ntribution from bottom and top interfaces, as shown in\nFig. 4(b). From Fig. 4(b), we can observe that the DMI5\n-12.00.012.0DOS\"\n#(a)Co\nTa\n-12.00.012.0DOS\"\n#(b)Co\nW\n-6 -4 -2 0 2 4\nEnergy, E-EF [eV]-12.00.012.0DOS\"\n# (c)Co\nIr\nFigure 5. Projected density of states (p-DOS) showing 3 d-\n5dband alignment between Co (black) and X (colored) in\nPt/GdCo/X. (a) X=Ta, (b) X=W, and (c) X= Ir. The red\nup (down) arrow represents the spin-up (spin-down) channel.\ncontribution from the top interface (L4) is large when Ir\nis used as a cap layer material while the DMI contribu-\ntions are smaller for the cases of W and Ta. The ob-\nserved trend of the DMI can be explained qualitatively\nby the Co 3 d-X 5dband alignment, which controls the\ncorresponding orbital hybridization. Figure 5 shows the\nprojected density of states (p-DOS) of Co-3 dand HM-5 d\norbitals. Clearly, in Co/Ir, the band alignment around\nthe Fermi level is higher than that of Co/W and Co/Ta,\nwhich in turn produce larger DMI contributions from L4\nfor Ir over W and Ta. The band alignment of Co/W and\nCo/Ta are close to each other. However, we note that\nthe sign of the DMI contribution from the top interfaceis di\u000berent for Ir than Ta and W. By analyzing the or-\nbital projected densities of states of the cap layer HM,\nwe \fnd that Ta and W behave in a similar way i.e, dxy\nanddx2\u0000y2have major contributions near the Fermi level\nwhile for Ir, the orbitals associated with the zcharacters,\nnamely dxz,dyz, and dz2are prominent, correlating with\nthe behavior shown in Fig. 4(b). Moreover, the variation\nof the DMI sign depending on the adjacent HM has pre-\nviously been seen in both theoretical and experimental\nstudies [16, 42]. Finally, adding the DMI contribution\nfrom both the interfaces (Fig. 4(b)) gives a smaller over-\nall DMI for Pt/GdCo/Ir because of the large negative\ncontribution from the top interface.\nConclusion. In summary, we demonstrate the impact of\nW composition in the cap layer of Pt =GdCo =Pt1\u0000xWx\ntrilayer structures using \frst principles calculations. We\n\fnd excellent tunability of the DMI that shows a ten-\ndency of saturation with increasing W composition. The\nsaturating trend of the DMI is attributed to the change\nof SOC energy at the top and the bottom intefacial HM\nlayers as a function of W composition. Moreover, we\n\fnd DMI sensitivity to the structural variation. We also\ndemonstrate the DMI variation in Pt/GdCo/(Ta, W or\nIr). We \fnd W in the cap layer provides a higher DMI\nthan Ta and Ir, due to the varying degree of orbital hy-\nbridization controlled by the band alignment between 3 d-\n5dorbitals at the cap layer interface. Our results provide\ncritical insights to the control mechanism of DMI in ferri-\nmagnetic GdCo based systems, providing a path towards\nmanipulating skyrmion properties for spintronic applica-\ntions.\nAcknowledgments. We thank Shruba Gangopadhyay,\nJianhua Ma, Hamed Vakilitaleghani, and S. Joseph\nPoon for insightful discussions. This work is funded\nby the DARPA Topological Excitations in Electronics\n(TEE) program (grant D18AP00009). The calculations\nare done using the computational resources from High-\nPerformance Computing systems at the University of\nVirginia (Rivanna) and XSEDE.\n[1] A. Fert, N. Reyren, and V. Cros, Nat. Rev. Mater. 2, 1\n(2017).\n[2] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8,\n152 (2013).\n[3] W. Koshibae, Y. Kaneko, J. Iwasaki, M. Kawasaki,\nY. Tokura, and N. Nagaosa, Jpn. J. Appl. Phys. 54,\n053001 (2015).\n[4] X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 1 (2015).\n[5] H. Vakili, Y. Xie, and A. W. Ghosh, Phys. Rev. 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Lett. 120, 157204 (2018)."    },    {        "title": "1905.02117v2.Ground_State_Phases_of_Distorted__S_1__Diamond_Chains.pdf",        "content": "arXiv:1905.02117v2  [cond-mat.str-el]  24 Jun 2019Journal of the Physical Society of Japan FULL PAPERS\nGround State Phases of Distorted S= 1 Diamond Chains\nKazuo Hida∗\nProfessor Emeritus, Division of Material Science, Graduat e School of Science and Engineering,\nSaitama University, Saitama, Saitama, 338-8570\n(Received June 26, 2019)\nThe ground states of distorted S= 1 diamond chains are investigated for two types of distorti on called\ntype A and B [J. Phys. Soc. Jpn. 79 (2010) 114703]. For the type A distortion, Haldane phases with and\nwithout spontaneous translational symmetry breakdown are present for large values of parameter λthat\nparametrize the strength of frustration. For small λ, the Haldane phase and two quantized ferrimagnetic\nphases in the undistorted chain remain stable even for stron g distortion. In contrast, for the type B distor-\ntion, the quantized ferrimagnetic phases with and without s pontaneous translational symmetry breakdown\nare present for large λ. The partial ferrimagnetic phases emerge between them. For smallλ, two quantized\nferrimagnetic phases remain and the partial ferrimagnetic phases also emerge between them. The Haldane\nphase between the two kinds of ferrimagnetic phases turns in to a topologically trivial double Haldane phase\nfor strong distortion.\n1. Introduction\nExotic ground state phases of low-dimensional quan-\ntum frustrated spin systems have been attracting broad\ninterest in recent condensed matter physics from exper-\nimental and theoretical viewpoints.1,2)To understand\nthe nature of these phases theoretically, the frustrated\nspin models with exact ground states are helpful starting\npoints. The Majumdar-Ghosh model3)and the Shastry-\nSutherland model4)are examples of such models that\nhave ground states consisting of singlet dimers.\nThe diamond chain discussed in the present work is\nanother frustrated spin chain with exact ground states.\nThe lattice structure is shown in Fig. 1. In a unit cell,\ntherearetwokindsofnonequivalentlatticesitesoccupied\nby spins with magnitudes Sandτ; we denote the set\nof magnitudes by ( S,τ) where spins with magnitude S\nare on the vertex sites and those with magnitude τare\non the apical sites. Two τ-spins are connected by the\nvertical bond with strength λ. The features common to\nall types of diamond chains are their infinite number of\nlocalconservationlawsandmorethantwodifferenttypes\nof exact ground states that are realized depending on the\nstrength of frustration.\nTakanoand coworkers5,6)introducedthis lattice struc-\nture and generallyinvestigatedthe case of( S,S). Partic-\nularly, in the case of (1/2, 1/2), they determined the full\nphase diagramofthe ground state by combiningrigorous\narguments with numerical calculations.\nThe ground states of spin-1 diamond chains (S1DC)\nwith (S,τ) = (1,1) are further studied by Hida and\nTakano.7)In the strongly frustrated regime, the ground\nstate of the S1DC is same as that of the mixed dia-\nmondchainwith( S,τ) = (1,1/2).8)Threedifferentpara-\n∗E-mail address: hida@mail.saitama-u.ac.jpSτ(1)\nτ(2)1 1\n11λ\nFig. 1. Structure of the diamond chain. Spin magnitudes in a\nunit cell are indicated by S.τ(1)andτ(2); we denote the set of\nmagnitudes by ( S,τ) where τ(1)=τ(2)=τ. We consider the case\nS=τ= 1 in the present paper.\nmagnetic phases accompanied by spontaneous transla-\ntional symmetry breakdown (STSB) and one paramag-\nnetic phase without STSB are found in this regime. This\nmodel also has a nonmagnetic Haldane phase and ferri-\nmagnetic phases with and without STSB in a less frus-\ntrated regime.7)\nIn the present paper, we study the effect of distortion\non the ground states of S1DC. We investigate the distor-\ntion patterns depicted in Figs. 2(a) and 2(b). Following\nRef. 9, we call the distortion patterns in Fig. 2(a) and\nFig.2(b)astype AandtypeB,respectively.Asdiscussed\nin Ref. 9, these types of distortion break the local conser-\nvation laws that hold in the undistorted S1DC. As a re-\nsult, they induce effective interactions between the clus-\nter spins, and form novel exotic phases such as Haldane\nphases with STSB, ferrimagnetic phases with STSB, and\npartial ferrimagnetic (PF) phase that can be regarded as\nspontaneously magnetized Luttinger liquid.10,11)In ad-\ndition, a double Haldane phase12–16)is found for strong\ntype B distortion. It should be remarked that the S1DC\nwith type A distortion is realized in real material,17,18)\n1J. Phys. Soc. Jpn. FULL PAPERS\nSτ(1)\nτ(2)1−δA\n1+δAλ1+δA\n1−δA(a)\nSτ(1)\nτ(2)1+δB\n1−δBλ1+δB\n1−δB(b)\nFig. 2. Structures of S1DC with (a) type A and (b) type B dis-\ntortions.\nalthoughthe exchangeparametersofthe materialcannot\nbe controlled freely to realize the most exotic Haldane\nphases with STSB.\nThis paper is organized as follows. In §2, the Hamilto-\nnians for the S1DCs with type A and type B distortions\nare presented, and the structure of the ground states of\ntheS1DCwithoutdistortionissummarized.Theground-\nstate phases for the S1DC with type A distortion are\ndiscussed in §3, and those for the S1DC with type B dis-\ntortion are discussed in §4. The last section is devoted to\nsummary and discussion.\n2. Hamiltonian\nThe S1DCs with type A and type B distortions are\ndescribed, respectively, by the following Hamiltonians:\nHA=N/summationdisplay\nl=1/bracketleftBig\n(1+δA)Slτ(1)\nl+(1−δA)τ(1)\nlSl+1\n+(1−δA)Slτ(2)\nl+(1+δA)τ(2)\nlSl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,\n(2.1)\nHB=N/summationdisplay\nl=1/bracketleftBig\n(1+δB)Slτ(1)\nl+(1+δB)τ(1)\nlSl+1\n+(1−δB)Slτ(2)\nl+(1−δB)τ(2)\nlSl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,\n(2.2)\nwhereSl,τ(1)\nlandτ(2)\nlarethespin-1operatorsinthe lth\nunit cell. The parameter δA(δB) represents the strength\nof type A (type B) distortion, and is taken to be non-\nnegative without spoiling generality. The number of unit\ncells is denoted by N, and then the total number of sites\nNsis 3N.\nForδA= 0andδB=0, botheqs. (2.1)and(2.2)reduceto the Hamiltonian of the undistorted S1DC as,\nH0=N/summationdisplay\nl=1/bracketleftbigg\nSlTl+TlSl+1+λ\n2/parenleftbigg\nT2\nl−3\n2/parenrightbigg/bracketrightbigg\n,(2.3)\nwhere the composite spin operator Tlis defined by Tl≡\nτ(1)\nl+τ(2)\nl. Before goinginto the analysisof the distorted\nS1DC, we briefly summarize the ground-state properties\nof the Hamiltonian (2.3) reported in Refs. 6 and 7 for\nconvenience.\n(i)T2\nlcommutes with the Hamiltonian H0for anyl.\nTherefore, the composite spin magnitude Tldefined\nbyT2\nl=Tl(Tl+1) is a good quantum number that\ntakes the values 0, 1, and 2. Hence, each energy\neigenstate has a definite set of {Tl}, i.e. a sequence\nof 0’s, 1’s and 2’s with length N. A pair of τ(1)\nland\nτ(2)\nlwithTl= 0 is called a dimer. A cluster includ-\ningnsuccessive pairs with Tl/ne}ationslash= 0 bounded by a pair\nof dimers is called a cluster- n.\n(ii) Forλ >2,Tl= 2 is not allowed in the ground state.\n(iii) There are 4 distinct paramagnetic ground-state\nphases called dimer-cluster- n(DCn) phases with\nn= 0,1,2,3. The DC nstate is an alternating\narray of dimers and cluster- n’s. There are also 3\nphases with a single infinite cluster that correspond\nto a Haldane phase (HDC ∞phase:∀l Tl= 1),\na ferrimagnetic phase with m= 1/6 (F1/6phase;\n∀l(T2l,T2l+1) = (1,2) or (2,1), andm= 1/3 (F1/3\nphase;∀l Tl= 2), where m=M/Nsis the sponta-\nneous magnetization Mper site. The phase bound-\naryλc(n,n′) between DC nand DCn′phases are\ngiven by\nλc(0,1) = 3, (2.4)\nλc(1,2)≃2.6604, (2.5)\nλc(2,3)≃2.5827. (2.6)\nThe DC3-Haldane, Haldane-F 1/6, and F 1/6-\nF1/3phase boundaries are denoted by λc(3,H),\nλc(H,F1/6) andλc(F1/6,F1/3), respectively. They\nare given by\nλc(3,H)≃2.5773, (2.7)\nλc(H,F1/6)≃1.0727, (2.8)\nλc(F1/6,F1/3)≃1.0182. (2.9)\n(iv) The DC nphases with 0 ≤n≤3 are realized for\nλ > λ c(3,H). Since λc(3,H)>2, we have Tl= 1\nwithin each cluster- n. This implies that a cluster- n\nis equivalent to the ground state of an antiferromag-\nnetic spin-1 Heisenberg chain of length 2 n+1 with\nopen boundary condition.\n(v) In the DC nphase with 1 ≤n≤3, (n+ 1)-fold\nSTSB takes place. In the F 1/6phase, two-fold STSB\n2J. Phys. Soc. Jpn. FULL PAPERS\n(a)\n(b)\n(c)\n(d)\n(e)\nFig. 3. Valence bond structures ofthe nonmagnetic ground state\nphases of S1DC with type A distortion for (a) HDC0 (uniform Ha l-\ndane), (b) HDC1, (c)HDC2, (d) HDC3, and (e) HDC ∞(uniform\nHaldane) phases. The big open circles indicate the spin-1 si tes. One\nspin-1 site consists of two spin-1/2’s depicted by the filled small cir-\ncles that are symmetrized within each open circle. Each gray oval\nencircles a singlet pair of two spin-1/2’s. The thick solid, thin solid,\nand dotted lines are the bonds with strength 1 + δA, 1−δA, and\nλ, respectively.\ntakes place. In the DC0, HDC ∞, and F 1/3phases,\nno translational symmetry is broken.\nIn what follows, we examine the effects of the type A and\ntype B distortions on the ground state of S1DC analyt-\nically and numerically. Because the DC3 phase is only\nrealized within a very narrow interval of λ, numerical\nanalysesare difficult in this phase. Hence, we do not con-\nsider the DC3 phase in the following numerical analyses.\n3. Ground-State Properties of the S1DC with\nType A Distortion\n3.1 Weak distortion regime (δA≃0)\nForλ >2, only the states with Tl= 0 and 1 are al-\nlowed. Hence, the argument proceeds in the same way\nas the case of ( S,τ) = (1,1/2).9)For the type A distor-\ntion, the total spins of the cluster- n’s on both sides of\na dimer tend to be antiparallel to each other. Namely,\nthe effective coupling between the spins of neighboring\ncluster-n’s is antiferromagnetic. As in Ref. 9, we have es-\ntimated the ratioofthe effective bilinear and biquadratic\ninteractionsbetween the spins ofcluster- nand confirmed\nthat the ground state of the whole chain is a Haldane\nstate. We call this state the Haldane DC n(HDCn) state.\nThe valence bond structures for the HDC nphases with\nn= 0,1,2,3 and∞are shown in Fig. 3. In the HDC n\nstate with 1 ≤n≤3, the (n+ 1)-fold translationalsymmetry is spontaneously broken unlike the conven-\ntional Haldane state. On the other hand, the Haldane\nstate in the undistorted S1DC is robust against distor-\ntion, since this ground state has an energy gap. Both\nthe HDC0 state for λ > λ c(0,1) and the Haldane state\nforλc(H,F1/6)< λ < λ c(3,H) are the Haldane states\nwithout STSB. The ferrimagnetic F 1/6and F1/3phases\nwith finite longrange ordersarealso robust againstweak\ndistortion.\n3.2 Strong distortion regime (δA≃1)\nForδA= 1 and λ= 0, the three spins τ(2)\nl−1,Sl, and\nτ(1)\nlform a three spin cluster. The cluster Hamiltonian\nis given by\nH3= 2J/bracketleftBig\nτ(2)\nl−1Sl+Slτ(1)\nl/bracketrightBig\n(3.1)\nThis can be regarded as a spin-1 antiferromagnetic\nHeisenberg chain with length 3. According to the\nMarshal-Lieb-Mattis theorem, the total spin of the\nground state of this Hamiltonian is unity. Hence, each\ncluster carries an effective spin ˜Sl(=τ(2)\nl−1+Sl+τ(1)\nl)\nwith magnitude 1. The three spin ground state/vextendsingle/vextendsingle/vextendsingleG;˜Sz\nl/angbracketrightBig\nwith˜Sz\nl= 1 is expressed using the basis/vextendsingle/vextendsingle/vextendsingleτ(2)z\nl−1Sz\nlτ(1)z\nl/angbracketrightBig\nas\n|G;1/an}bracketri}ht=/radicalbigg\n3\n20/bracketleftbigg\n−1\n3(|11¯1/an}bracketri}ht+|¯111/an}bracketri}ht)\n−2|1¯11/an}bracketri}ht+(|001/an}bracketri}ht+|100/an}bracketri}ht)−2\n3|010/an}bracketri}ht/bracketrightbigg\n(3.2)\nwhere¯1 stands for −1. The eigenstates |G;0/an}bracketri}htand|G;¯1/an}bracketri}ht\nare obtained by applying the descending operator on\n|G;1/an}bracketri}ht. Up to the first order in λand 1−δA, the ef-\nfective interaction between ˜Sls can be described by the\nHamiltonian\nHA\neff=N/summationdisplay\nl=1JA\neff˜Sl˜Sl+1, (3.3)\nwhere\nJA\neff=−3\n4(1−δA)+9\n16λ. (3.4)\nHence, the ground state is the Haldane state consisting\nof˜Sls forλ >4\n3(1−δA), while it is a ferrimagnetic state\nwithm= 1/3 (ferromagnetic in terms of ˜Sl) forλ <\n4\n3(1−δA). This is consistent with the numerical phase\ndiagram discussed in the next section around ( λ,δ) =\n(0,1) as shown by the dotted line in Fig. 4.\nThe Haldane ground state for λ >4\n3(1−δA) is\nnot accompanied by STSB. This nature is common to\nthe HDC0 and HDC ∞phases in the weak distortion\nlimit. Furthermore, the HDC ∞state is transformed into\nthe HDC0 state only by rearranging two valence bonds\nwithin each diamond unit, as can be seen from Figs. 3(a)\n3J. Phys. Soc. Jpn. FULL PAPERS\n−1 0 1 2 3012\nδA\nNo STSBUH\nF1/3 F1/6\nλHDC1HDC2\nHDC3\nFig. 4. Phase diagram of the S1DC with type A distortion. The\ndotted line is the phase boundary λ=4\n3(1−δA)obtained by the\nstrong distortion approximation in sect. 3.2. Enlarged figu res for\nHDC phases and F 1/6phase are shown in Figs. 5 and 6, respec-\ntively.\n2.6 2.8 300.010.020.03\nδA\nλλc(2,3)λc(3,H)2−fold STSB3−fold STSBNo STSB\nλc(1,2) λc(0,1)HDC1HDC2UH\nFig. 5. Enlarged phase diagram of the S1DC with type A dis-\ntortion for strongly frustrated regime (large λ) determined by the\nDMRG method. The triangles indicate the position of the phas e\nboundary for δA= 0.\nand 3(e). Also, considering that the Hamiltonian (3.1)\ncan be regarded as a spin-1 chain with length 3, the\nground state of the 3-spin cluster has the valence bond\nsolid structure.19,20)Thus the valence bond structure of\nthe Haldane phase consisting of ˜Sls is just as depicted in\nFig. 3(e). Therefore, the Haldane state consisting of ˜Sls,\nHDC0, and HDC ∞should be regarded as different parts\nof a single phase. The continuity of the three regimes will\nbe confirmed by the numerical analysis discussed in §3.3.\nIn what follows, we call this phase the uniform Haldane\n(UH) phase as a whole.\nThe ferrimagnetic ground state for λ <4\n3(1−δA) has\nm= 1/3. This phase is connected to the F 1/3phase in\nthe absence of distortion.0.8 0.9 1 1.100.20.4\nδA\nNo STSBUH\nF1/3\nF1/6\nλNs=18\nFig. 6. Enlarged phase diagram of the S1DC with type A dis-\ntortion for weakly frustrated regime (small λ) determined by nu-\nmerical diagonalization with Ns= 18.\n1 1.02 1.0400.20.4\nm\nδA=0.1 Ns=18\nλ\nFig. 7. λ-dependence of the spontaneous magnetization for δA=\n0.1 withNs= 18.\n3.3 Numerical phase diagram\nFollowing Ref. 9, we employ the DMRG calculation\nwiththeopenboundaryconditiontodeterminethephase\ndiagram for finite δAin the region λ > λ c(3,H). In the\nDMRG calculation, the number of states χkept in each\nsubsystem ranged from 200 to 400, and the system size\nNsfrom 60 to 288. From the valence bond structures for\nthe HDC nphases in Fig. 3, the HDC nground state has\ntranslationalinvarianceofperiod n+1.Hence,the( n+1)-\nfold STSB takes place at the HDC n-UH phase boundary.\nAs in Ref. 9, we carried out the size extrapolation of\nthe order parameter assuming the 2-dimensional ( n+1)-\nclock model universality class. The results are shown in\nthe phase diagrams of Figs. 4 and 5. The error bars are\nwithin the size of the symbols.\nIn the weakly frustrated and unfrustrated regime λ <\nλc(H,F1/6), we have carried out the exact diagonaliza-\ntion for the system sizes up to Ns= 18 to obtain the\nphasediagramsofFigs.4and6.Thespontaneousmagne-\ntization obtained by numerical diagonalization is shown\nin Fig. 7. We find no evidence supporting the presence\nof PF ground states in the thermodynamic limit within\nthe numerical accuracy.\n4J. Phys. Soc. Jpn. FULL PAPERS\n4. Ground-State Properties of the S1DC with\nType B Distortion\n4.1 Weak distortion regime\nIn the case of the type B distortion, the effective\ninteraction between the spins of two cluster- n’s sepa-\nrated by a dimer is ferromagnetic for small δBas dis-\ncussed in Ref. 9. Therefore, we expect the ferrimagnetic\nground state with spontaneous magnetization quantized\nasm= 1/(3(n+ 1)) per site for small δBin the range\nλc(n,n+1)< λ < λ c(n−1,n). Following Ref. 9, we call\nthis phase a ferrimagnetic DC nphase (FDC nphase).\nIn contrast, the ground state for λc(H,F1/6)< λ <\nλc(3,H) remains in the Haldane phase, since a nonmag-\nnetic gapped phase without STSB is generally robust\nagainst weak distortions. This phase is a symmetry pro-\ntected topological phase with half-integer edge spins.\n4.2 Strong distortion regime (δB≃1)\nForδB= 1andλ= 0, the whole system is decomposed\ninto two parts. One is a single spin-1 chain of length\n2Nconsisting of Slandτ(1)\nlwith exchange constant 2 J\ndescribed by the Hamiltonian\nH0=2N/summationdisplay\ni=12Jσlσl+1 (4.1)\nwhereσ2l=τ(1)\nlandσ2l+1=Sl. The ground state of\nthe Hamiltonian (4.1) is the nonmagnetic Haldane state\n|H/an}bracketri}htwith energy EH. The remaining part is Nisolated\nspinsτ(2)\nl.\nFor small 1 −δBandλ, the spins τ(2)\nlinteract with\neachothermediated by the fluctuation in the chain(4.1).\nSince the correlation within the Haldane chain (4.1) is\nshort ranged, we consider only the nearest-neighbour ef-\nfective coupling JB\neffbetween τ(2)\nlandτ(2)\nl+1. Then,JB\neff\ncan be estimated by the second order perturbation cal-\nculation as\nHB\neff=N/summationdisplay\ni=1JB\neffτ(2)\nlτ(2)\nl+1(4.2)\nJB\neff= 2(F(0)+2F(2)+F(4))(1−δB)2\n+4(F(1)+F(3))(1−δB)λ\n+2F(2)λ2(4.3)\nwhereF(l) is defined by\nF(l) =−/summationdisplay\nα/an}bracketle{tH|σz\ni|α/an}bracketri}ht/an}bracketle{tα|σz\ni+l|H/an}bracketri}ht\nEα−EH(4.4)\nHere,|α/an}bracketri}htandEαare the eigenstate and eigenenergy of\ntheαth excited state of H0. We estimated the values\nofF(l) (l= 1,...,4) for the finite length spin-1 chain\n(4.1) with 2 N= 8,10 and 12. After the extrapolation to−4 −2 0 2012\nM=0F1/3\nλδB Ns=18 M=0\nF1/6,PFF1/3(FDC0)\nF1/6(FDC1)\n     PFF1/6, PFF1/6,PF\nFig. 8. Phase diagram of the S1DC with type B distortion with\nNs= 18. The dotted lines are the boundaries determined by Eq.\n(4.5)\nN→ ∞, we find that JB\neffis positive for\n0.594/greaterorsimilar1−δB\nλ/greaterorsimilar0.416 (4.5)\nand negative otherwise. Hence, the ground state of the\nchain consisting of spins τ(2)\nlis a nonmagnetic Haldane\nstate in the region (4.5) and a ferrimagnetic state with\ntotal magnetization M=Notherwise. As a whole dia-\nmond chain, the former corresponds to the double Hal-\ndane phase12–16)and the latter to the ferromagnetic\nphase with m= 1/3. The double Haldane phase con-\nsists of two coupled chains with Haldane ground state.\nIn contrast to the previously studied examples of double\nHaldane phases that consist of two Haldane chains with\nequal length, the length of one Haldane chain H0is two\ntimes larger than the other one HB\neffin the present case.\nNevertheless, this ground state is topologically trivial,\nsince the the edge spins of two Haldane chains can lo-\ncally cancel out. The phase boundary (4.5) is consistent\nwith the numerical phase diagram presented in the next\nsection around ( λ,δB) = (0,1).\n4.3 Numerical phase diagram\nFor finite δB, we determined the ground-state phase\ndiagram by the numerical diagonalization for the sys-\ntem size Ns= 18, as shown in Fig. 8. Among system\nsizestractablebynumericaldiagonalization,onlythesize\nNs= 18 is compatible with all the ground-state struc-\ntures with n= 0,1, and 2. As expected, the FDC nquan-\ntized ferrimagnetic phases with m= 1/(3(n+ 1)) are\nfound for these values of n.\nBy inspecting numerical data for Ns= 18, we also\nfind narrow steps where the spontaneous magnetization\ndoes not satisfy m= 1/(3(n+1)) for any integer nbe-\n5J. Phys. Soc. Jpn. FULL PAPERS\n1 1.500.20.4\nδB=0.4 m\nλ\nFig. 9. Spontaneous magnetization for δB= 0.4. The exact diag-\nonalization results for Ns= 18 with periodic boundary condition\nare shown by thick solid lines and DMRG results for Ns= 72\nwith open boundary condition are shown by the open squares. T he\ndotted lines indicate the values of the spontaneous magneti zation\nm= 1/3, 1/6 and 1/9 in the FDC nphases. The left and right\ntriangles indicate the positions of other steps for Ns= 18 that\nsuggest the possibility of PF phases.\ntween quantized ferrimagnetic phases as shown in Fig. 9\nby thick solid lines for δB= 0.4. The positions of these\nsteps are indicated by the left and right triangles. These\nsteps suggest the presence of PF phases. The ferrimag-\nnetic phase of this kind has been found in various frus-\ntrated one-dimensional quantum spin systems.10,11)The\nDMRG calculation for larger Nsalso supports the pres-\nence of PF phases in addition to the quantized ferrimag-\nnetic phase with m= 1/1/3 and 1/6 as shown in Fig.\n9 by open squares for δB= 0.4. The origin of these PF\nphases can be understood by the same argument as that\nfor the mixed diamond chain with ( S,τ) = (1,1/2).9)In\ncontrast to the case of type A distortion, PF phases are\nalsopresentfor λ < λc(H,F1/6).Thephysicalmechanism\nto stabilize the latter PF phases remains unresolved.\nAs discussed in the subsections 4.1 and 4.2, a Haldane\nphase, which is a symmetry protected topological state,\nand the trivial double Haldane phase are present in the\nnonmagnetic phase. To distinguish these two phases, we\ncarry out the finite size DMRG calculation and estimate\nthe energy gap with structures H and DH depicted in the\nFigs. 10(a) and (b), respectively. The number of states χ\nkept in each subsystem ranged from 480 to 840. For the\nstructure H, additional spins SLandSRwith magni-\ntude 1/2 are added to both ends with exchange coupling\nJad(SLS1+SRSN) to compensate the edge spins with\nmagnitude 1/2 at both ends of the open Haldane chain.\nThen, the energygap ∆ Hin the Haldanephase should be\nfiniteinthethermodynamiclimit.Onthecontrary,inthe\nstructureDH,theinteraction(1 −δB)S1τ(2)\n1+λτ(1)\nNτ(2)\nNis\nreplaced by Jad(S1τ(2)\n1+τ(1)\nNτ(2)\nN) so that the edge spins\nof two Haldane chains are coupled antiferromagnetically\nto form a nonmagnetic singlet pair on each end. Then,\nthe energy gap ∆ DHin the double Haldane phase should\nbe finite.\nThe numerical results for δB= 0.5, 0.52 and 0.6 areSL=1/2\nJad JadSR=1/2(a)\nJadJad(b)\nFig. 10. Structures (a) H and (b) DH used for the calculation of\nthe energy gaps ∆ Hand ∆ DHin the nonmagnetic phase, respec-\ntively.\nshown in Fig. 11(a), (b) and (c), respectively. We set the\ncoupling Jad= 1. In the nonmagnetic phase, the scaled\ngapsNs∆HandNs∆DHare shown for several values of\nNsby open symbols. Here, Nsis the number of spins in-\ncluding the additional spins. In the ferrimagnetic phase,\nthe spontaneous magnetization per spin mis plotted for\nNs= 72. The quantized ferrimagnetic phase is so narrow\nthat it is numerically undetectable in this regime.\nForδB= 0.5, the whole nonmagnetic phase belongs\nto the Haldane phase since the scaled gap Ns∆Hin-\ncreases with the system size as shown in Fig. 11(a). For\nδB= 0.52, the transition between the Haldane and dou-\nble Haldane phase takes place as shown in Fig. 11(b).\nUnfortunately, even in the region where the finite size\nenergy gap ∆ DHis finite, the scaled gap Ns∆DHis al-\nmost independent of the system size. This means that\nthe correlation length is larger than the length of the\nsystem employed in the numerical analysis. Considering\nthe continuity to larger values of δB, we expect this re-\ngion is in the double Haldane phase with large correla-\ntion length. Actually, for δB= 0.53, we clearly found\nthatNs∆DHincreases with the system size in the corre-\nsponding region. Nevertheless, we have chosen to present\nthe data for δB= 0.52, since ∆ Hin the Haldane phase is\nnumerically undetectably small for δB= 0.53.\nForδB= 0.6, the scaled gap Ns∆DHincreases with the\nsystem size as shown in Fig. 11(c) for 0 .69/lessorsimilarλ/lessorsimilar0.826.\nHence, this region clearly belongs to the double Haldane\nphase. The spontaneous magnetization mis clearly fi-\nnite forλ/greaterorsimilar0.86 andλ/lessorsimilar0.69. For 0 .826/lessorsimilarλ/lessorsimilar0.86,\nhowever, the ground state has magnetization M= 1 for\nNs= 72. It is not clear whether this spontaneous mag-\nnetization remains finite in the thermodynamic limit.\nHence, we cannot rule out the possibility of a new non-\nmagnetic phase in this region.\n5. Summary and Discussion\nWe investigated the ground-state phases of S1DC with\ntwo types of distortion, type A and type B. In the re-\ngion where the ground states of the undistorted S1DC\nare DCnstates with finite n, the effective interaction be-\ntween the cluster spins is antiferromagnetic for the type\nA distortion and ferromagnetic for the type B distor-\n6J. Phys. Soc. Jpn. FULL PAPERS\n0.8 1 1.2 1.400.20.4\n00.51\nm Ns∆H∆H Ns=48\n60\n721/3m\nλNs=72(a)\n0.8 1 1.200.20.4\n00.51\nm Ns∆H,DH\n∆DH Ns=48\n60\n721/3m\nλNs=72\n∆H Ns=60\n72DHH(b)\n0.6 0.8 100.20.4\n00.51\nm Ns∆DH∆DH Ns=48\n60\n721/3m\nλNs=72(c)\n1/3\nλ\nFig. 11. Spontaneous magnetization m, scaled energy gaps\nNs∆HandNs∆DHfor(a)δB= 0.5,(b)δB= 0.52 and (c) δB= 0.6.\nThe spontaneous magnetization for structure H (DH) is shown by\nfilled squares (circles).\ntion. Hence, for the type A distortion, the DC nground\nstates are transformed into the HDC nground states.\nThe nature of the HDC nphase is essentially the same as\nthat of the mixed diamond chain with ( S,τ) = (1,1/2).\nHence, we have determined the phase diagram in thesame way as in Ref. 9. For the type B distortion, the\nDCnground states are transformed into the ferrimag-\nnetic FDC nground states. In addition to the FDC n\nphases with quantized spontaneous magnetization m=\n1/(3(n+ 1)), the PF phases are also found numerically\nbetween the FDC nand FDC( n+1) phases.\nIf the ground state of the undistorted S1DC is the uni-\nform Haldane or ferrimagnetic F 1/3or F1/6phases, they\nare robust against a weak distortion. The quantized fer-\nrimagnetic phases, however, shift to the small λregime\nwith the increase of distortion. For the type B distortion,\nthe PF phases emerge between the quantized ferrimag-\nnetic phases, while we found no such evidence for the\ntype A distortion within our numerical calculation. The\nphysical origin of this difference is left for future studies.\nFor the type B distortion, a nonmagnetic region is\npresent in the intermediate frustration regime between\ntwo types of ferrimagnetic phases. In this region, two\ntopologically distinct phases are identified, namely the\nHaldane phase, which is a symmetry protected topolog-\nical phase, and the double Haldane phase, which is a\ntrivial phase. The latter consists of two coupled Haldane\nchains. In contrast to the previously known examples of\ndouble Haldane phases for frustrated spin-1 Heisenberg\nchains that consists of two Haldane chains with equal\nlength,12–16)the present double Haldane phase consists\nof Haldane chains with lengths Nand 2N. Further in-\nvestigationis required to elucidate the nature ofthis new\ntype of double Haldane phase. The possibility of a non-\nmagnetic phase different from these two phases is also\nsuggested. Within our numerical data, however, it is not\npossible to conclude whether this is an artifact of the\nfinite size calculation or remains in the thermodynamic\nlimit. The investigation of these states is left for future\nstudies.\nIn contrast to the distorted mixed diamond chain with\n(S,τ) = (1,1/2), which has no experimental counterpart\nso far, the S1DC with type A distortion is already real-\nized as experimentalmaterial.17,18)Although the ground\nstate of this material is ferrimagnetic, the realization of\nthe materials with other exotic ground states such as\nthe Haldane phases with STSB or the double Haldane\nphase would be hopefully within the scope of experimen-\ntal studies in the near future.\nThe author thanks K. Takano, K. Okunishi, and T.\nHikihara for valuable discussion and comments. The nu-\nmerical diagonalization program is based on the package\nTITPACK ver.2 coded by H. Nishimori. Part of the nu-\nmerical computation in this work has been carried out\nusing the facilities of the Supercomputer Center, Insti-\ntute for Solid State Physics, University of Tokyo, and\nthe Yukawa Institute Computer Facility, Kyoto Univer-\nsity. This works is supported by JSPS KAKENHI Grant\nNumber JP25400389.\n7J. Phys. Soc. Jpn. FULL PAPERS\n1)Introduction to Frustrated Magnetism: Materials, Experi-\nments, Theory , ed. C. Lacroix, P. Mendels, and F. Mila\n(Springer Series in Solid-State Sciences, Springer, Heide lberg,\n2011).\n2)Frustrated Spin Systems , ed. H. T. Diep, (World Scientific,\nSingapore, 2005), Chaps. 5 and 6.\n3) C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 101399\n(1969).\n4) B. S. Shastry and B. Sutherland, Physica B+C 1081069\n(1981).\n5) K. Takano, J. Phys. A: Math. Gen. 27L269 (1994).\n6) K. Takano, K. Kubo, and H. Sakamoto, J. Phys.: Condens.\nMatter8, 6405 (1996).\n7) K. Hida and K. Takano, J. Phys. Soc. Jpn. 86, 033707 (2017).\n8) K. Takano, H. Suzuki, and K. Hida, Phys. Rev. B 80, 104410\n(2009).\n9) K. Hida, K. Takano, and H. Suzuki, J. Phys. Soc. Jpn. 79,\n114703 (2010).\n10) S. C. Furuya and T. Giamarchi, Phys. Rev. B 89, 205131(2014).\n11) K.Sekiguchi and K.Hida, J.Phys.Soc.Jpn. 86, 084706 (2017)\nand references therein.\n12) T. Hikihara, M.Kaburagi, H. Kawamura and T. Tonegawa, J.\nPhys. Soc. Jpn. 69, 259 (2000).\n13) T. Hikihara, J. Phys. Soc. Jpn. 71, 319 (2002).\n14) A. Kolezhuk, R. Roth, and U. Schollw¨ ock, Phys. Rev. B 55,\n8928 (1997).\n15) A.Kolezhuk, R.Roth, and U.Schollw¨ ock, Phys.Rev.Lett .77,\n5142 (1996).\n16) A. K. Kolezhuk and U. Schollw¨ ock, Phys. Rev. B 65, 100401\n(2002).\n17) K. Kunieda, Master Thesis, University of Fukui (2016)[i n\nJapanese].\n18) H. Kikuchi, private communication.\n19) I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki, Commn.\nMath. Phys. 115, 477 (1988).\n20) I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki, Phys. Rev.\nLett.59, 799 (1987).\n8"    },    {        "title": "2303.14809v1.Strong_lateral_exchange_coupling_and_current_induced_switching_in_single_layer_ferrimagnetic_films_with_patterned_compensation_temperature.pdf",        "content": "* zhaochu.luo@pku.edu.cn  \n**pietro.gambardella@mat.ethz.ch  \n***ales.hrabec@psi.ch  Strong  lateral exchange coupling and current -induced \nswitching in single -layer ferrimagnet ic films  with patterned \ncompensation temperature  \nZhentao Liu1,2, Zhaochu Luo1,2,3,4,*, Ivan Shorubalko5, Christof Vockenhuber6, Laura J. Heyderman1,2, \nPietro Gambardella7,**, Aleš Hrabec1,2,7 ,*** \n1Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland  \n2Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  \n3State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, 100871 Beijing, People’s \nRepublic of China  \n4Beijing Key Laboratory for Magnetoelectric Materials and Devices, 100871 Beijing, People’s Republic of \nChina  \n5Transport at Nanoscale Interfaces Laboratory, Empa - Swiss Federal Laboratories for Materials Science and \nTechnology, 8600 Dübendorf, Switzerland  \n6Laboratory of Ion  Beam Physics, ETH Zürich, 8093  Zürich, Switzerland  \n7Laboratory for Magnetism and Interface Physics, Department of Materials, ETH Zurich, 8093 Zurich, \nSwitzerland  \nStrong, adjustable magnetic couplings are of great importance  to all devices based on \nmagneti c materials . Controlling the coupling between adjacent regions of a single magnetic \nlayer, however, is challenging. In this work, we demonstrate strong exchange -based \ncoupling between arbitrarily shaped regions of a single ferrimagnetic layer. This is achi eved \nby spatially patterning the compensation temperature of the ferrimagnet by either \noxidation or He+ irradiation. The coupling originates at the lateral interface between regions \nwith different compensation temperature and scales inversely with their width. We show \nthat this coupling generates large lateral exchange coupling  fields  and we demonstrate its \napplication to control the switching of magnetically compensated dots with an electric \ncurrent.  \nIn spintronic architectures based on magnetic mu ltilayers [1-3], interlayer couplings  such as  \nthe Ruderman –Kittel –Kasuya –Yosida  interaction [4,5] , exchange  coupling leading to \nexchange  bias [6,7] , and the dipolar interaction [8-10] allow for the tuning  of the magnetic \nstability and electrical properties of the device [1-3]. The exchange interaction, composed of \nsymmetric and antisymmetric parts, provides the strongest coupling channel  in magnetic \nsystems . The symmetric part favors collinear magnetic conf igurations and is commonly \nexploited in multilayers to provide direct exchange coupling between,  for example , two \nferromagnets [11] or a ferromagnet and an antiferromagnet  [6], and indirect coupling \nbetween two ferromagnets separated by a nonmagnetic space r [4,5] . The antisymmetri c part, 2 \n known as  the Dzyaloshinskii –Moriya interaction (DMI) , favors non -collinear magnetic textures, \nbut is generally indirect and weaker in multilayer systems [12-14]. Controlling the coupling \nbetween adjacent regions of a single magnetic layer is more challenging [15-17]. In single \nlayers, the interfacial DMI provides a means to couple  planar structures [16,18] , which has \nenabled the realization of electrically -controlled magnetic logic devices [19-23]. However, the \nstrength of this coupl ing is limited by the interface  properties [24]. The stronger collinear \nexchange coupling in magnetic multilayers thus lacks a counterpart in the lateral  direction.   \nIn this work, we realize a strong lateral coupling based on the exchange interaction  in a single \nmagnetic layer . We take inspiration from an approach previously developed for synthetic \nferrimagnetic systems consisting of stacked layers of rare-earth transition -metal ferrimagnets  \nwith different magnetic compensation temperature ( 𝑇M) [25-31]. This type of multilayer is \nalso known as a n exchange  spring magnet and includes  a compensation wall. We further \ndevelop this method , apply ing it to a single -layer ferrimagnetic alloy.  In such an alloy , the \nstrong intra -lattice coupling between the transition -metal atoms and the weaker inter -lattice \ncoupling between the rare-earth and transition metal atoms can be separately tuned by  \naltering the  composition [32] or microstructure [33], and  through  reduction/oxidation \nreactions [34,35] . Recently, He+ irradiation has been used to modify 𝑇M in a Co/Tb multilayer \nin order to  create  multidomain configurations with dimensions of several m [30,36] . Here \nwe show that patterning of 𝑇M in a single GdCo film by either selective oxidation or He+ \nirradiation  leads to strong lateral exchange coupling  between regions with different  𝑇M. We \nshow how the coupling varies as a function of temperature and width of the pa tterned regions . \nWe discuss the strength of the coupling and compare the exchange interaction in  our planar \nstructure s with that found in multilayer systems. We further combine spin -orbit torques [3] \nand lateral coupling in a Pt|GdCo bilayer to demonstrate selective current -induced switching \nof adjacent magnetic domains , which results in reproduc ible manipulation of lateral exchange \nbias. \nOur Ta(1  nm)|Pt(5  nm)| Gd 0.3Co0.7(x nm)|Ta(2  nm) multilayers possess out-of-plane (OOP) \nmagnetization  with x in the range  from 3.2 to 6.2  nm, where 𝑇M can be tuned by changing the \nstoichiometric ratio or the thickness of the GdCo layer  as shown in the Supplemental Material \n[37] (see, also, reference  [38] therein) . To spatially modify  𝑇M, we use an oxygen plasma to \npartially oxidize the magnetic film  in specific regions . The effect of oxidation is verified by \nRutherford Backscattering  technique [37]. In particular , a GdCo  film with 𝑇M=𝑇c2 above \nroom temperature (RT) can be oxidized in order to lower the compensation point below room \ntemperature  (𝑇c1). Magneto -optic Kerr effect (MOKE) measurements, which are \npredominantly sensitive to the magnetization of the Co sublattice, show a reversal in the  \nhysteresis loop after oxidation of a GdCo film  [37], indicating  that  𝑇M is suppressed below \nroom temperature after  the oxidation .  \nFor a device containing  two regions with different 𝑇M (𝑇c2>𝑇c1), one can distinguish \nbetween three temperature scenarios  illustrated in Fig. 1(a-c): (i) the temperature is higher 3 \n than both compensation temperatures (𝑇>𝑇c2), (ii) t he t emperature is lower than both \ncompensation temperatures  (𝑇<𝑇c1), and (iii)  the t emperature is lower than the \ncompensation temperature  of the original film but higher than the compensation \ntemperature  of the partially oxidized film  (𝑇c2>𝑇>𝑇c1). \nIn the temperature scenarios with 𝑇>𝑇c2 and 𝑇<𝑇c1 [Fig. 1(a) and (b)] , all of the Co \nmagnetic moments are parallel to each other and all of the Gd moments are antiparallel to \nthe Co moments, m inimizing the exchange and magnetic anisotropy energy. The net \nmagnetization is then given by the sum  of the two sublat tice magnetizations,  𝑴𝐧𝐞𝐭=𝑴Co+\n𝑴Gd. At a temperature 𝑇>𝑇c2 ( 𝑇<𝑇c1), 𝑴𝐧𝐞𝐭 is parallel to 𝑴Co(𝑴Gd) in both the pristine \nand oxidized regions of the film [Fig. 1(a,b)]. Because the neighboring Co moments  are \nstrongly exchange -coupled and prefer to maintain a parallel alignment , not only within the \ntwo different regions but  also across the interface between them , the net magnetization in \nthe two regions  is effectively (trivially ) ferromagnetically coupled . \nIn the temperature range 𝑇c2>𝑇>𝑇c1 , however, 𝑴𝐧𝐞𝐭 is parallel to  𝑴Co in the region with \n𝑇c1 but parallel to 𝑴Gd in the region with  𝑇c2. Hence, the low -energy configuration that \nminimizes the exchange energy between the Co moments across the oxidation interface is an \nantiparallel state of the net magnetization [Fig.  1(c)]. At the same time , the dipolar energy is \nreduced since the net magn etization of left -hand and right -hand regions form a flux -closure \nconfiguration . A sufficiently high external magnetic field  can twist the antiparallel \nmagnetization state to the parallel state [Fig. 1(d)], so reducing the Z eeman energy  \n[26,27,30,36,39] . This switching process is accompanied by the creation  of a DW for the Co \nand Gd moments associated with an energy cost  𝐸DW. In the regime where  the dipolar energy \nis negligible  (see Supplemental  Material [37]), the DW energy determines the strength  of the \neffective antiparallel coupling 𝐽AP between the net magnetization in the regions with \ncompensation temperatures of 𝑇c1 and 𝑇c2. The antiparallel coupling gives rise to  an effective \nexchange coupling  field  𝐻EC: \n𝜇0𝐻EC=𝐽AP\n𝑀net≅𝜆DW\n𝑀net𝑤=4√𝐴eff𝐾eff−𝜋𝐷\n𝑀net𝑤 ,  (1) \nwhere 𝜆DW, 𝐴eff, 𝐾eff, 𝐷,𝑀net and 𝑤 are the DW energy density, effective exchange stiffness, \neffective magnetic anisotropy, DMI strength, net magnetization and the width of the switched \narea, respectively  [37]. \nThe impact of the coupling on 𝑴Co and 𝑴𝐧𝐞𝐭 [illustrated in Fig. 1(c) ] can be demonstrated by \nselectively oxidizing a check erboard  pattern with square width of 800  nm in a film with  𝑇M>\nRT, as schematically shown in the inset of Fig. 1(e). After  removal of a large magnetic field \nsaturating the sample, t he Kerr contrast, predominantly arising from the Co sublattice, \ndisplays  a uniform state  [Fig.  1(e)]. In contrast,  the magnetic force microscopy  image , where  \nthe stray fields produced by the net magnetization  are detected , reveals an alternating \ncontrast  [Fig. 1(f)]. The nanoscale  magnetization pattern is predominantly driven  by lateral 4 \n exchange coupling whereas , increasing the dimensions towards  a micrometer scale pattern \nwould lead to an increase in the influence of  the dipolar interaction [36]. To further \ndemonstrate this lateral exchange coupling  at ambient temperature, we patterned a \n50 μm×50 μm squar e with half of the square being oxidized  [light and dark grey regions of \nthe square in Fig. 1(g)] . The as -grown part of the square [white region  of the square in Fig. \n1(g)] is compensated with 𝑇M slightly above RT, such that  its magnetization is negligible . The \noxidized region has its 𝑇M far below RT, resulting in  a lateral exchange -biased structure. As \nshown in Fig. 1(g), by  warming up from 250 K to 300 K  in an applied magnetic  field 𝜇0𝐻z=\n±6 T in order to preset the state of  the compensated region, a switching of the exchange -\nbiased hysteresis loop (𝜇0𝐻EB=±24 mT) can be observed depending on the state of the \ncompensated region  [37].  \nIn order t o confirm  the interfacial origin  of the exchange coup ling, we selectively oxidized \ntrack s with width s in the range  from 50 to 200  nm in the original GdCo films . The electric \ndetection of the magnetic state  (𝑴Co) is performed via  1-μm-wide Hall bars [Fig. 2(a)]. Full \nand minor  hysteresis loop s at temperature s rang ing from 300  K down to 200  K are then \nrecorded  [37]. An example  set of  hysteresis loop s for a  150 nm-wide track measured at 300, \n220 and 2 00 K, corresponding to the three distinct temperature ranges , is presented in \nFig. 2(a). As expected, the hysteresis loops in the temperature range  with  𝑇>𝑇c2 [300 K loop \nin Fig. 2(a)] and 𝑇<𝑇c1[200 K loop in Fig. 2(a )] are trivial  since the net magnetization of both  \nthe 𝑇c1 and 𝑇c2 regions simply switch when applying a sufficient  magnetic field . In the \ntemperature range 𝑇c2>𝑇>𝑇c1, on a pplication of  a large enough magnetic field, the \nZeeman energy will cause the net magnetization of both regions to follow the applied field , \nleading to 𝑴Co in the oxidized and non-oxidized regions  pointing in opposite directions . On \nreducing the magnetic field, the exchange coupling overcomes  the Zeeman interaction  \nresulting in parallel orientation of the  two Co magnetic sublattices . This is accompanied by an \nenhancement of the Hall signal  [37]. After surpassing the coercive field of the surrounding \nnon-oxidized GdCo layer , the magnetization switches while maintaining the parallel Co \nmagnetic configuration. When the field is further increased, the net magnetization in the two \nregions again aligns in parallel.  \nThe systematic measurement  of the exchange coupling  strength  at different temperatures  is \nsummarized in Fig. 2(b). In lin e with the proposed mechanism, no exchange coupling field  is \nobserved when the  temperature is above or below both 𝑇c1 and 𝑇c2. However,  once the \ntemperature is below 𝑇M of the surrounding non-oxidized GdCo layer , an increase in the  \nexchange coupling field  can be observed as the magnetization of the track approaches  its \ncompensation point  on reducing the temperature . The increase in the exchange coupling  field \nis caused by the reduction of the net magnetization  of the tracks , which can be quant itatively \ndescribed by Equation  (1) and fitted to the experimental data . By reducing  the track width \nfrom 200 to 50 nm, the exchange coupling field  strength is further increased . This confirms \nthe interfacial origin of the coupling effect, which becomes stronger in devices  with  reduced  \nlateral  dimensions.  The exchange coupling fields  reach values as high as 2.5 T. It should be 5 \n noted  that, in contrast to the lateral  exchange coupling , the dipolar coupling  mechanism  \nreported previously [30,36]  decrease s in smaller devices  and is therefore not useful for \nminiaturization of devices . The micromagnetic simulations of the effective coupling field with \nand without dipolar field is shown in the Supplemental Material [37] (see, also, reference s \n[40,41]  therein).  \nTo provide  microscopic insight into the switching mechanism , a GdCo film with  𝑇M above RT \nis patterned into 40 µm long tracks of various  width. Imaging in a w ide field polar Kerr \nmicroscope reveals that the magnetization reversal  is driven by DW propagation along  the \ntrack  [Fig. 3(a)]. The reverse  domains [given by white contrast in Fig. 3 (a)] are created at the \nsharp tips at both ends of the track and propagate towards the center. Moreover, the \ncurvature of the moving DWs suggests that the DW is strongly dragged by the lateral interface. \nThe curvature angle of the DW respect to the DW moving d irection  is in stark contrast to  the \ncase where the DW motion is hindered by pinning at the edges  of a magnetic racetrack  [42]. \nThe DW can then serve as a probe of the magnetic coupling  [18]. We measure the s o-called \nstop ping fi eld at which the DW motion is arrested,  meaning that the exchange coupling is \nbalanced by the Zeeman energy . The stopping field can be deduced from the hysteresis loop , \nan example of which is shown in Fig. 3(a), and corresponds to  the field value where the Kerr \nrotation switches sign . A clear increase  of the stopping field is observed when the track width \nis reduced from 800 to 200 nm [Fig. 3(b)] , which again indicate s the interfacial origin of the \ncoupling . In addition , the stopping field s are more than one order of magnitude higher than \nthe ones originating from the DMI -driven chiral coupling mechanism , which opens routes to \nmore efficient , tunable lateral couplings  [18].  \nIn addition to  using  oxidation , we can realize lateral exchange coupling  using  He+ irradiation , \nwhere the spatial  modif ication of  𝑇M is achieved by introducing defects and oxygen atoms \ninto the film  [30,31,36,43] . The He+ irradiation technique offers a good  alternative to \npatterning using  oxidation , with exquisite control over the desired coupling strength by \nchanging the irradiation dose and a smaller achievable feature size  of 5 nm [44] (see \nSupplemental Material  [37] for the relevant experimental results ). \nTo illustrate the potential of the lateral  coupling  in a single f errimagnetic  film for applications , \nwe designed a planar  counterpart of exchange -biased ferromagnet/antiferromagnet  bilayers , \nin which  the exchange bias is manipulated  via the spin -orbit torques  (SOTs)  [45]. We have \npatterned a compensated GdCo film (𝑇M≥RT) into cross -like structures placed on a Pt \nconduit. The Pt layer is used as a source of sizeable DMI as well of spin current  [3,19] . Each  \ncross  is divided into five squares , where the four surrounding squares are selectively oxidized , \nwhereas  the central  square remains in its pristine state  [Fig.  4(a)]. By applying a large \n(±250 mT) OOP  magnetic  field, only the oxidized regions can be switched since they have a \nsizeable net magnetization [Fig.  4(c)]. In order to utilize  SOTs  to switch the magnetization, an \nIP external magnetic field (𝐻) is applied along the current direction  (𝐽) [3]. Starting from  the \ncase where 𝑴Co is parallel across the entire device , a series of current pulses causes the 6 \n magnetization in the entire structure to be switched [Fig.  4(d)] . This is caused  by the SOT -\ndriven switching of the uncompensated squares , and the central square switches with them \ndue to the strong lateral exchange coupling . From t he hysteresis loops in Fig.  4(g), we  then \nsee a clear exchange bias field of ±30 mT whose polarity depends on the orientation of the \ncompensated square  set by the SOT . The electric switching of lateral exchange bias  is highly \nreproducible (see Supplemental Material [37]). The unique combination of SOT switching and \nlateral exchange coupling therefore provides a means to achieve  magnetic states , which are \notherwise  only  accessible via a field cooling protocol [Fig.  1(g)]. To corroborate the proposed \nmechanism , we have also fabricated a complementary cross -structure where an oxidized \nsquare is placed in the center  of the cross while the outer four squares are non-oxidized  \n[Fig. 4(b)] . While a large magnetic field can be us ed to  switch the magnetization of the inner \nsquare only [Fig.  4(e)], the SOT is not able to induce switching of the central region because \nthe lateral exchange coupling to the surrounding regions is too strong  [Fig. 4(f)] .  \nIn conclusion, the lateral exchange coupling reported in single -layer  ferrimagnetic  devices \nwith sub -micron dimensions provides an important addition to  the family of intra -layer \ncoupling s. Unlike in vertical stacks, where the interfacial exchange coupling always contains \nan immobile compensation wall [25-29], the lateral interfacial e xchange coupling strength can \nbe easily tuned by modifying the  device geometry , altering the He+ irradiation  dose , changing \nthe oxidation exposure, or changing the temperature. The coupling is given by the local \nexchange interaction between the transition metal atoms across the interface between \nregions with  different compensation temperatures , which allow s for device downscaling \nwithout the loss of the coupling  strength . The estimated  strength  of the  lateral interfacial \nexchange coupling is much larger than the volume -like dipolar coupling  in ferrimagnetic \nmultilayers [30,36]  and it is one order of magnitude stronger than the DMI -mediated coupling  \nin single -layer ferromagnets [16,18] . Furthermore, by combin ing the interfacial exchang e \ncoupling with current driven SOT switching, we are  able to access both magnetization states \nof a compensated ferrimagnet , which can be otherwise only be access ed by a field -cooling \nprotocol . The lateral exchange coupling, where the coupling strength is m aintained on \ndownscaling of the device , serve s as a n important  counterpart to the coupling in vertical \ndevices  and opens up the possibility for new functionalities in planar devices.  \n \nAcknowledgement s:  \nThis project received funding from the Swiss National Science Foundation (Grant Agreements \nNo. 200021_182013  and 200020_200465). A.H. was funded by the European Union's Horizon \n2020 research and innovation program under Marie Skłodowska -Curie grant agreeme nt No. \n794207 (ASIQS).  Z.L. and L.J.H. acknowledge funding from the European Union's Horizon 2020 \nFET-Open program under Grant Agreement No. 861618 (SpinEngine) . Z.L. also acknowledge \nfunding from the National Natural Science Foundation of China (No. 52271 160).  We thank \nMax Doebeli for help with the analysis of the ERDA measurements . 7 \n  \nData Availability:  \nThe data that support this study are avail able via the Zenodo repository , \n10.5281/zenodo.6936908 , Ref.  [46].  8 \n  \nFIG.1  Demonstration of the lateral exchange coupling principle.  (a-d) Schematic of interfacial \nexchange coupling at different temperatures. 𝑴net=𝑴Co+𝑴Gd. The dark and light blue \nbackground correspond to the non -oxidized and oxidized regions , respectively. The 𝐽Co−Co \n(green),  𝐽Gd−Gd (light green) and 𝐽Co−Gd (yellow ) represent exchange coupling between \ndifferent elements, respectively.  (e) Kerr  image  and (f) MFM  image of the checkboard pattern \nspontaneously formed in 800-nm-wide  GdCo  squares  at zero field . The symbols in the insets \nrepresent the orientation of the magnetization of the  Co sublattice (red) and the net \nmagnetization (black)  while the dark and light blue background correspond to the non -\noxidized and oxidized sections, respectively . (g) Hysteresis loop s measured on  a 50 μm×\n50 μm GdCo square with one half being magnetically compensated (in white) and the other \nhalf being  oxidized  (in light or dark  gray). The dark and light gra y data points  correspond to  \nmeasurements taken after  field  cooling the sample from T=250 K to 300  K at −6 T and +6 T \n(Hz), respectively .  \n9 \n  \nFIG.2  Temperature dependence of the lateral exchange coupling.  (a) A set of hysteresis  loop s \nmeasured via Hall resistance as a function of applied magnetic field of a 150 -nm-wide  track \nat different temperatures . The plots are shifted by 0.3  and 0.6  Ohm  (for 220 and 300 K)  for \nclarity.   In the inset is a schematic of the oxidized track (in gray) on a 1-μm-wide  Hall bar (in \nblue) used for anomalous Hall resistance measurements. The relevant magnetization states \nof oxidized|non -oxidized regions  are depicted  (black for 𝑴net and red for 𝑴Co). (b) Exchange \ncoupling  field  versus  temperature plots for different track width s in the range  50 to 200  nm. \nThe lines are fits according to Eq.  (1). The thickness of the GdCo film is 4.6 nm.  Note that the \ndata points outside the shaded regions, where the exchange coupling field is null, overlap.   \n10 \n  \nFIG.3  Determination of the stopping field associated with the lateral exchange coupling.  (a) \nKerr contrast as a function of decreasing magnetic field obtained from a  2-μm-wide track . The \nthickness of GdCo is  6.5 nm. Examples of Kerr micrographs at different fields and enlargement \nof DW profile are displayed in the inset.  (b) Stopping field versus  track width in GdCo \npatterned by oxidation . 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Hrabec, \nDataset for Strong lateral exchange coupling and current -induced switching in single -layer \nferrimagnetic films with patterned compensation temperature, Zenodo.  \n \n  15 \n Supplementary Note 1: Sample fabrication  \nThe magnetic stack of Ta  (1 nm)|Pt  (5 nm)|GdCo  (x nm)|Ta  (2 nm) is deposited via \nmagnetron sputtering on Si 3N4|Si substrates at a base pressure below 3×10−8 Torr and an \nAr sput tering pressure of 3  mTorr using a commercial sputtering system. We deposited the \nGdCo layer by co -sputtering from Gd and a Co targets. We varied the thickness of the GdCo \nlayer while keeping the atomic Co fraction of 69.9% fixed. This allows us to modify the \ncompensation temperature  𝑇M of the ferrimagnetic GdCo layer (see Supplementary Note 2).   \nThe oxidation process is performed in  a commercial Oxford plasma chamber with  an oxygen \nplasma power between 30 and 40  W for a time period of 60  – 90 s. The oxidation of the layer \nhas been confirme d quantitatively by measuring the oxygen concentration in non -oxidized \nand oxidized samples via 13 MeV 127I heavy ion elastic recoil detection analysis (ERDA). \nExample depth profiles for O are shown in Supplementary Fig. 1. Although, the sample \nthickness i s slightly below the depth resolution of the technique, there is clear evidence that \nthe total oxygen content in the Ta capping layer and ferrimagnetic CoGd layer increases by  \n40% after the oxygen plasma treatment, while the profiles of the other elements are not \ninfluenced.  \n \nSupplementary Figure 1 : 13 MeV 127I heavy ion ERDA measurement to determine the \ndifference in the atomic fraction of oxygen between Ta|Pt|CoGd|Ta thin films before (fresh) \nand after (ox) oxygen plasma treatment.   \nThe designed patterns  are prepared by electron  beam lithography (EBL). We use 330  nm thick \n4% PMMA in 950K molecular weight ethyl lactate as an electron -beam resist . The same resist \nis also used as a protection mask  for the oxidation process.  \n \nSupplementary Note 2: GdCo 𝑻𝐌 thickness dependence  \n16 \n By changing the thickness of the GdCo layer, the compensation temperature of the GdCo film \ncan be tuned. The change in  𝑇M with thickness is a result of the variation in the composition \nacross the film thickness due to the formation o f microstructures such as island and voids \n[38,47] . As shown in Supplementary Fig.  2, by decreasing the thickness of GdCo layer from \n6.3 nm to 3.3  nm, a reversal of the hysteresis loop measured by polar -MOKE at roo m \ntemperature is observed, which indicates that  𝑇M changes from being above 300  K to being \nbelow 300  K. This is also accompanied by divergence in the coercive field at the thickness \nwhere the sample is magnetically compensated at room temperature.  \n \nSupplementary Figure 2 : Room temperature coercive fields  measured using polar -MOKE  for \nGdCo films with variable thickness. The insets depict two representative hysteresis loops \ntaken for thicknesses smaller (in black) and larger (in red) than the thickness  at which the \nmagnetic film is compensated at room temperature.  \n \nSupplementary Note 3 : Effect of oxidation on the 𝑻𝐌 of the GdCo films  \nBy introducing oxygen into the GdCo layer, we can reduce the 𝑇M of the original film. As \nshown in Supplementary Fig.  3, the 𝑇M of the as -grown film of 4.6 nm thick GdCo changes \nfrom 260 K to 190 K on oxidation. The hysteresis loops, with the coercive field diverging at \nthe compensation temperatures, are recorded via Hall resistance measurement in 1  μm Hall \nbar devices.  \n17 \n  \nSupplementary Figure 3 : Anomalous Hall effect measurement of the coercivity of GdCo \n(4.6 nm) before (dark blue) and after (light blue) oxidation.  \nSimilarly, by oxidizing a 6.3 nm thick GdCo film whose 𝑇M is higher than room temperature, \nwe could bring the 𝑇M of the film below room temperature. This scenario can be verified by \npolar -MOKE measurement at room temperature as shown in Supplementary Fig.  4, where \nthe sign of the hysteresis loops is reversed.  \n \nSupplementary Figure 4 : Polar MOKE hysteresis l oop measurement of a GdCo (6.3 nm) film \nat room temperature (a) before and (b) after oxidation.  \nThe film topography is also affected by the oxygen absorption. As shown in Supplementary \nFig. 5, the height profile along the blue line in the atomic force micrograph reveals a 1 nm \nincrease in the thickness after the oxidation.  \n18 \n  \nSupplementary Figure 5  (a) Atomic force micrograph of the checkerboard pattern in GdCo \n(6.3 nm). The bright and dark reg ions correspond to the measured height of oxidized and non -\noxidized regions, respectively . (b) Height profile along the line shown in panel (a) reveals a \nheight difference of 1  nm between oxidized and non -oxidized regions.  \n \nSupplementary Note 4 : Transport and wide -field Kerr measurements  \nThe anomalous Hall electrical measurements are performed in a commercial Physical \nProperty Measurement System (PPMS). The samples are measure d with a sensing current of \n100 μA under standard DC drive mode with  an average of 10 readings per data point.  \nThe Kerr images are recorded using  a commercial wide -field Kerr microscope  in the polar \nconfiguration . The sequence of Kerr images in Fig.  3(a) are captured while continuously \ndecr easing the field to zero at 0.5  mT/sec. The magnetic contrast is visualized using  \ndifferential Kerr imaging  where the images are obtained by subtraction from the background \nimage. The background image used in Fig.  3 was taken at 5  mT, while the background images \nin Fig.  1 and Fig.  4 are cap tured at 0 mT after applying field of 250  mT.  \n \nSupplementary Note 5 : Domain wall driven exchange bias  \nFor the 50 μm ×50 μm square design and the corresponding exchange bias loop \nmeasurements shown in Fig.  1(g), the switching is due to interfacial exchange  coupling \ninduced domain nucleation and corresponding domain wall motion as shown in \nSupplementary Fig.  6. \n19 \n  \nSupplementary Figure 6 : A series of Kerr images of GdCo(6.3 nm) recorded during the \nexchange bias measurement  [see Fig. 1(g) of the main text ], (a) with negative exchange bias \nand (b) with positive exchange bias. The straight vertical line in the middle of the square \nseparates the oxidized (right) and non -oxidized (left) regions.  \n \nSupplementary Note 6 : Anomalous Hall effect measurements of the later al exchange \ncoupling field  \nAnomalous Hall effect (AHE) measurements of selectively oxidized GdCo(4.6  nm) tracks (see \nFig. 2a of the main text) reveal a composite hysteresis loop consisting of a narrow square loop \ncentered around zero field (larger signal f rom surrounding non -oxidized region) and two side \nloops at higher field (smaller signal from oxidized track region). The side loops arise from the \nlateral exchange coupling between the regions with 𝑇𝑀larger and smaller than the \nmeasurement temperature (ranging from 220K to 260K). Depending on the temperature, we \nobserve two types of loops: as shown in Supplementary Fig.  7: (a) the exchange coupling field  \nminor loop s are separa ted from the central hyste resis loop; (b ) the exchange coupling field  \nminor loop s are merged with the central hysteresis loop . The shape of the loops depends on \nthe relative strength of exchange coupling field  between the non -oxidized region and the \noxidized track as we change the temperature.  \n20 \n  \nSupplementary Figure 7 : Full hysteresis loops of a 200 -nm-wide oxidized track in CoGd \n(4.6 nm) at 230  K (a) and 240  K (b). The black and red lines correspond to the field sweep \ndirection from +1  T to -1 T and vice versa , respectively. The relevant magnetization states of \noxidized|non -oxidized regions are depicted (black for 𝑴net and red for 𝑴co ). \nIn Supplementary Fig.  7(a) is shown the full hysteresis loop measured via Hall resistance as a \nfunction of OOP magnetic field ranging from + 1T to -1T (black line) and back (red line). The \nloops are recorded at 230 K using a Hall bar as shown in the inset of Fig. 2(a) in the main text. \nAt a field of +1  T, the net magnetization of the oxidized track and the surrounding non -\noxidized region are pa rallel to each other. The AHE resistance has an intermediate value \nbecause it reflects the overall antiparallel alignment of the Co sublattice magnetization in the \ntwo regions. On reducing the applied field from +1 T to 𝐻1, the interfacial exchange coupli ng \novercomes the Zeeman energy. Considering that the non -oxidized region is much larger than \nthe oxidized region, only the oxidized DW track can be switched by the interfacial exchange \ncoupling via a DW propagation mechanism.  Thus, the net magnetization of  the oxidized track \nswitches to the exchange -favored state in which it is antiparallel to that of the surrounding \nregion. This configuration corresponds to the maximum amplitude of the AHE resistance \nbecause of the parallel alignment of the Co magnetic mom ents. At 𝐻2, the magnetization of \nthe surrounding region and that of the oxidized track switch simultaneously due to application \nof a reversed magnetic field that exceeds the coercivity of the non -oxidized region.  Finally, \nonce the Zeeman energy again ove rcomes the exchange coupling energy at 𝐻3, the \nmagnetization of the oxidized track switches so that its net magnetization follows the applied \nmagnetic field direction. This corresponds to the net magnetization (Co magnetic moments) \ninside and surrounding the oxidized track being aligned parallel (antiparallel) to each other, \nanalogous to the initial configuration at +1 T but with opposite magnetization direction. The \nfield 𝐻1′ , 𝐻2′  and 𝐻3′ marked in Supplementary Fig. 7 (a) refe r to the similar situati on when \nthe field is swept from -1 T to +1  T (red line). The exchange coupling field is defined as the \nfield value at the center of the minor hysteresis loop , which is calculated as  𝐻EC=\n|𝐻1+𝐻3′|+|𝐻3+𝐻1′|\n4. In Supplementary Fig.  7(b) is shown the hysteresis loop of the same sample \n21 \n recorded at 240 K. In this case, the interfacial exchange coupling strength is smaller than the \ncoercivities of the surrounding region 𝐻2 and 𝐻2′. Thus, the minor loops merge into the \ncentral loop, and 𝐻3 and 𝐻3′ are not distinguishable anymore.  \nMoreover, the relative signal arising from the oxidized track depends on its width. As shown \nin Supplementary Fig.  8, given the same Hall bar width of 1  μm, with different track widths of \n200 nm and 100  nm, the minor loop s ignal height of the 100  nm track is roughly half the signal \nheight of the 200  nm track. This further demonstrates that the minor loops are associated \nwith the oxidized tracks.  \n \nSupplementary Figure 8 : Normalized full hysteresis loop measurement of 200 -nm and 100 -\nnm-wide oxidized tracks at 230  K. The height of the minor loop scales with the size of the \noxidized track  \n \nSupplementary Note 7 : Details of FIB He+ irradiation  \nThe He+ irradiation is performed with a Focused Ion Beam (FIB) using a  He+ microscope with \na pattern generator. Cr  (5 nm)|Au  (20 nm) alignment markers are patterned on the pristine  \nGdCo films using electron -beam lithography  followed by lift -off. Then the track patterns are \nirradiated  with a  He+ current with dose s rang ing from 3×1015 He+/cm2 to 24×\n1015 He+/cm2. The He+ beam can produce features down to 5 -10 nm and a 100 -500 nm \nimplantation depth at 30kV acceleration voltage (5 -20 pA current) [44,48] . On injecting He+ \nions into the GdCo films , the magnetic properties such as  the anisotropy and  the 𝑇M of the \nferrimagnetic m aterial are modified. These changes are due to vacancies induced by ion \npenetration, which modify the chemical short -range order, and an increase of oxygen atoms \ndiffusing into the film that cause preferential oxidation of the Gd atoms [31].  \n22 \n The stopping fields are measurable only in the dose range 9 to 18×1015 He+/cm2 \n[Supplementary Fig. 9 (a)]. While doses smaller than 9×1015 He+/cm2 do not bring the \ncompensation temperature of the irradiated region below room temperature, doses higher \nthan 20×1015 He+/cm2 result in sample damage. Within the irradiation dose range that \nresults in lateral exchange coupling, a decrease in the stopping field is observed when \nincreasing the He+ dose.  Different track widths ranging from 200  nm to 2  μm are irradiated \nwith a dose of  9×1015 He+/cm2. The corresponding stopping fields shown in Supplementary \nFig. 9 (b) increase as the track width  is reduced , which reflects the interfacial ori gin of the \ncoupling effect. The He+ irradiation technique therefore offers an alternative patterning \nmethod to oxidation with exquisite control over the desired coupling strength by changing \nthe irradiation dose.   \n \nSupplementary Figure 9 : (a) Stopping field versus  irradiation dose in 6.5 nm GdCo patterned \nby He+ irradiation . The gray shading indicates the dose range where the stopping field is \nmeasurable . (b) Stopping field versus  track width in 6.5 nm GdCo patterned by He+ irradiation. \nThe lines in (b) are fits according to Eq.  (1) in the main text.  The thickness of GdCo is 6.5 nm.  \n \nSupplementary Note 8 : Micromagnetic simulations of the effective couplin g field with and \nwithout dipolar field  \nTo evaluate the effect of the dipolar field on the ground state, a 2D micromagnetic simulation \nis performed via Mumax3 [40]. The material parameters used in the simulation are as follows: \n𝐷=0.2×10−3 J/m2, the saturation magnetization is 8.2×104 A/m in the non -oxidized \nregion and 2.3×105 A/m in the oxidized region, as obtained from SQUID measurements. \nThe exchange stiffness is 𝐴ex=7×10−12 J/m and OOP uniaxial anisotropy constant 𝐾u=\n86 kJ/m3. We al so considered periodic boundary conditions. To quantify  the magnitude of \nthe effective OOP magnetic field acting on the central stripe, we determined  the energies of \n⊙⊙⊙ and ⊙⊗⊙ states  ( 𝐸⊙⊙⊙ and 𝐸⊙⊗⊙ ) from 1D micromagnetic simulations  [18]. As \nsketched in t he inset of Supplementary Fig. 10, the central oxidized track is \n23 \n antiferromagnetically coupled to the surrounding film via two cells with a negative exchange \ncoupling ( JAP) at both interface.  By performing systematic simulations, we find that the dipolar \nfield contributes marginally to the effective field. We conclude that the coupling is mostly \nmediated by the proposed lateral exchange coupling mechanism.  \nThe simulations also show that JAP is of the order of 10% of the exchange coupling Aex. In this \nsimplified model, a single homogenous exchange parameter between the Co -Co magnetic \nmoments over the whole GdCo film is considered. While the Co -Gd coupling is strong enough \nto maintain ferrimagnetic coupling, the Gd -Gd coupling strength is much smaller than \nbetween Co -Co magnetic atoms. The coupling strength estimated from the simulations (see \nEq. 1 of the main text) is  𝐽AP≈0.7×10−12 J/m [41]. This number can be compared to the \nanalytically estimated number, which is 𝐽AP=4√𝐴𝐾eff−π𝐷≈ 16.1×10−12 J/m. The \nestimated numbers are susceptible to the uncertainty in the value of DMI and magnetic \nanisotropies used in the simulation, which affect the DW energy but not the effective fields \nin micromagnetic simulations. The computed effective fields are also in a relatively good \nagreement with the measured stopping fields presented in Fig.  3, which are r eplotted in \nSupplementary Fig.  10. \n \nSupplementary Figure 10 : Experimental data (dots) and simulated effective field versus  track \nwidth in the presence (full lines) or absence (dashed lines) of dipolar field. A schematic of the \nsimulated geometry is shown in the inset.  \n \nSupplementary Note 9: Reproducible electrical switching of exchange bias  \nTo illustrate the statistical significance of the data presented in Fig.  4, we probed the repeated reversal \nof exchange bias by the application of current pulses of fixed polarity while reversing the pola rity of \nthe longitudinal in -plane magnetic field. A highly -reproducible switching of exchange bias can be \nobserved in Supplementary Fig.  11. \n24 \n  \nSupplementary Figure 11 : Exchange bias measurement of cross -shape structure after \napplying 50×150 -ns-long current pulses at a current density of 1.38×1011 A/m2 while \nalternating an in -plane magnetic field between −250  mT (red) and +250  mT (blue) applied \nalong the current direction.  \n \nReference:  \n[1] R. Malmhäll and T. Chen, Thickness depe ndence of magnetic hysteretic properties of rf ‐  \nsputtered amorphous Tb –Fe alloy thin films, Journal of Applied Physics 53, 7843 (1982).  \n[2] B. Hebler, A. Hassdenteufel, P. Reinhardt, H. Karl, and M. Albrecht, Ferrimagnetic Tb –Fe Alloy thin \nfilms: composition and thickness dependence of magnetic properties and all -optical switching, \nFrontiers in Materials 3, 8 (2016).  \n[3] I. Shorubalko, K. Choi, M.  Stiefel, and H. G. Park, Ion beam profiling from the interaction with a \nfreestanding 2D layer, Beilstein journal of nanotechnology 8, 682 (2017).  \n[4] I. Shorubalko, L. Pillatsch, and I. Utke, in Helium Ion Microscopy  (Springer, 2016), pp. 355.  \n[5] M. Krupinski, J . Hintermayr, P. Sobieszczyk, and M. Albrecht, Control of magnetic properties in \nferrimagnetic GdFe and TbFe thin films by He+ and Ne+ irradiation, Physical Review Materials 5, \n024405 (2021).  \n[6] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia -Sanchez, and B. Van Waeyenberge, \nThe design and verification of MuMax3, AIP advances 4, 107133 (2014).  \n[7] Z. Liu, Z. Luo, S. Rohart, L. J. Heyderman, P. Gambardella, and A. Hrabec, Engineering of Intrinsic \nChiral Torques in Magnetic Thin Films Based on the Dz yaloshinskii -Moriya Interaction, Physical \nReview Applied 16, 054049 (2021).  \n[8] L. Caretta, M. Mann, F. Büttner, K. Ueda, B. Pfau, C. M. Günther, P. Hessing, A. Churikova, C. Klose, \nand M. Schneider, Fast current -driven domain walls and small skyrmions in a co mpensated \nferrimagnet, Nature nanotechnology 13, 1154 (2018).  \n \n"    },    {        "title": "2204.09776v1.Ferrimagnet_GdFeCo_characterization_for_spin_orbitronics__large_field_like_and_damping_like_torques.pdf",        "content": " \n \n1 \n Ferrimagnet GdFeCo  characterization for spin-orbitronics:  large field -like and \ndamping -like torques  \n \nHéloïse Damas1*, Alberto Anadon1, David Céspedes -Berrocal1,2, Junior Alegre -Saenz1,2, Jean -\nLoïs Bello1, Aldo Arriola -Córdova1,2, Sylvie Migot1, Jaafar Ghanbaja1, Olivier Copie1, Michel \nHehn1, Vincent Cros3, Sébastien Petit -Watelot1* and Juan -Carlos Rojas -Sánchez1* \n \n1Université de Lorraine,  CNRS, Institute Jean Lamour, F -54000 Nancy, France  \n2Universidad Nacional de Ingeniería, Rímac 15333, Peru  \n3Unité Mixte de Physique, CNRS, Thales, Université Paris -Saclay, 91767 Palaiseau, France  \n \nCorresponding authors:  heloise.damas@univ -lorraine.fr, sebastien.petit@univ -lorraine.fr,  \nJuan-Carlos.ROJAS -SANCHEZ@univ -lorraine.fr  \n \nH. Damas, Dr. A. Anadon, D. Céspedes -Berrocal, A.Y. Arriola -Córdova, J -L Bello, S. Migot, \nJ. Ghanbaja, Dr. O. Copie, Prof. M. Hehn, Dr. S. Petit -Watelot,  Dr. J. -C. Rojas -Sánchez.  \nUniversité de Lorraine,CNRS, Inst itute Jean Lamour, F -54000 Nancy, France  \nE-mail:  heloise.damas@univ -lorraine.fr , sebastien.petit@univ -lorraine.fr  , Juan-\nCarlos.ROJAS -SANCHEZ@univ -lorraine.fr   \n \nD. Céspedes -Berrocal, A.Y. Arriola -Córdova , J. Alegre -Saenz  \nUniversidad Nacional de Ingeniería, Rímac 15333, Peru  \n \nDr. V. Cros,  \nUnité Mixte de Physique, CNRS, Thales, Université Paris -Saclay, 91767 Palaiseau, France  \n \nKeywords:  \nFerrimagnet GdFe Co; spin -orbit torque; spin -torque ferromagnetic resonance; spin anomalous \nHall effect; spin Hall effect  \n \nSpintronics is showing promising results in the search for new materials and effects to reduce \nenergy consumption in information technology. Among these materials, ferrimagnets are of \nspecial interest, since they can produce large spin currents that trigge r the magnetization \ndynamics of adjacent layers or even their own magnetization. Here, we present a study of the \ngeneration of spin current by GdFeCo in a GdFeCo/Cu/NiFe trilayer where the  FeCo  sublattice \nmagnetization is domina nt at room temperature. Magn etic properties such as the saturation \nmagnetization are deduced from magnetometry measurements while damping constant is  \n \n2 \n estimated from spin -torque ferromagnetic resonance (ST -FMR). We show that the overall \ndamping -like (DL)  and field -like (FL) effective fields as well as the associated spin Hall angles \ncan be reliably obtained by performing the dependence o f ST-FMR by an added dc current. The \nsum of the spin Hall angles  for both  the spin Hall effect (SHE) and the spin anomalous Hall \neffect ( SAHE ) symmetries  are: 𝜃𝐷𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐷𝐿𝑆𝐻𝐸=−0.15±0.05 and 𝜃𝐹𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐹𝐿𝑆𝐻𝐸=\n0.026±0.005. From the symmetry of ST -FMR signals we find that 𝜃𝐷𝐿𝑆𝐻𝐸 is positive and \ndominated  by the negative 𝜃𝐷𝐿𝑆𝐴𝐻𝐸. The present study paves the way f or tuning the different \nsymmetries in spin conversion in highly efficient ferrimagnetic systems.  \n \n1. Introduction  \n \nIn the last years, ferrimagnets have attracted growing interest for their potential utility in \nspintronic devices  [1]. In particular, GdFeCo ferrimagnetic alloy is extensively studied  as it \nexhibits  a wide diversity of phenomena arising from the specific properties of rare earth -\ntransition metal (RE -TM) ferrimagnets. Furthermore , the two antiferromagnetically coupled \nsublattices have a different response to external stimuli and the spin -orbit couplin g (SOC) of \nthe Gd 5d state allows the interplay between charge, spin , and orbital transport. The different \nrelaxation times of these two coupled sublattices are thought to be responsible for the all -optical \nhelicity -independent switching (AO -HIS) in GdFeCo  demonstrated for almost a decade  [2,3] . \nAO-HIS has also been recently observed in TbCo  [4]. Nowadays, GdFeCo is used to perform \nthe AO -HIS of Co/Pt  [5–7] or CoNi/Pt  [8] ferromagnetic multilayers. Moreover, it is possible \nto tune the Dzyaloshinskii -Moriya interaction  in thin GdFeCo ferrimagnetic alloys  [9], a \nrelevant property for skyrmions formation.  It has been shown that GdFeCo ferrimagnet  also \nhosts  large self-induced spin -orbit torque, or self -torque  [10,11] , with recent theoretical \nadvances  [12,13] . Ferro and ferrimagnetic materials  are the source of spin currents with  \ndifferent symmetries  [10,12]  coming from the s pin anomalous Hall effect (SAHE)  [14–16] and \nthe spin Hall effect (SHE)  [17]. In the SAHE, the spin polarization of the spin current 𝐽sSAHE is  \n \n3 \n parallel to the magnetization while in the SHE it is perpendicular to both the injected charge \ncurrent and the produced spin current 𝐽sSHE. A giant overall spin Hall angle  for SAHE -like and \nSHE -like symmetries  has been reported in a Gd -rich GdFeCo/Cu at r oom temperature  [10]. \nSizable i nterconversion efficiencies ha ve also been reported for  other magnetic materials such \nas NiFe  [18–20] and CoFeB [15,21] . In the present work, we study  room temperature  FeCo -\nrich GdFeCo in a //Gd 25Fe65.6Co9.4(8 nm)/Cu(4 or 6 nm)/Ni 81Fe19(4 nm) trilayer by structural, \nmagnetic and spintronics characterization. W e use two complementary ST -FMR techniques to \nreveal the signs and ma gnitudes of the contributions coming from the different spin current \nsymmetries in GdFeCo. Namely, the modulation of the damping along with the  shift of \nresonance field to extract the overall parameters (sum of the SHE -like and SAHE -like \ncontributions) and  the symmetry of the ST -FMR signal which is sensitive only to the SHE -like \nparameters. We found that damping -like (DL) SAHE spin Hall angle, 𝜃𝐷𝐿𝑆𝐴𝐻𝐸, is negative  for \nFeCo -rich GdFeCo . In contrast, the DL SHE -like symmetry , is positive.   \n \n2. Structur al and chemical characterization  \n \nSamples were grown on thermally oxidized Si wafers using dc magnetron sputtering at room \ntemperature with  an Ar gas pressure of 3 mTorr and base pressure of 1x10-7 Torr. GdFeCo \n(Gd 25Fe65.6Co9.4) was co -deposited using separate Gd, Co , and Fe targets. All the samples in the \npresent study were capped with 3 nm of naturally oxidized Al. The c omposition was controlled \nby varying the sputter gun power on each target. The d eposition rate was calibrate d by X -ray \nreflectivity and lift -off and profilometer measurements of the thickness.  In order to perform \nstructural characterization, a thin lamella was extracted by focused ion beam (FIB) milling \nusing an FEI Helios Nanolab dual -beam 600i.  \nTransmission e lectron microscopy (TEM) investigations were carried out using a JEM - ARM \n200F Cold FEG TEM/STEM  (Scanning TEM)  operating at 200 kV, coupled with a GIF \nQuantum 965 ER and equipped with a spherical aberration (Cs) probe and image correctors  \n \n4 \n (point resoluti on 0.12 nm in TEM mode and 0.078 nm in Scanning TEM  (STEM)  mode) . High -\nResolution TEM (HRTEM) micrographs were performed to study the atomic structure of the \ndeposit layers  as shown in  Figure 1 a. The Fast Fourier Transformation (FFT) patterns in Figure \n1b,c confirm that the Cu/NiFe layers are [111] textured along the growth direction while \nGdFeCo is amorphous as evidenced by the diffuse rings.  Electron Energy Loss Spectroscopy \n(EELS) maps were carried out systematically on the different samples and confirm the nominal \ncomposition and thickness of the different materials  (Figure 1 d). We also evidence a slight  \nGdFeCo composition variation along the growing direction as usually observed in RE -TM \nferrimagnets  [10,11,22] . The EELS maps displayed in Figure 1 d were performed with \n1ev/channel and  a step of 0.3 nm.   \n \n \nFigure 1.  TEM/STEM characterization of  Gd 25Fe65.6Co9.4(8)/Cu(6)/Ni 81Fe19(4)/AlOx . (a) \nHRTEM micrograph of the deposit ed layers. The yellow (blue) square shows where the FFT \nanalysis have been performed on Cu/NiFe (GdFeCo). The FFT patterns ( b) and ( c) indicate the \n[111] growth direction of textured Cu/NiFe and that GdFeCo is amorphous. ( d) High Angle \nAnnular Dark Field (H AADF) -STEM micrograph and the corresponding individual EELS \nelemental maps obtained from the green rectangle area in the HAADF micrograph. Co (yell ow), \nFe (gr een), Gd (orange), O (red), Ni (cyan), Cu (pink).  \n \n \n5 \n  \n \n \n \n3. Magnetic characterization: magnetic anisotropies in GdFeCo  \n \n3. 1. SQUID magnetometry  \n \nMagnetization loops were performed at room temperature on a //GdFeCo (8)/Cu(6)/NiFe (4) \nstack with the applied field  parallel and perpendicular to the field plane. Both 𝑀(𝐻) \nmeasurements  are display ed in Figure 2a show ing an open hysteresis loop . This confirm s that \nthe NiFe magnetization  direction  𝒎̂NiFe lies in the plane of the sample while that of GdFeCo , \n𝒎̂GdFeCo ,  is spontaneously perpendicular to the film plane  as shown in the inset . We assume  \nthat the 6  nm thick Cu layer decouples the two magnetic layers to extract their distinct saturation \nmagnetization and saturation magnetic field . For NiFe, the saturation magnetization 𝑀sNiFe is \n625 kA/m  and the saturation field 𝜇0𝐻sat−zNiFe to place 𝒎̂NiFe out of the plane of the film is 0.85 \nT. In the case of GdFeCo, the  saturation magnetization 𝑀sGdFeCo is 115 kA/m  and the saturation \nfield 𝜇0𝐻sat−xyGdFeCo to align 𝒎̂GdFeCo along the plane is about  0.13 T which are typical values  for \nboth NiFe and Gd25Fe65.6Co9.4 at room temperature   [23–25]. From the  saturation field  and \nmagnetizations , we can also estimate the effective saturation magnetization for both magnetic \nmaterials, it results  𝑀effNiFe=676  kA/m  and 𝑀effGdFeCo=103  kA/m. The relatively  low \nperpendicular magnetic anisotropy of GdFeCo allows its magnetization to be easily placed \nalong t he plane  of the film  which is useful for ST-FMR measurements.  The trilayer used in the \nnext section has a 4 nm Cu spacer and displays a lower saturation field 𝜇0𝐻sat−xyGdFeCo to align \n𝒎̂GdFeCo along the plane,  which is about  ~0.047 T.  \n \n \n  \n \n6 \n  \nFigure 2. Bulk magnetization data using a SQUID magnetometer. (a) Magnetic \nhysteresis loop of the Gd25Fe65.6Co9.4(8)/Cu(6)/Ni 81Fe19(4) trilayer. Magnetization \nvalues are normalized by the surface sample. To identify the different saturation fields \nand effective saturation magnetization, we consider that the magnetic layers are \ndecoupled  by the 6 nm of Cu . The inset shows a s chematic of the sample with the \nspontaneous magnetization alignment of GdFeCo (out -of-plane) and NiFe (in -plane) \naccording to SQUID results.  \n \n \n \n3. 2. Spin -torque FMR study  \nWe perform ST -FMR measurements  [26–31] on a GdFeCo (8)/Cu(4)/ NiFe (4) trilayer  to extract \nproperties such as the damping constant 𝛼 and the Landé g -factor of the magnetic layers. From \nMagneto optic Kerr effect measurements we have verified that this GdFeCo (8) is FeCo -rich at \nroom temperature.  The experimental setup is described in Figure 3a. A radiofrequency (rf) \ncharge current , 𝑖rf,  is applied  along the 𝒙̂ direction and generates an oscillating Oersted field \nwhich  triggers the magnetization precession  at the resonance condition . A sweeping dc \nmagnetic field 𝐻dc is applied in the xy plane of the device, at an angle of 𝜑𝐻 with respect to  the \ncurrent line. At the resonance field 𝐻res, a dc voltage 𝑉mix composed of a mix ing of a \nsymmetric and antisymm etric Lorentzians of amplitude Vsym and Vanti respectively can be \nmeasured using a bias tee. The measured mixed voltage  display ed in Figure 3b can be fitted \nwith the following general expression:  \n𝑉mix=𝑉offset+𝑉symΔ𝐻2\nΔ𝐻2+(𝐻−𝐻res)2+𝑉anti(𝐻−𝐻𝑟𝑒𝑠)Δ𝐻\nΔ𝐻2+(𝐻−𝐻𝑟𝑒𝑠)2, (1) \n \n \n7 \n where  we consider an additional offset 𝑉offset  and where Δ𝐻 is the linewidth. In Figure 3b, we \nobserve the two resonance lines corresponding to the NiFe resonance  (lower resonance field)  \nand the GdFeCo resonance  (higher resonance field ). For the sake of clarity,  it is only show n at \n8, 12 and 14  GHz . Then, from  broadband frequency dependence  ST-FMR  we can extract the \neffective saturation magnetization  Meff (it results negative f or system s where  perpendicular \nmagne tic anisotropy dominate s over shape anisotropy ), and the Landé g -factor considering the  \nfollowing  expression : \n𝑓=𝛾\n2π√(𝐻+𝐻uni)(𝑀eff+𝐻+𝐻uni) ,          (2) \nwhere 𝛾=𝑔μB\nℏ is the gyromagnetic ratio and where  Huni stands for a small in -plane uniaxial \nmagnetic anisotrop y. Equation 2  applies for a thin film ferromagnetic  layer with  a magnetic \nfield applied in the plane . We fix the NiFe  Landé g -factor to 2.10 . We  determine the effective \nsaturation magnetization of NiFe , 𝑀effNiFe=569±1kA/m . The difference with previous \nSQUID results  comes from  the difference in Cu thickness which affect s the NiFe anisotropy . \nWe also evaluate  a rather small  𝐻uni=−7±1 Oe. We exploit the same Equation 2 for \nGdFeCo resonance condition to determine the GdFeCo  Landé g -factor  and its effective \nsaturation magnetization 𝑀effGdFeCo. We obtain  g=2.87±0.04, and 𝑀effGdFeCo=−37±4 kA/\nm (−46.5 mT). The fitted experimental data is shown in Figure 3c. Finally, from  the frequency \ndependence  of the linewidth  H, we calculate the Gilbert -type magnetic damping constant  𝛼: \n𝛥𝐻=𝛥𝐻0+2π𝑓\n𝛾𝛼 , (3) \nwhere H0 is the f-independent contribution due to inhomogeneity . We have fixed g -Landé \nfactor for both NiFe and GdFeCo. The fits on the measurements are shown in Figure 3d.  The \ndamping of in-plane NiFe is estimated  as 𝛼𝑁𝑖𝐹𝑒=0.012±0.001 which  is about 8  times \nsmaller than the damping of our out -of-plane Gd25Fe65.6Co9.4 𝛼GdFeCo=0.085±0.006 but \ncomparable with in -plane Gd12.5Fe76.1Co11.4 [32]. Recently, it has been point ed out that actual   \n \n8 \n or intrinsic damping in ferrimagnet s is lower than th at measured  directly  due to different spin \ndensity for each magnetic sublattice in GdFeCo and  determine d by domain wall mobility  [33].  \nIn the next section , we show how we c an estimate the effective field s that drive the spin -orbit \ntorque from Gd25Fe65.6Co9.4(8)/Cu(4)  to Ni81Fe19(4). \n \n \n \n \n \nFigure 3. Determination of Meff and damping using broadband ST -FMR in \nGd 25Fe65.6Co9.4(8)/Cu(4)/Ni 81Fe19(4), and g -Landé factor for GdFeCo . (a) Illustration of a \ntypical ST -FMR device along with  the dc magnetic field applied at 𝜑𝐻 to the trilayer slab  which  \nis along 𝑥̂. (b) Typical ST -FMR spectr a at 8, 12 and 14  GHz. At a higher field s, the GdFeCo \nresonance line  is observed . The symmetrical (orange) and antisymmetrical (green) voltage \ncontribution s are shown  for NiFe at 8 GHz. Broadband frequency  dependence of Hres (c) and \nlinewidth H (d) are used to determine Meff and , respectively. Equation 2 is used for NiFe \nand GdFeCo layer  in (c). For Gd 25Fe65.6Co9.4 (8 nm), we estimate   𝑔=2.87±0.04 and 𝛼=\n0.085±0.006. Black (green ) experimental data are obtained for the resonance of NiFe \n(GdFeCo) as depicted in (b). \n \n \n \n \n9 \n  \n4. Damping -like and field -like efficiencies  determination  by ST -FMR techniques  \n4. 1. Spin torque  symmetries  and ST -FMR signal   \nAs discussed, there are two symmetries for spin current generation in magnetic materials, \nSAHE -like and SHE -like. When these spin currents are absorbed by a nother  magnetic layer, \nthey contribute to the total torque on the magnetization :  \n𝚪tot=𝚪SHE+𝚪SAHE . (4)  \nIn the geometry of our ST -FMR measurements , the GdFeCo and NiFe magnetizations are both \naligned with an angle of  𝜑𝐻 with respect to the 𝑥̂ axis. The sp in polarization corresponding to \nthe SHE -like symmetry, 𝜎̂SHE, lies along the  𝑦̂ axis regardless of the direction  of both \nmagnetizations . The spin polarization direction related to the SAHE-like spin current , 𝜎̂SAHE , \nlies along the direction of 𝑚̂GdFeCo  which in turn is aligned with the equilibrium direction of \nthe NiFe magnetization , 𝑚̂NiFe . The different contributions to the total torque  can further be \ndivided into two contributions, coming from the  damping -like (ℎDL) and the field-like (ℎFL) \neffective  fields:  \n𝚪SHE\n𝛾𝑀SNiFe= ℎDLSHE𝒎̂NiFe×(𝝈̂SHE⏟\n𝒚̂×𝒎̂NiFe)+ℎFLSHE 𝒎̂NiFe×𝝈̂SHE⏟\n𝑦̂,   (5𝑎) \n𝚪SAHE\n𝛾𝑀SNiFe= ℎDLSAHE  𝒎̂NiFe ×( 𝝈̂SAHE⏟  \n𝒎̂GdFeCo×𝒎̂NiFe)+ℎFLSAHE 𝒎̂NiFe×𝝈̂SAHE⏟  \n𝒎̂GdFeCo.(5𝑏) \nThe efficiency of the charge -to-spin current  conversion is described  by the spin Hall angles \n𝜃DL(FL)SAHE and 𝜃DL(FL)SHE. They are related to the SAHE and SHE -like effective fields generated by \nGdFeCo and acting on NiFe layer as follow s  [14,15,34,35] : \nℎDL(FL)SHE=ℏ\n2|𝑒|𝑗cGdFeCo\n𝜇0𝑀sNiFe𝑡NiFe 𝜃DL(FL) SHE, (6𝑎) \nℎDL(FL)SAHE=ℏ\n2|𝑒|(𝒎̂GdFeCo×𝑱cGdFeCo).𝒛̂\n𝜇0𝑀sNiFe𝑡NiFe  𝜃DL(FL)SAHE , (6𝑏) \nwith (𝒎̂GdFeCo×𝑱cGdFeCo).𝒛̂=sin(𝜑𝐻) 𝐽cGdFeCo in our geometry , as depicted in Figure 3a .  \n \n10 \n  \nWe show in the following subs ections that ST -FMR techniques can be  useful tool s to further \nstudy the sign and quantification of the different contributions.  Indeed, t he analytical expression \nfor the lon gitudinal voltage obtained by ST-FMR measurements reads   [15,27,36,37] :  \nVdc=−∆𝑅AMRNiFe\n2sin(2𝜑𝐻)𝐼rf(𝜒𝜑𝜃′𝛿ℎ𝜃+𝜒𝜑𝜑′𝛿ℎ𝜑), (7) \nwhere Δ𝑅AMRNiFe is the anisotropic magnetoresistance amplitude , 𝜒𝜑𝜃′ and 𝜒𝜑𝜑′ are respectively \nthe real part of the 𝜑𝜃 and the 𝜑𝜑 components of the susceptibility matrix  of NiFe . And, 𝛿ℎ𝜃 \nand 𝛿ℎ𝜑 are respectively the polar and azimuthal component of the exciting fi eld 𝛿ℎ (whose \nexpression is discussed in the next subsection). We can see that only the transverse components \nof the excitation fields contribute to the ST -FMR voltage . We will discuss  the different \ncontributions to the total torque: i) first considering the symmetries  of Equation 7 , and  ii) \nadding a dc current which will modify the susceptibility  components.  \nWe highlight here that all the equations and signs that our model describe s have been verified \nby considering the results obtained in a //Pt(5)/NiFe (4) reference system. Namely, in this system , \n𝜃𝐷𝐿=𝜃𝐷𝐿𝑆𝐻𝐸>0 and 𝜃𝐹𝐿=𝜃𝐹𝐿𝑆𝐻𝐸>0 (with a negative Oersted field).    \n \n4. 2. Symmetry of the ST -FMR signal  \nThe NiFe magnetization resonance is triggered by the rf current induced Oersted field,  and the \nspin torques described in Equation 5a  and Equation 5b . We gather the different contributions  \nunder the general term of the exciting field, 𝛿ℎ. Here, the delta means  that the excitation is \nweak.  The dynamics around the equilibrium position, which takes place in the (𝑒̂𝜃,𝑒̂𝜑) plane \nin spherical coordinates, is only sensitive to the polar and azimuthal components of the exciting \nfield 𝛿ℎ𝜃 and 𝛿ℎ𝜑. Since 𝝈̂SAHE lies along the NiFe magnetizatio n equilibrium position \n𝒎̂𝑁𝑖𝐹𝑒=𝒆̂r, the associated SAHE effective fields do not contribute to the magnetization \ndynamics. On the contrary, the effective fields associated to the SHE -like symmetry contribute  \n \n11 \n to the dynamics since 𝝈̂𝑆𝐻𝐸∥ 𝒚̂ and 𝛿ℎ𝜃=ℎDLSHEcos (𝜑𝐻) and 𝛿ℎ𝜑=cos (𝜑𝐻)(ℎOe−\nℎFLSHE)  [36,37] . Since at the resonance  𝜒𝜑𝜑′ is an antisymmetric function of the app lied \nmagnetic field and 𝜒𝜑𝜃′ is a symmetric function, we can express th e symmetrical voltage \n𝑉𝑠𝑦𝑚 amplitude  and the antisymmetrical amplitude  𝑉𝑎𝑛𝑡𝑖 introduced in Equation 1  by replacing \nthe suitable expressions  in Equation 7 :    \n𝑉sym= −sin(𝜑𝐻)1\n4𝐼𝑟𝑓 Δ𝑅AMRNiFe\n𝜇0(2𝐻+𝑀effNiFe) 2π𝑓\n𝛾 ℎDLSHE\nΔ𝐻 , (8) \n𝑉anti= −sin(𝜑𝐻)1\n4𝐼rf Δ𝑅AMRNiFe\n𝜇0(2𝐻+𝑀effNiFe) 2π𝑓\n𝛾 [1+𝑀eff\n𝐻res]1\n2\n ℎOe−ℎFLSHE\nΔ𝐻 .(9) \n \n𝑉𝑠𝑦𝑚 (𝑉𝑎𝑛𝑡𝑖) only depends on ℎ𝐷𝐿𝑆𝐻𝐸(ℎ𝑂𝑒−ℎ𝐹𝐿𝑆𝐻𝐸) but the extraction of the effective fields using \nEquation 8 and Equation 9 is not trivial since  the rf current has to be evaluate d. Nevertheless, \nwe can discuss the signs of the SHE effective fields. As depicted in Figure 3b , 𝑉𝑠𝑦𝑚 is positive  \nwhich me ans that ℎ𝐷𝐿𝑆𝐻𝐸>0. 𝑉𝑎𝑛𝑡𝑖 is negative , and thus  ℎ𝐹𝐿𝑆𝐻𝐸>0 assuming that the Oersted \nfield is lower than the FL effective field.  \n \n4. 3. Adding  a dc bias in ST -FMR: damping modulation and shift of 𝑯𝐫𝐞𝐬 \n \n \nWhen adding a dc bias to the previous ST -FMR measurement, a constant torque is applied on \nthe oscillating magnetization which results in a change in the expression of its dynamical \nsusceptibility matri x. This change induces a modulation of the linewidth and a shift in the \nresonant field, which can be both probed by the ST -FMR technique with an added dc bias. \nBecause the susceptibility is related to the effective field along which the magnetization lies, \nonly the spin polarizations with a projection along this effective field induce a change in the \nsusceptibility. The modulation of damping technique is thus sensitive to both the SHE and \nSAHE -like symmetries and allows to extract overall parameters.   \n \n12 \n In the limit of low current densities where we can neglect strong heating contribution that \ndeformed the linear behavior, we can modify the expressions developed for magnetic tunnel \njunctions  [38,39] , to apply it in our system  [10,15,26] . For the modulation of the NiFe linewidth , \nit reads : \n𝜕𝛥𝐻NiFe\n𝜕𝑖dc=−𝑓\n𝛾𝑁𝑖𝐹𝑒2\n(2𝐻𝑟𝑒𝑠𝑁𝑖𝐹𝑒+𝑀𝑒𝑓𝑓𝑁𝑖𝐹𝑒)𝑆GdFeCo\n𝑊𝑡GdFeCo   (𝝈̂SAHE.𝒎̂NiFe ⏟        \n1𝜕ℎ𝐷𝐿𝑆𝐴𝐻𝐸\n𝜕𝐽𝑐𝐺𝑑𝐹𝑒𝐶𝑜+ 𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟        \nsin(𝜑𝐻)𝜕ℎ𝐷𝐿𝑆𝐻𝐸\n𝜕𝐽𝑐𝐺𝑑𝐹𝑒𝐶𝑜), (10) \nwhere the left -hand term in the equation is the slope of the modulation of NiFe linewidth,  \n𝛾NiFe=𝑔𝑁𝑖𝐹𝑒𝜇𝐵\nℏ . 𝑆GdFeCo  accounts for the shunting of the  GdFeCo layer  by the other conductive \nlayer s, i.e., the current density flowing in GdFeCo layer is 𝐽𝑐GdFeCo=𝑆GdFeCo\n𝑊𝑡GdFeCo𝑖dc with 𝑊 the \nwidth of the slab (10 m). For simplicity, Equation 10 can also be written in terms of the Hall \nangles  using Equation 6a,b  in the following way : \n𝜕𝛥𝐻NiFe\n𝜕𝑖dc=−𝑓\n𝛾NiFe2\n(2𝐻resNiFe+𝑀effNiFe)𝑆GdFeCo\n𝑊𝑡GdFeCo ℏ\n2|𝑒|sin(𝜑𝐻)  𝜃DLSAHE+ 𝜃DLSHE\n𝜇0𝑀𝑠NiFe𝑡NiFe,(11) \nThe slope s 𝜕𝛥𝐻NiFe\n𝜕𝑖dc that account for the linewidth modulation at 8 GHz are displayed in Figure \n4b for 𝜑𝐻=135°  and 𝜑𝐻=−45°. The resistivities were determined independently through \nthe dependence of the GdFeCo and Cu thicknesse s for the different layers  obtaining  𝜌𝐶𝑢=15 \ncm, 𝜌𝐺𝑑𝐹𝑒𝐶𝑜=175 cm, and 𝜌𝑁𝑖𝐹𝑒=40 cm. It follows 𝑆GdFeCo=0.11. We note \nthat the slope s obtained when 𝜑𝐻=135° , for all the different frequencies measured, are \nopposite  than the one s measured for  the //Pt/NiFe reference sample  (not shown) . It indicates \nthat the DL overall spin Hall angle,   𝜃DLSAHE+ 𝜃DLSHE,  is negative and opposite  to the one  of Pt  \nwhere only the SHE is present . From the average of positive and negative dc fields , or 135° and \n-45°, and  for 8, 12 and 14 GHz , we evaluate the overall DL efficiency  𝜃DL𝑆𝐴𝐻𝐸+𝜃𝐷𝐿𝑆𝐻𝐸=\n−0.15±0.05 for the FeCo -rich GdFeCo interfaced with Cu .   \n \n13 \n Furthermore, the same experiment also allows  us to obtain the corresponding field -like \nvalue s, ℎFL and 𝜃FL. Based on the work of ref . [38,39] , we also obtain the following expression  \nthat account s for the linear displacement of the resonance f ield with an added dc current :  \n𝜕𝐻resNiFe\n𝜕𝑖dc=𝑆GdFeCo\n𝑊𝑡GdFeCo  (𝝈̂SAHE.𝒎̂NiFe ⏟        \n1𝜕ℎFLSAHE\n𝜕𝐽𝑐GdFeCo+ 𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟        \nsin(𝜑𝐻)𝜕ℎFLSHE\n𝜕𝐽𝑐GdFeCo )−𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟        \nsin(𝜑𝐻)𝜕ℎOe\n𝜕𝑖dc, (12) \nwhere ℎ𝑂𝑒 is the Oersted field which lies along the −𝒚̂ direction in the geometry of our system. \nIts amplitude can be approximated with ℎOe=−1\n2(𝑗cGdFeCo𝑡GdFeCo+𝑗cCu𝑡Cu). Equation 12 \nreads in terms of the FL Hall angles  (Equation 6a,b ): \n𝜕𝐻resNiFe\n𝜕𝑖dc=sin(𝜑𝐻)[𝑆GdFeCo\n𝑊𝑡GdFeCo (ℏ\n2|𝑒|𝜃FLSAHE+ 𝜃FLSHE\n𝜇0𝑀𝑠NiFe𝑡NiFe)− 𝜕ℎOe\n𝜕𝑖dc] , (13) \nThe slope  obtain ed from the shift of the resonance field  vs. 𝑖dc is displayed in Figure 4c for \ndifferent frequencies . We observe that the slope is frequency -independent in agreement with \nEquation 1 3. Moreover, the slope has the same sign  as the one in  the //Pt/NiFe reference system . \nThat implies that if there is any FL contribution on  the GdFeCo/Cu/NiFe system studied here it \nhas the same sign as for  the //Pt/NiFe. The slope is  evaluate d as 𝜕𝐻resNiFe\n𝜕𝑖dc=0.037 T/A. The \nOersted field is approximated  as  𝜕ℎOe\n𝜕𝑖dc=−0.0476  T/A. Finally, considering  Equation 13, the \noverall FL efficiency is assessed a s 𝜃FLSAHE+𝜃FLSHE=0.026±0.005. This value  has the same \nsign and is comparable  to the one measure d in NiFe/Pt  [37,40] . We have independently \nmeasured a control //Cu/NiF e sample with out a  sizable effect. We can therefore exclude the \nCu/NiFe interface  as the origin behind the FL measured in GdFeCo/Cu/NiFe . The sizable  \noverall FL value would indicate that even though  GdFeCo is not in contact with NiFe, a \nsignificant  FL contribution can  still be detected. The origin of the FL effect in the trilayer is not \nclear at this sta ge. \n  \n \n14 \n  \nFigure 4. Damping modulation and Resonance field shift . (a) Schematic of the NiFe \nresonance condition  with additional 𝑖dc current injected . (b) 𝑖dc dependence of the NiFe \nlinewidth  for a rf frequency of 8 GHz . (c) Resonance field shift vs. 𝑖dc for three frequencies. \nUnlike the damping or linewidth modulation, we can see that the resonance field shift is \nfrequency independent.  \n \n5. Discussion and conclusions  \nThe overall efficiencies for FeCo -rich GdFeCo/Cu/NiFe are evaluated 𝜃DL𝑆𝐴𝐻𝐸+𝜃𝐷𝐿𝑆𝐻𝐸=\n−0.15±0.05 and 𝜃FLSAHE+𝜃FLSHE=0.026±0.005. For sake of comparison, the SAHE \nefficiency of a ferromagnet such as CoFeB is 𝜃𝑆𝐴𝐻𝐸𝐶𝑜𝐹𝑒𝐵=−0.14  [15], and the SHE efficiency of \nPt heavy metal is  𝜃𝑆𝐻𝐸𝑃𝑡=0.056−0.076   [29,41,42] . Seki et al . show in FePt that DL \n𝜃𝑆𝐴𝐻𝐸+𝑆𝐻𝐸𝐹𝑒𝑃𝑡=0.25 from the linewidth modulation  [43].  \nThe damping -like SAHE contribution dominates over the SHE one: |𝜃𝐷𝐿𝑆𝐴𝐻𝐸|>|𝜃𝐷𝐿𝑆𝐻𝐸| with a \nnegative SAHE contribution  for FeCo -rich GdFeCo , and a positive SHE contribution . We also \nshow that the field -like SHE contribution is positive. However, we cannot  estimate the \nindividual value of each contribution.  We perform the same experiments at 15 K where our \nferrimagnet is Gd -rich and its magnetization aligns in -plane with a field above 0.4 T.  From the \nsign of the symmetric contribution we confirm that SHE remains positive when crossing the \nmagnetic compensation temperature. This is consistent with the fact that the SHE does not \ndepend on the GdFeCo magnetic propert ies.  In contrast, we cannot conclude of any 𝜃𝐷𝐿𝑆𝐴𝐻𝐸 sign \nchange  because the modulation of linewidth experiments at 15 K is hidden by others effect that \n \n \n15 \n are out of the scope of this study.  However, the large variation in absolute value between these \nresults and the one previously reported,  for a Gd-rich GdFeCo at room temperature,  |𝜃𝐷𝐿𝑆𝐴𝐻𝐸+\n𝜃𝐷𝐿𝑆𝐻𝐸|=0.80±0.05  [10], suggest that the si gn of 𝜃𝐷𝐿𝑆𝐴𝐻𝐸 changes between FeCo -rich and Gd -\nrich samples . If so, t he opposite DL -SAHE sign for  FeCo -rich GdFeCo might indicate that the \nSAHE spin polarization comes always from the same magnetic sublattice . Despite that , further \nstudies could  be carried out to confirm that.     \nGdFeCo can thus generate efficient spin currents and the different symmetries allow this \nmaterial to be used in a wide variety of devices for spintronics. For instance, the SHE spin \ncurrent can generate self -torque  [10] and can be used for the electrical switching of the \nmagnetization, as shown in epitaxial FePt   [44] or CoTb  [45]. Also, the total spin current \n(SAHE+SHE) can be used to induce a torque on another magnetic layer or for the manipulation \nof skyrmions.  \n \nIn summary, w e have studied  FeCo -rich GdFeCo/Cu/NiFe  heterostructure  at room t emperature . \nFirst, structural,  and chemical analyses were performed by HRTEM and EELS. T hen, the \nmagnetic properties and the relevant spin -orbitronics  parameters  were determined by \ncombining magnetometry, spin-torque ferromagnetic resonance and additional dc current \ndependence . The overall damping -like and field -like efficienc ies, which include the SHE -like \nand the SAHE-like symmetries, are 𝜃𝐷𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐷𝐿𝑆𝐻𝐸=−0.15±0.05 and 𝜃𝐹𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐹𝐿𝑆𝐻𝐸=\n0.026±0.005 at room temperature . We show that  SAHE  dominates over SHE contribution on \nthe DL torque. Furthermore , this study show s that the SHE contribution does not change sign \nwhen crossing the magnetic compensation temperature while SAHE may change sign \ndepending on the dominant  sublattice of the ferrimagnet . All this underlines the importance of \nGdFeCo, and RE -TM ferrimagnets in general, as promising materials in spintronics for the \nexploitation of their strong spin-orbit  torque .  \n  \n \n16 \n  \nData availability  \nThe data that support the fi ndings of this study are available from the corresponding author on \nreasonable request.  \n \nAcknowledgements  \nWe acknowledge A. Fert for fruitful discussions. This  work was supported from Agence \nNationale  de la Recherche (France) under contract ANR -19-CE24 -0016 -01 (TOPTRONIC) , \nANR -20-CE24 -0023 (CONTRABASS),  and ANR -17-CE24 -0025 (TOPSKY), from the \nFrench PIA project “Lorraine Universit é d’Excellence ”, reference ANR -15IDEX -04-LUE  and \nby the « SONOMA» projec t co-funded by FEDER -FSE Lorraine et Massif des Vosges 2014 -\n2020, a European Union Program . DCB and JAS also thanks 2019 and 2021 Master -LUE \nprogram internship. Devices in the present study were patterned at MiNaLor  clean -room \nplatform which is partially s upported by FEDER and Grand Est Region through the RaNGE \nproject.  \n \n \nReferences  \n[1] S. K. Kim, G. S. D. Beach, K. -J. Lee, T. Ono, T. Rasing, and H. Yang, “Ferrimagnetic \nspintronics” Nat. Mater.  21, 24 (2022).  \n[2] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxitia, O. Chubykalo -\nFesenko, S. El Moussaoui, L. Le Guyader, E. Mengotti, L. J. Heyderman, F. Nolting, \nA. 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Oppeneer1\n1Department of Physics and Astronomy, Uppsala University, Box 516, Uppsala, SE-75120, Sweden and\n2Department of Spintronics and Nanoelectronics, Institute of Physics of the Czech Academy of Sciences,\nCukrovarnická 10, CZ - 162 00 Praha 6, Czech Republic\n(Dated: July 20, 2021)\nSpin-nutation resonance has been well-explored in one-sublattice ferromagnets. Here, we investi-\ngate the spin nutation in two-sublattice antiferromagnets as well as, for comparison, ferrimagnets\nwith inter- and intra-sublattice nutation coupling. In particular, we derive the susceptibility of the\ntwo-sublattice magnetic system in response to an applied external magnetic field. To this end, the\nantiferromagnetic and ferrimagnetic (sub-THz) precession and THz nutation resonance frequencies\nare calculated. Our results show that the precession resonance frequencies and effective damping\ndecrease with intra-sublattice nutation coupling, while they increase with inter-sublattice nutation\nin an antiferromagnet. However, we find that the THz nutation resonance frequencies decrease with\nboth the intraandinter-sublattice nutation couplings. For ferrimagnets, conversely, we calculate\ntwo nutation modes with distinct frequencies, unlike antiferromagnets. The exchange-like precession\nresonance frequency of ferrimagnets decreases with intra-sublattice nutation coupling and increases\nwith inter-sublattice nutation coupling, like antiferromagnets, but the ferromagnetic-like precession\nfrequency of ferrimagnets is practically invariant to the intraandinter-sublattice nutation couplings.\nI. INTRODUCTION\nEfficientspinmanipulationatultrashorttimescalesholds\npromise for applications in future magnetic memory tech-\nnology [1–5]. Introduced by Landau and Lifshitz, the time\nevolutionofmagnetization M(r;t),canbedescribedbythe\nphenomenological Landau-Lifshitz-Gilbert (LLG) equation\nof motion, which reads [6–9]\n_M=\u0000\r(M\u0002H) +\u000b\nM0\u0010\nM\u0002_M\u0011\n;(1)\nwith the gyromagnetic ratio \r, constant magnetization am-\nplitudeM0, and Gilbert damping parameter \u000b. The LLG\nequation consists of the precession of spins around a field\nHand transverse damping that aligns the spins towards\nthe field direction. While the spin precessional motion can\nbe explained by Zeeman-like field-spin coupling, there are\nseveral fundamental and microscopic mechanisms leading\nto Gilbert damping [10–22].\nWhen one approaches the femtosecond regime, however,\nthe spin dynamics can not only be described by the tra-\nditional LLG dynamical equation of motion [23, 24], but\nit has to be supplemented by a fast dynamics term due\nto magnetic inertia [25–27]. Essentially, the inclusion of\nmagnetic inertia leads to a spin nutation at ultrashort\ntimescalesandcanbedescribedbyatorqueduetoadouble\ntime-derivative of the magnetization i.e., M\u0002M[26, 28].\nThe inertial LLG (ILLG) equation of motion has the form\n_M=\u0000\r(M\u0002H) +\u000b\nM0\u0010\nM\u0002_M\u0011\n+\u0011\nM0\u0010\nM\u0002M\u0011\n;\n(2)\nwith the inertial relaxation time \u0011. In general, the Gilbert\ndamping\u000band the inertial relaxation time \u0011are ten-\nsors [29], however, for an isotropic system, these param-\neters can be considered as scalars. The emergence of spin\n\u0003mondal@fzu.cznutation has been attributed to an extension of Kamberský\nbreathing Fermi surface model [30, 31], namely, an s\u0000d-\nlike interaction spin model between local magnetization\nand itinerant electrons [32, 33]. Moreover, the ILLG equa-\ntion has been derived from the fundamental Dirac equa-\ntion [20, 29]. Note that the Gilbert damping and inertial\nrelaxation time are related to each other as the Gilbert\ndamping is associated with the imaginary part of the sus-\nceptibility, while the inertial dynamics are associated with\nthe real part of the susceptibility [20, 34]. The characteris-\ntic timescales of the nutation have been predicted to be in a\nrangeof 1\u0000100fs[25,32,35,36]and 1\u000010ps[36,37]. More\nrecently, it has been demonstrated that simple classical me-\nchanical considerations superimposed with Gilbert dynam-\nics naturally lead to magnetic inertial dynamics [38, 39].\nTheoretically, the spin nutation has recently been exten-\nsively discussed for one-sublattice ferromagnets [25, 36, 40–\n44]. The nutation resonance has also been observed in ex-\nperiments, however for two-sublattice ferromagnets [37]. A\nrecent theoretical investigation predicts that the precession\nand nutation resonance frequencies may overlap in two-\nsublattice ferromagnets [45]. The spin nutation resonance\nhas been observed at a higher frequency than ferromagnetic\nresonance, e.g., while the ferromagnetic resonance occurs in\nthe GHz regime, the nutation resonance occurs in the THz\nregime [37, 46]. Moreover, the spin nutation shifts the fer-\nromagnetic resonance frequency to a lower value. Although\nthis shift is very small, the line-width of the resonance de-\ncreases, however, and thus the effective damping decreases,\ntoo.\nSpin nutation effects have not yet comprehensively been\ndiscussed in two-antiparallel aligned sublattice magnetic\nsystems (e.g., antiferromagnets, ferrimagnets). In a recent\ninvestigation, it has been predicted that the spin nutation\nin antiferromagnets may have much significance [46]. Due\nto sublattice exchange interaction, the antiferromagnetic\nresonance frequency lies in the THz regime, while the nu-\ntation resonance frequency has similar order of magnitude.\nThis helps to detect the antiferromagnetic precession andarXiv:2107.07939v2  [cond-mat.mtrl-sci]  19 Jul 20212\nnutation resonances experimentally as they fall in the same\nfrequency range. Moreover, the calculated shift of the anti-\nferromagnetic resonance frequency is stronger than that of\na ferromagnet. Additionally, the nutation resonance peak\nis exchange enhanced [46], which is beneficial for detection\nin experiments. However, the previous investigation only\nconsiders the intra-sublattice inertial dynamics, while the\neffect of inter-sublattice inertial dynamics is unknown.\nIn a previous work, the LLG equation of motion with\ninter-sublattice Gilbert damping has been explored by\nKamra et al.[47]. It was found that the introduction\nof inter-sublattice Gilbert damping enhances the damp-\ning [47–49]. In this study, we formulate the spin dynamical\nequations in a two-sublattice magnetic system with both\nintraandinter-sublattice inertial dynamics as well as in-\nterandintra-sublattice Gilbert damping, extending thus\nprevious work [46]. First, we derive the magnetic suscepti-\nbility with the inter-sublattice effects and compute the pre-\ncession and nutation resonance frequencies. We find that\ntheprecessionresonancefrequencyandtheeffectiveGilbert\ndamping decrease with the intra-sublattice nutation cou-\npling in antiferromagnets, however, they increase with the\ninter-sublattice nutation. Unlike antiferromagnets, we find\nfor ferrimagnets that the change of precession resonance\nfrequencies is more pronounced with both intra and inter-\nsublattice nutation coupling constants in the exchange-like\nmode, but nearly negligible for the ferromagnetic mode.\nThe article is organized as follows. First, in Sec. II, we\ndiscuss the linear-response theory of spin dynamics to cal-\nculate the magnetic susceptibility with the intra and inter-\nsublattice nutation effects. In Sec. III, the precession reso-\nnance frequencies have been calculated with analytical and\nnumerical tools for antiferromagnets (Sec. IIIA) and ferri-\nmagnets (Sec. IIIB). We summarize the obtained results in\nSec. IV.\nII. LINEAR-RESPONSE SUSCEPTIBILITY IN\nTWO-SUBLATTICE MAGNETS\nFor two-sublattice magnetic systems, namely AandB\nrepresentingthetwosublattices, theILLGequationsofmo-\ntion read\n_MA=\u0000\rA(MA\u0002HA) +\u000bAA\nMA0\u0010\nMA\u0002_MA\u0011\n+\u000bAB\nMB0\u0010\nMA\u0002_MB\u0011\n+\u0011AA\nMA0\u0010\nMA\u0002MA\u0011\n+\u0011AB\nMB0\u0010\nMA\u0002MB\u0011\n; (3)\n_MB=\u0000\rB(MB\u0002HB) +\u000bBB\nMB0\u0010\nMB\u0002_MB\u0011\n+\u000bBA\nMA0\u0010\nMB\u0002_MA\u0011\n+\u0011BB\nMB0\u0010\nMB\u0002MB\u0011\n+\u0011BA\nMA0\u0010\nMB\u0002MA\u0011\n: (4)\nIn the above dynamical equations, the first terms relate\nto the spin precession, the second and third terms repre-sent the intraandinter-sublattice Gilbert damping, and\nthe last two terms classify the intraandinter-sublattice\ninertial dynamics. The intra-sublattice magnetization dy-\nnamics has been characterized with the Gilbert damping\nconstants\u000bAA,\u000bBBand inertial relaxation time \u0011AAor\n\u0011BB, while the inter-sublattice dynamics is characterized\nby Gilbert damping \u000bABor\u000bBAand inertial relaxation\ntime\u0011ABor\u0011BA. Note that the Gilbert damping parame-\ntersaredimensionless, however, inertialrelaxationtimehas\na dimension of time [25, 26, 29]. The extended equations\nof motions in Eqs. (3) and (4) represent general magneti-\nzation dynamics for two-sublattice magnets (e.g., antifer-\nromagnets, ferrimagnets, two-sublattice ferromagnets, and\nso on).\nThe free energy of the considered two-sublattice system\nreads\nF(MA;MB) =\u0000H0(MAz+MBz)\u0000KA\nM2\nA0M2\nAz\n\u0000KB\nM2\nB0M2\nBz+J\nMA0MB0MA\u0001MB:(5)\nHere, the first term defines the Zeeman coupling of two\nsublattice spins with an external field H0=H0^z. The\nsecondandthirdtermsrepresenttheanisotropyenergiesfor\nthe sublattice AandB, respectively. The last term can be\nidentified as the Heisenberg exchange energy between the\ntwo sublattices. Note that the Heisenberg coupling energy,\nJ >0for antiferromagnets and ferrimagnets, however J <\n0for ferromagnetic-like coupling.\nWe calculate the effective field in the ILLG equation as\nthe derivative of free energy in Eq. (5) with respect to the\ncorresponding magnetization\nHA=\u0000@F(MA;MB)\n@MA\n=\u0012\nH0+2KA\nM2\nA0MAz\u0013\n^z\u0000J\nMA0MB0MB;(6)\nHB=\u0000@F(MA;MB)\n@MB\n=\u0012\nH0+2KB\nM2\nB0MBz\u0013\n^z\u0000J\nMA0MB0MA:(7)\nFirst, in the ground state, we consider that the Asub-\nlattice magnetization is MA=MA0^z, while the Bsub-\nlattice magnetization is antiparallel MB=\u0000MB0^z, such\nthat we can describe the antiferromagnets ( MA0=MB0)\nand ferrimagnets ( MA0> MB0). We then expand the\nmagnetization around the ground state in small deviations,\nMA=MA0^z+mA(t)andMB=\u0000MB0^z+mB(t). The\nsmall deviations mA=Bare induced by the transverse ex-\nternal field hA=B(t).\nFor convenience, we work in the circularly polar-\nized basis, i.e., mA=B\u0006=mA=Bx\u0006imA=By; hA=B\u0006=\nhA=Bx\u0006ihA=By, and define \nA=\rA=MA0(J+ 2KA+\nH0MA0);\nB=\rB=MB0(J+ 2KB\u0000H0MB0). With\nthe time-dependent harmonic fields and magnetizations\nhA=B\u0006; mA=B\u0006/e\u0006i!t, we obtain the magnetic suscep-\ntibility tensor [46]3\n\u0012\nmA\u0006\nmB\u0006\u0013\n=1\n\u0001\u00060\nBB@1\n\rBMB0\u0000\n\nB\u0006i!\u000bBB\u0000!2\u0011BB+!\u0001\n\u00001\n\rBMA0\u0012\rB\nMB0J\u0006i!\u000bBA\u0000!2\u0011BA\u0013\n\u00001\n\rAMB0\u0012\rA\nMA0J\u0006i!\u000bAB\u0000!2\u0011AB\u00131\n\rAMA0\u0000\n\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!\u00011\nCCA\u0012\nhA\u0006\nhB\u0006\u0013\n=\u001fAB\n\u0006\u0012\nhA\u0006\nhB\u0006\u0013\n; (8)\nwiththedefinitionofthedeterminant \u0001\u0006= (\rA\rBMA0MB0)\u00001\u0000\n\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!\u0001\u0000\n\nB\u0006i!\u000bBB\u0000!2\u0011BB+!\u0001\n\u0000\n(\rA\rBMA0MB0)\u00001\u0010\n\rA\nMA0J\u0006i!\u000bAB\u0000!2\u0011AB\u0011\u0010\n\rB\nMB0J\u0006i!\u000bBA\u0000!2\u0011BA\u0011\n.\nAs one expects, the inter-sublattice Gilbert damping and\ninertial dynamical contributions arise in the off-diagonal\ncomponents of the susceptibility tensor, while the intra-\nsublattice contributions are in the diagonal component of\nthe susceptibility [46]. Note that without inertial dynamics\nterms, the expression for the susceptibility is in accordance\nwith the one derived by Kamra et al.[47].\nTo find the resonance frequencies, the determinant \u0001\u0006\nmust go to zero, thus one has to solve the following fourth-\norder equation in frequency\nA\u0006!4+B\u0006!3+C\u0006!2+D\u0006!+E\u0006= 0;(9)\nwhere the coefficients have the following forms\nA\u0006=\u0011AA\u0011BB\u0000\u0011AB\u0011BA; (10)\nB\u0006=\u0007i (\u000bAA\u0011BB+\u000bBB\u0011AA)\u0000(\u0011AA\u0000\u0011BB)\n\u0006i (\u000bAB\u0011BA+\u000bBA\u0011AB); (11)\nC\u0006=\u00001\u0006i (\u000bAA\u0000\u000bBB)\u0000(\nA\u0011BB+ \nB\u0011AA)\n\u0000\u000bAA\u000bBB+\u0012\rA\nMA0\u0011BA+\rB\nMB0\u0011AB\u0013\nJ\n+\u000bAB\u000bBA; (12)\nD\u0006= (\nA\u0000\nB)\u0006i (\nA\u000bBB+ \nB\u000bAA)\n\u0007i\u0012\rA\nMA0\u000bBA+\rB\nMB0\u000bAB\u0013\nJ; (13)\nE\u0006= \nA\nB\u0000\rA\rB\nMA0MB0J2: (14)\nThe solutions of the above equation (9) result in four dif-\nferent frequencies in the presence of a finite external field.\nTwo of those frequencies can be associated with the mag-\nnetization precession resonance, !p\u0006(positive and negative\nmodes) that exists even without nutation. The other two\nfrequencies dictate the nutation resonance frequencies, !n\u0006\n(positive and negative modes).\nIII. RESULTS AND DISCUSSION\nTheintrinsicintra-sublatticeinertialdynamicshavebeen\ndiscussedextensivelyinRef.[46]. Essentially, theresonance\nfrequencies and effective damping decrease with increasing\nintra-sublattice inertial relaxation time for antiferromag-\nnets and ferrimagnets. Therefore, we consider a constant\nintra-sublatticeinertialrelaxationtimeinthiswork. Inthis\nsection, we specifically discuss the effects of inter-sublattice\nnutation in both antiferromagnets and ferrimagnets.\n-3-2-10123!/2º(THz)0.00.20.40.60.81.01.21.4PAB£ÆAA∞AMA0|hA|2¥= 0,¥0=0¥= 100 fs,¥0=0¥= 100 fs,¥0= 50 fs\n-0.6-0.4-0.200.20.40.600.010.02Figure 1. The calculated dissipated power vs. frequency for an\nantiferromagnet with MA0=MB0= 2\u0016B, and various values\nof the intra- and inter-sublattice nutations parameters, \u0011and\n\u00110. The inset shows the dissipated power close to the precession\nresonance frequencies. The other used parameters are \rA=\n\rB= 1:76\u00021011T\u00001s\u00001,J= 10\u000021J,KA=KB= 10\u000023J,\nH0= 1T,\u000bAA=\u000bBB= 0:05,\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\n\u0011and\u0011AB=\u0011BA=\u00110.\nA. Antiferromagnets\nTo start with, we calculate the frequency-dependent\ndissipated power of an antiferromagnet. Using the\nexpressions for the susceptibility in Eq. (8), we cal-\nculate the dissipated power in the inertial dynamics\nwith the following definition PAB=_mA\u0001hA+_mB\u0001\nhB=1\n2( _mA+hA\u0000+ _mA\u0000hA++ _mB+hB\u0000+ _mB\u0000hB+)\nwhich leads to a complicated expression (not given). For\nconvenience, we define \u000bAA=\u000bBB=\u000b,\u0011AA=\u0011BB=\u0011.\nTo focus on the inter-sublattice nutation \u0011AB=\u0011BA=\u00110,\nwe set the inter-sublattice Gilbert damping to zero, i.e.,\n\u000bAB=\u000bBA= 0, and choose MA0=MB0= 2\u0016B. The\nexchange and anisotropy energies, magnetic moments used\nin the here-presented computations are comparable to a\ntypical antiferromagnetic NiO [23, 50, 51] or CoO [52, 53]\nsystem. However, we mention that NiO or CoO bulk crys-\ntals have biaxial anisotropy. Also, the Gilbert damping of\nNiO is very small \u000b\u001810\u00004, i.e., less than the here-used\nvalue. In contrast, a large spin-orbit coupling in antifer-\nromagnetic CrPt (that has \u00182\u0016BCr moments) leads to a4\n10−1100101102\nη/prime(fs)0.160.20.240.280.32ωp±/2π(THz)\nη= 100 fs,η/prime= 0\nη= 100 fs,η/prime= 0η=η/prime= 0\nη=η/prime= 0(a)\nRe(ωp+)\nRe(ωp−)\nEq. (17)\n10−1100101102\nη/prime(fs)0.160.20.24Im(ωp±)/Re(ωp±)\nη=η/prime= 0 η=η/prime= 0\nη= 100 fs,η/prime= 0 η= 100 fs,η/prime= 0(b)\nIm(ωp+)/Re(ωp+)\nIm(ωp−)/Re(ωp−)\nEq. (18)\nFigure 2. The calculated precession frequencies as a function of inter-sublattice nutation \u00110for an antiferromagnet, setting MA0=\nMB0= 2\u0016B. The data points denote the numerical solution of Eq. (9) and the black lines correspond to the analytical solution\nin Eq. (16). (a) The real part of the resonance frequency, and (b) the ratio of imaginary and real part of the frequency have been\nplotted. The other used parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,J= 10\u000021J,KA=KB=K= 10\u000023J,H0= 1T,\n\u000bAA=\u000bBB=\u000b= 0:05,\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\u0011= 100fs and\u0011AB=\u0011BA=\u00110. The horizontal lines correspond to\nsolutions with zero inter-sublattice nutation ( \u00110= 0). Note that we show Re(!p\u0000)as\u0000Re(!p\u0000).\nhigherGilbertdamping \u000b\u001810\u00002[54,55]. Importantly, the\ninertialrelaxationtimes \u0011and\u00110arenotknowninthesean-\ntiferromagnetic systems. Our simulations pertain therefore\nto typical, selected model systems. We show the evaluated\ndissipated power with and without inertial dynamics for\nsuch antiferromagnet in Fig. 1. Note that the dissipated\npower has already been calculated in Ref. [46], however,\nwithout the inter-sublattice inertial dynamics. We can ob-\nserve that while the intra-sublattice inertial dynamics de-\ncreases the precessional resonance frequencies (see the cyan\nlines in Fig. 1), the inter-sublattice inertial dynamics works\noppositely. Note that the nutation resonance frequencies\ndecrease with the introduction of inter-sublattice inertial\ndynamics.\nTo understand the effect of the inter-sublattice nuta-\ntion terms, first, we solve the Eq. (9), considering again\n\u000bAA=\u000bBB=\u000b,\u0011AA=\u0011BB=\u0011,\u0011AB=\u0011BA=\u00110,\nand\u000bAB=\u000bBA= 0. As the nutation in antiferromagnets\nis exchange enhanced [46], we calculate the effect of inter-\nsublatticetermsontheprecessionandnutationfrequencies,\nsetting\rA=\rB=\randMA0=MB0=M0for antifer-\nromagnets. Therefore, the fourth-order equation in Eq. (9)\nreduces to an equation with AAFM\n\u0006 =\u00112\u0000\u001102,BAFM\n\u0006 =\n\u0007i2\u000b\u0011,CAFM\n\u0006 =\u00001\u0000(\nA+ \nB)\u0011+ 2\r\nM0\u00110J,DAFM\n\u0006 =\n(\nA\u0000\nB)\u0006i (\nA+ \nB)\u000b, andEAFM\n\u0006 = \nA\nB\u0000\u0010\n\r\nM0J\u00112\n.\nThesolutionoftheaboveequationresultsinprecessionand\nnutation resonance frequencies for the two modes (positive\nandnegative). Insertingtherealandimaginarypartsofthe\nsolutions!\u0006=Re(!\u0006)+iIm(!\u0006), we numerically calculate\nthe precession resonance frequencies and effective damping\n(the ratio of imaginary and real frequencies) for an anti-\nferromagnet as a function of inter-sublattice nutation. The\nresults are shown in Fig. 2, where the data points corre-\nspond to the numerical solutions.\nOntheotherhand, thefourth-orderequation, AAFM\n+!4+\nBAFM\n+!3+CAFM\n+!2+DAFM\n+!+EAFM\n+ = 0can analytically\nbe solved using the considerations that KA=KB=K,\nJ\u001dK,M0H0and\u000b\u001c1. Therefore, one has \nA=\nB\u0019\r(J+2K)=M0. Essentially,thefourth-orderequation\nreduces to\n\u0000\n\u00112\u0000\u001102\u0001\n!4\u0000\u0014\n1 + 2\r\u0011(J+ 2K)\nM0\u00002\r\u00110J\nM0\u0015\n!2\n\u00002i\u000b\u0011!3\n(0)+ 2\rH0!(0)+2i\r\u000b\nM0(J+ 2K)!(0)\n+\r2\nM2\n0(J+ 2K)2\u0000\r2J2\nM2\n0\u0000\r2H2\n0= 0; (15)\nwith!(0)being the solution of the above equation for \u000b= 0\nandH0= 0. The solutions of the above equation are rather\nsimple and provide the two precession frequency modes\n(positive and negative) for antiferromagnets. Expanding\nthe solutions of Eq. (15) up to the first order in \u000band\nH0, and also in first order in K=J\u001c1, the precession reso-\nnance frequencies are obtained (neglecting the higher-order\nin!(0)-terms) as\n!p\u0006\u0019\u0006\r\nM0p\n4K(K+J)r\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\n+\rH0+ i\r\u000b\nM0(J+ 2K)\nr\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)j!(0)j\n\r\nM0p\n4K(K+J):\n(16)\nNowsubstitutingthe j!(0)jfromtheleadingterminthefre-\nquency expression into the perturbative terms in Eq. (16),\nthe approximate precession frequencies are obtained as\n!p\u0006\u0019\u0006\r\nM0p\n4K(K+J)r\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\n+\rH0+ i\r\u000b\nM0(J+ 2K)\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110): (17)5\nThis equation has been plotted in Fig. 2 as black lines.\nNote that, for !p\u0000we show for convenience \u0000Re(!p\u0000)in\nFig. 2(a) and in the following. Due to the presence of \u0011\u0000\u00110\nin the denominator of the frequency expressions, the pre-\ncession resonance frequency increases when inter-sublattice\nnutation is taken into account ( \u00110<\u0011), which explains the\nincrease in frequency in Fig. 2(a). At the limit \u0011!\u00110, the\nnutation (intra and inter-sublattice) does not play a signifi-\ncant role as the precession resonance frequency is decreased\nby a factorq\n1 +4\r\u0011K\nM0which is very small due to K\u001cJ.\nNote that the two resonance frequencies are approximately\n0.332 THz and 0.276 THz with \u000b= 0and\u0011=\u00110= 0, while\nthese two frequencies are 0.322 THz and 0.266 THz with\n\u000b= 0:05and\u0011=\u00110= 0. The latter has been shown in Fig.\n2(a) as dashed lines. Therefore, the Gilbert damping has\nalready the effect that it reduces the resonance frequencies.\nThe effective Gilbert damping can be calculated using\nthe ratio between the imaginary and real parts of the fre-\nquencies, i.e., the line width. From Eq. (17) one arrives\nat\nIm(!p\u0006)\nRe(!p\u0006)\u0019\u000b(J+ 2K)p\n4K(K+J)1r\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110):\n(18)\nNote that the two resonance modes have the same effec-\ntive damping. For ferromagnets, the exchange energies do\nnot contribute and thus the effective damping remains the\nsame as\u000b, in the absence of magnetic inertial terms (see\n[45]). However, in antiferromagnets the effective damping\nis enhanced due to the exchange interaction by a factor\n(J+2K)p\n4K(K+J), even without any inertial terms. As investi-\ngated earlier [46], the effective damping decreases with the\nintra-sublattice relaxation time. However, similar to the\nincrease in frequency, the effective damping also increases\nwith the inter-sublattice inertial relaxation time, as seen in\nFig. 2(b). The analytical solution in Eq. (18) agrees ex-\ncellently with the numerical solutions. Close to the limit\n\u00110!\u0011, the effective Gilbert damping in Eq. (18) one ex-\npects the effective damping to be increased by a factor\u0010\n1 +4\r\u0011K\nM0\u0011\u00001=2\n, as can be seen in Fig. 2(b).\nNext, we discuss the field dependence of the reso-\nnance frequencies. The precession resonance frequencies\nand effective damping have been plotted as a function\nof the applied field H0for several inter-sublattice relax-\nation times in Fig. 3. As can be observed, at zero ap-\nplied field, the two modes (positive and negative) coin-\ncide in antiferromagnets, a fact that can be seen from\nEq. (17). However, the applied field induces the splitting\nof these two modes. The frequency splitting scales with\u0014\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\u0015\u00001\n\rH0, meaningthatthesplit-\nting is linear in the applied field, H0. On the other hand,\nat a constant field, the splitting also depends on the inter-\nand intra-sublattice nutation. From Eq. (17), it is clear\nthat the splitting is reduced with intra-sublattice nutation,\nwhile it is enhanced with inter-sublattice nutation. Such a\nconclusion can also be drawn from the numerical solutions\nin Fig. 3(a). The effective damping of the antiferromagnet\n0.160.20.240.280.32Re(ωp±)/2π(THz)\n+\n–+\n–+\n–+\n–(a)\n0.0 0.2 0.4 0.6 0.8 1.0\nH0(T)0.150.20.25Im(ωp±)/Re(ωp±)(b)\nη=η/prime= 0\nη= 100 fs,η/prime= 0\nη= 100 fs,η/prime= 10 fs\nη= 100 fs,η/prime= 50 fsFigure 3. The calculated precession frequencies at several inter-\nsublattice relaxation times as a function of applied field for anti-\nferromagnets using MA0=MB0= 2\u0016B. The solid and dashed\nlinesrepresentthepositiveandnegativemodes, respectively. (a)\nThe real part of the resonance frequencies and (b) the ratio of\nimaginary and real part of the frequency have been plotted. The\nother used parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,J=\n10\u000021J,KA=KB=K= 10\u000023J,\u000bAA=\u000bBB=\u000b= 0:05,\n\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\u0011= 100fs and\u0011AB=\u0011BA=\u00110.\nremains field independent which can be observed in Fig.\n3(b).\nProceeding as previously, we obtain the following nuta-\ntion frequencies\n!n\u0006\u0019\u00061\n\u0011vuuuuut1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\n1\u0000\u001102\n\u00112 \n1\u0000\n(\u00112\u0000\u001102)\r2\nM0\u00024K(J+K)\n2\u0014\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\u00152!\n\u0000\rH0\u0000i\u000b\u0014\u0011\n\u00112\u0000\u001102+\r\nM0(J+ 2K)\u0015\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110):(19)\nNote that at the limit \u00110!0, the nutation frequen-\ncieswithouttheinter-sublatticecouplingarerecovered[46].\nThe dominant term in the calculated frequency is the first\nterm in Eq. (19). With the introduction of inter-sublattice\ncoupling\u00110, both the numerator and denominator of the\ndominant frequency term decrease and therefore, the nuta-\ntion frequencies approximately stay constant (with a slow6\n2.533.54ωn±/2π(THz)(a)\nη= 100 fs,η/prime= 0\nη= 100 fs,η/prime= 0\nRe(ωn+)\nRe(ωn−)\n10−1100101102\nη/prime(fs)0.020.040.060.08Im(ωn±)/Re(ωn±)(b)\nη= 100 fs,η/prime= 0\nη= 100 fs,η/prime= 0\nIm(ωn+)/Re(ωn+)\nIm(ωn−)/Re(ωn−)\nFigure 4. The calculated nutation frequencies as a function\nof inter-sublattice nutation for antiferromagnets using MA0=\nMB0= 2\u0016B. (a) The real part of the nutation resonance fre-\nquencies and (b) the ratio of imaginary and real part of the nu-\ntation resonance frequency have been plotted. The other used\nparameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,J= 10\u000021J,\nKA=KB=K= 10\u000023J,H0= 1T,\u000bAA=\u000bBB=\u000b= 0:05,\n\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\u0011= 100fs and\u0011AB=\u0011BA=\u00110.\ndecrease) with inter-sublattice nutation when \u0011 > \u00110as\nplotted in Fig. 4. However, in the limit \u00110!\u0011, the denom-\ninator vanishes, and thus the nutation frequencies diverge\nas can be seen in Fig. 4. It is interesting to note that\nthe inter-sublattice inertial dynamics increase the preces-\nsion resonance frequencies, however, decrease the nutation\nfrequency. Such observation is also consistent with the dis-\nsipatedpowerinFig.1. Thedampingoftheinertialdynam-\nics also shows a similar behavior: it stays nearly constant\nwith a divergence at the limit \u00110!\u0011.\nAs mentioned before, the inertial relaxation times \u0011and\n\u00110are not known in for typical antiferromagnetic systems.\nNotwithstanding, we obtain the general result that the pre-\ncession resonance frequencies decrease with intra-sublattice\ninertial dynamics, however, increase with inter-sublattice\ninertial dynamics. Thus, to experimentally realize the sig-\nnature of inertial dynamics, an antiferromagnet with a\nhigher ratio of intra to inter-sublattice inertial relaxation\ntime (\u0011=\u00110\u001d1) is better suited.\nB. Ferrimagnets\nNext, we consider a ferrimagnetic system where the mag-\nnetic moments in the two sublattices are different, i.e.,\nMA06=MB0. In this case, the analytical solution of\nEq. (9) becomes cumbersome. The main reason is that\n0.00.20.40.60.81.01.2ωp±/2π(THz)\nη= 0,η/prime= 0η= 0,η/prime= 0\nη= 100 fs,η/prime= 0η= 100 fs,η/prime= 0(a)\nRe(ωp+)\nRe(ωp−)\n10−1100101102\nη/prime(fs)2345ωn±/2π(THz)\nη= 100 fs,η/prime= 0η= 100 fs,η/prime= 0(b)\nRe(ωn+)\nRe(ωn−)Figure 5. The calculated precession and nutation frequencies\nas a function of inter-sublattice nutation for ferrimagnets using\nMA0= 5MB0= 10\u0016B. The real part of the (a) precession reso-\nnance and (b) nutation resonance frequencies have been plotted.\nThe other used parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,\nJ= 10\u000021J,KA=KB=K= 10\u000023J,H0= 1T,\u000bAA=\n\u000bBB=\u000b= 0:05,\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\u0011= 100fs\nand\u0011AB=\u0011BA=\u00110.\n\nA6= \nBfor ferrimagnets, in fact, we calculate \nA\u0000\nB=\n\r(J+2K)(MA0\u0000MB0)\nMA0MB0+2\rH0. For antiferromagnets, the mag-\nnetic moments in the two sublattices are exactly the same,\ni.e.,MA0=MB0and thus, within the approximation of\nJ\u001dM0H0, we find \nA= \nBwhich simplifies the ana-\nlytical solution of Eq. (9). Thus, we numerically solve the\nEq. (9) to calculate the precession and nutation resonance\nfrequencies for ferrimagnets. We consider the case where\nMA0= 10\u0016BandMB0= 2\u0016B, reminiscent of rare-earth–\ntransition-metal ferrimagnets as GdFeCo [2, 3] or TbCo\n[56–58]. However, we emphasize that the inertial relaxation\ntimes\u0011and\u00110are not known for these materials. The\ncalculated precession frequencies are shown in Fig. 5(a).\nThe effect of intra-sublattice inertial dynamics has already\nbeenstudiedinRef.[46]. Forferrimagnets, thenegativefre-\nquencymodeappearstohaveahigherfrequency(i.e.,larger\nnegative) than the positive one. However, both precession\nfrequenciesdecreasewithintra-sublatticerelaxationtime, \u0011\n[46]. We, therefore, have set the intra-sublattice relaxation\ntime\u0011to 100 fs and vary the inter-sublattice relaxation\ntime\u00110< \u0011. The upper precession resonance mode !p\u0000–\nthe exchange-like mode – increases with the inter-sublattice\nrelaxation time \u00110, while the ferromagnetic-like mode !p+\nshows a very small increase. Thus, for ferrimagnets, the\nchange in precession frequencies is more significant in the7\nexchange-like mode than in the ferromagnetic-like mode.\nAtthelimit \u00110!\u0011, theprecessionresonancefrequenciesal-\nmost coincide with the resonance frequencies calculated at\n\u0011=\u00110= 0, meaning that the inertial dynamics do not play\nany role for the precession resonance frequency. The lat-\nter can clearly be seen in Fig. 5(a). These observations are\nsimilar to the antiferromagnet as discussed earlier. The nu-\ntation resonance frequencies in Fig. 5(b) again decline with\nthe inter-sublattice relaxation time showing a divergence\nat the limit \u00110!\u0011. However, one can notice here two dis-\ntinguishable nutation resonance frequencies unlike almost\na single-valued nutation frequencies of antiferromagnets.\nIV. SUMMARY\nIn summary, we have formulated a linear-response theory\nof the ILLG equations for antiferromagnets with inter- and\nintra-sublattice inertial dynamics. The calculation of the\nsusceptibility tensor shows that the intra-sublattice terms\nappear in the diagonal elements, while the inter-sublattice\nterms appear in the off-diagonal elements. The dissipated\npower contains a precession resonance peak in the sub-THz\nregime for antiferromagnets, however, the introduction of\ninertial dynamics causes another peak, a nutation reso-\nnance peak at a higher, few THz frequency. Moreover, we\nobserve that the inter-sublattice inertial dynamics work op-\npositely to the intra-sublattice inertial one. By finding the\npoles of the susceptibility, we calculate the precession andnutation resonance frequencies. While the precession reso-\nnance frequencies decrease with intra-sublattice relaxation\ntime, the inter-sublattice inertial dynamics have the op-\nposite effect. In fact, we observe that the magnetic inertia\ndoesnothaveanyeffectontheantiferromagneticprecession\nresonance at the limit \u00110!\u0011. On the other hand, the THz\nnutation resonance frequency decreases slightly with the\nintroduction of inter-sublattice inertial dynamics, however,\nshowing a divergence at the limit \u00110!\u0011. Our derived an-\nalytical theory explains such inter-sublattice contributions.\nFinally, for ferrimagnets, we find a similar behavior for the\ninter-sublattice inertial dynamics. 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Barry,1,∗Reed A. Irion,1Matthew H. Steinecker,1Daniel K.\nFreeman,1Jessica J. Kedziora,1Reginald G. Wilcox,1, 2and Danielle A. Braje1\n1MIT Lincoln Laboratory, Lexington, Massachusetts 02421, USA\n2Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA\n(Dated: June 1, 2023)\nQuantum sensors offer unparalleled precision, accuracy, and sensitivity for a variety of measure-\nment applications. We report a compact magnetometer based on a ferrimagnetic sensing element in\nan oscillator architecture that circumvents challenges common to other quantum sensing approaches\nsuch as limited dynamic range, limited bandwidth, and dependence on vacuum, cryogenic, or laser\ncomponents. The device exhibits a fixed, calibration-free response governed by the electron gy-\nromagnetic ratio. Exchange narrowing in the ferrimagnetic material produces sub-MHz transition\nlinewidths despite the high unpaired spin density ( ∼1022cm−3). The magnetometer achieves a\nminimum sensitivity of 100 fT/√\nHz to AC magnetic fields of unknown phase and a sensitivity be-\nlow 200 fT/√\nHz over a bandwidth ≳1 MHz. By encoding magnetic field in frequency rather than\namplitude, the device provides a dynamic range in excess of 1 mT. The passive, thermal initial-\nization of the sensor’s quantum state requires only a magnetic bias field, greatly reducing power\nrequirements compared to laser-initialized quantum sensors. With additional development, this de-\nvice promises to be a leading candidate for high-performance magnetometry outside the laboratory,\nand the oscillator architecture is expected to provide advantages across a wide range of sensing\nplatforms.\nFor the published version, refer to Physical Review Applied, DOI: doi.org/10.1103/PhysRevApplied.19.044044\nI. INTRODUCTION\nIn recent years, tremendous experimental effort has ad-\nvanced quantum sensors [1] using unpaired electron spins\nembedded in solid-state crystals. These solid-state sen-\nsors employ electron paramagnetic resonance to achieve\nmeasurement precision and accuracy comparable to their\natomic counterparts, but with advantages such as smaller\nsensing volumes, compatibility with a wide range of ambi-\nent conditions, and fixed sensing axes provided by a rigid\ncrystal lattice. The most-developed solid-state quan-\ntum sensing platform uses negatively-charged nitrogen-\nvacancy (NV) centers in diamond as sensitive magnetic\nfield probes [2, 3]. Such sensors have been used to detect\nor image biological targets [4–9], single proteins [10, 11],\nNMR species [12–16], individual spins [17–20], and con-\ndensed matter phenomena [21–25].\nThough recent efforts have focused on optically-active\nparamagnetic defects [26–30], ferrimagnetic materials of-\nfer distinct advantages for quantum sensors. Ferrimag-\nnetic materials provide higher unpaired electron spin\ndensities than their solid-state paramagnetic counter-\nparts [31], for example ∼1022cm−3versus ∼1016−1019\ncm−3, while the strong coupling of the exchange interac-\ntion mitigates the dipolar resonance broadening observed\nin high-defect-density paramagnetic materials [32, 33].\nImportantly, initialization of ferrimagnetic spins into the\ndesired quantum state requires only a bias magnetic field,\nwithout the need for active optical initialization.\n∗john.barry@ll.mit.eduConsequently, magnetic sensors employing spin-wave\ninterferometry in ferrimagnetic films [34, 35] or ferrimag-\nnetic resonance (FMR) in spheres [36–38] or films [39–42]\nhave been investigated, including demonstrations with\npT/√\nHz-level sensitivity. Using ferrimagnetic materials,\nclassical sensors such as fluxgates [43–45] and Faraday-\nrotation-based devices [46, 47] have achieved sensitivities\ndown to 40 fT/√\nHz and 10 pT/√\nHz, respectively. Ad-\nditionally, ferrimagnetic materials have long found com-\nmercial use in tunable microwave filters [48, 49] and\noscillators [50–52]. Despite these well-developed com-\nmercial technologies however, magnetometry schemes for\nferrimagnetic materials have not previously employed a\nself-sustaining oscillator architecture to encode magnetic\nfields in the output waveform frequency rather than am-\nplitude [53]. We find this architecture provides crucial\nadvantages in performance, capabilities, and simplicity\nof a magnetometer device.\nHere we report a magnetometer using FMR as the\nmagnetically-sensitive frequency discriminator in an elec-\ntronic oscillator. With this construction, the frequency\nof the output voltage signal tracks the FMR frequency,\nwhich varies linearly with the applied magnetic field.\nThis ferrimagnetic oscillator magnetometer exhibits a\nminimum sensitivity of 100 fT/√\nHz to magnetic fields\nnear 100 kHz and sensitivities below 200 fT /√\nHz from 3\nkHz to 1 MHz. As the device encodes the measured mag-\nnetic field directly in frequency, superior dynamic range\nis achieved relative to devices employing amplitude en-\ncoding. In addition, the sensor head is simple, compact,\nand lower power than existing quantum magnetometers\nof comparable sensitivity.arXiv:2305.19938v1  [quant-ph]  31 May 20232\nII. OSCILLATOR ARCHITECTURE\nQuantum sensors based on atomic vapors or electron\nspins in solid-state crystals operate by localizing reso-\nnances which vary with a physical quantity of interest.\nFor example, the ambient magnetic field may be de-\ntermined by measuring a ferrimagnetic material’s uni-\nform precession frequency [54], the paramagnetic reso-\nnance frequency of NVs in diamond [55], or the hyper-\nfine resonance frequency of an alkali vapor [56]. Sev-\neral experimental techniques have been developed for\nthis task, from continuous-wave absorption [27] or dis-\npersion [28, 57] measurements to pulsed protocols such\nas Ramsey [58] or pulsed ESR [59] schemes. In all these\nmethods, externally-generated electromagnetic fields ma-\nnipulate the spin system, and the resonance location is\ndetermined from the system’s resulting response.\nAs an alternative to probing the spin system with ex-\nternal signals, however, an oscillator architecture can be\narranged to generate a microwave (MW) signal that di-\nrectly encodes the spin resonance location. Such an os-\ncillator consists of two main components: a frequency\ndiscriminator and a gain element, arranged in a feedback\nloop.\nThe frequency discriminator can be constructed by\ncoupling input MW signals to the discriminator’s output\nthrough the quantum spins. If the discriminator’s input\nand output are each coupled to the quantum spin reso-\nnance, but not directly to each other, the resulting fre-\nquency discriminator will pass frequencies near the spin\nresonance ωywhile rejecting all others.\nThe needed gain can be provided by an ordinary RF\namplifier; by amplifying the frequency discriminator’s\noutput and returning a fraction of this signal to the dis-\ncriminator’s input, sustained self-oscillation can be re-\nalized [60]. Because only frequencies near the spin res-\nonance ωyare transmitted through the frequency dis-\ncriminator, the resultant oscillation frequency ωcclosely\ntracks the spin resonance.\nThus, the oscillator architecture eliminates the need for\nan external RF source. The limited component count of\nthe oscillator architecture is advantageous for compact-\nness and design simplicity. In addition, the oscillator ar-\nchitecture encodes the spin resonance in frequency, which\ncan offer greater dynamic range and improved linearity\ncompared to amplitude-encoded measurements [61]; dy-\nnamic range is particularly important for a magnetome-\nter, where, for example, detection of a 100 fT signal in\nEarth’s ∼0.1 mT field requires a dynamic range ∼109.\nFor an oscillator to operate at steady state, losses\nthrough the frequency discriminator and other elements\nmust be exactly compensated by the amplifier, produc-\ning unity gain around the oscillator loop. Additionally,\nthe phase-length around the oscillator loop must equal\nan integer number of wavelengths at the steady-state os-\ncillator output frequency. Together, these requirements\nconstitute the Barkhausen criterion, and with reasonable\nassumptions the requirements result in Leeson’s equation[60, 62–64], an empirical model of phase noise amplitude\nspectral density applicable to a wide range of oscillators.\nLeeson’s equation is given by\nL1\n2(fm) =s\n1\n2\u0014f2\nL\nf2m+ 1\u0015\u0014fc\nfm+ 1\u0015\u0014FkBT\nPs\u0015\n,(1)\nwhere L1\n2(fm) is the single-sideband phase noise ampli-\ntude spectral density at offset frequency fmfrom the\ncarrier, fLis the Leeson frequency (equal to the fre-\nquency discriminator’s loaded half width at half maxi-\nmum linewidth), fcis the 1 /fflicker noise corner [60,\n62, 65], Psis the input power to the sustaining ampli-\nfier,Tis the temperature, kBis Boltzmann’s constant,\nandFis the oscillator’s measured wideband noise factor.\nRoughly, Leeson’s equation expresses the phase noise cre-\nated by amplified white thermal noise (the final bracketed\nfactor), enhanced within the bandwidth of the frequency\ndiscriminator via regeneration (the first bracketed fac-\ntor), and further enhanced by flicker noise below the noise\ncorner of the amplifier (the second bracketed factor). Ad-\nditional details of oscillator phase noise are discussed in\nSupplemental Material (SM) Sec. B [66] In Sec. III we\nshow the magnetometer’s noise floor is proportional to\nfm× L1\n2(fm), establishing the oscillator’s phase noise as\nthe principal determinant of magnetometer sensitivity.\nIII. FERRIMAGNETIC RESONANCE\nThe material with the narrowest known ferrimagnetic\nresonance linewidth and lowest known spin-wave damp-\ning is yttrium iron garnet (YIG), a synthetic, insulating\ncrystal ferrimagnet with chemical composition Y 3Fe5O12.\nOther attractive aspects of YIG are low acoustic damp-\ning, less than that of quartz, and well-developed growth\nprocesses which yield samples of high crystal quality [67].\nConsequently, YIG is the prototypical material for cav-\nity spintronics research, and is used in magnon-cavity\ncoupling experiments [68–73], magneto-acoustic coupling\nstudies [74, 75], hybrid quantum circuits [76, 77], and ax-\nion searches [57].\nIn crystallographically-perfect YIG, five of every\ntwenty lattice sites, equivalent to one unit formula\nY3Fe5O12, are populated by trivalent iron (Fe3+, elec-\ntronic spin S= 5/2). The five trivalent iron atoms oc-\ncupy three tetrahedral lattice sites and two octahedral\nlattice sites. Strong superexchange interactions, medi-\nated by oxygen ions between the iron ions, align the\nthree tetrahedral Fe3+antiparallel to the two octahedral\nFe3+in the absence of thermal excitation. The strong\ncoupling between nearby electronic spins results in col-\nlective spin behavior, including resonances between col-\nlective spin states which are observed as ferrimagnetic\nresonances. The strong spin-spin coupling also results\nin exchange-narrowing of the ferrimagnetic resonances,\nallowing sub-MHz transition linewidths despite the high\nunpaired spin density ∼1022cm−3. Narrower resonances3\n Phase \nshifterBias\nmagnetsSustaining\namplifier\nDirectional\ncouplerPhase\nnoise\nanalyzer\nMixer\nLocal oscillatorAnalog front end\n and digitizer\nBuffer\namplifierSwitchYIG sphere\n& coupling loops\nθ\nd)b) c) a)\nOscillator output waveform\nApplied magnetic fieldFerrimagnetic\nresonance frequency\nValue\nTime (s)External magnetic field\nFIG. 1. Oscillator magnetometer principles of operation. (a) In the presence of a uniform external magnetic field, the\nspins of a ferrimagnetic sphere precess in phase. (b) The resonance frequency of the uniform precession mode varies linearly with\napplied magnetic field. (c) By using the ferrimagnetic resonance as a frequency discriminator, an oscillator can be constructed\nwhere the oscillation frequency tracks the ferrimagnetic resonance frequency. (d) Experimental schematic as described in the\nmain text.\nare desired to achieve better oscillator phase noise per-\nformance. Additional relevant properties of YIG are de-\ntailed in SM Sec. C [66].\nKittel’s formula for the uniform precession\nfrequency of ferromagnetic resonance [54] is\nωy=p\n[γBz+(Ny−Nz)γµ0Mz][γBz+(Nx−Nz)γµ0Mz],\nwhere γis the electron gyromagnetic ratio; B=Bzˆz\nis the applied magnetic field and defines the system’s\nˆzaxis; Mzis the magnetization along ˆ z, where Mz\nis assumed equal to the saturation magnetization Ms\nwith no MW power applied; Nx,Ny, and Nzare\nthe demagnetization factors; and all quantities are in\nSI units. Demagnetization factors characterize the\nshape-dependent reduction in internal magnetic field\ndue to the magnetization [78]. For a spherical sample,\nNx=Ny=Nz=1\n3, and Kittel’s formula becomes\nωy=γBz. (2)\nKittel’s above formula neglects crystal anisotropy, but\nsuch effects can be treated perturbatively if needed, as\ndetailed in SM Sec. L [66].\nA ferrimagnetic resonance can be used to implement\nthe frequency discriminator discussed in Sec. II, as shown\nin Fig. 1. Consider two orthogonal circular coupling\nloops with a small ferrimagnetic sphere centered at\nthe intersection of the coupling loop axes, as shown in\nFig. 1d. In the presence of an externally applied DC mag-\nnetic bias field B=B0ˆz, the magnetic domains within\nthe sample align along ˆ z, producing a net magnetization.\nA MW drive signal with angular frequency ωd≈ωy, ap-plied to the input coupling loop, causes the sphere’s mag-\nnetization to precess about the ˆ zaxis [79], as shown in\nFig. 1a. This precessing magnetization then inductively\ncouples to the output coupling loop, and the transmission\nscattering parameter S21obeys\nS21=√κ1κ2\ni(ωd−ωy) +κ0+κ1+κ2\n2e−iπ\n2, (3)\nwhere κ0,κ1, and κ2are the unloaded FMR linewidth,\ninput coupling rate, and output coupling rate, respec-\ntively, in angular frequency units (see SM Sec. D [66]),\nand the π/2 phase retardation arises from the gyrator ac-\ntion of the ferrimagnetic material. The power transmis-\nsion|S21|2exhibits a Lorentzian line shape, with a max-\nimum at the FMR frequency ωyand a loaded full-width\nhalf-maximum (FWHM) linewidth κL≡κ0+κ1+κ2.\nAs discussed above, changes in the external DC mag-\nnetic field alter the FMR frequency ωyand therefore the\noscillator output frequency. The FMR frequency also re-\nsponds to AC magnetic fields; the mechanism by which\nAC fields alter the magnetometer output waveform is dis-\ncussed in SM Sec. F [66]. Operationally, AC magnetic\nfields are encoded as frequency modulation of the oscil-\nlator’s output waveform. For example, an AC magnetic\nfield with root-mean-square (rms) amplitude Brms\nsenand\nangular frequency ωmproduces sidebands at ±ωmrela-\ntive to the oscillator carrier frequency when applied par-\nallel to B0. These two sidebands each exhibit a carrier-4\nnormalized amplitude of\ns=γBrms\nsen√\n2ωm. (4)\nThe oscillator magnetometer’s sensitivity can then be\ndetermined from the sideband amplitude and the mea-\nsured phase noise L1\n2(fm), which represents the back-\nground against which the sidebands are discerned; see\nSM Sec. F and G [66]. The expected sensitivity is\nη(fm) =√\n2fm\nγ/(2π)L1\n2(fm). (5)\nWe note a striking feature of the oscillator magnetome-\nter architecture: assuming the oscillator phase noise is\nwell-described by Leeson’s equation (Eqn. 1), the signal\ns∝1/ωm= 1/(2πfm) and the phase noise amplitude\nspectral density L1\n2(fm) are expected to exhibit nearly\nidentical scaling within a range of frequencies between\nthe noise corner fcand the Leeson frequency fL. Thus,\nthe sensitivity of the device versus frequency fmis ex-\npected to be approximately flat for fc≲fm≲fL, as\ndiscussed in SM Sec. G [66].\nIV. EXPERIMENTAL SETUP\nWhile all presently commercially-available YIG oscil-\nlators [80, 81] employ a reflection architecture, the de-\nvice here employs a transmission (feedback) architec-\nture [82–86]. The transmission oscillator is constructed\nfrom four components connected in a serial feedback loop\nas shown in Fig. 1d: the FMR frequency discriminator\nwhich only passes signals near the ferrimagnetic reso-\nnance ωy, a directional coupler to sample the oscillator\nwaveform for device output, a sustaining amplifier to pro-\nvide the needed gain, and a mechanical phase shifter to\nensure the Barkhausen criterion is satisfied [60].\nThe device’s sensing element is a 1-mm-diameter YIG\nsphere mounted on the end of an insulating ceramic rod.\nAs shown in Fig. 1, two circular coupling loops in the\nxzandyzplanes inductively couple input and output\nMW signals to the YIG sphere. The coupling loops are\nmounted orthogonal to each other so that S21transmis-\nsion occurs only at the FMR frequency and is suppressed\nelsewhere. The values of κ0,κ1, and κ2are determined by\nsimultaneously measuring the S-parameters S11andS21\nof the FMR frequency discriminator using a vector net-\nwork analyzer; see SM Sec. E [66]. We find κ0= 2π×790\nkHz, κ1=2π×315 kHz, and κ2=2π×405 kHz. The total\nloaded linewidth is then κL≡κ0+κ1+κ2= 2π×1.510 MHz,\ncorresponding to a loaded quality factor QL=ωy\nκL=3300\nand a predicted Leeson frequency of fL=1\n2κL\n2π= 755\nkHz.\nTwo cylindrical permanent magnets positioned sym-\nmetrically relative to the YIG sphere create a uniform\nbias magnetic field B0=B0ˆzof approximately 0 .178 T,as depicted in Fig. 1. This value of B0is more than suffi-\ncient to saturate the sphere’s magnetization, so that the\nresponse is governed by ωy(t) =γB(t). The YIG sphere\nis aligned along the zero temperature compensation axis,\nas discussed in SM Sec. L [66]. With this bias magnetic\nfield, the oscillation frequency is ≈2π×5 GHz.\nThe YIG sphere’s precessing magnetization continu-\nously induces a sinusoidal voltage on the output cou-\npling loop at the magnetization’s precession frequency.\nThis MW voltage signal is first amplified and then me-\nchanically phase shifted before passing through a 6 dB\ndirectional coupler, as shown in Fig. 1d. The directional\ncoupler’s through port directs the MW signal back to the\ninput coupling loop, inductively coupling the MW sig-\nnal back to the YIG’s precessing magnetization and clos-\ning the oscillator feedback loop. The mechanical phase\nshifter is adjusted to minimize the device phase noise,\nwhich is measured in real time.\nUnder operating conditions, the input power to the\nsustaining amplifier is Ps≈3 dBm. The sustaining am-\nplifier has a measured gain of 10 dB at Ps= 3 dBm\nso that, after accounting for ≈2 dB of additional loss,\n≈11 dBm of MW power is delivered to the input cou-\npling loop. This MW power is estimated to tip the mag-\nnetization by ≈0.1 radians from the z axis; see SM Sec. H\n[66].\nThe signal sampled by the 6 dB directional coupler\nis first amplified by a buffer amplifier and then sent to\neither a phase noise analyzer for diagnostics and device\noptimization, or to a mixer which downconverts the sig-\nnal to an intermediate frequency, ωi, in the MHz range\nappropriate for a digitizer. The downconverted signal is\ndemodulated to recover the magnetic field time series as\ndescribed in SM Sec. I [66]. All experiments are per-\nformed with the device in an unshielded laboratory envi-\nronment.\nV. EXPERIMENTAL RESULTS\nThe sensitivity of a magnetometer can be determined\nfrom the response to a known applied magnetic field\nalong with the measured noise. As AC magnetic fields\nare frequency-encoded in the oscillator magnetometer’s\n≈5 GHz output waveform, the measured phase noise\nsets the magnetic sensitivity of the device; see SM\nSec. F and G [66]. The oscillator magnetometer’s single-\nsideband phase noise power spectral density L(fm) is\nshown in Fig. 2a. The device realizes a phase noise of\n-132.8 dBc/Hz at 10 kHz offset and -154.4 dBc/Hz at\n100 kHz offset.\nFitting Leeson’s equation (Eqn. 1) to the oscillator’s\nmeasured phase noise above 3 kHz gives fL= 600 kHz,\nF= 8, and fc= 6.6 kHz with the measured Ps≈3 dBm.\nThis value of fL= 600 kHz is in reasonable agreement\nwith the value of fL= 755 kHz expected from the FMR\nfrequency discriminator’s loaded linewidth.\nTo verify the device’s response matches that predicted5\n1001 k1 0k100k1 M-200-180-160-140-120-100-80-60-40-200Single sideband phase  \nnoise L(fm) (dBc/Hz)O\nffset frequency (Hz)\na)\n1001 k1 0k100k1M-60-40-20020406080RMS Signal (dBc)F\nrequency (Hz) b)\n1001 k1 0k100k1 M10-1410-1310-1210-1110-1010-9RMS magnetic field (T/Hz1/2)F\nrequency (Hz) c)\nFIG. 2. Performance of the ferrimagnetic oscillator magnetometer. (a) Single-sideband phase noise power spectral den-\nsityL(fm) of the ferrimagnetic oscillator magnetometer. The single-sideband phase noise is −132.8 dBc/Hz and −154.4 dBc/Hz\nat 10 kHz and 100 kHz offsets from the carrier, respectively. Red depicts smoothed data while the raw phase noise data is\ngray. (b) Measured response ( ■) to a 2.12 µT rms AC magnetic field applied along ˆ z, in agreement with that predicted by\nEqn. 4 ( ). (c) Single-sided magnetic field amplitude spectral density of the ferrimagnetic oscillator magnetometer. The\ndevice achieves a minimum sensitivity of approximately 100 fT /√\nHz at frequencies near 100 kHz, with sensitivities below 200\nfT/√\nHz from 3 kHz to 1 MHz. Blue depicts smoothed data while the raw data is gray. We note that by convention the\nsingle-sideband quantity L(f) is the positive-frequency half of the double-sided phase noise spectral density, distinct from a\nsingle-sided spectrum, which is the sum of positive- and negative-frequency components; see Ref. [87].\nby Eqn. 4, a sinusoidal magnetic field with rms am-\nplitude Brms\nsen= 2.12µT is applied to the sensor, the\nangular frequency ωmof this field is varied, and the\ncarrier-normalized amplitude of the resulting sidebands\nis recorded. The measured data are in excellent agree-\nment with the theoretical prediction of Eqn. 4, as shown\nin Fig. 2b.\nHaving confirmed the device’s frequency response is\nindeed governed by Eqn. 4, the measured phase noise\nspectrum can be converted to a sensitivity spectrum by\nEqn. 5, and the result is shown in Fig. 2c. As dis-\ncussed previously, the sensitivity is expected to be ap-\nproximately flat in the region between the amplifier noise\ncorner at fc≈6.6 kHz and the Leeson frequency fL≈\n600 kHz. The measured data are consistent with this ex-\npectation; for AC signals of unknown phase we observe a\nminimum sensitivity of approximately 100 fT /√\nHz and a\nsensitivity below 200 fT /√\nHz over the band from 3 kHz\nto 1 MHz.\nTo operate the device as a practical magnetometer,\nthe oscillator output is mixed down and digitized. The\nmagnetic field time series is recovered from the digitized\nvoltage waveform as described in SM Sec. I [66]. To con-\nfirm device performance, a 35 kHz sinusoidal test field\nBrms\nsen= 0.9 pT is applied along the sensor’s z axis. The\nresultant amplitude spectral density with the test field\non and off is shown in Figs. 3a and 3b respectively, and\na time series of the 35 kHz signal size with the test field\nchopped on and off is shown in Fig. 3c. All data are\nconsistent with the expected device response and a min-\nimum sensitivity of 100 fT/√\nHz. Supplemental Material\nSec. K [66] details calibration of the test field.VI. DISCUSSION\nThe device phase noise of -132.8 and -154.4 dBc/Hz\nat 10 and 100 kHz offset frequencies compares favor-\nably to the lowest-phase-noise commercial YIG oscilla-\ntors presently available [80, 81]. The best commercial\ndevice we have measured achieves -112 and -134 dBc/Hz\nat 10 and 100 kHz from its 5 GHz carrier frequency.\nThe improved performance of our device is likely mainly\nattributable to a difference in QL; whereas we observe\nfL= 600 kHz ≈1\n2πωy\n2QL, the best commercial oscilla-\ntor measured exhibits fL= 5.2 MHz. This difference in\nfLshould translate to an 18.8 dB improvement in phase\nnoise at offset frequencies below fL, similar to the ob-\nserved difference of approximately 20 dB.\nThe device demonstrated here provides the best AC\nsensitivity achieved to date for a solid-state quantum\nmagnetometer [7, 28, 88–92]. Among quantum magne-\ntometers, this sensitivity performance is surpassed only\nby cryogenic SQUID magnetometers and vacuum-based,\noptically-pumped vapor cell magnetometers. Additional\nsensitivity improvement may be attained by increasing\nthe frequency response to magnetic fields or by decreas-\ning the phase noise. For example, the frequency response\ncould possibly be increased above γ= 2π×28 GHz/T\nin Eqn. 2 using strong cavity coupling schemes [28].\nReference [93] describes such a scheme for a ferrimag-\nnetic system with a predicted frequency response of\n≈2π×500 GHz/T. Cavity-enhanced ferrimagnetic os-\ncillator magnetometers are currently under investigation\nand may be explored in future work.\nOn the other hand, methods to improve oscillator\nphase noise would also improve sensitivity and are well-\nestablished. Increasing the sustaining power Psis a com-\nmon method to improve oscillator phase noise. However,6\n30000320003400036000380004000010-1410-1310-1210-11RMS magnetic field (T/Hz1/2)F\nrequency (Hz)\na)\n30000320003400036000380004000010-1410-1310-1210-11RMS magnetic field (T/Hz1/2)F\nrequency (Hz) b)\n02 04 06 08 00.00.20.40.60.81.01.2RMS magnetic field at 35 kHz (pT)T\nime (s) c)\nFIG. 3. Magnetometer sensitivity determined from magnetic field time series. (a) Single-sided magnetic field\namplitude spectral density in 1 Hz bins with test field Brms\nsen= 0.9 pT applied at 35 kHz. The spectrum is obtained by dividing\na 10-s magnetic field time series into ten 1-s segments, computing the discrete Fourier transform for each segment, adding\ncomponents at positive and negative frequencies in quadrature to convert to single-sided spectra, and rms-averaging the ten\nspectra together. (b) Single-sided magnetic field amplitude spectral density in 1 Hz bins without the test signal applied,\nobtained as in (a). (c) Value of the 35 kHz frequency bin calculated from 1-s of data per point as the Brms\nsen= 0.9 pT signal is\nchopped on and off. As the device is tested in an unshielded environment, a Tukey window with α= 0.01 prevents spectral\nleakage of low-frequency noise. This window is nearly rectangular, as α= 0 and α= 1 correspond to rectangular and Hann\nwindows respectively. The observed 100 fT/√\nHz sensitivity is consistent with that calculated from measured phase noise,\nshown in Fig. 2c.\nthis approach would likely improve phase noise only at\nfrequencies above the flicker noise corner fc, and fcit-\nself may increase with larger values of Ps[60, 65]. Fur-\nther, the maximum usable sustaining power is presently\nbelieved to be limited by instabilities arising from non-\nlinear coupling of the uniform precession mode to spin-\nwave modes [79, 94]. Under some conditions not far\nabove current operating powers, we have seen indications\nthat applying additional power to the YIG causes a bi-\nnary change in the phase noise spectrum of the oscillator,\nwith substantially deteriorated performance.\nOther approaches to improving overall phase noise\nmight focus on the amplifier’s additive phase noise.\nAmplifier-induced phase noise can be mitigated using\noscillator-narrowing techniques such as Pound-Drever-\nHall locking [95–99], carrier suppression interferometric\nmethods [100–105], careful design [106, 107] and other ap-\nproaches [60]. However, even in the ideal case, oscillator-\nnarrowing techniques cannot reduce the oscillator’s phase\nnoise to the thermal noise limit of -177 dBm/Hz expected\nin the absence of Leeson gain. While lowering the Leeson\nfrequency fLwill improve phase noise performance, the\nnoise gain introduced by the Leeson effect appears to be\nfundamental to the oscillator architecture, as discussed\nin SM Sec. G [66].In conclusion, the magnetometer design reported here\noffers a unique combination of state-of-the-art sensitivity\nwith realistic prospects for improvement, high dynamic\nrange, compactness, and low power requirements. These\nadvantages could drive widespread adoption of similar\nquantum sensing devices in the near future. The os-\ncillator architecture can be adapted to simplify high-\nperformance ensemble sensing with a range of quantum\nmaterials and in a variety of sensing applications, such\nas sensing of electric fields [108–110], temperature [111],\nor pressure [112].\nVII. ACKNOWLEDGEMENTS\nThe authors acknowledge Peter F. Moulton, Kerry A.\nJohnson, Liam J. Fitzgerald, and Erik R. Eisenach for\nhelpful discussions. 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Johnson, Optimum thermal com-\npensation axes in YIG and GaYIG ferrimagnetic\nspheres, IEEE Transactions on Magnetics 7, 267 (1971).\n[130] W. A. Yager, J. K. Galt, F. R. Merritt, and E. A. Wood,\nFerromagnetic resonance in nickel ferrite, Phys. Rev. 80,\n744 (1950).\n[131] J. I. Masters, Instability of magnetic resonance in sin-\ngle crystal spheres of yttrium iron garnet, Journal of\nApplied Physics 31, S41 (1960).\n[132] J. Clark, J. Brown, and D. E. Tribby, Temperature sta-\nbilization of gyromagnetic couplers (correspondence),\nIEEE Transactions on Microwave Theory and Tech-\nniques 11, 447 (1963).\n[133] D. J. Craik, Magnetic Oxides (John Wiley and Sons,\n1975).\n[134] P. W. Anderson and H. Suhl, Instability in the motion\nof ferromagnets at high microwave power levels, Phys.\nRev.100, 1788 (1955).\n[135] D. Budker and M. Romalis, Optical magnetometry, Na-\nture Physics 3, 227 (2007).\n[136] D. Budker and M. G. Kozlov, Sensing: Equation one,\narXiv:2011.11043 (2020).\n[137] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert,\nM. B. Plenio, and J. I. Cirac, Improvement of frequency\nstandards with quantum entanglement, Phys. Rev. Lett.\n79, 3865 (1997).\n[138] W. F. Brown, Thermal fluctuations of a single-domain\nparticle, Phys. Rev. 130, 1677 (1963).\n[139] N. Smith, Fluctuation dissipation considerations for\nphenomenological damping models for ferromagnetic\nthin films, Journal of Applied Physics 92, 3877 (2002).\n[140] N. Smith, Modeling of thermal magnetization fluctua-\ntions in thin-film magnetic devices, Journal of Applied\nPhysics 90, 5768 (2001).\n[141] N. Smith, Comment on “Fluctuation-dissipation consid-\nerations and damping models for ferromagnetic materi-\nals”, Phys. Rev. B 74, 026401 (2006).\n[142] N. Smith and P. Arnett, White-noise magnetization\nfluctuations in magnetoresistive heads, Applied Physics\nLetters 78, 1448 (2001).\n[143] V. L. Safonov and H. N. Bertram, Fluctuation-\ndissipation considerations and damping models for fer-\nromagnetic materials, Phys. Rev. B 71, 224402 (2005).12\nSUPPLEMENTAL MATERIAL FOR “A\nFERRIMAGNETIC OSCILLATOR\nMAGNETOMETER”\nA. Variable names and symbols\nA list of variables and constants referenced in this work\nis given in Table I. Where possible we adopt the notation\nof Ref. [60].\nB. Qualitative details of oscillator phase noise\nThe phase noise predicted by Leeson’s equation can be\ninterpreted as follows: Additive phase noise in the oscil-\nlator loop below the Leeson frequency fLpasses through\nthe FMR frequency discriminator and is regeneratively\namplified into the oscillator’s output waveform. Con-\nsequently, the system’s aggregate phase noise depends\nstrongly on both the Leeson frequency fLand additive\nphase noise from components within the oscillator loop.\nThe sustaining amplifier, with both flicker phase noise\nbelow approximately fcand wideband phase noise char-\nacterized by the amplifier noise factor F[65], is a pri-\nmary contributor to this in-loop phase noise. In contrast,\nadditive phase noise introduced by components outside\nthe oscillator loop, such as buffer amplifiers, mixers, and\ndigitizers, experiences no such regenerative gain [60] and\ncontributes much less to the total oscillator phase noise.\nC. Additional details of YIG\nFor a single magnetic domain at absolute zero, YIG ex-\nhibits a net magnetic moment equal to that of one Fe3+\natom per every 20 lattice atoms [113], resulting in a po-\nlarized electron spin density of 2 .1×1022/cm3. Magne-\ntization at room temperature retains 72% of the zero-\ntemperature magnetization [114], equal to a polarized\nelectronic spin density of 1 .5×1022/cm3. For compari-\nson, typical paramagnetic spin systems exhibit spin den-\nsities within a few orders of magnitude of 1017/cm3[29],\nwhile alkali vapor cells operate with a spin density\n∼1013/cm3[115].\nMagnetometer performance depends upon localizing\nthe ferrimagnetic resonance with precision, accuracy and\nspeed. The precision with which the FMR resonance\ncan be localized, and therefore the ambient magnetic\nfield determined, depends on the FMR intrinsic linewidth\nκ0. Single-crystal YIG exhibits the lowest linewidth of\nall known ferromagnetic or ferrimagnetic materials, with\nhighly polished YIG spheres [116] exhibiting measured\nlinewidths of 2 π×560 kHz or below [117–119]. The ma-\nterial employed in this work exhibits a FWHM linewidth\nofκ0≈2π×790 kHz at ωy≈2π×5 GHz.Minimal values of κ0occur when the YIG crys-\ntal’s magnetic domains are uniformly oriented, which is\nachieved by applying an external bias magnetic field with\nsufficient strength to saturate the magnetization [79]. For\npure YIG an external bias field B0≈0.178 T is more\nthan sufficient to saturate the magnetization. Device\noperation in the saturated magnetization regime is im-\nportant to ensure the ferrimagnetic resonance displays a\nconstant response to changes in the externally applied\nmagnetic field, namelydωy\ndB=γ[79, 120].\nThe high spin density and strong coupling between\nspins, which prevents deleterious broadening, allows\n∼100 fT/√\nHz sensitivities to be achieved using crystal\nvolumes ≲1 mm3. Magnetic gradients can compromise\nsensitivity when the gradient broadening becomes com-\nparable to the intrinsic linewidth. Assuming an intrinsic\nlinewidth of κ0= 2π×1 MHz results in a gradient tol-\nerance ∼0.4 mT/cm before substantial degradation of\nsensor performance is expected. This gradient tolerance\ncompares favorably to the ∼30 nT/cm gradient toler-\nance characteristic of alkali vapor magnetometers, which\ntypically have sample volume length scales ∼10×larger\nand intrinsic linewidths ∼103×smaller than the device\nreported here.\nD. Equivalent circuit and S-parameters for a YIG\ntransmission filter\nThe equivalent circuit for a transmission-topology YIG\nwith orthogonal coupling loops (e.g., two coupling loops\nin planes rotated from each other by π/2, the geometry of\nthis device) is shown in Fig. 4 and closely resembles the\nequivalent circuit for a series RLC circuit with inductive\ncoupling [120–123]. The only difference is that the gyra-\ntor action of the YIG introduces a direction-dependent\n(i.e. non-reciprocal) phase shift: S21is retarded by π/2\nwhile S12is advanced by π/2. The Sparameters describ-\ning this system are\nS11= 1−κ1\ni(ωd−ωy) +κ0+κ1+κ2\n2, (6)\nS21=√κ1κ2\ni(ωd−ωy) +κ0+κ1+κ2\n2e−iπ\n2, (7)\nS12=√κ1κ2\ni(ωd−ωy) +κ0+κ1+κ2\n2eiπ\n2, (8)\nS22= 1−κ2\ni(ωd−ωy) +κ0+κ1+κ2\n2, (9)\nwhere ωdis the drive frequency, ωyis the FMR frequency,\nκ0=R\nLis the intrinsic YIG linewidth, κ1=Z0\nn2\n1Lis the\ninput coupling rate, and κ2=Z0\nn2\n2Lis the output cou-\npling rate; the resistance R, the inductance L, the line\nimpedance Z0, and the transformer turns ratios n1and\nn2are parameters of the RLC model. The parameters\nωd,ωy,κ0,κ1, and κ2are all in angular units. The intrin-\nsic linewidth of the FMR filter is given by κ0=ωy\nQ0where13\nTABLE I. Partial list of variables\nName Symbol Approx. value Units\nGyromagnetic ratio ge 2 unitless\nBohr magneton µB 9.274×10−24J/T\nVacuum permeability µ0 1.257×10−6H/m\nBoltzmann constant kB 1.381×10−23J/K\nGyromagnetic ratio γ 2π×28×109rad/(s ·T)\nSystem temperature T 300 K\nOscillator’s measured wideband noise factor F 8 unitless\n1/fflicker noise corner fc 6.6×103Hz\nAmplifier input (sustaining) power Ps 0.002 W\nIntrinsic linewidth κ0 2π×7.90×105rad/s\nInput coupling rate κ1 2π×3.15×105rad/s\nOutput coupling rate κ2 2π×4.05×105rad/s\nYIG resonant frequency ωy 2π×5×109rad/s\nMW drive frequency ωd 2π×5×109rad/s\nOscillator carrier frequency ωc 2π×5×109rad/s\nIntermediate frequency ωi - rad/s\nLoaded linewidth κL=κ0+κ1+κ22π×1.51×106rad/s\nLoaded quality factor QL=ωy/κL 3300 unitless\nUnloaded quality factor Q0=ωy/κ0 6300 unitless\nLeeson frequency fL=κL/(2×2π)∼6.0×105Hz\nS-parameters S11, S12, S21, S22 - unitless\nTotal magnetic field B=B0+Bsen - T\nBias magnetic field B0=B0ˆz 0.1780 T\nTest sensing magnetic field Bsen(t) - T\nTest sensing magnetic field rms amplitude Brms\nsen 2.12×10−6T\nTest sensing magnetic field angular frequency ωm= 2πfm - rad/s\nTest sensing magnetic field frequency fm=ωm/(2π) - Hz\nDemagnization factors Nx, Ny, Nz1\n3,1\n3,1\n3unitless\nSaturation magnetization Ms 1.42×105A/m\nSingle-sideband phase noise power spectral density L(fm) - dBc/Hz\nSingle-sideband phase noise amplitude spectral density L1\n2(fm) - dBc/√\nHz\nTime t - s\nOscillator voltage waveform v(t) - V\nOscillator voltage waveform amplitude V0 - V\nOscillator instantaneous phase ϕ(t) - rad\nOscillator additive phase (noise or signal) φ(t) - rad\nOscillator additive amplitude noise α(t) - unitless\nLine impedance Z0 - Ω\nTransformer turn ratios n1, n2 - unitless\nMagnetic sensitivity at frequency fm η(fm) - T/√\nHz\nYIG sphere volume Vy 5.2×10−10m3\nQ0is the unloaded quality factor of the YIG sphere, ex-\ntracted from measurements performed by sweeping ωd\nwhile B0is fixed. The intrinsic linewidth can also be de-\ntermined by sweeping the value of B0for a fixed value\nofωd, and literature values of YIG linewidths are of-\nten given in magnetic field units rather than frequency\nunits [79]. Eqns. 6-9 are valid only near resonance, both\nbecause of the inclusion of the ideal transformer and be-\ncause the equations have been symmetrized about the\nresonance frequency.\nWhile changing the coupling loop diameters allows ad-\njustment of κ1andκ2, the value of κ0is less easily varied\nand depends in part on the chemical purity of the YIG\nmaterial, the sphere’s surface finish, and the uniformity\nof the bias magnetic field.In practice, we found that the requirement of orthog-\nonal coupling loops is not stringent; minor twisting and\npositional variation of the coupling loops did not produce\nproblematic off-resonant coupling.\nE. Determination of intrinsic linewidth\nWe wish to determine the intrinsic linewidth κ0of the\nuniform mode ferrimagnetic resonance from S-parameter\ndata measured by a vector network analyzer (VNA) on a\ntwo-port YIG transmission filter, as shown in Fig 4. From\nEqns. 7 and 8, we determine the maximum and minimum14\nof the power transmission and reflection, respectively,\n|S11|2\nmin=\u0012\n1−κ1\n(κ0+κ1+κ2\n2)\u00132\n, (10)\n|S21|2\nmax=κ1κ2\u0000κ0+κ1+κ2\n2\u00012. (11)\nThe value of the loaded linewidth,\nκL=κ0+κ1+κ2, (12)\ncan be determined from the distance in frequency be-\ntween the points where |S21|2is reduced by 3 dB from\nits peak value. The system can then be solved for κ0,κ1,\nandκ2, as there are three equations and three unknowns.\nThus, we have\nκ0=κL\n2 \n1 +q\n|S11|2\nmin−|S21|2\nmax\n1−p\n|S11|2\nmin!\n,(13)\nκ1=κL\n2\u0012\n1−q\n|S11|2\nmin\u0013\n, (14)\nκ2=κL\n2 \n|S21|2\nmax\n1−p\n|S11|2\nmin!\n. (15)\nF. Ferrimagnetic oscillator magnetometer theory\nof operation\nFor the uniform mode of ferrimagnetic resonance in a\nspherical sample with saturated magnetization, the time\nderivative of the instantaneous phase ϕ(t) obeys\ndϕ(t)\ndt=γB(t), (16)\nwhere B(t) is the externally applied magnetic field and\nγ=geµB\nℏ(with the electron g-factor ge≈2, the Bohr\nmagneton µB, and the reduced Planck’s constant ℏ, so\nthatγ≈2π×28 GHz/T). For the purpose of this discus-\nsion, we neglect crystal anisotropy (see SM Sec. L) which\nintroduces higher-order terms into Eqn. 16.\nThe precessing magnetization of the sphere described\nby Eqn. 16 inductively couples to the output coupling\nloop, producing a voltage signal which is then ampli-\nfied by the sustaining amplifier and inductively coupled\nback to the precessing magnetization; this closed feed-\nback loop produces sustained self-oscillation as described\nin the main text. The oscillator output voltage is then\nv(t) =V0cos[ϕ(t)] where V0represents the oscillator’s\nvoltage amplitude [124].\nThe oscillator phase ϕ(t) is continuous in time and\ngiven by\nϕ(t) =Zt\n0dϕ(τ)\ndτdτ=Zt\n0γB(τ)dτ, (17)\nwhere we have set ϕ(t= 0)≡0. The total magnetic\nfieldB(t) seen by the sensor is the vector sum of a static\nZ0\nn1:1 1:n2R L CZ0Non-reciprocal\nphase shifter\nπ\n2π\n2FIG. 4. The YIG transmsision filter can be modelled as a\nseries RLC circuit with idealized inductive coupling and a\nnon-reciprocal π/2 phase shifter to account for the gyrator\naction of the ferrimagnetic resonance.\nfieldB0=B0ˆ zcreated by the permanent magnets and\nthe ambient field Bsen(t) external to the sensor. For sim-\nplicity we assume Bsen(t) lies along ˆ z, but the case of\narbitrary Bsen(t) is easily worked out (see SM Sec. N).\nThe value of B0is assumed to exhibit only slow tem-\nporal variation (e.g., due to thermal drift of the mag-\nnets or vibration of the mechanical structure holding the\nmagnets) on time scales below the frequencies of inter-\nest, so that B0can be treated as constant. Then the\ntotal magnetic field seen by the ferrimagnetic sphere is\nB(t) =B0+Bsen(t), allowing the oscillator phase to be\nexpressed as\nϕ(t) =Zt\n0γ\u0002\nB0+Bsen(τ)\u0003\ndτ. (18)\nAn arbitrary real waveform Bsen(t) can be decomposed\ninto its Fourier series as\nBsen(t) =a0\n2+∞X\nn=1ancos(ωnt) +∞X\nn=1bnsin(ωnt).(19)\nFor simplicity we assume Bsen(t) consists of a single spec-\ntral component such that Bsen(t) =√\n2Brms\nsen\u0002\ncosωmt\u0003\n,\nwhere ωmis the angular frequency of the ambient exter-\nnal magnetic field and Brms\nsenis the rms field amplitude.\nWith this simplification, the oscillator phase is\nϕ(t) =Zt\n0γ\u0002\nB0+√\n2Brms\nsen\u0002\ncosωmτ\u0003\u0003\ndτ\n=γB0t+√\n2γBrms\nsen\nωm\u0002\nsinωmt\u0003\n.\nThe oscillator waveform is then\nv(t) =V0cos\u0002\nγB0t+√\n2γBrms\nsen\nωmsin[ωmt]\u0003\n. (20)\nWe now outline the steps to manipulate Eqn. 20 to a\nmore convenient form. A similar derivation can be found\nin Ref. [62]. From the Jacobi-Anger expansion [125], we\ncan derive the Bessel function identity\ncos(ωct+βsin(ωmt)) =∞X\nk=−∞Jk(β) cos\u0002\n(ωc+kωm)t\u0003\n.\n(21)15\nFork≥0,Jk(β) may be expressed to leading order in β\nas [125]\nJk(β)≈\u0012β\n2\u0013k1\nΓ(k+ 1), (22)\nwhere Γ denotes the gamma function, and for integer k\nthe Bessel functions satisfy [125]\nJ−k(x) = (−1)kJk(x). (23)\nForβ≪1, only terms with small |k|contribute. The\nseries in Eqn. 21 can then be approximated using only\nthek=−1,0,1 terms, giving\ncos(ωct+βsin(ωmt))≈cos(ωct)\n+β\n2cos[(ωc+ωm)t]−β\n2cos[(ωc−ωm)t)] (24)\nComparing Eqns. 20 and 24, we identify ωc=γB0and\nβ=√\n2γBrms\nsen\nωm. Then if γBrms\nsen≪ωm, we have β≪1 and\nEqn. 20 can be rewritten as\nv(t)≈V0\u0014\ncos[γB0t] +γBrms\nsen√\n2ωmcos[(ωc+ωm)t] (25)\n−γBrms\nsen√\n2ωmcos[(ωc−ωm)t]\u0015\n.\nFor external fields satisfying γBrms\nsen≪ωm, the B-field\nFM-modulation results in two antisymmetric sidebands\nat±ωm, each with amplitude γBrms\nsen/(√\n2ωm). For ex-\nample, a 1 pT RMS magnetic field at 100 kHz produces\ntwo sidebands each with power -134 dBc.\nG. YIG magnetometer noise\nAs detailed in the preceeding section, a single-\nfrequency AC magnetic field applied to the sensor results\nin frequency modulation of the oscillator’s carrier at an-\ngular frequency ωm. In the frequency domain, this mod-\nulation manifests as two sidebands offset by ±ωmfrom\nthe oscillator’s carrier frequency, each with a carrier-\nnormalized amplitude of\ns=γBrms\nsen√\n2ωm. (26)\nMagnetic field detection then reduces to resolving these\ntwo sidebands from the oscillator’s measured phase noise.\nThe magnetic sensitivity η(fm) may be written as a ra-\ntio between the phase noise amplitude spectral density\nL1\n2(fm) and the signal due to these FM sidebands, each\nwith carrier-normalized amplitude s. Thus the sensitiv-\nity is\nη(fm) =√\n2fm\nγ/(2π)× L1\n2(fm), (27)where L1\n2(fm) is the single-sided phase noise spectral\ndensity of the oscillator. With optimal synchronous\ndetection of a magnetic field with known phase, the\nsensitivity is improved by√\n2 over that expected from\nEqn. 27, which assumes the phase of the B field is un-\nknown. We also note that for realistic oscillators the\nphase noise power spectral density is symmetric about\nthe carrier, and thus there is no improvement to be\ngained by processing both the upper and lower sideband\n(see SM Sec. J).\nTo predict the frequency dependence of the sensitivity,\nwe can apply Leeson’s model of oscillator phase noise to\nEqn. 27. Leeson’s equation for the single-sideband phase\nnoise of an oscillator as a function of the offset frequency\nfmfrom the carrier is [64]\nL1\n2(fm) =s\n1\n2\u0014f2\nL\nf2m+ 1\u0015\u0014fc\nfm+ 1\u0015\u0014FkBT\nPs\u0015\n,(28)\nwhere fL≡1\n2κL\n2πdenotes the Leeson frequency [60], fcis\nthe 1 /fflicker noise corner [60, 62, 65], Psis the input\npower to the sustaining amplifier, Tis the temperature,\nkBis Boltzmann’s constant, and Fis the oscillator’s mea-\nsured wideband noise factor. We note that κLas used in\nthis work is an angular frequency FWHM while fLis a\nnon-angular frequency half-width. Combining Eqns. 27\nand 28 yields an expected sensitivity of\nη(fm) =√\n2fm\nγ/(2π)s\n1\n2\u0014f2\nL\nf2m+ 1\u0015\u0014fc\nfm+ 1\u0015\u0014FkBT\nPs\u0015\n.\n(29)\nEquation 29 suggests that best sensitivity will occur for\nfrequencies fmsatisfying fc< fm< fL. In this region,\nboth the signal and the noise scale as ≈1/fm, resulting\nin an approximately flat frequency response. For the re-\nsults reported here, we find fc= 6.6 kHz and fL≈600\nkHz, and the device’s best sensitivity is observed between\nthose two frequencies as expected, as shown in Fig. 2c. At\nfrequencies below fcor above fL, sensitivity is reduced.\nAt frequencies near or below fc, the flicker noise of the\namplifier (as well as other effects such as thermal drift of\nthe ferrimagnetic resonance or vibration) increases the\noscillator phase noise relative to the signal. For frequen-\ncies near or above the Leeson frequency fL, sensitivity is\ncompromised because the phase noise amplitude spectral\ndensity is independent of fmwhile the signal response\ncontinues to decrease as fmincreases.\nIffmadditionally satisfies fc≪fm≪fL, Eqn. 29\nsimplifies to\nη≈1\n2κL\nγr\nFkBT\nPs. (30)\nWe note that setting F= 1 in Eqn. 30 yields a sensi-\ntivity equivalent to that of an idealized (that is, ther-\nmal noise limited and with ideal amplifier), optimally-\ncoupled ( κ1=κ2=κ0/2, assuming an ideal amplifier\n[126]) transmission interferometer.16\nThe above discussion reflects what appear to be fun-\ndamental limits of oscillator phase noise. Quantitatively\nthe Leeson effect [60, 127] dictates that\nSφ(fm) =\u0014\n1 +f2\nL\nf2m\u0015\nSψ(f), (31)\nwhere Sψ(fm) is the single-sided power spectral density\nof additive phase shifts inside the oscillator loop, and\nSφ(fm) denotes the single-sided power spectral density\nof the oscillator’s output phase noise. As thermal noise\nsets a lower bound on Sψ(fm) and this noise is effec-\ntively enhanced in Sφ(fm) for frequencies below fL, an\noscillator would appear to be forbidden from reaching the\nnaive thermal phase noise limit [ L(fm) =−177 dBm/Hz,\nequivalent to Sφ(fm) =−174 dBm/Hz] for frequencies\nbelow fL, regardless of any oscillator narrowing tech-\nniques that may be used. The same limits are observed\nin inferometric frequency discriminators [101].\nH. Tip angle\nThe RF magnetic field Brftips the precessing mag-\nnetization away from the applied DC magnetic field B0.\nGiven the 11 dBm of MW power applied to the YIG\nsphere, Brf≈2×10−6T is estimated from the known\ngeometry. The tip angle is then calculated using\nθtip= arccos\u00141\n1 +1\n4γ2B2\nrfT1T2\u0015\n. (32)\nForT2=1\nπ×790 kHz= 400 ns, and approximating T1=\nT2/2, we expect θtip≈0.1 radians.\nI. Demodulation\nThe magnetic field B(t) applied to the sensor is en-\ncoded in frequency modulation of the oscillator’s output\nwaveform. By demodulating the output waveform, the\noriginal time-domain magnetic field signal B(t) may be\nrecovered. We describe that process here. The oscilla-\ntor’s instantaneous phase ϕ(t) is governed by Eqn. 18,\nϕ(t) =Zt\n0γ[B0+Bsen(τ)]dτ,\nwhere we assume operation with a sufficiently large bias\nfield to saturate the YIG’s magnetization. Differentiating\nthe oscillator’s instantaneous phase yields\ndϕ(t)\ndt=γ[B0+Bsen(t)]. (33)\nThe time domain magnetic field waveform Bsen(t) is then\ndetermined by calculating\nBsen(t) =1\nγdϕ(t)\ndt−B0. (34)As a practical matter, we note the demodulation process\nis facilitated by applying the Hilbert transform to the\n(real-valued) voltage waveform of the oscillator, produc-\ning a complex signal that allows the instantaneous phase\nϕ(t) to be determined in a simple manner. The phase\nis then “unwrapped” if necessary so that it is continuous\nand free from 2 πjumps, and finallydϕ(t)\ndtis calculated nu-\nmerically using the difference in phase between successive\npoints in the digitized signal.\nJ. Hilbert transform properties\nThe demodulation scheme described above determines\nthe value of the magnetic field from the instantaneous\nphase of the oscillator. We now detail how the Hilbert\ntransform allows the instantaneous phase ϕ(t) to be de-\ntermined, and in particular how ϕ(t) is isolated from\nvariations in the instantaneous amplitude. Use of the\nHilbert transform to achieve this objective requires two\nconditions be met: first, the additive phase noise φ(t)\nmust be small, i.e. |φ(t)| ≪1, and second, the additive\nphase noise φ(t) and additive amplitude noise α(t) must\nvary slowly compared to the intermediate frequency out\nof the mixer ωi(that is, both must have negligible fre-\nquency components above ωi). Both conditions hold for\nthe device in this work.\nThe output of the mixer is digitized and may be written\nas a real-valued waveform,\nv(t) =V0[1 +α(t)] cos[ ωit+φ(t)]. (35)\nGiven |φ(t)| ≪ 1 (the first condition), trigonometric\nidentities and the small angle approximations (cos φ(t)≈\n1 and sin φ(t)≈φ(t)) allow Eqn. 35 to be rewritten as\nv(t)≈V0[1 +α(t)]\u0002\ncos[ωit]−φ(t) sin[ωit]\u0003\n. (36)\nFrom Bedrosian’s theorem, the second condition (that\nα(t) and φ(t) vary slowly compared to ωi) allows the\nHilbert transform of v(t) to be calculated by transforming\nonly the high-frequency components cos[ ωit] and sin[ ωit]\n[128]. Denoting the Hilbert transform of v(t) as ˆv(t), we\nhave\nˆv(t)≈V0[1 +α(t)]\u0002\nsin[ωit] +φ(t) cos[ωit]\u0003\n. (37)\nAgain using small angle approximations and trigonomet-\nric identities, we obtain the resulting signal,\nv(t) +iˆv(t)≈V0[1 +α(t)]ei(ωit+φ(t)). (38)\nThe instantaneous phase of the mixed-down signal ωit+\nφ(t) is easily determined by taking the argument of the\nabove. As the quantity [1 + α(t)] is common to both\nthe real and imaginary components, the additive ampli-\ntude noise α(t) is thereby isolated from the instantaneous\nphase. This approach should be compared to the real-\nvalued waveform of Eqn. 35 where there is no direct way17\nto isolate the instantaneous phase from additive ampli-\ntude noise.\nNote that φ(t) is real, so its double-sided power spec-\ntrum is symmetric about zero frequency. Therefore, in\nthe frequency domain picture, it is clear we cannot gain\nany sensitivity by processing both the positive and neg-\native frequency sidebands, as their information is redun-\ndant.\nK. Test field calibration\nThe test field Brms\nsenis created by a single-turn coil near\nthe sensor. The coil is connected in series with a 50 Ω\nresistor and driven by a function generator. The value\nofBrms\nsen= 0.9 pT used to evaluate sensitivity is checked\nby increasing the function generator’s voltage by 103×\nand measuring the field with a commercial magnetic field\nprobe (Beehive Electronics, 100C EMC Probe). The\ncommercial probe measures an rms field of 1 .2±0.4 nT,\nwith nearly all uncertainty attributed to the probe’s spec-\nified calibration uncertainty. This measurement suggests\nthe field during testing is 1 .2±0.4 pT rms, consistent\nwith the expected value of 0.9 pT rms using our sensor’s\nknown response tied to the electron gyromagnetic ratio.\nL. Crystal anisotropy and frequency shifts\nDue to the Coulomb interaction, the wavefunctions of\nunpaired electrons within a crystal lattice deviate from\nthose of an isolated atom. The distorted spatial wave-\nfunctions couple to the electron spin via the spin-orbit\ninteraction, breaking the isotropy of the spin Hamilto-\nnian. This anisotropy causes the crystal to magnetize\nmore easily along certain directions, giving rise to easy\nand hard magnetization axes, and introduces a crystal-\norientation-dependent term into the FMR frequency. Al-\nthough calculating the value of this term for the general\ncase of an arbitrary-direction magnetic field is somewhat\ninvolved [129], the calculation simplifies for external mag-\nnetic fields confined to lie in the {110}plane. Under these\nconditions, the uniform mode resonant frequency differs\nfrom ωy=γBand is instead given to good approxima-\ntion by [130–132]\nωy≈γ\u0014\nB+K1\nµ0Ms\u0012\n2 +15\n2sin4θ−10 sin2θ\u0013\u0015\n,(39)\nwhere θis the angle in the {110}plane between the\n<100>crystallographic axis and the externally applied\nmagnetic field, andK1\nµ0Ms≈ −4.2 mT for YIG [133].\nEquation 39 suggests that aligning the <111>axis par-\nallel to Bwill result in a resonant frequency higher\nthan γBby≈2π×160 MHz, while alignment of the\nhard axis <100>would result in a resonance lower\nby≈2π×240 MHz. The dependence of the FMRfrequency on the value of K1/Msis approximately re-\nmoved for θ= arcsin\u0014q\n(10−2√\n10)/15\u0015\n≈29.7◦; this\nangular alignment, known as zero temperature compen-\nsation (ZTC), is employed in this work.\nAs the anisotropic contribution to the FMR frequency\nin Eqn. 39 is additive, anisotropy-induced frequency\nshifts are not expected to alter the device response to\nAC magnetic fields to first order in Bsen/B0. Higher-\norder anisotropic effects also exist beyond those included\nin Eqn. 39, but these effects are considered to be negligi-\nble for YIG [129].\nThe ZTC alignment described above can mitigate\ntemperature-induced frequency shifts of the FMR; for\nexample, the YIG sphere’s temperature might vary as\nthe power applied to the YIG sphere fluctuates. How-\never, fluctuations in applied power could shift the YIG\nfrequency in other ways, for example by reducing Mzas\nthe magnetization is tipped away from the z-axis. Here,\nKittel’s formula in the main text illustrates a key advan-\ntage for a sphere over other geometries such as a rod or\nplane [134]; only for a sphere is the FMR frequency in-\ndependent of the magnetization. Thus a sphere protects\nagainst frequency shifts due to changes in the magneti-\nzation along ˆ zfrom fluctuations in applied power.\nM. Fundamental limits for a spin magnetometer\nThe spin-projection-limited magnetic sensitivity ηspl\nfor a spin-based DC magnetometer is [135, 136]\nηspl≈ℏ\ngeµB1p\nNT∗\n2, (40)\nwhere Nis the number of total spins and T∗\n2is the\nfree induction decay time (i.e. dephasing time). Im-\nportantly, Eqn. 40 assumes the Nspins are indepen-\ndent. In YIG, there are 4 .22×1021unit formula of\nY3Fe5O12per cm3, with each unit formula contributing\n5 unpaired electrons. For a 1 mm diameter YIG sphere\nat room temperature [114], the number of unpaired spins\nisN= 8×1018. For a FWHM unloaded linewidth of\n2π×560 kHz ( T∗\n2≈570 ns), we have ηspl= 2.7 aT√s.\nThe relevance of this expression as a measure of the\nfundamental limits of a ferrimagnetic magnetometer re-\nmains unclear, as the strong coupling of nearby spins\nin a ferrimagnet violates the assumption of indepen-\ndent spins. Indeed, the extremely low spin-projection\nlimit calculated for YIG highlights that the limits of this\ntype of magnetometer likely must be understood quite\ndifferently than those of its paramagnetic counterparts.\nWhile the coupling present in ferrimagnets may allow\nentanglement-enhanced sensing schemes which surpass\nthe limit imposed by Eqn. 40 [31, 137], other expres-\nsions may emerge with further study that produce less\noptimistic fundamental limits.\nWe now examine one such additional limit on device\nperformance that may prove to be fundamental. Ther-18\nmal variations in the YIG sphere’s magnetization are ex-\npected to translate to variations in the sphere’s magnetic\nfield. If large enough, such variations could limit the sen-\nsitivity of the device. Using the fluctuation-dissipation\ntheorem, the expected thermal magnetic noise can be es-\ntimated as [138–143]\nηthe≈s\nkBT\nγMsVyQ0, (41)\nwhere Vyis the volume of the YIG sphere. Evaluation\nat room temperature with Ms= 142000 A/m for a 1\nmm diameter YIG sphere with Q0= 5 GHz /560 kHz\n= 8900 gives ηthe≈190 aT√s. Under these conditions,\nthis noise is approximately 70 times higher than the spin\nprojection noise calculated above. We note that several\npapers employ a similar formula [34, 44] to Eqn. 41 but\nwith different prefactors of order unity, so that the ex-\npression above is best understood as a rough estimate of\nthe thermal noise limit. By convention the theoretical\nlimits for ηsplandηtheare referenced to a 1 second mea-\nsurement rather than a 1 Hz bandwidth (as indicated by\nthe units); see Ref. [90].\nFinally, regardless of the fundamental limits, more re-\nstrictive practical limits may emerge. For example, the\ncoupling between spins may produce limits on the power\nthat can be applied to probe the resonance, as this cou-\npling gives rise to degenerate spin-wave modes coupled\nto the uniform precession mode [94].\nN. Errors introduced by the finite bias field\nmagnitude\nNominally, the sensor measures the projection of the\nexternal magnetic field Bsenalong the direction of the\nbias magnetic field B0created by the permanent magnet\npair, so that the device operates as a vector magnetome-\nter. However, slight errors are introduced by components\nofBsenorthogonal to the bias magnetic field B0. We an-\nalyze the origin and magnitude of this error here.\nThe total field seen by the sensor arises both from the\npermanent magnet which is defined to produce field B0\nalong the ˆ zdirection (i.e. B0≡B0ˆz) and the field to\nbe sensed, Bsen. The MW magnetic field applied at the\nFMR frequency is assumed to oscillate rapidly comparedto the sensing bandwidth, allowing its contribution to be\nneglected. With this approximation, the total field seen\nby the sensor is\nB=B0+Bsen. (42)\nNeglecting the perturbative effects of crystal anisotropy\n(see SM Sec. L), the FMR frequency depends only on\n|B|. In the limit where B0≫Bsen, the device responds\nlinearly to the component of Bsenparallel to B0and re-\nsponds weakly to the component of Bsenperpendicular\ntoB0, as will be shown. The external field may be de-\ncomposed as Bsen=B∥\nsen+B⊥\nsen, where B∥\nsenandB⊥\nsenare\nthe external field components parallel and perpendicular\ntoB0, respectively. The scalar field is then\nB=√\nB·B (43)\n=r\u0010\nB0+B∥\nsens\u00112\n+ (B⊥sens)2(44)\n=B0vuut \n1 +B∥\nsens\nB0!2\n+\u0012B⊥sens\nB0\u00132\n(45)\n=B0vuut1 + 2B∥\nsens\nB0+ \nB∥\nsens\nB0!2\n+\u0012B⊥sens\nB0\u00132\n.(46)\nTaylor expanding the final expression above with\nBsen≪B0, we have\nB≈B0+B∥\nsen+\u0000\nB⊥\nsen\u00012\n2B0. (47)\nThe third term in the above expansion provides an esti-\nmate of the error,\nerror≈\u0000\nB⊥\nsen\u00012\n2B0, (48)\nThe maximum error occurs when Bsenis oriented per-\npendicular to the z axis. The error is eliminated when\nBsenis parallel to the z-axis. For a 0 .05 mT external field\nand|B0|= 0.178 T, the maximal error is 7 nT.\nThese errors may be suppressed (by ∼103or more)\nby constructing a full vector magnetometer out of three\nsensors. In this configuration, the measured values from\neach of the three sensors would be combined to refine the\nreconstructed magnetic field vector."    },    {        "title": "1206.6672v1.The_Landau_Lifshitz_Bloch_equation_for_ferrimagnetic_materials.pdf",        "content": "The Landau-Lifshitz-Bloch equation for ferrimagnetic materials\nU. Atxitia, P. Nieves and O. Chubykalo-Fesenko\nInstituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: November 15, 2016)\nWe derive the Landau-Lifshitz-Bloch (LLB) equation for a two-component magnetic system valid\nup to the Curie temperature. As an example, we consider disordered GdFeCo ferrimagnet where\nthe ultrafast optically induced magnetization switching under the action of heat alone has been\nrecently reported. The two-component LLB equation contains the longitudinal relaxation terms\nresponding to the exchange \felds from the proper and the neighboring sublattices. We show that\nthe sign of the longitudinal relaxation rate at high temperatures can change depending on the\ndynamical magnetization value and a dynamical polarisation of one material by another can occur.\nWe discuss the di\u000berences between the LLB and the Baryakhtar equation, recently used to explain\nthe ultrafast switching in ferrimagnets. The two-component LLB equation forms basis for the large-\nscale micromagnetic modeling of nanostructures at high temperatures and ultrashort timescales.\nPACS numbers: 75.78.Jp, 75.40.Mg, 75.40.Gb\nI. INTRODUCTION\nThe Landau-Lifshitz-Bloch (LLB) dynamical equation\nof motion for macroscopic magnetization vector [1] has\nrecommended itself as a valid micromagnetic approach at\nelevated temperatures [2], especially useful for tempera-\nturesTclose to the Curie temperature TC(T >3TC=4)\nand ultrafast timescales. In several exciting novel mag-\nnetic phenomena this approach has been shown to be a\nnecessary tool. These phenomena include laser-induced\nultrafast demagnetization [3{6], thermally driven domain\nwall motion via the spin-Seebeck e\u000bect [7], spin-torque ef-\nfect at elevated temperatures [8, 9] or heat-assisted mag-\nnetic recording [10].\nIn the area of laser-induced ultrafast demagnetization,\nthe LLB equation has been shown to describe adequately\nthe dynamics in Ni [5] and Gd [6]. The main feature of\nthe LLB equation allowing its suitability for the ultra-\nfast magnetization dynamics is the presence of longitu-\ndinal relaxation term coming from the strong exchange\ninteraction between atomic spins. Because the exchange\n\felds are large (10 \u0000100 T), the corresponding character-\nistic longitudinal relaxation timescale is of the order of\n10-100 femtoseconds and thus manifests itself in the ul-\ntrafast processes. The predictions of the LLB equations\nrelated to the linear reversal path for the magnetization\ndynamics [4] as well as to the critical slowing down of the\nrelaxation times at high laser pump \ruency [5] have been\ncon\frmed experimentally.\nIn ferrimagnetic GdFeCo alloys not only the longitudi-\nnal change of magnetization but also a controllable opti-\ncal magnetization switching has been observed, and this\nhas stimulated a great deal of e\u000bort to attempt on many\nlevels to explain this process, see review in Ref. [11].\nThe ferrimagnetic materials consist of at least two an-\ntiferromagnetically coupled magnetic sublattices. The\nmagnetic moments of each sublattice are di\u000berent, lead-\ning to a net macroscopic magnetization M(T) de\fned\nas the sum of magnetization coming from each sublat-\ntice. The main feature of the ferrimagnetic materials isthat at some temperature, called magnetization compen-\nsation temperature TM, the macroscopic magnetization\nis zeroM(TM) = 0, although the magnetization of each\nsublattice is not. The angular momentum compensation\ntemperature, at which the total angular momentum TA\nis zero is also of interest. Simpli\fed considerations of the\nferromagnetic resonance of two-sublattice magnets [12]\npredict that at this temperature the e\u000bective damping is\nin\fnite and this stimulated investigation of the magneti-\nzation reversal when going through angular momentum\ncompensation point [13, 14].\nRecently, K. Vahaplar et al. [4], suggested that the op-\ntically induced ultrafast switching in GdFeCo involves a\nlinear reversal mechanism, proposed theoretically in Ref.\n[15]. This is an especially fast mechanism since it is gov-\nerned by the longitudinal relaxation time, which can be\ntwo orders of magnitude faster than the transverse relax-\nation time governing precessional switching. The mod-\neling of Ref.[4] was based on macrospin LLB approach,\nessentially treating a ferrimagnet as a ferromagnet. The\nmodel showed that in order to have the magnetization\nswitching a strong \feld around 20 T was necessary. This\n\feld can, in principle, come in the experiment with cir-\ncularly polarized light from the inverse Faraday e\u000bect.\nMore recently, T. Ostler et al. [16] used a multi-spin\natomistic approach based on the Heisenberg model show-\ning that the switching occurs without any applied \feld\nor even with the \feld up to 40 T applied in the oppo-\nsite direction. The predictions for the heat-driven re-\nversal were con\frmed in several experiments in magnetic\nthin \flms and dots using linearly polarized pulses. More-\nover, I. Radu et al. [17] used the same atomistic model\nfor the magnetization dynamics to simulate GdFeCo and\ncompared the simulation results to the experimental data\nmeasured by the element-speci\fc x-ray magnetic circular\ndichroism (XMCD). They unexpectedly found that the\nultrafast magnetization reversal in this material, where\nspins are coupled antiferromagnetically, occurs by way of\na transient ferromagnetic-like state.\nThe latter experiments demonstrate the de\fciency inarXiv:1206.6672v1  [cond-mat.mtrl-sci]  28 Jun 20122\napplication of the macrospin ferromagnetic LLB model to\nthe description of the ultrafast dynamics in a ferrimag-\nnetic material GdFeCo. It is clear that the situation of a\nferromagnetic-like state in a ferrimagnetic material can-\nnot be described in terms of a macrospin LLB equation\nin which a ferrimagnet is essentially treated as a ferro-\nmagnet. In a ferromagnetic LLB equation the sublattices\ncannot have their own dynamics and thus the processes\nsuch as the angular momentum transfer between them\nare essentially ignored. In this situation the only pos-\nsible reversal mode is the linear relaxation requiring a\nstrong applied magnetic \feld as was the case of Ref.[4].\nOn a general basis, atomistic models are convenient to\nmodel ferrimagnetic materials but for modeling of larger\nspatial scales, a macroscopic equation similar to ferro-\nmagnetic LLB equation is desirable. This will open a\npossibility to a correct micromagnetic modeling of ferri-\nand antiferromagnetic nano and micro structures at ul-\ntrafast timescales and and/or high temperatures. Addi-\ntionally, this can also allow more correct understanding\nof longitudinal relaxation in two-component (for exam-\nple, ferrimagnetic) compounds, taking into account the\ninter-sublattice exchange.\nIn this article we derive a macroscopic equation for\nthe magnetization dynamics of a two-component system\nvalid at elevated temperatures in the classical case. As\na concrete example, we consider the disordered GdFeCo\nalloy, the cases of two-component ferromagnets as well as\nordered ferrimagnets and antiferromagnets can be easily\ndeduced. Fig.1 shows a sketch of an atomistic model for a\nferrimagnetic material and the corresponding micromag-\nnetic approximation. The atomistic model is based on\nthe classical Heisenberg model for a crystallographically\namorphous ferrimagnetic alloy [18] and the Langevin dy-\nnamics simulations of a set of the Landau-Lifshitz-Gilbert\n(LLG) equations for localized atomistic spins. In the\nmacroscopic approach each sub-lattice is represented by a\nmacrospin with variable length and direction. We use the\nmean \feld approximation (MFA) to derive a macroscopic\nequation of motion for the magnetization of each sublat-\ntice. It contains both transverse and longitudinal relax-\nation terms and interpolates between the Landau-Lifshitz\nequation at low temperatures and the Bloch equation at\nhigh temperatures. We investigate the signs of the relax-\nation rates of both transition (TM) and rare-earth (RE)\nmetals as a function of temperature. We conclude that\nit is a good starting point for performing large scale sim-\nulations in multi-lattice magnetic systems as the LLB\nequation is for ferromagnetic materials [3, 19].\nII. ATOMISTIC MODEL FOR A DISORDERED\nFERRIMAGNET.\nThe models for binary ferrimagnetic alloys of the type\nAxB1\u0000x, randomly occupied by two di\u000berent species ( A\nandB) of magnetic ions have been previously extensively\ninvestigated theoretically [20{22]. In such models Aand\nAtomisticdescription MicromagneticFIG. 1: (Left) Sketch of atomistic regular ferrimagnetic lat-\ntice. Each point represents a magnetic moment associated\nwith an atomic site. Magnetic moments of blue points are\npointing downwards and red ones upwards. (Right) A macro-\nscopic view of partial average magnetization mA=hsAiand\nmB=hsBiby two macrospins in each sublattice as described\nby the Landau-Lifshitz-Bloch equation.\nBions have di\u000berent atomic quantum spin values SAand\nSB(SA6=SB). In the present article we use the classical\ncounterpart of these models by considering the classical\nspins with magnetic moments \u0016A6=\u0016B. We denote A\nspecie as TM and B specie as RE. A further but non\nessential simpli\fcation is to assume that the interactions\nbetween spins in the disordered binary alloy are of the\nHeisenberg form with the exchange interactions di\u000berent\nfor di\u000berent pairs of spins (AA, BB or AB).\nLet us start with the model for a ferrimagnet described\nby the classical Hamiltonian of the type\nH=\u0000NX\ni\u0016iH\u0001si\u0000NX\niDi(sz\ni)2\u0000X\nhijiJijsi\u0001sj;(1)\nwhereNis the total number of spins, ( i; j) are lattice\nsites,\u0016iis the magnetic moment located at lattice site\ni. The external applied \feld is expressed by H. The\nanisotropy is considered as uniaxial with Dibeing the\nanisotropy constant of site i. The third sum is over all\nnearest and next-to-nearest neighbor pairs and we have\nconsidered unit length classical vectors for all lattice sites\njsij= 1. Heisenberg exchange interaction parameter be-\ntween adjacent sites is Jij=JAA(BB)>0 if both sites\n(i;j) are occupied by A(B) type magnetic moments and\nJij=JAB<0 if the sites ( i;j) are occupied by Aand\nBrespectively. We consider that the ordered TM al-\nloy is represented by the fcc-type lattice. To simulate\nthe amorphous character of the TM-RE alloy, x\u0001100%\nlattice sites are substituted randomly with RE magnetic\nmoments.\nThe magnetization dynamics of this model interact-\ning with the bath is described by the stochastic Landau-\nLifshitz-Gilbert (LLG) equation\n_ si=\ri[si\u0002Hi;tot+\u0010i]\u0000\ri\u0015i[si\u0002[si\u0002Hi;tot]] (2)3\nwhere\u0015iis the coupling to the heat bath parameter and\n\riis the gyromagnetic ratio. In what follows and for sim-\nplicity we use the same values for TM and RE, \rTM=\n\rRE=\r= 1:76\u0001107rad s\u00001Oe\u00001,\u0015TM=\u0015RE=\u0015= 0:1.\nThe stochastic thermal \felds \u0010iare uncorrelated in time\nand on di\u000berent lattice sites. They can be coupled to\ndi\u000berent heat baths (via temperature of phonon or elec-\ntron) and could have di\u000berent strength of coupling (via\n\u0015iand\u0016i) for each atom type ( AorB). The correlators\nof di\u000berent components of thermal \feld can be written\nas:\nh\u0010i;\u000b(t)\u0010j;\f(t0)i=2\u0015ikBT\n\u0016i\ri\u000eij\u000e\u000b\f\u000e(t\u0000t0) (3)\nwhere\u000b;\fare Cartesian components and Tis the tem-\nperature of the heat bath to which the spins are coupled.\nThe e\u000bective \felds are given by\nHi;tot\u0011\u00001\n\u0016i@H\n@si=H+2Di\n\u0016isz\niez+1\n\u0016iX\nj2neig(i)Jijsij\nThe particular values for exchange parameters and the\nanisotropy constants (see Table I) are chosen in such a\nway that the static properties coincide with experimental\nmeasurements in GdFeCo [18].\n\u0016=\u0016BD[Joule] J[Joule]\nTransition Metal (TM) 2 :217 8:0725\u000210\u0000244:5\u000210\u000021\nRare-Earth (RE) 7 :63 8:0725\u000210\u0000241:26\u000210\u000021\nTM-RE \u0000 \u0000 \u0000 1:09\u000210\u000021\nTABLE I: Table with parameters of transition metal (TM)\nand rare-earth (RE) compounds. Anisotropy constant\nDTM(RE) is taken equal for both lattices. Exchange parame-\ntersJTM(RE) /per link are taken in order to give correct Curie\ntemperature of pure compounds ( x= 0 pure TM or x= 1\npure RE). Antiferromagnetic exchange parameter JRE-TM is\nchosen so that the temperature dependence of the TM and RE\nsublattices agrees qualitatively with results of XMCD mea-\nsurements of static magnetization [18].\nIII. LLB EQUATION FOR CLASSICAL\nFERRIMAGNET\nA. Equation derivation\nThe idea of the two-component LLB model is presented\nin Fig. 1. Namely, our aim is to evaluate the dynamics of\nthe macrosopic classical polarization m=hsiconf, where\nthe average is performed over temperature as well as the\nmicroscopic disorder con\fgurations.\nThe dynamics of the mean magnetization can be ob-\ntained through the Fokker-Planck equation (FPE) fornon-interacting spins [1]. The FPE for the distribution\nfunction of an ensemble of interacting spins can be de-\nrived in the same way as in the ferromagnetic case [1].\nThe FPE has as the static solution the Boltzmann distri-\nbution function f0(fsig)/exp [\u0000\fH(fsig)], whereHis\ngiven by Eq. (1) and \f= 1=(kBT). Since the exact solu-\ntion is impossible even in the simple ferromagnetic case,\nthen, we resort to the mean \feld approximation (MFA)\nwith respect to spin-spin interactions and random aver-\nage with respect to disorder con\fgurations. In the MFA\nthe distribution function is multiplicative and we can use\nthe same strategy as in the ferromagnetic case [1], we take\nthe distribution function fiof each lattice site i, which\nsatisfy the FPE for a non-interacting spin and perform\nthe substitution H)\nHMFA\n\u0017\u000bconf, where\u0017=TM or RE\nindicates the sublattices. Thus, we start with the para-\nmagnetic LLB equation which was derived in the origi-\nnal article by D. Garanin [1] and is equally valid for the\npresent purpose and substitute the external \feld by the\nMFA one in each sublattice. The corresponding set of\ncoupled LLB equations for each sublattice magnetization\nm\u0017has the following form:\n_ m\u0017=\r\u0017[m\u0017\u0002\nHMFA\n\u0017\u000bconf]\u0000\u0000\u0017;k\u0012\n1\u0000m\u0017m0;\u0017\nm2\u0017\u0013\nm\u0017\n\u0000\u0000\u0017;?[m\u0017\u0002[m\u0017\u0002m0;\u0017]]\nm2\u0017; (4)\nwhere\nm0;\u0017=B(\u00180;\u0017)\u00180;\u0017\n\u00180;\u0017;\u00180;\u0017\u0011\f\u0016\u0017\nHMFA\n\u0017\u000bconf:(5)\nHere\u00180;\u0017\u0011\f\f\u00180;\u0017\f\f,B(\u0018) = coth (\u0018)\u00001=\u0018is the Langevin\nfunction,\n\u0000\u0017;k= \u0003\u0017;NB(\u00180;\u0017)\n\u00180;\u0017B0(\u00180;\u0017);\u0000\u0017;?=\u0003\u0017;N\n2\u0012\u00180;\u0017\nB(\u00180;\u0017)\u00001\u0013\n(6)\ndescribe parallel and perpendicular relaxation, respec-\ntively, \u0003\u0017;N= 2\r\u0017\u0015\u0017=\f\u0016\u0017is the characteristic di\u000busion\nrelaxation rate or, for the thermo-activation escape prob-\nlem, the N\u0013 eel attempt frequency.\nNext step is to use in Eqs. (4) and (5) the MFA ex-\npressions. The MFA treatment for the disordered ferri-\nmagnet has been presented in Ref. [18]. The resulting\nexpressions for the \felds have the following forms:\nhHMFA\nREiconf=H0\ne\u000b,RE+J0;RE\n\u0016REmRE+J0;RE-TM\n\u0016REmTM (7)\nhHMFA\nTMiconf=H0\ne\u000b,TM+J0;TM\n\u0016TMmTM+J0;TM-RE\n\u0016TMmRE(8)\nwhereJ0;TM=qzJTM-TM ,J0;RE=xzJTM-TM ,J0;RE-TM =\nqzJTM-RE ,J0;TM-RE =xzJTM-RE ,zis the number of nearest4\nneighbors between TM moments in the ordered lattice,\nxandq= 1\u0000xare the RE and TM concentrations.\nThe \feld H0\ne\u000b;\u0017contains the external applied and the\nanisotropy \felds acting on the sublattice \u0017=TM,RE.\nThe equilibrium magnetization of each sublattice me;\u0017\nwithin the MFA approach can be obtained via the self-\nconsistent solution of the Curie-Weiss equations\nmRE=B(\u0018RE)\u0018RE\n\u0018RE;mTM=B(\u0018TM)\u0018TM\n\u0018TM:(9)\nThe resulting equation (4) with expressions (7) and (8)\nconstitutes the LLB equation for a ferrimagnet and can\nbe already used for numerical modeling at large scale\nsince in what follows some approximations will be used.\nThe use of these approximations is necessary for under-\nstanding the relaxation of a ferrimagnetic system from\ntheoretical point of view. We will also get the LLB equa-\ntion in a more explicit and compact form.\nWe treat the most general case where the continuous\napproximation in each sub-lattice can be used. Basically,\nin the spirit of the MFA approximation, in each sub-\nlattice we treat the k= 0 mode. In order to handle the\nproblem analytically we decompose the magnetization\nvector m\u0017into two components m\u0017=\u0005\u0017+\u001c\u0017, where\n\u0005\u0017is perpendicular to m\u0014, so that it can be expressed\nas\u0005\u0017=\u0000[m\u0014\u0002[m\u0014\u0002m\u0017]]=m2\n\u0014, and \u001c\u0017is parallel to\nm\u0014, and it can be expressed as \u001c\u0017=m\u0014(m\u0017\u0001m\u0014)=m2\n\u0014,\nwhere\u00146=\u0017.\nWe can shorten the notation by de\fnition of the fol-\nlowing new variable \u0002 \u0017\u0014\n\u0002\u0017\u0014=m\u0017\u0001m\u0014\nm2\u0014=)m\u0017=\u0005\u0017+ \u0002\u0017\u0014m\u0014: (10)\nAs a consequence, the MFA exchange \feld hHMFA\nEX;\u0017iconf\nin Eqs. (7) and (8) can be written as the sum of the ex-\nchange \felds parallel and perpendicular to magnetization\nof the sublattice \u0017.\nhHMFA\nEX;\u0017iconf=\u0012J0;\u0017\n\u0016\u0017+J0;\u0017\u0014\n\u0016\u0017\u0002\u0014\u0017\u0013\nm\u0017+J0;\u0017\u0014\n\u0016\u0017\u0005\u0014\n=eJ0;\u0017\n\u0016\u0017m\u0017+J0;\u0017\u0014\n\u0016\u0017\u0005\u0014\n=Hk\nEX;\u0017+H?\nEX;\u0017 (11)\nwhere we have de\fned a new function eJ0;\u0017(m\u0014;m\u0017) as\neJ0;\u0017=J0;\u0017+J0;\u0017\u0014\u0002\u0014\u0017(m\u0014;m\u0017), we remark that eJ0;\u0017is\nnot a constant but a function of both sublattice magneti-\nzation. The exchange \feld is, therefore, separated in two\ncontributions, a longitudinal one Hk\nEX;\u0017= (eJ0;\u0017=\u0016\u0017)m\u0017\nand a transverse one H?\nEX;\u0017= (J0;\u0017\u0014=\u0016\u0017)\u0005\u0014.\nIn the following we will consider that the transverse\ncontribution is small in comparison to longitudinal one,\ni.e.jHk\nEX;\u0017j\u001djH?\nEX;\u0017j. Finally,\nHMFA\n\u0017\u000bconf'Hk\nEX;\u0017+\nH00\ne\u000b;\u0017where H00\ne\u000b;\u0017=H+HA;\u0017+H?\nEX;\u0017. We now expandm0;\u0017up to the \frst order in H00\ne\u000b;\u0017, under the assumption\f\f\fHk\nEX;\u0017\f\f\f\u001djH00\ne\u000b;\u0017j. Similar to the ferromagnetic case,\nfrom Eq. (5) we get (see Appendix A)\nm0;\u0017'B\u0017\nm\u0017m\u0017+B0\n\u0017\f\u0016\u0017\u0010\nm\u0017\u0001H00\ne\u000b;\u0017\u0011\nm\u0017\nm2\u0017\n\u0000B\u0017\u0016\u0017\nm\u0017eJ0;\u0017h\n[H00\ne\u000b;\u0017\u0002m\u0017]\u0002m\u0017i\nm2\u0017; (12)\nsubstituting this into Eq. (4) and repeating the same\ncalculations as in the ferromagnetic case we get the fol-\nlowing equation of motion\n_ m\u0017=\r\u0017[m\u0017\u0002H00\ne\u000b;\u0017]\n\u0000\r\u0017\u000b\u0017\nk\u00121\u0000B\u0017=m\u0017\n\u0016\u0017\fB0\u0017\u0000m\u0017\u0001H00\ne\u000b;\u0017\nm2\u0017\u0013\nm\u0017\n\u0000\r\u0017\u000b\u0017\n?h\nm\u0017\u0002h\nm\u0017\u0002H00\ne\u000b;\u0017ii\nm2\u0017(13)\nwhereB\u0017=B\u0017\u0010\n\feJ0;\u0017(m\u0017;m\u0014)m\u0017\u0011\ndepends on the\nsublattice magnetizations ( m\u0017;m\u0014) and the damping pa-\nrameters are:\n\u000b\u0017\nk=2\u0015\u0017\n\feJ0;\u0017; \u000b\u0017\n?=\u0015\u0017 \n1\u00001\n\feJ0;\u0017!\n: (14)\nB. Temperature dependence of damping\nparameters\nThe temperature dependence of the damping param-\neters is obtained in the \frst order in deviations of mag-\nnetization from their equilibrium value. Note that in\nEq. (13) all terms are of the \frst order in the parame-\nterH00\ne\u000b;\u0017=HEX;\u0017so that the damping parameters should\nbe evaluated in the zero order in this parameter. Conse-\nquently, we can use the following equilibrium expression:\neJ0;\u0017'J0;\u0017me;\u0017+jJ0;\u0017\u0014jme;\u0014\nme;\u0017(15)\nwhere the sign of the second term does not depend on\nthe sign of the interlattice exchange interaction, J0;\u0017\u0014.\nThe e\u000bective damping parameters depend on tempera-\ntureTvia temperature-dependent equilibrium magneti-\nzation. The temperature dependence of damping param-\neters (14), normalized to the intrinsic coupling parame-\nter, are presented in Fig. 2 for a GdFeCo RE-TM ferri-\nmagnet and for various concentrations of RE impurities.\nLet us consider some limiting cases. First we consider\nthe simplest case of a completely symmetric antiferro-\nmagnet (AFM). In the AFM all the relevant parameters5\nare equal for both lattices, they have the same magnetic\nmoments\u00161=\u00162and the same intra-lattice exchange\nparameters J0;\u0017, the inter-lattice exchange parameter is\nalso the same J0;\u0017\u0014=J0;\u0014\u0017in contrast to our disor-\ndered ferrimagnet. In this case the equilibrium mag-\nnetizations as a function of temperature are the same\nme;\u0017(T) =me;\u0014(T) and the e\u000bective exchange param-\neter reduces to eJ0;\u0017=J0;\u0017+jJ0;\u0017\u0014j,i.e. the sum of\nthe two interactions coming from the intra-lattice and\ninter-lattice exchange. The N\u0013 eel temperature in the MFA\nreadskBTN=eJ0;\u0017=3 and the damping parameters re-\ncover the ferromagnetic type expression\n\u000b\u0017(AFM)\nk=\u0015\u00172T\n3TN; \u000b\u0017(AFM)\n?=\u0015\u0017\u0012\n1\u0000T\n3TN\u0013\n:(16)\nThe use of the critical temperature provides an expres-\nsion in which the damping parameters do not depend\nexplicitly on the interlattice exchange, the implicit de-\npendence comes from the change of the N\u0013 eel tempera-\nture as the exchange parameter J0;\u0017\u0014varies. There is\na more simple AFM, with nearest neighbor interactions\nonly and one inter-lattice exchange parameter J0;\u0017\u0014, it\ngives the same result as above and exactly the same as\nfor the ferromagnet.\nNext interesting case is when one of the three exchange\nparameters can be neglected. We can consider, for exam-\nple, a negligible exchange between the rare-earth mag-\nnetic moments, it is a good approximation if the impurity\ncontent is low. Then we can write the e\u000bective exchange\nas\neJ0;TM=J0;TMme;TM+jJ0;TM-REjme;RE\nme;TM'J0;TM(17)\neJ0;RE=jJ0;RE-TMjme;TM\nme;RE: (18)\nIn this case the TM damping parameters can be approx-\nimately expressed with the antiferromagnetic or ferro-\nmagnetic (TN!TC) formula (16) because in the limit\nx!0 the Curie temperature of the disordered ferri-\nmagnet is close to kBTC=J0;TM=3 [16]. The damp-\ning parameter for the the RE lattice, however, is dif-\nferent. It strongly depends on the polarization e\u000bect of\nthe TM lattice on the RE magnetization. In this case\nclose toTCthe polarization e\u000bect can be expressed us-\ning the expansion, B\u0019\u0018=3, which for this case reads\nme;RE\u0019\fJ0;RE-TMme;TM, thus,eJ0;RE\u00191=(3\f). There-\nfore, we have the following expressions\n\u000bTM\nk=\u0015TM2T\n3TC; \u000bRE\nk=2\n3\u0015RE: (19)\n\u000bTM\n?=\u0015TM\u0012\n1\u0000T\n3TC\u0013\n; \u000bRE\n?=2\n3\u0015RE:(20)\nThis relation becomes quite important above TC. We\nobserve in Fig. 2 that even for quite large amounts of\nRE of 25% and 50%, the above approximation holds quite\nwell.\nRE(x= 0.25)TM(x= 0.25)FM(x= 0)2\n3αν⊥/λν αν/bardbl/λν1.0\n0.8\n0.6\n0.4\n0.2\n0.0\nRE(x= 0.5)TM(x= 0.5)FM(x= 0)2\n3αν⊥/λν αν/bardbl/λν1.0\n0.8\n0.6\n0.4\n0.2\n0.0\nRE(x= 0.75)TM(x= 0.75)FM(x= 0)2\n3αν⊥/λν αν/bardbl/λν\nϑ[T/TC]10.80.60.40.201.0\n0.8\n0.6\n0.4\n0.2\n0.0FIG. 2: Damping parameters \u000b\u0017\nk(?)(#) (normalized to the\ncorresponding intrinsic values) for a pure ferromagnet (FM),\nrare earth (RE) component in a GdFeCo ferrrimagnet and\na transition metal (TM) in a ferrimagnet as a function of\nreduced temperature #=T=TCfor three di\u000berent rare earth\n(RE) concentrations x. The blue solid line represents the\nx= 0 limit which corresponds to a pure ferromagnet (FM).\n(Up) The corresponding curves for a 25% concentration of\nRE. (Middle) The corresponding damping parameters for a\n50% alloy. (Bottom) Damping values for 75% RE amount.\nIt can be also seen as a RE doped with a 25% of transition\nmetal (TM).\nIf the inter-lattice exchange is large in comparison to\nthe intra-lattice one then the equilibrium magnetization\nof both lattices is similar and the damping parameters\nbehave similar to those of the FM damping parameters,\npresented above. This case is in agreement with a con-\ncentration of 75% of RE in Fig. 2 (down). As predicted,\nwe observe that the damping parameters are very similar6\nfor both sublattices.\nNote that these damping parameters should be dis-\ntinguished from those of the normal modes (FMR and\nexchange) with more complicated expressions which can\nbe obtained via linearization of the set of two-coupled\nLLB equations [28], similar to the LLG approach.\nC. Longitudinal relaxation parameters\nThe function 1\u0000B\u0017=m\u0017in Eq. (13) is a small quan-\ntity proportional to the deviation from the equilibrium in\nboth sublattices. It can be further simpli\fed as a func-\ntion of the equilibrium parameters after some algebra.\nSimilar to the ferromagnetic case, the ferrimagnetic LLB\nequation can be put in a compact form using the notion\nof the longitudinal susceptibility.\nThe initial longitudinal susceptibility can be evalu-\nated on the basis of the Curie-Weiss equations (9). Let\nus assume that in the absence of an external \feld, the\nequilibrium sublattice magnetizations mTMandmREare,\nrespectively, parallel and antiparallel to the z-axis (a\nstronger condition of the smallness of the perpendicular\ncomponents can be also applied). The z-axis is chosen\nsuch that it is the easy axis of the magnetic crystal. To\nevaluate the longitudinal susceptibility, the \feld should\nbe applied parallel to the easy direction, then in the ap-\nproximation of small perpendicular components (large\nlongitudinal exchange \feld) we can neglect in the \frst\napproximation the possible change of directions of mREandmTM. In order to calculate the susceptibility, we\nexpand the right-hand side of Eq. (9) in terms of the\nexternal \feld:\nm\u0017(T;Hz)\u0019m\u0017(T;0) +\u0016\u0017Hz\fB0\n\u0017\u0012\n1 +@Hz\nEX;\u0017\n@Hz\u0013\n;\n(21)\nwhereB\u0017=B\u0017(\f\u0016\u0017HEX;\u0017) and its derivative B0\n\u0017=\nB0\n\u0017(\f\u0016\u0017HEX;\u0017) are evaluated in absence of applied and\nanisotropy \felds. Then,\ne\u001f\u0017;jj=\u0012@m\u0017(T;Hz)\n@Hz\u0013\nHz=0=\u0016\u0017\fB0\n\u0017\u0012\n1 +@Hz\nEX;\u0017\n@Hz\u0013\n;\n(22)\nwhere\n@Hz\nEX;\u0017\n@Hz=\fJ0;\u0017e\u001f\u0017;jj+\fjJ0;\u0017\u0014je\u001f\u0014;jj:\nThus, the longitudinal susceptibility of one sublattice is\nexpressed in terms of another:\ne\u001f\u0017;jj=\u0016\u0017\nJ0;\u0017J0;\u0017\fB0\n\u0017\n1\u0000J0;\u0017\fB0\u0017\u0014jJ0;\u0017\u0014j\n\u0016\u0017e\u001f\u0014;jj+ 1\u0015\n:(23)\nFinally, we obtain two coupled equations for e\u001fRE;jjand\ne\u001fTM;jj, solving them, we get the MFA expression for the\nsusceptibilities:\ne\u001f\u0017;jj=\u0012\u0016\u0014\njJ0;\u0014\u0017j\u0013jJ0;\u0014\u0017j\fB0\n\u0017jJ0;\u0017\u0014j\fB0\n\u0014+ (\u0016\u0017=\u0016\u0014)jJ0;\u0014\u0017j\fB0\n\u0017(1\u0000J0;\u0014\fB0\n\u0014)\n(1\u0000J0;\u0017\fB0\u0017) (1\u0000J0;\u0014\fB0\u0014)\u0000(jJ0;\u0014\u0017j\fB0\u0017) (jJ0;\u0017\u0014j\fB0\u0014)=\u0012\u0016\u0014\njJ0;\u0014\u0017j\u0013\nG\u0017(T) (24)\nThe longitudinal susceptibility e\u001f\u0017;jjis, therefore, a func-\ntion of temperature which we have called G\u0017(T). It\ntends to zero at low temperature and diverges approach-\ning Curie temperature TCof the magnetic system, sim-\nilar to the ferromagnetic case. The function G\u0017=\n(jJ0;\u0017\u0014j=\u0016\u0017)e\u001f\u0017;jjcan be seen as a reduced longitudinal\nsusceptibility.\nNow we derive an approximate expression for the small\nquantity 1\u0000B\u0017=m\u0017as a function of equilibrium quan-\ntities and the deviation of each sublattice magnetization\nfrom its equilibrium. In the \frst approximation, we ex-\npand the function B\u0017=m\u0017near the equilibrium, as was\ndone for the ferromagnet. The function B\u0017in the zero\norder in perpendicular \feld components, H00\neff;\u0017=HEX;\u0017,\ncan be written as a function of m\u0017andm\u0014as follows\nB\u0017\u0019B\u0017(\f[J0;\u0017m\u0017+jJ0;\u0017\u0014j\u001c\u0014]) (25)\nwhere\u001c\u0014=j(m\u0017\u0001m\u0014)j=m\u0017is the length of the projectionof the magnetization of the sublattice \u0014onto the sublat-\ntice\u0017. We expand the function B\u0017=m\u0017in the variables\nm\u0017andm\u0014near the equilibrium :\nB\u0017\nm\u0017\u0019Be;\u0017\nme;\u0017+\u00141\nm\u0017\u0012@B\u0017\n@m\u0017\u0013\n\u00001\nm2\u0017B\u0017\u0015\neq\u000em\u0017 (26)\n+\u00141\nm\u0017@B\u0017\n@\u001c\u0014\u0015\neq\u000e\u001c\u0014\n= 1\u0000[1\u0000\fJ0;\u0017B0\n\u0017]eq\u000em\u0017\nme;\u0017+ [\fjJ0;\u0017\u0014jB0\n\u0017]eq\u000e\u001c\u0014\nme;\u0017;\nhere\u000em\u0017=m\u0017\u0000me;\u0017, withme;\u0017=B\u0017(\f\u0016\u0017HEX;\u0017),\nwhereHEX;\u0017is evaluated at the equilibrium, and \u000e\u001c\u0014=\n\u001c\u0014\u0000\u001ce;\u0014, where\u001ce;\u0014=j(me;\u0017\u0001me;\u0014)j=me;\u0017and it corre-\nsponds to the projection of the equilibrium magnetization\nme;\u0014onto the other sublattice magnetization direction.\nIt is easy to show that @\u001c\u0014=@m\u0017= 0. Similar to the fer-\nromagnetic case, we would like to arrive to a simpli\fed7\nexpression as a function of sublattice susceptibilities. For\nthis purpose, we divide the above expression by \u0016\u0017\fB0\n\u0017\n1\u0000B\u0017=m\u0017\n\u0016\u0017\fB0\u0017=1\ne\u001f\u0017;jj\u000em\u0017\nme;\u0017+\n+G\u0014\u00141\ne\u001f\u0017;jj\u000em\u0017\nme;\u0017\u00001\ne\u001f\u0014;jj\u000e\u001c\u0014\nme;\u0017\u0015\n(27)\nwhere we have used Eq. (23) and the function G\u0014=\njJ0;\u0017\u0014je\u001f\u0014;jj=\u0016\u0017has now more sense. Thus, the contribu-\ntion to the dynamical equation (4) of the exchange in-\nteraction (the LLB equation with longitudinal relaxation\nonly) given by Eq. (27) reads\n_m\u0017\n\r\u0017jEX=\u0000\u000b\u0017\nk\nme;\u0017\u00121 +G\u0014\ne\u001f\u0017;jj\u000em\u0017\u0000jJ0;\u0017\u0014j\n\u0016\u0017\u000e\u001c\u0014\u0013\nm\u0017(28)\nNote that the \frst term de\fnes the intralattice relaxation\nof the sub-lattice (for example, TM) to its own direction.\nThe second term describes the angular momenta trans-\nfer between sublattices driven by the temperature. This\nequation has the form\n_m\u0017\n\r\u0017=e\u0000\u0017m\u0017 (29)\nand it gives the exact LLB equation for the case when\nthe average magnetization of the two sublattices remain\nalways antiparallel.\nD. Final forms of the LLB equation\nIn order to be consistent with the ferromagnetic LLB\nequation (and the Landau theory of phase transitions),\nwe expand the deviations \u000em\u0017(\u000e\u001c\u0014) aroundm2\ne;\u0017(\u001c2\ne;\u0017)\nup to the quadratic terms. Similar to FM case we write:\n\u000em\u0017\nm\u0017;e\u00191\n2m2e;\u0017\u0000\nm2\n\u0017\u0000m2\ne;\u0017\u0001\n(30)\nTherefore we can write the e\u000bective longitudinal \felds as\nH\u0017\ne\u000b;jj=\u00141\n2\u0003\u0017\u0017\u0012m2\n\u0017\nm2e;\u0017\u00001\u0013\n\u00001\n2\u0003\u0017\u0014\u0012\u001c2\n\u0014\n\u001c2e;\u0014\u00001\u0013\u0015\nm\u0017\n(31)\nwhere in order to shorten the notations we have de\fned\nthe longitudinal rates as:\n\u0003\u00001\n\u0017\u0017=1\ne\u001f\u0017;jj(1 +G\u0014);\u0003\u00001\n\u0017\u0014=\u001ce;\u0014\nme;\u0017jJ0;\u0017\u0014j\n\u0016\u0017with\u00176=\u0014;\n(32)\nwhereG\u0014is also expressed in terms of the longitudinal\nsusceptibility via Eq.(24).Form 1\nFinally, we collect all the above derived approximate\nexpressions and we \fnish up with the compact form of\nthe LLB equation for the reduced magnetization vector,\nm\u0017=M\u0017=M\u0017(T= 0K)\n_ m\u0017=\r\u0017[m\u0017\u0002He\u000b;\u0017]\u0000\r\u0017\u000b\u0017\nk(m\u0017\u0001He\u000b;\u0017)\nm2\u0017m\u0017\n\u0000\r\u0017\u000b\u0017\n?[m\u0017\u0002[m\u0017\u0002He\u000b;\u0017]]\nm2\u0017(33)\nwhere the e\u000bective \feld He\u000b;\u0017for sublattice \u0017is de\fned\nas\nHe\u000b;\u0017=H+HA;\u0017+J0;\u0017\u0014\n\u0016\u0017\u0005\u0014\n+\u00141\n2\u0003\u0017\u0017\u0012m2\n\u0017\nm2e;\u0017\u00001\u0013\n\u00001\n2\u0003\u0017\u0014\u0012\u001c2\n\u0014\n\u001c2e;\u0014\u00001\u0013\u0015\nm\u0017(34)\nand the relaxation parameters \u000b\u0017\nkand\u000b\u0017\n?are given by\nEqs. (14).\nOr in a more explicit form, as a function of sub-lattice\nmagnetizations m\u0017and its values at the equilibrium\nme;\u0017:\n_ m\u0017=\r\u0017[m\u0017\u0002He\u000b;\u0017]\u0000\r\u0017\u000b\u0017\nk\u0010\nm\u0017\u0001Hk\ne\u000b;\u0017\u0011\nm2\u0017m\u0017\n\u0000\r\u0017\u000b\u0017\n?[m\u0017\u0002[m\u0017\u0002He\u000b;\u0017]]\nm2\u0017(35)\nwhere we have de\fned the longitudinal \feld, Hk\ne\u000b;\u0017, as\nHk\ne\u000b;\u0017=h1\n2\u0003\u0017\u0017\u0012m2\n\u0017\nm2e;\u0017\u00001\u0013\n\u00001\n2\u0003\u0017\u0014 \u0012m\u0017\u0001m\u0014\nme;\u0017\u0001me;\u0014\u00132\n\u00001!i\nm\u0017(36)\nand the e\u000bective \feld, He\u000b;\u0017, reads\nHe\u000b;\u0017=H+HA;\u0017+J0;\u0017\u0014\n\u0016\u0017m\u0014:\nIn Eq. (35) also the temperature dependent damping\nparameters are given by Eqs. (14).\nForm 2\nIt is also interesting to put the LLB equation in a more\nsymmetric form in terms of the macroscopic magnetiza-\ntion,M\u0017=x\u0017\u0016\u0017m\u0017=\u001d\u0017, wherex\u0017stands for the concen-\ntration of sites of type \u0017=TM or RE ( x\u0017=xfor RE and\nx\u0017=qfor TM),\u0016\u0017is the atomic magnetic moment of8\nthe lattice\u0017and\u001d\u0017is the atomic volume. We multiply\neach sublattice LLB equation (35) by the corresponding\nfactor, for example, in the case of TM by q\u0016TM=\u001dTMand\nwe obtain\n_M\u0017=\r\u0017[M\u0017\u0002He\u000b;\u0017]\u0000Lk;\u0017\u0010\nM\u0017\u0001Hk\ne\u000b;\u0017\u0011\nM2\u0017M\u0017\n\u0000L?;\u0017[M\u0017\u0002[M\u0017\u0002He\u000b;\u0017]]\nM2\u0017(37)\nwhere the e\u000bective \felds read:\nHk\ne\u000b;\u0017=h1\n2e\u0003\u0017\u0017\u0012M2\n\u0017\nM2e;\u0017\u00001\u0013\n\u00001\n2e\u0003\u0017\u0014 \u0012M\u0017\u0001M\u0014\nMe;\u0017\u0001Me;\u0014\u00132\n\u00001!i\nM\u0017;(38)\nThe rate parameters are e\u0003\u0017\u0014=\u001d\u0017\u0003\u0017\u0014=\u0016\u0017x\u0017and the ef-\nfective \feld, He\u000b;\u0017, has the following form:\nHe\u000b;\u0017=H+HA;\u0017+AM\u0014:\nHere the exchange parameter is introduced as A=\nzJTM-RE=\u0016RE\u0016TM. The damping coe\u000ecients Lk;\u0017and\nL?;\u0017read\nLk;\u0017=\r\u0017x\u0017\u0016\u0017\u000b\u0017\nk=\u001d\u0017; L?;\u0017=\r\u0017x\u0017\u0016\u0017\u000b\u0017\n?=\u001d\u0017:\nIV. RELAXATION OF MAGNETIC\nSUBLATTICES\nThe rate of the longitudinal relaxation is temperature\ndependent through the parameters such as the damping\nparameters \u000b\u0017\nk, see Eq. (14) and Fig. 2, and the longi-\ntudinal susceptibilities. The sign of the rate, e\u0000\u001770, de-\npends on the instantaneous magnetization values. From\nEq. (28) we can consider the following lines separating\ndi\u000berent relaxation signs:\n\u000em\u0017=jJ0;\u0017\u0014j\n\u0016\u0017e\u001f\u0017;jj\nG\u0014+ 1\u000e\u001c\u0014=e\u001f\u0017\u0014;jj\u000e\u001c\u0014; (39)\nwhere we have de\fned the dimensionless variable e\u001f\u0017\u0014;jj,\nwhich describes the e\u000bect of the change in one sublat-\ntice on the other. This variable can be interpreted as a\nsusceptibility e\u001f\u0017\u0014;jj=\u000em\u0017=\u000em\u0014. Indeed, we can expand:\nm\u0017(T;\u000em\u0017;\u000em\u0014)\u0019m\u0017(T;0;0) +\fJ0;\u0017B0\n\u0017\u000em\u0017\n+\fjJ0;\u0017\u0014jB0\n\u0017\u000em\u0014 (40)\nNow using that by de\fnition \u000em\u0017=m\u0017(T;\u000em\u0017;\u000em\u0014)\u0000\nm\u0017(T;0;0), we obtain\ne\u001f\u0017\u0014;jj=jJ0;\u0017\u0014j\u0012\fB0\n\u0017\n1\u0000J0;\u0017\fB0\u0017\u0013\n(41)Next, we substitute Eq. (23) into Eq. (41) and we get the\nrelation between the susceptibilities, exactly described by\nEq. (39).\nThe problem of relaxation sign is, therefore, reduced to\nthe study of the sign of the function \u000em\u0017\u0000e\u001f\u0017\u0014;jj\u000e\u001c\u0014. Let\nus assume the equilibrium state that is close to TC, de-\nscribing the situation during the ultrafast laser-induced\ndemagnetization [17]. Fig. 3 shows three possible instan-\ntaneous rates for T= 0:95TC, depending on the relative\nstate of both sublattice magnetizations. The lines sep-\narating di\u000berent relaxation types are straight lines with\nthe slopee\u001f\u0017\u0014;jj(T).\nIn the following we use atomistic LLG Langevin sim-\nulations described in Sec. II as well as the integra-\ntion of the LLB equation (4) for the same material pa-\nrameters, see Table I. In order to compare MFA based\nLLB equation and the atomistic simulations, we have\nre-normalized exchange parameters, as described in Ref.\n[18]. In the atomistic simulation the system size is taken\nasN= 603,i.e.3Ncoupled di\u000berential equations has to\nbe solved simultaneously within this approach, whereas\nonly 6 (two sublattices and three components for each) in\nthe macrospin LLB approach. We compare the di\u000berent\nrelaxation regions depending on the instantaneous mag-\nnetic state with those predicted by the LLB equation and\ndepicted in Fig. 3. The initial conditions in the simula-\ntions are the following: in all three cases we start from\na an equilibrium state at T= 600 K (for the considered\nconcentration x= 0:25 we getTC= 800 K). After that\nfor the situations of Fig.4(a) and (e) we put one of the\nsublattice magnetizations equal to zero, mTM(RE) = 0. In\nthe atomistic approach this is done by totally disordering\nthe system. Finally, the temperature is set to T= 0:95TC\nand the relaxation of both sublattices is visualized. The\nresults are presented in Fig. 4.\nFor the region mRE\u001dmTMabove the green line in\nFig.3 the rate for the TM is positive, e\u0000TM>0, thus\nthe TM magnetization will increase while e\u0000RE<0 and\nthe RE magnetization will decrease. Thus, we have ini-\ntially a dynamical polarization of TM by RE. As it can\nbe seen in Fig.4(a),(b) initially the TM magnetic order\nincreases from a totally disordered state, while the RE\nrelaxes directly to the equilibrium, i.e. the sign of the\nRE rate is always the same. In the central region of\nFig.3, between green and red lines, both magnetizations\ngo to the equilibrium by decreasing their value, see Fig.\n4(c)(d). Finally, in the low region of Fig.3 the situation\nis symmetric to the upper region but now TM magne-\ntization decreases and the RE magnetization increases\ninitially, see Fig. 4(e)(f). Thus, the predictions of the\nLLB equation are in agreement with full atomistic simu-\nlations which also provides a validation for our analytic\nderivation.\nAs a representative example, in GdFeCo near the mag-\nnetization reversal the situation is the following [17]: the\nTM magnetization is almost zero, mTM\u00190 and the\nRE has \fnite magnetization value mTM>0. This hap-\npens due to the fact that the Gd sublattice is intrin-9\neΓRE>0eΓTM<0,eΓRE<0 eΓTM<0,eΓRE<0eΓTM>0\nmTMmRE\n0.60.50.40.30.20.100.60.50.40.30.20.10.0\nFIG. 3: Di\u000berent longitudinal relaxation regions for T=TC=\n0:95 for parameters of the GdFeCo alloy with x= 0:25.\nsically slower than the FeCo one due to a larger mag-\nnetic moment. This situation corresponds to the up-\nper region in Fig. 3 where the rates are e\u0000TM>0 and\ne\u0000RE<0. Under these circumstances the RE magne-\ntization dynamically polarizes the TM sublattice mag-\nnetization through the interlattice exchange interaction\nHEX,TM-RE\u0019jJ0;TM-REjmRE>0. Consequently, the TM\nmagnetization goes opposite to its equilibrium position\nmTM\ne= 0 [see Fig. 4(a)-(b)]. The existence of opposite re-\nlaxation signs in TM and RE is consistent with a recently\nreported ferromagnetic state in a ferrimagnetic materials\nRef. [17], however it does not necessary lead to it. Nor\nit necessary means the switching of the TM magnetiza-\ntion, as was suggested in Ref.[24]. To have a switching\none should cross the line mTM\nz= 0 which cannot be done\nwithin the approach of longitudinal relaxation only which\nonly describes the relaxation to the equilibrium. The\ncrossing of the line mTM\nz= 0 can be only provided by a\nstochastic kick which is always present in the modeling\nusing stochastic atomistic approach [16, 17]. This topic\nwill be the subject of future work.\nV. THE LLB EQUATION AND THE\nBARYAKHTAR EQUATION\nIn this section we would like to discuss the di\u000berences\nbetween the LLB equation and the equation derived by\nV. Baryakhtar [25] and used in Ref. [24] to explain\nthe ultrafast magnetization reversal and the transient\nferromagnetic-like state in ferrimagnets. The Baryakhtar\nequation was derived from the Onsager principle which\nin general is valid near the thermodynamic equilibrium\nonly. The general derivation is based on the symme-\ntry approach. Another strong supposition made in its\nderivation is the separation of the timescales: the ex-\nchange interaction timescale and the relativistic interac-\ntion timescale (de\fned in our case by the parameter \u0015)\nare assumed to be separated. The resulting equation has\nthe following form:\n \nFIG. 4: Comparison between atomistic LLG-Langevin and\nmacrospin LLB calculations of the longitudinal relaxation of\nthe GdFeCo alloy ( x= 0:25) corresponding to the three di\u000ber-\nent relaxation cases in Fig.3. In the left column we show atom-\nistic LLG-Langevin multispin simulations and in the right\none- the LLB macrospin calculations. The graphs (a) and\n(b) correspond to the region with e\u0000TM>0 and e\u0000RE<0. The\ngraphs (c) and (d) correspond to the region with e\u0000TM<0 and\ne\u0000RE<0. The graphs (e) and (f) correspond to the region with\ne\u0000TM<0 and e\u0000RE>0.\n1\n\r\u0017dM\u0017\ndt=\u0015e(H\u0017\u0000H\u0014) +\u0015\u0017H\u0017 (42)\nHere\u0017= TM, RE, \u0015\u0017describes transfer of the angu-\nlar momentum from sublattices to the environment, \u0015e\nis of the exchange origin and stems from spin-spin in-\nteractions, conserving the total angular momentum but\nallowing for the transfer of angular momentum between\nthe sublattices. The e\u000bective \felds de\fned as H\u0017=10\n\u0000\u000eW=\u000eM\u0017are derived from the magnetic energy W. In\nRef. [24] the authors used the Landau type free energy\nexpansion near the critical temperature, corresponding\nto the form Eq. (30).\nIn comparison to the Baryakhtar equation, the LLB\nequation, derived here includes the transverse exchange\nmode and allows the transfer of the energy or momentum\nbetween the longitudinal and transverse motion. The fer-\nrimagnetic LLB equation has three terms among which it\nis the precession term which conserves the total angular\nmomentum. The precession in the interlattice exchange\n\feld given by [ mTM\u0002mRE] allows the transfer of an-\ngular momentum between sublattices. The longitudinal\nand transverse relaxation terms which are related to the\ncoupling to the heat bath are both proportional to \u0015.\nDi\u000berently to ferromagnets, both the transverse motion\ngiven by precession and transverse relaxation terms are\nnot negligible on the femtosecond timescale in compari-\nson to longitudinal motion because in both cases the \feld\nacting on both motions is of the exchange origin.\nIn principle the ferrimagnetic LLB equation can be cast\nin a form, similar to the Baryakhtar equation if we re-\nstrict ourselves to longitudinal motion only, considering\nthe antiparallel sublattices alignment. For the longitudi-\nnal relaxation only (see Eq. (28)) we have the following\nexpression\n_m\u0017\nz\n\r\u0017=\u000b\u0017\nkH0\n\u0017+\u000bk\n\u0017\u0014(H0\n\u0017+H0\n\u0014) (43)\nwhereH0\n\u0017=\u0000(\u000em\u0017\ne\u001f\u0017;jj)m\u0017\nz=m\u0017, stands for the \felds coming\nfrom interaction of each lattice with itself and H0\n\u0014-with\nthe opposite sublattice. One can see that the sign of the\ne\u000bective \feld coming from the other sublattice is opposite\nfor the LLB Eq. (43) and the Baryakhtar equation Eq.\n(42). In order to illustrate the consequence of this, we\ncan compare the equations for the limiting case close to\nTC. In this case the Baryakhtar equation (see Eq. (1.33)\nin Ref. [25]) reads:\n_m\u0017\nz\n\r\u0017=\u0000\u0015\u0017m\u0017\nz\ne\u001f\u0017;jj\u0000\u0015e\u0012m\u0017\nz\ne\u001f\u0017;jj+m\u0014\nz\ne\u001f\u0014;jj\u0013\n(44)\nwherem\u0017\nzis the absolute value of the z-component of\nthe magnetization in the sub-lattice \u0017and we explicitly\nconsidered that the sign of z-components is opposite for\nthe sublattice \u0017and\u0014. In the same limit, considering\nmTM(RE) =me;TM(RE) +\u000emTM(RE) , and following Eq. (28)\nthe LLB equation takes a similar form:\n_m\u0017\nz\n\r\u0017=\u0000\u000b\u0017\nkm\u0017\nz\ne\u001f\u0017;jj\u0000\u000b\u0017\nkjJ0;\u0017\u0014j\n\u0016\u0017\u0012e\u001f\u0014;jj\ne\u001f\u0017;jjm\u0017\nz\u0000m\u0014\nz\u0013\n(45)\nNote that for the LLB equation the contribution of the\nopposite sublattice is negative while for the Baryakhtar\nequation it is positive. This has important consequencesin the longitudinal inter-lattice relaxation of the sub-\nlattices, changing the results of Fig. 3.\nIn Fig. 5 we show the temperature dependence of the\nratio of partial susceptibilities, e\u001f\u0014;jj=e\u001f\u0017;jjappearing in\nEq.(45). We can see that at temperatures not very close\ntoTC:e\u001fTM;jj=e\u001fRE;jj\u001c1 and the contrary behavior close\ntoTC. Thus for the TM and temperatures close to TCthe\nsecond term in the r.h.s. of Eq.(45) could be neglected\nand the third term with the opposite sign can compete\nwith the \frst one, leading either to slowing down of the\nrelaxation rate or even to changing its sign, as presented\nin Fig. 4(a) and (b). The behavior of RE on the contrar-\nily is dominated by this term and the sign of relaxation\ncannot be changed, as is seen in the same \fgure. Obvi-\nously this behavior cannot be described by Eq.(44) where\nall terms have the same sign. In order to have the oppo-\nsite relaxation sign, one has to assume for this equation\na priori that the signs of the z-components of magnetiza-\ntion in both sub-lattices are the same, i.e. to start with\nthe ferromagnetic-like state without specifying its origin.\nx= 0.75x= 0.5x= 0.25\nT/T C/tildewideχTM//tildewideχRE\n1 0.750.50.25010\n1\n0.1\n0.01\nFIG. 5: Temperature dependence of the ratio between longi-\ntudinal susceptibilities for parameters of the GdFeCo alloy.\nFinally, we would like to note that because we have\ntreated the spin-spin interaction in MFA we have lost\ncorrelation contribution. Consequently, both LLB and\nBaryakhtar equations do not describe the energy trans-\nfer from the uniform modes into nonlinear spin waves and\nvice versa. In ferromagnets [26] this contribution is usu-\nally two or three orders of magnitude smaller than the\ncontribution to relaxation through the coupling to the\nbath. At this stage we do not know how large this contri-\nbution can be in ferrimagnets. In Ref. [26] the contribu-\ntion of nonlinear spin waves was arti\fcially incremented\nby using a random anisotropy to cause non-coliniarities.\nIn principle, in ferrimagnets one can see a small amount\nof RE as precursor of non-coliniarities, with the strength\nof the order of interlattice exchange parameter JTM-RE .\nFor completeness, a microscopic treatment of the spin\nwave contribution would be desirable, we let this task for\nthe future.11\nVI. CONCLUSIONS\nWe have derived the Landau-Lifshitz-Bloch equation\nfor a two-sublattice system such as a GdFeCo ferrimag-\nnet for which an ultrafast switching has been reported\n[14, 17]. Although in our derivation we refer to a TM-\nRE alloy, it is equally valid for a two-component ferro-\nmagnet, as well as for an antiferromagnet. The general-\nization to more components is straightforward. The new\nequation constitutes an important step forward in clas-\nsical description of the dynamics of ferrimagnets which\nis traditionally based on two-coupled macroscopic LLG\nequations. For example, the FMR and exchange modes\nhave recently attracted attention due to possibility to\noptically excite them [13, 27]. Their temperature de-\npendence can be now correctly understood in terms of\nour approach [28]. Furthermore, recent ultrafast dynam-\nics experiments using XMCD showed di\u000berent sublattice\ndynamics on ultrafast timescale in a two-sublattice mag-\nnets such as GdFeCo [17] or FeNi [29], which can be mod-\neled using this new approach. Finally, this equation can\nserve in the future as a basis for multiscale modeling in\ntwo-component systems at high temperatures and/or ul-\ntrafast timescales, the same way as the LLB equation\nfor ferromagnets [19]. This also opens a possibility for\nmicromagnetic modeling of ultrafast dynamics in large\nstructures, such as sub-micron and micron-size ferrimag-\nnetic dots, whose dimensions do not allow modeling by\natomistic approach. Similarly, it will be useful for static\nmicromagnetic modeling at high temperatures, such as\nthermally-driven domain wall motion in nanostructures.\nThe LLB equation correctly shows the possibility to\nreverse the sign of relaxation at high temperatures and,\ntherefore, is consistent with the existence of a recently\nreported ferromagnetic state in a ferrimagnet [17]. The\nvalidity of the approach has been checked against full-\nscale atomistic simulations presented in Fig. 4. However,\nunlike the equation, derived by Baraykhtar and used re-\ncently to describe the GdFeCo switching [24], it is not\nbased on the separation of timescales and on the On-\nsager principle. Instead, both the coupling to the exter-\nnal bath and the exchange interaction form part of the\nsame longitudinal and transverse relaxation terms. We\nshow important di\u000berences in the resulting form of the\nequation.\nUnfortunately, at the present time the compact deriva-\ntion was possible only under some assumptions. The\nemployed conditions certainly allow to describe the nor-\nmal modes such as ferromagnetic resonance and anti-\nferromagnetic exchange precessional modes in ferrimag-\nnets [28]. The same way the approximation is su\u000e-\ncient to describe the switching of ferrimagnet if it occurs\nthrough a linear reversal path [4, 24] or if sublattices\nnon-collinearities are not too large. Weather the applied\napproximation completely describes the situation of the\nultrafast reversal is an open question which we will in-\nvestigate in the future. For modeling, the initial param-\nagnetic equation (4) with the MFA \feld (7) and (8) canalways be used, providing the check for the approxima-\ntion. Finally, up to now we were not able to derive a\ncompact expression for the equation above TCwhich is\nalso a necessary step for the full modeling of the ultrafast\nswitching.\nVII. ACKNOWLEDGEMENT\nWe gratefully acknowledge funding by the Spanish\nMinistry of Science and Innovation under the grant\nFIS2010-20979-C02-02.\nAppendix A\nIn this appendix we present detailed derivation of Eq.\n(12). We start from Eq.(5):\nm0;\u0017=B(\u00180;\u0017)^ u\u0017;\u00180;\u0017\u0011\f\u0016\u0017\nHMFA\n\u0017\u000bconf;(A1)\nwhere ^ u\u0017=\u00180;\u0017=\u00180;\u0017and\nHMFA\n\u0017\u000bconf=Hk\nEX;\u0017+H00\ne\u000b;\u0017.\nHere H00\ne\u000bcontains the anisotropy, applied and the per-\npendicular component of the exchange \feld (see section\nIII.A). In the case of a strong homogeneous exchange \feld\f\f\fHk\nEX;\u0017\f\f\f\u001d\f\f\fH00\ne\u000b;\u0017\f\f\fthe MFA \feld can be expanded up to\n\frst order in H0\ne\u000b;\u0017as\n\f\f\f\nHMFA\n\u0017\u000bconf\f\f\f'Hk\nEX;\u0017+Hk\nEX;\u0017\u0001H00\ne\u000b;\u0017\u0010\nHk\nEX;\u0017\u0011 (A2)\nTherefore, \u00180;\u0017=\f\u0016\u0017\f\f\f\nHMFA\n\u0017\u000bconf\f\f\fcan be writ-\nten as\u00180;\u0017=\u0018EX;\u0017+\u000e\u0018\u0017with\u0018EX;\u0017\u001d\u000e\u0018\u0017,\nwhere we identify \u0018EX;\u0017=\f\u0016\u0017Hk\nEX;\u0017and\u000e\u0018\u0017=\n\f\u0016\u0017\u0010\nHk\nEX;\u0017\u0001H00\ne\u000b;\u0017\u0011\n=Hk\nEX;\u0017. Expanding the Langevin\nfunction around \u0018EX;\u0017we get\nB(\u00180;\u0017)'B\u0017+B0\n\u0017\u000e\u0018\u0017 (A3)\nand\n^ u\u0017'Hk\nEX;\u0017+H00\ne\u000b;\u0017\nHk\nEX;\u00170\nB@1\u0000Hk\nEX;\u0017\u0001H00\ne\u000b;\u0017\u0010\nHk\nEX;\u0017\u001121\nCA;(A4)\nwhereB\u0017=B(\u0018EX;\u0017) andB0\n\u0017=B0(\u0018EX;\u0017). Substituting\nEq. (A3) and Eq. (A4) in Eq. (A1) and neglecting the\nterms quadratic in H00\ne\u000b;\u0017=jHk\nEX;\u0017jwe get\nm0;\u0017'B\u00172\n64Hk\nEX;\u0017+H00\ne\u000b;\u0017\nHk\nEX;\u0017\u0000\u0010\nHk\nEX;\u0017\u0001H00\ne\u000b;\u0017\u0011\nHk\nEX;\u0017\n\u0010\nHk\nEX;\u0017\u001133\n75\n+B0\n\u0017\f\u0016\u0017\u0010\nHk\nEX;\u0017\u0001H00\ne\u000b;\u0017\u0011\nHk\nEX;\u0017\n\u0010\nHk\nEX;\u0017\u00112: (A5)12\nUsing the vector calculus identity ( a\u0002b)\u0002c=b(a\u0001c)\u0000\na(b\u0001c) Eq.(A5) can be written as\nm0;\u0017'B\u0017Hk\nEX;\u0017\nHk\nEX;\u0017+B0\n\u0017\f\u0016\u0017\u0010\nHk\nEX;\u0017\u0001H00\ne\u000b;\u0017\u0011\nHk\nEX;\u0017\n\u0010\nHk\nEX;\u0017\u00112\n\u0000B\u0017\nHk\nEX;\u0017[h\nH00\ne\u000b;\u0017\u0002Hk\nEX;\u0017i\n\u0002Hk\nEX;\u0017]]\n\u0010\nHk\nEX;\u0017\u00112: (A6)\nFinally, we use Hk\nEX;\u0017=\u0010\neJ0;\u0017=\u0016\u0017\u0011\nm\u0017[see Eq. (11)] in\nEq. (A6) and obtain Eq. 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Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov,\nA. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson,\nand Th. Rasing, Phys. Rev. Lett. 108057202 (2012).\n[25] V. G. Baryakhtar, Zh. Exp Teor. Fiz. 94196 (1988).\n[26] D. A. Garanin and H. Kachkachi, Phys. Rev. B 80,\n014420 (2009).\n[27] A.Mekonnen, M.Cormier, A.V.Kimel, A.Kirilyuk,\nA.Hrabec, L.Ranno and Th.Rasing, Phys. Rev. B 107\n117202 (2011).\n[28] F. Schlickeiser Computer Simulation of the Dynamics of\nFerrimagnets , Master thesis, University of Konstantz,\n2011; F. Schlickeiser, U. Atxitia, S. Wienholdt, D.\nHinzke, O. Chubykalo-Fesenko and U. Nowak, to be pub-\nlished.13\n[29] I. Radu et al. unpublished"    },    {        "title": "1903.04432v3.Giant_spin_orbit_torque_in_a_single_ferrimagnetic_metal_layer.pdf",        "content": "Giant spin-orbit torque in a single ferrimagnetic metal layer\nSimon Lenne,1Yong-Chang Lau,1Ajay Jha,1Gwenal Y. P. Atcheson,1Roberto E.\nTroncoso,2Arne Brataas,2J.M.D. Coey,1Plamen Stamenov,1and Karsten Rode1,\u0003\n1CRANN, AMBER and School of Physics, Trinity College Dublin, Ireland\n2Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\nAntiferromagnets and compensated ferrimagnets o\u000ber opportunities to investigate spin dynamics\nin the `terahertz gap' because their resonance modes lie in the 0 :3 THz to 3 THz range. Despite some\ninherent advantages when compared to ferromagnets, these materials have not been extensively\nstudied due to di\u000eculties in exciting and detecting the high-frequency spin dynamics, especially\nin thin \flms. Here we show that spin-orbit torque in a single layer of the highly spin-polarized\ncompensated ferrimagnet Mn 2RuxGa is remarkably e\u000ecient at generating spin-orbit \felds \u00160He\u000b,\nwhich approach 0 :1\u000210\u000010T m2=A in the low-current density limit { almost a thousand times\nthe Oersted \feld, and one to two orders of magnitude greater than the e\u000bective \felds in heavy\nmetal/ferromagnet bilayers.\nWe depend on fast, reliable exchange of information\nacross long distances through intercontinental optical \f-\nbres, as well as short-distance connections between the\ncentral processing unit of a computer and its memory.\nThe latter is the bottleneck to the powerful computing fa-\ncilities needed in a future where machine learning and al-\ngorithms aid our daily lives. This bottleneck is di\u000ecult to\novercome because electronics lack a practical chip-based\nsolution to produce and detect electromagnetic waves in\nthe spectral range between 0 :3 THz and 30 THz known\nas the terahertz gap.\nSlonczewski1realised that angular momentum could\nbe transferred from one magnetic layer (a polariser) to\nanother (the analyser) by a spin polarised current.2This\nspin-transfer torque has enabled the scaling of devices\nthat depend on the relative magnetic orientation of two\nferromagnetic layers.3\nSpin electronics exploiting the orbital degree of free-\ndom of the electron is a recent development. A ma-\njor advance was the discovery that the angular momen-\ntum could be supplied by a di\u000busive spin current4,5cre-\nated via the spin Hall e\u000bect6,7in a non-magnetic heavy\nmetal layer adjacent to the ferromagnet. Devices based\non these bilayers require a bare minimum of two lay-\ners on a substrate. Earlier, Dresselhaus8and Bychkov\nand Rashba9had shown that in crystalline or patterned\nstructures lacking inversion symmetry, a current-induced\nspin polarisation (CISP) is a direct consequence of the\nsymmetry of the band structure. This idea was devel-\noped by \u0014Zelezn\u0013 y et al.1011to predict the form of the\ntensor relating the charge current to the CISP in crys-\ntals of di\u000berent symmetry. 90\u000eswitching of the metal-\nlic antiferromagnets CuMnAs12and Mn 2Au13was subse-\nquently observed. These ground-breaking results estab-\nlished the existence of a current-induced, \feld-like (or\nreactive) torque, and allow an estimate of the strength\nof the e\u000bective magnetic \feld by comparing it to the in-\nplane magnetic anisotropy of the material.\nHitherto there has been no quantitative measurement\nof the anti-damping (or dissipative) spin-orbit torquein homogeneously magnetised ferrimagnetic or antifer-\nromagnetic single-layer samples. Here we show, via har-\nmonic analysis of the anomalous Hall e\u000bect,14that in a\nsingle layer of the prototype half-metallic compensated\nferrimagnet Mn 2RuxGa withx= 0:7 (MRG),15{21both\nthe \feld- and damping-like components of the torque\nreach record values, almost two orders of magnitude\nstronger than those obtained in the bilayer ferromag-\nnet/heavy metal systems, or in metallic ferromagnets22\nand semimagnetic semiconductors23. The record values\nof the single-layer SOT and the dominance of the dissi-\npative torque, open a path to sustaining magnetic oscil-\nlations in the terahertz gap.\nThin-\flm samples of MRG grown on MgO by DC-\nmagnetron sputtering from stoichiometric targets crys-\ntallise in a Heusler-like structure, space group F\u001643mil-\nlustrated in FIG. 1a, where the conduction bands orig-\ninate predominantly from Mn in 4 csites.20The \flms\nare patterned into the micron-sized Hall bar structures\nshown in FIG. 1b, where the bias current jis parallel to\nthe MRG [010] axis. Further details on sample growth\ncan be found in the supplementary material and [19].\nWe determine the current-induced e\u000bective \felds via the\nanomalous Hall e\u000bect (AHE), assuming it is proportional\nto thezcomponent of the magnetisation of the Mn4csub-\nlattice. Due to the substrate-induced biaxial strain, the\npoint group of Mn in this position is reduced from \u001643m\nto\u001642m. Here we restrict our analysis to the e\u000bect on one\nsublattice, as the other will follow via inter-sublattice ex-\nchange with a phase-lag. We treat all e\u000bective torques as\nequivalent to external applied \felds. For an in-plane ap-\nplied \feldH, the magnetisation is described by the polar\nand azimuthal angles \u00120and\u001em, with the latter taken to\nbe equal to the azimuthal angle \u001eHof the applied \feld\nbecause the four-fold in-plane anisotropy is weak com-\npared with the uniaxial perpendicular anisotropy. The\ncoordinates describing the magnetic state are shown in\nFIG. 1c. In the presence of a unit charge current density\njk[010], the CISP produces a SOT e\u000bective \feld (seearXiv:1903.04432v3  [cond-mat.mes-hall]  29 Apr 20192\n(a)\nMn4a\nGa4b\nMn4c\nRu4d(b)\nI+\nI\u0000V\u0000\nxy V+\nxy\nV\u0000\nyyV+\nyy\n60µm\n(c)\nxyzm\n\u00160Hmxmy\n\u001eH\u00120(d)\nxyzm\nmxmy\u0000xfl\nmz\u0001xdlmx\u0001xdl\n\u001em\u00120\n1\nFIG. 1. (a) MRG crystal structure. The current is carried\nmainly by electrons in bands originating from Mn in the 4 c\nposition, which has point group symmetry \u001643m. Arrows show\nthe direction of the magnetic moment on each site. (b) Micro-\ngraph of a Hall bar with the contacts labelled. (c) Illustration\nof the coordinate system: \u00120is the polar angle of the magneti-\nsation vector in the absence of the SOT \feld, and \u001em=\u001eH\nis the azimuthal angle of the magnetisation and applied \feld\nvectors, respectively. (d) Illustration of the e\u000bective SOT\n\felds acting on the magnetisation with a bias current along\nMRG [010]ky, the \feld-like (reactive) component in blue,\nand the two damping-like (dissipative) components in green.\nFIG. 1d):\n\u00160hSOT=mzxdlex\u0000xfley+mxxdlez (1)\nwhere eiare unit vectors, miare the components of the\nunit magnetisation vector, and xfl,xdlare the coe\u000e-\ncients of the \feld- and damping-like contributions to the\nspin-orbit \feld, respectively. The units of \u00160hSOT,xfl\nandxdlare then T A\u00001m2. Henry is an equivalent unit.\nWhen the bias current has an alternating component\n(j=jdc+jacsin!t) we detect the e\u000bect of the CISP on\nboth the second and the third harmonic response using\nlock-in demodulation. The conversion from the voltages\ndetected at the di\u000berent harmonics to the magnitude of\nthe e\u000bective \felds is detailed in the supplementary ma-\nterial. In TABLE I we indicate the symmetry of the dif-\nferent contributions to Vxyin the \frst, second and third\nharmonic responses. Contributions from the anomalous\nNernst e\u000bect (ANE) are suppressed by measuring V3!\nxyor\nby taking the di\u000berence of V2!\nxymeasured with positive\nand negative DC bias. The contribution from the homo-\ngeneous temperature variation \u0001 Toscillating at twice\nthe applied frequency is determined from data in FIG. 2,\nas explained in the supplementary material.\nFIG. 2 shows the temperature dependence of the lon-TABLE I. Linear contributions to Vxyup to third order in\ncurrent density. - means no contribution, /a contribution odd\ninjdc,veven injdcand oindependent of current direction.\n\u001bxycontributes implicitly to all four e\u000bects.\nContribution/Harmonic !2!3!\nAnomalous Hall E\u000bect: \u001bxy o - -\nAnomalous Nernst E\u000bect: @T=@z - v -\nHomogeneous \u0001 Toscillating at 2 !:@\u001b=@T v / o\nCurrent-induced \felds: hSOT / v / o\n550575600625650(a)\n12141618(b)\n-600-400-2000200\n0 100 200 300(c)\n-20-15-10-50\n0 100 200 300(d)\nσxx(kS m−1)\nσxy(kS m−1)∂σxx\n∂T(S m−1K−1)\nT(K)\n∂σxy\n∂T(S m−1K−1)\nT(K)\nFIG. 2. Temperature dependence of the longitudinal and\ntransverse conductivity of MRG (a and b), and their tem-\nperature derivatives (c and d). The variation of \u001bxyfollows\nthe variation of the magnetisation of the 4 csublattice.20The\ncompensation temperature Tcomp where the net magnetisa-\ntion changes sign is 175 K (vertical dashed line). Since the\ndata were recorded at remnance, the direction of the sublat-\ntice moments does not change at Tcomp.\ngitudinal and transverse conductivity of MRG, recorded\nin the remnant state after saturation in a positive\n\feld at room temperature. The conductivity \u001bxx\n(FIG. 2a) increases with decreasing T, and its satura-\ntion value of 630 kS m\u00001or [159 µ\n cm]\u00001corresponds\nto the minimum metallic conductivity of a bad metal\nwhere the mean free path is comparable to the inter-\natomic spacing.24The Hall conductivity \u001bxy(FIG. 2b)\nclosely follows the Mn4csublattice magnetisation.25The\nlower panels show the temperature-derivatives of \u001b.\nWe now turn to the SOT. FIG. 3 a and b shows the\nexperimentally observed V3!\nxyand its calculated values\nbased on the experimental \u00120. There is excellent agree-\nment between experiment and the model, which is based\nonly on the site symmetry, the data in FIG. 2, and the\n\frst harmonic response (used to determine \u00120and the\nanisotropy constants). All the features in both the \u00120\nand\u001edependencies are well reproduced: two deep min-\nima around \u00160Hx;max, four maxima that align with the\nfour-fold in-plane anisotropy due to the small value of\nthe in-plane anisotropy constant K0\n2, as well as a weaker\ncentral minimum at small \felds. Qualitatively, the \feld-\ndependence of the SOT can be understood by compar-\ning equation (1) with FIG. 3 (a), and noting that the3\n-2-1012-2-1012(a) (b)\n(c) (d)-2-1012-2-1012(a) (b)\n(c) (d)\n02468\n0 2 4 6 8(a) (b)\n(c) (d)\n00.10.20.3\n0 2 4 6 8(a) (b)\n(c) (d)\nµ0Hx(T)µ0Hy(T)-2-101\nV3ω\nxy(µV)\nµ0Hx(T)µ0Hy(T)-2-101\nV3ω\nxy(µV)V2ω\nxy(µV)\nIac(mA)DC diff.\nDC sum\nIdc(mA)\nFIG. 3. (a) Surface plot and its 2D colour map projection\nof the experimentally observed voltage at the third harmonic\nV3!\nxy. (b) Calculated response based on the experimental val-\nues of cos\u00120. (c) and (d) show AC (with Idc= 3 mA) and\nDC (Iac= 1 mA) current-dependence of the second harmonic\nV2!\nxysignal. By making the di\u000berence and sum of records made\nwith positive and negative DC o\u000bset we isolate the SOT (the\ndi\u000berence) from the anomalous Nernst e\u000bect ANE (the sum).\nmagnitude of \u0001 \u0012depends not only on \u00120itself, but ex-\nplicitly on the competition between the SOT \feld, the\nanisotropy \feld and the applied \feld. At low applied\n\felds (\u00120\u00190) the current-driven wobble of mis de-\ntermined by a combination of the anisotropy and SOT\n\felds. \u0001\u0012is small, however, because cos \u00120\u00191, hence\nthe central minimum is shallow. At higher applied \felds,\n\u00120deviates from 0, but the SOT \feld now has to com-\npete with both the (higher) anisotropy \feld and the Zee-\nman torque provided by the applied \feld acting on the\nnet magnetisation. This gives rise to the characteristic\nfour-fold signal. An exceptional feature appears around\n\u00160Hx\u0019\u00062 T where the damping-like \feld in the zdi-\nrection scaling as mxproduces a \feld strong enough to\ndwarf both the anisotropy \feld and the applied \feld. We\nemphasise not only the qualitative agreement, but also\nthat the absolute magnitude of the signal agrees very well\nwith the model when we \ft the coe\u000ecients of the \feld-\nand damping-like \felds; xfl=\u000015\u000210\u000013T A\u00001m2\nandxdl= 50\u000210\u000013T A\u00001m2.\nWe then determine the dependence of the e\u000bective \feld\nmagnitude on bias current, (FIG. 3 c and d). A cur-\nrent of 1 mA in our \flms is equivalent to jof about\n2:5\u0002109A m\u00002. We expect V2!\nxydue to SOT to scale\nwithIdcandI2\nac, while the e\u000bects due to thermal gradi-\nents should be independent of Idcand scale as I2\nac. In-\ndeed, the DC di\u000berence is quadratic in Iac(FIG. 3c) and\nlinear inIdc(FIG. 3d), while the DC sum is quadratic in\nIacand practically independent of Idc.\nIt is instructive to compare the e\u000bective \felds due tointrinsic SOT with those recorded on conventional bilay-\ners of a heavy metal (typically Pt, Ta or W) and a 3 d\nferromagnet (typically Co, Fe, CoFe or CoFeB). For bi-\nlayers, the damping-like e\u000bective \feld per current density\ncan be written: \u00160Hdl=j= (\u0012SH\u0016h)=(2eMst), where\u0012SHis\nthe spin-Hall angle of the heavy metal, \u0016 his the Planck's\nconstant,eis the electron charge, Msthe magnetisa-\ntion of the ferromagnet and tits thickness. For 1 nm\nof CoFeB (Ms\u00191 MA m\u00001), which has a magnetic mo-\nment equivalent to that of \u001930 nm nearly compensated\nMRG, and \u0012SH= 40 % we obtain an e\u000bective, damping-\nlike \feld of 1 :3\u000210\u000013T A\u00001m2(0:13 pH). We would\nneed a \fctitious spin-Hall angle of 400 % to match the\nvalue of the \feld-like term in MRG and 1200 % to match\nthe damping-like term.\nThis comparison highlights the inherent advantage of\nusing ferrimagnets in combination with intrinsic SOT. In\na bilayer, increasing the thickness of the ferromagnet be-\nyond the spin di\u000busion length (typically <10 nm), does\nnot produce any additional torque. If the ferromagnet is\n2 nm rather than 1 nm thick, the e\u000bective \feld may be\nreduced to half, whereas the \feld in single-layer MRG is\nunchanged with thickness. The volume of MRG can be\nscaled up or down without changing the torque, provid-\ning the current density is constant. The nature of the\nintrinsic torque is staggered acting directly on the Mn4c\nsublattice, hence a more correct comparison might be\nbe to normalise the spin Hall angle using the sublattice\nmagnetisation, which is approximately ten times greater\nthan the net magnetisation at room temperature for the\npresent sample. Furthermore, the torque is maintained\neven in the absence of any net magnetisation at the fer-\nrimagnetic compensation temperature, thus permitting\nGMR- and TMR-based device structures to be excited\nby SOT even in the absence of any net moment of the\nfree layer. This enables targeted control of the dynamics,\nand the excitation of both in-and out-of phase resonance\nmodes.\nThe high e\u000bective \felds found above assuming small,\nlinear, current-driven variations in \u00120, imply that the ac-\ntion of the SOT should also be observable in the non-\nlinear transfer characteristics of our Hall bar device. We\ntherefore proceeded as follows:\nWe \frst recorded a full \feld-in-plane hysteresis loop\nfrom\u000014 T to 14 T to determine the relation between\nmzand the applied \feld and invert this relation numeri-\ncally, to be able to deduce, from mz, the value of the total\ne\u000bective \feld at any given applied current. Then, a con-\nstant external \feld \u00160H= 0:4 T is applied in the sample\nplane and rotated around ez, changing the azimuthal an-\ngle\u001eHfrom 0\u000eto 360\u000ewhile recording mz(inferred from\nVxy). This measurement is repeated for range of current\ndensities from 0 :2\u00021010A m\u00002to 2:5\u00021010A m\u00002. As\nthe action of the SOT \feld depends directly on its direc-\ntion relative to the direction of the magnetisation ( \u0012M\nand\u001eM\u0019\u001eH), we can subtract any variation that is \u001e-\nindependent . This\u001e-independent e\u000bective \feld contains\nall variations that are due to heating. The result, after4\n-2-1012-2-1012\njysinφH(1010A m−2)jycosφH(1010A m−2)-250255075\nLeff(pH)\nFIG. 4. High-current-density e\u000bective induction Lin pH =\npT m2A\u00001as a function of the bias current density and\nthe angle\u001eHbetween exand the applied magnetic \feld\n\u00160Happ= 0:4 T. The e\u000bective inductance reaches \u001875 pH,\ncorresponding to an e\u000bective \feld at j= 2:5\u00021010A m\u00002of\n\u00160He\u000b\u00181:9 T. The interpretation of this striking result is\ndiscussed in the main text.\nsubtraction, is shown in FIG. 4, where we give the e\u000bec-\ntive \feld in terms of the e\u000bective, current-induced induc-\ntance in pH = 1 \u00021012T m2A\u00001. A current density of\nj= 2:5\u00021010A m\u00002can produce an e\u000bective inductance\nLe\u000b\u001975 pH, equivalent to an e\u000bective in-plane \feld of\n1:9 T! We note that this \feld is su\u000ecient to magnetically\nswitch\u00192 % of the sample.\nWe make two important comments on this analysis.\nFirst, by removing the \u001e-independent part of the sig-\nnal, we also remove any SOT that behaves the same\nway. If we again assume the SOT \felds can be de-\nscribed by the tensors reported by \u0014Zelezn\u0013 y et al.11and\nTroncoso et al.26, and expand the relation between he\u000b\nand \u0001\u0012to second order in \u0001 \u0012, we \fnd that we have re-\nmoved a damping-like contribution along ezthat varies\nasm2\nx+m2\ny, which may be considerable. Second, as we\nare normalising with respect to the action of the exter-\nnal,in-plane , \feld, the SOT e\u000bective \feld directed along\nez, remaining after the procedure outlined above, con-\ntributes to our signal /1=cos\u0012Mas seen by the upturn at\n\u001eH= 270\u000ein FIG. 4.\nThe strong e\u000bective SOT \felds in MRG are related to\nits high anomalous Hall angle.16The value is unusual in\nthe sense that MRG does not contain any elements heav-\nier than Ru; in any case the AHE angle does not scale\nwith Ru content x. Furthermore the conduction electrons\nin MRG are predominantly d-like, although it has been\nsuggested that Ga in the Mn-containing Heuslers lends\nsomepcharacter to the bands at the Fermi-level through\nhybridisation, increasing the spin-orbit coupling of the\nconduction electrons27. We have already seen above that\nMRG is at the limit of metallic conductivity. From our\nmeasurements of \u001bxxand\u001bxy, we can deduce the spin-\norbit scattering cross-section and \fnd that it correspondsto 60 % of the unit cell surface area. The very large scat-\ntering cross section is consistent with the very short mean\nfree path.\nSo far we have demonstrated high current-induced ef-\nfective \felds as well as a high ratio ( \u00183) of the dissipative\n(anti-damping) to the reactive (\feld-like) torques. This\nwill allow for the realization of more e\u000ecient magnetic\nswitching28, exchange-bias manipulation29as well as low-\ncurrent control of magnetic textures30. The key question\nis, whether sustained self oscillation can be driven by\nSOT. We address this from two di\u000berent angles, \frst by\nconsidering the results established by Troncoso et al.26,\nnoting that the e\u000bective \felds will act distinctly on the\nmagnetisation and the Nel vectors. Using the numeri-\ncal values of the e\u000bective \felds found in the linear, low-\ncurrent regime, self-oscillations will emerge for current\ndensities that provide a reactive torque which is su\u000ecient\nto overcome the in-plane anisotropy \u00180:1 T for MRG,\nwhich corresponds to j>\u00187\u00021010A m\u00002. The second\nnecessary condition is that the dissipative torque must\novercome the Gilbert damping \u000b. Taking\u000b\u00190:01 we\n\fnd the condition j>\u001810\u00021010A m\u00002. An alternative\napproach is to compare directly the e\u000bective inductance\ncreated by the SOT and the self inductance of the os-\ncillating element. In a shorted Hall bar device, a crude\nestimate of the self inductance for a 500 nm thick \flm\nwith an active length of 20 µm is 0:1 pH { the dimensions\nare chosen to enhance impedance matching to free space\nin a real oscillator. We saw in FIG. 4 that the e\u000bective\ninductance reaches values two orders of magnitude higher\nthan this, ensuring that oscillatory behaviour is possible,\neven in the low-current-density region. The natural fre-\nquency of the oscillator will be determined by the larger\nof the two e\u000bective inductances, that is by the SOT and\nthe magnetic resonance frequency of the material, which\nwe previously estimated as 0 :75 THz.31\nIn summary, we \fnd that current-induced spin orbit\ntorque reaches record values in single-layers of the com-\npensated, half-metallic ferrimagnet Mn 2RuxGa, well in\nexcess of those achieved in bilayer structures. With real-\nistic values of damping, this should allow sustained mag-\nnetic oscillations that could be detected by magnetore-\nsistive e\u000bects, or free-space emission using a suitable an-\ntenna. A cheap, compact, and tunable oscillator operat-\ning in the terahertz gap would break new ground in spin\ndynamics, and could potentially unlock a new realm of\ninformation transfer at bandwidths three orders of mag-\nnitude higher than those of the present day.\nACKNOWLEDGMENTS\nThis project has received funding from the Euro-\npean Union's Horizon 2020 research and innovation pro-\ngramme under grant agreement No 737038, and from Sci-\nence Foundation Ireland through contracts 12/RC/2278\nAMBER and 16/IA/4534 ZEMS as well as the Research\nCouncil of Norway through its Centres of Excellence5\nfunding scheme, Project No. 262633 \\QuSpin\". The authors declare no competing \fnancial interests.\n\u0003Corresponding author:rodek@tcd.ie\n1J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n2J. Slonczewski, J. Mag. Magn. Mat. 159, L1 (1996).\n3C. Chappert, A. Fert, and F. N. Van Dau, Nature Mate-\nrials6, 813 (2007), review Article.\n4I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. 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B 98, 220406 (2018)."    },    {        "title": "1303.1372v1.Change_in_the_Magnetic_Domain_Alignment_Process_at_the_Onset_of_a_Frustrated_Magnetic_State_in_Ferrimagnetic_La2Ni_Ni1_3Sb2_3_O6_Double_Perovskite.pdf",        "content": "10-56 \n 1\nChange in the Magnetic Domain Alignment Process at the Onset of a \nFrustrated Magnetic State in Ferrimagnetic La 2Ni(Ni 1/3 Sb 2/3 )O6 \nDouble Perovskite \n(Revised ..) \n \nDiego G. Franco 1,2 , Raúl E. Carbonio 1, and G. Nieva 2,3 \n \n1INFIQC-CONICET, Depto. de Físico Química, Facultad de Cienc ias Químicas, Universidad Nacional de Córdoba, Ciudad \nUniversitaria. X5000HUA Córdoba, Argentina \n2Laboratorio de Bajas Temperaturas. Centro Atómico Baril oche –CNEA. 8400 Bariloche, R. N., Argentina. \n3Instituto Balseiro, CNEA and Universidad Nacional de Cuy o.  8400 Bariloche, R. N., Argentina. \n \nWe have performed a combined study of magnetization  hysteresis loops and time dependence of the magnet ization in a broad \ntemperature range for the ferrimagnetic La 2Ni(Ni 1/3 Sb 2/3 )O6 double perovskite.  This material has a ferrimagne tic order transition \nat ~100 K and at lower temperatures ( ~ 20 K) shows the signature of a frustrated state du e to the presence of two competing \nmagnetic exchange interactions. The temperature dep endence of the coercive field shows an important up turn below the point \nwhere the frustrated state sets in. The use of the magnetization vs. applied magnetic field hysteresis  data, together with the \nmagnetization vs. time data provides a unique oppor tunity to distinguish between different scenarios f or the low temperature \nregime. From our analysis, a strong domain wall pin ning results the best scenario for the low temperat ure regime. For \ntemperatures larger than 20K the adequate scenario seems to correspond to a weak domain wall pinning.   \n \nIndex Terms — Ferrimagnetic materials, Magnetic analysis, Magneti c domain walls, Magnetic hysteresis.  \n \nI.  INTRODUCTION  \nANY of the magnetic interactions found in transition \nmetal oxide perovskites are due to superexchange \nand/or super-superexchange interactions mediated through \nthe O2- p orbitals.  In some materials the relative strength of \nthese interactions determines the magnetic structure, range  \nof the ordering temperatures and the possibility of \nfrustration [1]-[5].  In the perovskite structure, the t ypical \nbond angles and distances usually favor antiferromagnetic \nsuperexchange interactions [6],[7]. However, in some \nspecial cases, due to disorder or differences in the magne tic \nstate of the cations, a ferrimagnetic state is develope d with \nmacroscopic characteristics similar to a ferromagnetic state. \nCoercivity and remanence are an indication of the \nmetastability in ferromagnetic samples. Their magnitudes \nindicate how far the system is from equilibrium. They ar e \nrelated, therefore, with the relaxation to the equilibriu m \nstate, the anhysteretic [8] curve in the ferromagnetic st ate.  In  \nbulk ferromagnets the energy barriers that determine the \ntime evolution of the magnetization are related to local \ninteractions within a domain, the nucleation and the \nmovement of domain walls (DW). The DW movement \ndepends on the applied magnetic force, wall thickness and \ntype and density of pinning centers.   \nIn bulk ferromagnetic samples, a local frustration is \nnormally hard to visualize due to the magnetic history \ndependence of the metastable states. However, the magneti c \nmoments alignment within a domain and the movement of \nthe DW have characteristic energies [9]  that could be \nmodified if some degree of magnetic frustration occurs at a  microscopic level. This normally results in a strong DW \npinning effect and causes an increase in the coercivity. \nThis article will present a detailed magnetic study of t he \nferrimagnetic double perovskite La 2Ni(Ni 1/3Sb 2/3)O 6.  We \nwill show that the material behaves as a ferrimagnet below  \n100 K and that there is a change in the magnetic domains \nalignment process at 20 K.  We will show that below 20 K \nthe hysteretic magnetic behavior is characteristic of a strong \ndomain wall pinning regime due to the onset of a frustrated \nmagnetic interaction. \nII.  RESULTS  \nWe prepared polycrystalline samples of \nLa 2Ni(Ni 1/3 Sb 2/3 )O6 by conventional solid-state reaction at \n1400 oC [10].  X ray diffraction data from powders at room \ntemperature showed the crystalline symmetry to be \nmonoclinic, space group P2 1/n. This space group \naccommodates a rock salt arrangement of BO 6 and B'O 6 \noctahedra described by the a -b-c+ system of three octahedral \ntilts in the Glazer's notation. The (Ni/Sb) 2dO6 and \n(Ni/Sb) 2cO6 octahedra are rotated in phase (along the \nprimitive c axis) and out-of phase (along the primitive a and \nb axes). We performed a Rietvelt refinement of the structur e \nusing the FULLPROF program [11], resulting in lattice \nparameters of a = 5.6051(3) Ǻ, b = 5.6362(3) Ǻ, c = \n7.9350(5) Ǻ and β = 89.986(4) o. We refined the two \ncrystallographic sites 2 d and 2 c with different occupancies \nNi 2+ /Sb 5+  to model the octahedral site disorder. The 2 d \ncation site is almost fully occupied by Ni 2+  whereas the 2 c \nsite has occupancy close to 1/3 of Ni 2+  ions and 2/3 of Sb 5+ . \nThe resulting crystallographic formula can be written as \nLa 2(Ni 0.976 Sb 0.024 )2d(Ni 0.357 Sb 0.643 )2cO6. M\nManuscript received January XX, 2013 (date on which  paper was \nsubmitted for review). Corresponding author: G. Nie va (e-mail: \ngnieva@cab.cnea.gov.ar). \nDigital Object Identifier inserted by IEEE 10-56 \n 2\nThe magnetic measurements were performed on \npolycrystalline pellets with a QD-MPMS SQUID \nmagnetometer in the range 2 to 300K and -5 to 5T. In the \nmain panel of Fig. 1 we show the magnetization, M, as a \nfunction of temperature, T, while cooling in a very low \napplied field, H.  There is a transition to a magnetic \npolarized state at TC = 98(2) K. \nWe extrated the low temperature value of the saturation \nmagnetization, Ms, from M vs. H curves, from the \nasymptotic extrapolation of the high field behavior with a \nLangevin function.  This saturation magnetization, Ms, has a \nlower value than the one expected for the complete \npolarization of the Ni 2+  magnetic moments, 2.67 µB/f.u.. \nInstead, the experimental Ms value was 1.19  µB/f.u., \nimplying that the system behaves as a ferrimagnet, with two \nNi 2+  magnetic sublattices antiferromagnetically coupled, one \nat the 2 d site and another at the 2 c site. The near 1/3 Ni 2+  \nrandom occupation of the 2 c sites sublattice give as a result \nuncompensated Ni 2+ magnetic moments that order at 100 K. \nFor a perfectly stoichiometric ferrimagnetic sample and ful l \nNi 2+  occupancy of the 2 d site Ms should be 1.33   µB/f.u., and \nlower values are expected if Sb 5+  partially occupies also the \n2d site. The expected value for Ms with the refined \noccupancies is 1.24   µB/f.u., very close to the experimental \none. \nWe measured hysteresis loops, M vs H, for several \ntemperatures below 100 K. We show in the inset of Fig. 1  a \ndetail of the loops for 2 K and 20 K.  \nFIG. 1 HERE \nWe have also measured the time evolution of the \nmagnetization at the coercive field (i.e. near the field f or \nzero magnetization)  after saturation at 1, 3 and 5 T for each \ntemperature. We show in Fig. 2 typical M vs time data, for \nthree different temperatures at their corresponding coercive \nfields. \n \nFIG. 2 HERE \nIII.  DISCUSION \nWe show in Fig. 3(a) and (b) the temperature dependence \nof the coercive field, Hc, and the ratio between remanent \nmagnetization and saturation magnetization, ( Mr / M s). The general feature observed in Fig. 3 is that Mr and Hc increase \nsteeply when the temperature is lowered below 20 K \nindicating an increase in the energy absorbed by the materi al \nto change the direction of M. \nThe measured values of the coercive field, Hc, display two \ndifferent regimes as can be seen in Fig. 3(a). For T > 20 K a \nlinear behavior of Hc was found. This linear behavior is \ncharacteristic of weak DW pinning (WDWP), produced by a \nrandom distribution of individual weak pinning sites [9].  In \nthis case the coercive field is given by \n \n\n\n\n\n\n\n\n− =2 031 25 1\nbTkH HB\nW cγ                       (1) \n \nwhere H0W  is the zero temperature extrapolated reversion \nfield, kB is the Boltzmann constant, γ is the DW energy per \nunit area and b is a measure of the DW thickness. The \nobtained values are shown in Table I. \nIn the low temperature regime, T < 20 K, two models \ndescribe reasonably well the data. One corresponds to stron g \nDW pinning (SDWP), \n \n23 / 2\n0475 1\n\n\n\n\n\n\n\n− =bf TkH HB\nS c                    (2) \n \nwhere H0S is the coercive field at zero temperature and f is \nthe magnetic force needed to depin a domain wall.  The \nfitted values are shown in Table I. \nThe other model corresponds to the freezing of single \ndomain large particles (SDLP) or clusters [11], [12]. In t his \nscenario, \n \n\n\n\n\n\n\n− =2/ 125 1KV TkH HB\nK c                              (3) \n \nwhere HK is the anisotropy field of a particle or cluster, V is \nits volume and K is the uniaxial anisotropy energy density. \nThe fitted values are shown in Table I.  \nFIG. 3 HERE \n \n \nFig. 1.  (color online) Magnetization as a function of temperature cooling \nwith an applied field of 1 Oe.  Inset: Magnetizatio n as a function of ap plied \nfield, detail of the magnetization loops for two fi xed temperatures T = 2 K \nand 20 K. \n \nFig. 2. (color online) Difference between the measured magnetization and \nthe initial one, M 0, as a function of time. The shown mag netization time \ndependence was taken at H c (M ~ 0) after saturation in the opposite \ndirection.  We indicate the fixed temperatures for each experiment.  10-56 \n 3\nIn Fig. 3(a) and the inset we show the lines (solid and \ndash doted) corresponding to each model. The best fit is \nobtained with the SDWP model but the freezing of SDLP \nmodel is also in fair agreement with the Hc data. \nThe time evolution of the magnetization could be used to \ndiscern between the two scenarios at low temperature.  I f a \ndistribution of activation energies is present in the mater ial \n[9] a logarithmic behavior is expected for M(t): \n \n)ln( 0 t S M M − =                                (4) \n \nwhere M0 is the starting value of the magnetization and S the \nmagnetic viscosity coefficient. The above relation holds  \napproximately for our polycrystalline pellets samples a s we \nshow in Fig. 2. \nIn a tipical ferromagnet, the time dependence of M is \nirreversible and this behavior has been connected with the \nirreversibility caused by a small change in field, the so \ncalled irreversible susceptibility, χirr . Both irreversibilities \nare related by a fictitious field, the fluctuation field,  Hf, in \nthe theory introduced by Neel [13] that represents an \naverage of the thermally activated, time dependent processes \n[14] leading to equilibrium by reversing the metastable  \nmagnetization. In terms of the magnetization derivatives, a t \na given field and temperature, \n \nf irr \nMi\nt iirr \nHirr HtH\nHM\ntMS\nirr iχ=\n\n\n\n∂∂\n\n\n\n\n∂∂=\n\n\n\n∂∂=)ln( )ln(       (5) \n \nwhere Mirr  is the irreversible magnetization and Hi is the \ninternal field. In the case of a time independent viscosity coeficient S, the fluctuation field is equivalent to the \nmagnetic viscosity parameter Sv, that can be written in terms \nof the activation energy, E, necessary for magnetization \nreversal [15], \n \nirr MB\nvS\ndH dE TkS\nirr χ=−=) /(.                       (6) \n \nTo determine the temperature dependence of Sv, a value of \nH equal to the coercive field is chosen [15] ( Mirr = 0). The \nactivation energy for the SDWP, WDWP and clusters or \nSDLP freezing are given by [16]: \n \n2/32/1\n01) 3 / 4 (\n\n\n\n\n\n\n\n− =\nSSHHbf E                   (7) \n\n\n\n\n\n\n\n− =\nWWHHb E\n021) 31 (γ                         (8) \n2\n1\n\n\n\n\n\n\n\n− =\nKFHHKV E                           (9) \n \nand in each case the magnetic viscosity parameters are g iven \nrespectively by, \n \n\n\n\n\n\n\n\n−\n\n\n=3/2 3/2\n0475 1475 \n75 4)(bf Tk\nbf TkH SB B\nS Sv         (10) \n2 031 25 \n25 1)(bTkH SB\nW Wvγ=                             (11) \n2/125 \n50 1)( \n\n=KV TkH SB\nK Fv.                        (12) \n \nFrom the experimental data (such as those of the inset of \nFig. 4) the values of χirr (H = Hc) can be extracted. They can \nbe approximated as those of the total χ at Hc, neglecting the \nreversible contribution to χ [14]. These values are shown in \nFig. 4. Also from the experimental data the S values can be \ncalculated from (4) at Hc, since the linear behavior holds, \nFig. 2. In this case we take Mirr  as the measured M, \nneglecting the reversible component. \nFIG. 4 HERE \nTABLE 1 HERE \nThe experimental values of Sv were obtained by using (6). \nThey are displayed in Fig. 5 together with the fits for \ndifferent models in different temperature ranges using (10) - \n(12). At low temperature, the best fit to the data is given  \nTABLE  I \n γb2 \n(10 -14 erg) H0W  \n(Oe)  4bf \n(10 -13  erg) H0S  \n(Oe) KV \n(10 -14  erg) H0F  \n(Oe) \nHc 1.26 53.5 3.07 780 7.4 760 \nSv 1.4 53.5 2.13  780 7.4 760  \n \nFitted parameters for the WDWP model above 20 K and  the two models \ncompared below 20K, SDWP and freezing of SDLP. The first column \nindicates whether the coercive field or the magneti c viscosity parameter \nwere used in the parameters determination.  In the case of the viscosity \nparameter, the zero temperature fields H0W , H0S  and H0F  were not fitted but \ntaken from the Hc fits. \n \n \nFi g. 3.  (color online) (a) Coercive field Hc vs T for La 2Ni(Ni 1/3Sb 2/3)O 6\npolycrystalline samples. The solid lines (SDWP mode l), dash- dotted lines \n(SDLP freezing model) at low temperature and the da shed line (WDWP \nmodel) at high temperature are fits described in th e text. Inset: H  c 1/2  vs T 2/3 \nshowing the linear behavior expected in the SDWP mo del, solid lines are \nthe SDWP model and dash- dotted lines are the SDLP freezing model. (b) \nNormalized remanent magnetization (at H = 0) vs T.  The different \nsymbols indicate different samples in both panels.  10-56 \n 4\nusing (10) (a SDWP scenario) provided the nonmonotonic \nbehavior of Sv. The parameters obtained are shown in Table \nI. Clearly no good agreement is found for the freezing of \nclusters or SDLP scenario. In the high temperature region \nthe experimental Sv vs T is in agreement with the linear \nbehavior calculated in (11). However, a non-zero Sv (T = 0) \nvalue was found, not present in the model. \nFIG. 5 HERE \nTherefore, based on combined data extracted from the \nhysteresis loops and time dependence of the magnetization,  \nwe depicted two regimes in La 2Ni(Ni 1/3Sb 2/3)O 6 pelletized \npolycrystalline samples: Weak pinning of DWs at T >2 0 K \nand strong pinning of DWs below that temperature. The \nmicroscopic origin of this change of regime could be related \nwith the onset of a super-superexchange antiferromagnetic \ninteraction among Ni 2+  via O 2- -Sb 5+  - O2- paths [10] that \ncreates a frustrated magnetic interaction. IV.  CONCLUSION  \nThe ferrimagnetic state in La 2Ni(Ni 1/3Sb 2/3)O 6 was found \nto be characterized by two different regimes for domain w all \nmovement, a strong and a weak domain wall pinning regime \nat low and high temperatures respectively. The temperature \nrange of the strong domain wall pinning regime coincides  \nwith that of the existence of a proposed frustrated stat e. The \nscenario of clusters or large single domain particles f reezing \nwas discarded based in the coercive field and magnetic \nviscosity parameter temperature dependence analysis.  \nACKNOWLEDGMENT  \nWe thank E.E. Kaul for fruitfull discussions.  R.E.C, and \nG.N. are members of CONICET. D.G.F. has CONICET \nscholarship. Work partially supported by ANPCyT PICT07-\n819, CONICET PIP 11220090100448 and SeCTyP-\nUNCuyo 06/C313.  R.E.C. thanks FONCYT (PICT2007-\n303), CONICET (PIP 11220090100995) and SECYT-UNC \n(Res. 214/10) for finantial support. \nREFERENCES  \n[1]  H.-J. Koo and M.-H. Whangbo, “Importance of the O-M -O bridges \n(M = V5+, Mo 6+) for the spin-exchange interactions in the magneti c \noxides of Cu 2+ ions bridged by MO 4 tetrahedra: Spin-lattice models of \nRb 2Cu 2(MoO 4)3,BaCu 2V2O8, and KBa 3Ca 4Cu 3V7O28 ,” Inorg. 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Wohlfarth, “The coefficient of magnetic visco sity,”  J. Phys. F. \nMet. Phys.  14, pp. L155-L159 (1984). \n[14]  O. V. Billoni, E. E. Bordone, S. E. Urreta, L. M. Fab ietti, H. R. \nBertorello, “Magnetic viscosity in a nanocrystalline  two phase \ncomposite with enhanced remanece,”  J. of Magn. Magn. Mater.  208, \npp. 1-12 (2000). \n[15]  D. C. Crew, P.G. McCormick, R. Street, “Temperature D ependence \nof the Magnetic Viscosity Parameter,”  J. of Magn. Magn. Mater.  177-\n181, pp. 987-988 (1998). \n[16]  J. F. Liu and H. L. Luo, “On the coercive force and  effective \nactivation volume in magnetic materials,”  J. of Magn. Magn. Mater.  \n94, pp. 43-48 (1991). \nFig. 5.  (color online) Magnetic viscosity paramete r Sv vs T for \nLa 2Ni(Ni 1/3Sb 2/3 )O 6 polycrystalline pellets. Different symbols indicate \ndifferent samples, the solid symbols corresponds to  Sv values cal culated \nwith the data displayed in Fig.2 . The lines represent the models described in \nthe text, SDWP model (solid line), freezing of SDLP  model (dash dot ted \nline) and WDWP model (dash line). In the inset , a zoom of the low \ntemperature region is shown.  \nFig. 4.  (color online) Total susceptibility χ= dM/dH measured at Hc vs T \nfor the polycrystalline pellets. Different symbols indicate different samples. \nThe line is a guide to the eye. The inset shows a t ypical χ vs H curve taken \nat 10 K.  "    },    {        "title": "2307.04669v2.Reversal_of_the_skyrmion_topological_deflection_across_ferrimagnetic_angular_momentum_compensation.pdf",        "content": "Reversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation\nReversal of the skyrmion topological deflection across ferrimagnetic\nangular momentum compensation\nL. Berges,1R. Weil,1A. Mougin,1and J. Sampaio1\nUniversité Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France\n(*Electronic mail: joao.sampaio@universite-paris-saclay.fr)\n(Dated: 6 October 2023)\nDue to their non-trivial topology, skyrmions describe deflected trajectories, which hinders their straight propagation\nin nanotracks and can lead to their annihilation at the track edges. This deflection is caused by a gyrotropic force\nproportional to the topological charge and the angular momentum density of the host film. In this article we present\nclear evidence of the reversal of the topological deflection angle of skyrmions with the sign of angular momentum\ndensity. We measured the skyrmion trajectories across the angular momentum compensation temperature ( TAC) in\nGdCo thin films, a rare earth/transition metal ferrimagnetic alloy. The sample composition was used to engineer the\nskyrmion stability below and above the TAC. A refined comparison of their dynamical properties evidenced a reversal\nof the skyrmions deflection angle with the total angular momentum density. This reversal is a clear demonstration of\nthe possibility of tuning the skyrmion deflection angle in ferrimagnetic materials and paves the way for deflection-free\nskyrmion devices.\nThe discovery of efficient driving of chiral magnetic tex-\ntures by current-induced spin-orbit torques1–3has opened\nthe possibility of energy-efficient and high-performance spin-\ntronic devices4,5, with applications in digital6or neuromor-\nphic7–10computation, ultra-dense data-storage11,12, and sig-\nnal processing13,14. Chiral textures are stable in magnetic thin\nfilms with a significant Dzyaloshinskii-Moriya interaction\n(DMI), typically induced with an adjacent heavy-metal layer\n(e.g. Pt/Co). Additionally, the heavy-metal layer, through the\nspin Hall effect, converts an applied charge current into a spin\ncurrent that drives the magnetic textures by spin orbit torque\n(SOT). Very promising mobility of chiral magnetic domain\nwalls (DW) has been observed1,15, with nonetheless a saturat-\ning mobility at large current densities2. Another archetypal\nchiral magnetic texture is the skyrmion, a small (down to few\ntens of nm) radially symmetric whirling texture. Although\nhighly mobile16–19, their non-trivial topology induces a trans-\nverse deflection of their trajectory, a phenomenon known as\ngyrotropic deflection or skyrmion Hall effect18,20,21. This re-\nduces the velocity in the forward direction and can lead to the\nannihilation of the skyrmion at the edges of the hosting mag-\nnetic track, and is thus highly undesired.\nThe gyrotropic deflection can be mitigated in magnetic sys-\ntems with anti-parallel lattices22, such as antiferromagnets or\nferrimagnets, where the overall angular momentum density of\nthe double skyrmion can be suppressed. In particular, fer-\nrimagnetic alloys of the rare-earth/transition-metal (RETM)\nfamily, where the RE and TM moments are antiferromagnet-\nically coupled23,24, are a promising example. In a previous\nwork by our team, it was shown that skyrmions in GdCo thin\nfilms attained the high-mobility linear regime beyond pinning,\nand that their velocity and deflection followed the predictions\nof the Thiele model25. However, there is still only little ex-\nperimental evidence of the advantages of these systems26,27,\nespecially regarding the control of the gyrotropic deflection.\nIn RETMs, The balance between the moments of differ-\nent nature can be changed with alloy composition or temper-\nature which leads to two points of interest for skyrmions. At\nthe first one, the magnetic compensation temperature TMC, themagnetization of the two sub-lattices are equal, the total mag-\nnetization ( Ms=MTM−MRE) vanishes, and the size of the\nskyrmions is minimal due to the absence of dipolar fields27.\nAs RE and TM have different gyromagnetic ratios ( γREand\nγTM), the total angular momentum density ( Ls=MTM\nγTM−MRE\nγRE)\nwill vanish at a different temperature, the angular compensa-\ntion temperature TAC. Both TMCandTACdepend on composi-\ntion. The reduction and reversal of the total angular momen-\ntum, which is the root cause of magnetic precession, leads to\ninteresting dynamical properties near TAC, such as e.g. the re-\nversal of the deflection angle of chiral domain wall fingers28\nor the precessionless motion of magnetic domains walls29.\nHowever, the reversal of the skyrmion gyrotropic deflection\natTAChas not yet been demonstrated.\nIn this letter, we measure the velocity and deflection angle\nof skyrmions driven by spin-orbit torques in two Pt/GdCo/Ta\nfilms of different composition, above and below their TAC. We\nshow the dependence of the deflection with angular moment\ndensity, and in particular its reversal by changing sample com-\nposition or temperature. A quantitative analysis with a rigid\ntexture model based on the Thiele equation is used to char-\nacterize the role of the material parameters on the skyrmion\ndynamics.\nThe skyrmion dynamics were measured in two samples.\nSample 1 is composed of a film of (Si/SiOx(100))/ Ta(1)/\nPt(5)/ Gd 0.32Co0.68(5)/ Ta(3) and sample 2 of (Si/SiOx(300))/\nTa(3)/ Pt(5)/ Gd 0.3Co0.7(8)/ Ta(5)/ Pt(1) (thicknesses in nm)\nas presented in the insets in Fig. 1a. The samples were pat-\nterned into 10 µm- or 20 µm-wide tracks in order to apply\ncurrent pulses (Fig.1b). The magnetization as a function of\ntemperature was measured by SQUID magnetometry on un-\npatterned samples and is presented in Fig. 1(a). Sample 1\npresents a TMCaround 360 K whereas sample 2 presents a\nTMCaround 200 K. Therefore, at room temperature, sam-\nple 1 is RE-dominated whereas sample 2 is TM-dominated,\nwhere RE or TM domination refers to which sublattice has\nthe higher magnetic moment and therefore aligns with an ex-\nternal magnetic field. It is useful to use the effective ferromag-arXiv:2307.04669v2  [cond-mat.mtrl-sci]  5 Oct 2023Reversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation 2\nFIG. 1. a) Msversus temperature for sample 1 (top panel,\nblack points) and sample 2 (bottom panel, gray points) measured\nby SQUID (Superconducting Quantum Interference Device) magne-\ntometry, and mean-field-computed curves of Ms(solid line) and Ls\n(dashed line) for both samples. Samples stacks are presented in in-\nsets and skyrmion temperature stability regions in colored bands. b)\nTypical magnetic device studied for sample 2. c) Examples of differ-\nential MOKE images obtained in sample 2 at 350 K.\nnet model of ferrimagnets30, which assumes a signed mag-\nnetization and angular momentum density that are positive,\nby convention, when TM-dominated: Ms=|MCo|−MGd|and\nLs=|LCo|−LGd|. The exact determination of the TACis not\nstraightforward. It was therefore deduced for both samples,\nusing the mean field model described in ref.25. The calculated\nLS(T)are shown by the dashed lines in Fig. 1(a), and yield\nTAC=416 K for sample 1 and TAC=260 K for sample 2.\nThese results are consistent with the empirical law described\nin ref.31which gives TACfor GdCo between 40 to 60 K above\ntheTMC.\nThe magnetic textures are observed in each sample as\na function of temperature by magneto-optical-Kerr-effect\n(MOKE) microscopy. A typical differential MOKE image is\npresented in Fig. 1(c). Skyrmions are observed in the tem-perature ranges indicated by the color bands in Fig. 1(a). In\nthese ranges, starting from a saturated state and lowering the\napplied external magnetic field, skyrmions with a core of op-\nposing magnetization will naturally nucleate at small enough\nfield (−30 to 0 mT for an initial saturation at large negative\nmagnetic field). Skyrmions can also be nucleated by applying\nelectrical pulses25,32. A typical phase diagram (versus tem-\nperature and field) of these samples is presented in a previ-\nous work25. In the studied temperature range, sample 1 only\npresents one skyrmion stability range around 290 K, whereas\nsample 2 presents two skyrmion stability ranges, one around\n90 K and a second around 350 K. In sample 1, the skyrmion\nstability range is below TMC(and TAC), where the film LS<0,\nand so these are dubbed RE-dominated skyrmions. In sam-\nple 2, the skyrmions at 90 K are RE-dominated as well, while\nthe skyrmions at 350 K are TM-dominated (above TMCand\nTACwith therefore LS>0). Note that in the MOKE images,\nthe signal is proportional to the Co sublattice, independently\nof the temperature33. Thus, skyrmions with a core Co mo-\nment pointing along the same direction will appear with the\nsame color (black for −zwith our experimental conditions),\nwhether they are RE- or TM-dominated (Fig. 1c).\nOnce skyrmions are nucleated, electrical pulses of 3 to\n10 ns are applied and MOKE images are acquired in order\nto study the skyrmions dynamics. The skyrmion motion\nis tracked over several pulses using a partially-automated\nprocess described in ref.25, and their velocity and deflection\nare calculated considering the pulse duration and the traveled\ndistance. Typical images of skyrmions displacements are\nshown in Fig. 2, in the case of sample 2 at low temperature\nandLs<0 (a) and high temperature and Ls>0 (b). The\naverage skyrmion diameter was similar for the three studied\ncases, 0.86 ±0.28 µm. An example of the observed skyrmion\ndynamics in sample 1 is presented in Fig. 2(c) with a super-\nposition of successive MOKE images where the skyrmion\ncolor refers to the MOKE image number.\nThe skyrmion deflection ( θsk) and velocity ( v) versus\napplied current density ( j) are presented in Fig. 3(a,b) for\nthe three cases: RE-dominated skyrmions in sample 1, and\nRE- and TM-dominated skyrmions in sample 2. Videos of\nsuccessive displacements in both samples are shown in S.I. In\nthe three cases, the velocity shows a clear depinning transition\nabove a current threshold (different for each case), and then\nfollows a linear regime. The mobility in the linear regime\n(i.e.∆v/∆j) is much higher in sample 2 than in sample 1. In\nsample 2, the mobility of TM-dominated skyrmions is slightly\nhigher than RE-dominated skyrmions. These differences in\nmobility will be discussed later. The linear regime extends up\nto 190 m/s in sample 1 and to 450 m/s in sample 2. At highest\nj, skyrmions are nucleated by the pulse, which hinders the\ntracking analysis and thus limits the maximum jthat can\nbe examined. In the linear regime, the deflection angle θsk\nis approximately constant with the current density, and its\nabsolute value is about 40◦for the three cases. The deflection\nangle is clearly reversed between the TM- and RE-dominated\nskyrmions: it is positive for TM-dominated skyrmions (in\nsample 2) and negative for RE-dominated skyrmions (in bothReversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation 3\nFIG. 2. a)-b) Example of three successive MOKE images separated by 10-ns (6 ns) pulses showing the displacement of skyrmions in sample\n2 with at a) at T = 110 K and j=120 GA/m2and b) T = 350 K and 150 GA/m2(field around 0 mT.) The colored circles identify the same\nskyrmions in the three images. As the images in a) were obtained using a cryostat and are of lower resolution, a different temperature from\nthe one used in the dynamical studies (90 K) was used to render the skyrmions larger and more visible. c) Superposition of four consecutive\nMOKE images, in the case of sample 1 with Ls<0, showing the propagation of three skyrmions for 2 and 4 images. The different forces\ndefined in the Thiele equation acting on the skyrmions are sketched around the black dot.\nFIG. 3. a) Averaged skyrmion deflection θsk, and b) velocity, mea-\nsured in sample 1 at 290 K and 2 at 90 K and 350 K. The error bars\nrepresent the standard deviation of the measurements. The dashed\nlines are obtained with the Thiele based model (with fitted θSHE).\nThe color bands represent the expected error due to the experimental\nerror of θSHE, as described in the main text.\nsamples). The deflection also reverses with core polarity,\ni.e. with the Co moment pointing along +z(which appear\nas white skyrmions in the MOKE images; see Supplemental\nMaterials). The θskin the pining regime is measured to belarger than in the flow regime in sample 1, whereas it is lower\nin sample 2. This is perhaps a bias induced by the different\nnucleation protocol used in these measurements. For sample\n1, skyrmions were only nucleated by current pulses, mostly\nnear one of the edges due to the Oersted field25, whereas\nfor sample 2 they were first nucleated homogeneously by\nmagnetic field. As skyrmions can be annihilated at the\nedges, only the skyrmions that deviate towards the center are\naccounted for, which biases the measurement of the mean θsk.\nThe skyrmion dynamics in the linear regime can be quanti-\ntatively analyzed using a rigid-texture formalism based on the\nThiele equation34. It expresses the equilibrium of all forces\napplied on the magnetic texture that reads in our case as:\nFG+FSOT+αDv=0, where FSOTis the SOT force, FGthe\ngyrotropic force and αD is, in general, a tensor describing\nthe dissipation. This formalism can be applied to skyrmions\nin double-lattice systems as presented in refs.25,35. These\nforces are depicted in Fig. 2 c), on a black dot representing\na skyrmion in the case of Ls<0. The norm of the skyrmion\nvelocity |v|and its deflection θskcan be deduced to be:\n|v|=v0p\n1+ρ2(1)\nθsk=arctan (ρ) (2)\nIn the limit of skyrmions larger than the domain wall width\nparameter ∆, the parameters v0andρare:\nv0≈ −π∆\n2Lα¯h jθSHE\n2et(3)\nρ≈∆\n2πRLS\nLαn (4)Reversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation 4\nT<TMC TMC<T<TAC T>TAC\nMs − + +\nLs − − +\ncore magnetic moment +z⊙ − z⊗ − z⊗\ncore cobalt moment −z⊗ − z⊗ − z⊗\npCo −1 −1 −1\nFG·y − − +\nFSOT +x +x +x\nTABLE I. Signs versus temperature of the material parameters,\nof the skyrmion core configuration parameters and of the expected\nforces acting on a skyrmion with negative polarity (black in the\nMOKE images) driven by a positive ( +x) current.\nParameter Sample 1 (290 K) Sample 2 (350 K)\nγ/2π[GHz/T]a18.3 40.8\nαa0.15 0.175\n|MS|[kA/m]b78 125\n|LS|[kg/(ms)]b6.8×10−74.9×10−7\n|Lα|[kg/(ms)]b1.02×10−78.5×10−8\nµ0Hk[mT]c200 60\nKu[kJ/m3]c11.5 13.6\nD[ mJ/m2]a−0.22 −0.14\nA[pJ/m]a4.6 (4.6)d\nθSHEe0.04 0.09\n2R[µm]f0.85±0.28 0 .86±0.28\nTABLE II. Measured parameters used in the model.aγ,α, the ex-\nchange stiffness Aand the DMI strength D(not used in the model)\nwere determined with Brillouin light scattering (BLS) at 290 K (sam-\nple 1) or 350 K (sample 2).bMs(T)(Fig. 1a) was used to determine\nLS=MS/γandLα=LSα.cHkwas measured from hysteresis cy-\ncles and BLS, from which Kuwas deduced.dAwas measured only\non sample 1 and assumed to be the same in sample 2.eθSHEwas\ndetermined by transport measurements using the double-harmonic\ntechnique25,36(see Supplementary Materials).fThe shown variation\nis the standard deviation of the observed radius and not the error of\nthe average value.\nwhere ¯his the Planck constant, ethe fundamental charge,\ntthe magnetic film thickness, θSHE is the effective SHE\nangle in the Pt layer, Lα=LSαthe energy dissipation rate,\nn=pCo4π=±4πthe topological charge of the skyrmion, R\nits radius, and pCo=±1 is the orientation along zof the core\nCo moment. Because Lαis always positive, the sign of the\ndeflection is given by the sign of the product of Ls(positive\nforT>TAC) and pCo. This sign is presented in Table I as\na function of temperature for pCo=−1, which is the case\nshown here (black skyrmions).\nThe parameters needed for the model were measured on\nboth samples (see Table II). Hk(T)was obtained by analyz-\ning hysteresis loops at various temperatures, which yielded a\nvalue for Ku(Ku=µ0HkMs/2−µ0M2\ns/2) with negligible ther-\nmal variation. The domain wall width parameter was calcu-\nlated using ∆=p\nA/Keff, where Keff=µ0HkMs/2 is the effec-\ntive anisotropy. For Sample 2 at 90 K ( MS=135 kA/m), the\nthermal variation of KuandAwas neglected (as it is smallerthan the precision of the other parameters) and the values at\n350 K were used; LS=12.1×10−7kg/(ms) and Lα=10.7\n×10−8kg/(ms) were deduced using a mean-field model as de-\nscribed in ref.25, assuming constant sub-lattice Gilbert damp-\ning parameters ( Lα=αCo|LCo\ns(T)|+αGd|LGd\ns(T)|). The\nskyrmion diameter was taken from the average observed di-\nameter, which is very similar for the three studied cases37.\nThese measured parameters allow to constrain the model\nand obtain curves for the velocity and deflection angle (dashed\nlines in Fig. 3). A constant deflection is predicted, and its\nvalue is obtained with no fitting parameters. The velocity is\npredicted to be linear with j, and its slope is obtained with\na single fitting parameter, θSHE. The fitted values ( θSHE =\n0.03 for sample 1 and 0.09 for sample 2) are consistent with\nthe precision of the measured θSHE(see Table II and Suppl.\nMat.). The model prediction range, calculated with the esti-\nmated error θSHE, is shown in the figure as a color band38.\nAbove the depinning threshold, where the model is expected\nto be valid, it both reproduces the qualitative behavior (con-\nstant deflection and linear velocity) and agrees quantitatively\nwith the experimental data, within the estimated error margin.\nIn particular, the sign of the deflection angle observed in the\nexperiments agrees with Eq (3b) taking into account the LSof\nthe film ( LS<0 for RE-dominated skyrmions and LS>0 for\nTM-dominated skyrmions).\nThe skyrmion mobility, given by the slope of the velocity\nversus j(Fig. 3(b)), is much higher in sample 2 than in sample\n1 (1.80 at 350 K vs 0.6 m ·s−1/GA·m−2, respectively). This\ndifference in mobility cannot be ascribed to a difference in\nskyrmion diameter (see eq. 1), as the two conditions present\nvery similar average sizes (Table II). This large difference has\nmultiple origins. First, the Lαof sample 2 is lower by 20%.\nThe second major cause is the difference of the film stacks, in\nparticular the thickness of the Ta capping layer. The measured\nθSHEis more than twice higher in sample 2 than in sample 1\n(Table II). This can be expected to be due a better passivation\nof the Ta layer in sample 2 which can therefore contribute\nmore to the SOT than the thinner (3 nm) Ta cap of the sample\n1 which is probably fully oxidized.\nFinally, comparing the skyrmion velocity curves for the\ntwo conditions in sample 2 (at 90 and 350 K), it can be seen\nthat both the depinning current and the mobility in the linear\nregime are significantly different. The depinning current is\nhigher at 90 K, which can be attributed by the thermal nature\nof the depinning process39. The difference in mobility is not\ndue to a difference in skyrmion diameter (which again is very\nsimilar in all three studied conditions). It can be expected that\nseveral magnetic parameters vary between 90 and 350 K, but\nthe experimental mobility can be understood by considering\nonly the variation of Lα(Lα(90 K )\nLα(350 K )≈1.25). This result and the\nThiele model suggest that Lαis a more pertinent parameter\nthan αto characterize the role of dissipation in the skyrmion\nmobility. Interestingly, Lαcan be more easily optimized than\nαto increase mobility, by increasing the sample temperature\n(as was the case here) or by decreasing the material’s Curie\ntemperature (all other parameters remaining equal). A recent\nwork39on skyrmions measured at relatively high temperature\nalso seems to point toward such an effect which seems to beReversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation 5\nan interesting path to increase skyrmion mobility.\nIn conclusion, we observed the propagation of skyrmions\nin the flow regime, i.e., beyond the effects of pinning in two\nGdCo samples, below and above the angular compensation\ntemperature. The observed mobilities were very large, with\na velocity up to 450 m/s. The skyrmion dynamics was stud-\nied in three cases, two in RE-dominated films and one in a\nTM-dominated film. The deflection angle was constant with\ndriving current and its sign was opposite between RE- and\nTM-dominated cases, both when comparing two samples of\ndifferent composition and when comparing two temperatures\n(above and below TAC) in the same sample. This confirms the\nmodulation of deflection angle θskwith LS.\nThese experiments demonstrate the effects of the angular\nmomentum density LSof the host material on the deflection of\nskyrmions. They show that θskcan be reversed in GdCo ferri-\nmagnetic thin films across their angular compensations, either\nby changing the alloy stoichiometry or simply its temperature.\nIn particular, the reversal of sign of θskacross compensation\nstrongly supports that θskshould be zero at angular moment\ncompensation. The engineering of magnetic parameters that\nwas done to produce the two presented skyrmion-hosting sam-\nples could be repeated rather straightforwardly to engineer a\nfilm with stable skyrmions at TACwith no deflection.\nACKNOWLEDGMENTS\nThe authors thank Stanislas Rohart for fruitful discussions,\nand André Thiaville for the study of the sample properties by\nBLS. This work was supported by a public grant overseen by\nthe French National Research Agency (ANR) as part of the\n“Investissements d’Avenir” program (Labex NanoSaclay, ref-\nerence: ANR-10-LABX-0035, project SPICY). Magnetome-\ntry and Anomalous Hall effect measurements were performed\nat the LPS Physical Measurements Platform.\nSUPPLEMENTARY MATERIAL\nSee supplementary material for videos of successive\nMOKE images showing the skyrmion motion for the three\ntemperature regions discussed in the text. Motion of\nskyrmions of opposite polarity (i.e., pCo= +1; white in the\nMOKE images) is also shown for sample 1. A note on the\nanalysis of the images and the selection of relevant textures\nis also included. Experimental data of θSHEare also included\nfor both samples at several temperatures.\nDATA AVAILABILITY STATEMENT\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.1T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. 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Ohno, “Quantitative char-\nacterization of the spin-orbit torque using harmonic Hall voltage measure-\nments,” Physical Review B 89, 144425 (2014).\n37The skyrmions show a large dispersion of diameter (Table II), which we\nestimate to lead to a ±14% dispersion of velocity and to a ±10º dispersion\nof the deflection angle. However, as the values of velocity and deflection\nwere averaged over many skyrmions and many displacements, the error\ndue to the size dispersion is drastically reduced and is neglected.\n38We estimate that the main sources of error of ρandvis the spin Hall effi-\nciency ( θSHE). which was estimated to be 25% (sample 1) and 12% (sample\n2); see supplemental materials.\n39K. Litzius, J. Leliaert, P. Bassirian, D. Rodrigues, S. Kromin, I. Lemesh,\nJ. Zazvorka, K.-J. Lee, J. Mulkers, N. Kerber, D. Heinze, N. Keil, R. M.\nReeve, M. Weigand, B. Van Waeyenberge, G. Schütz, K. Everschor-Sitte,\nG. S. D. Beach, and M. Kläui, “The role of temperature and drive current\nin skyrmion dynamics,” Nature Electronics 3, 30–36 (2020)."    },    {        "title": "2206.13593v1.Bridging_atomistic_spin_dynamics_methods_and_phenomenological_models_of_single_pulse_ultrafast_switching_in_ferrimagnets.pdf",        "content": "Bridging atomistic spin dynamics methods and phenomenological models of single pulse ultrafast\nswitching in ferrimagnets\nFlorian Jakobs and Unai Atxitia\nDahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit ¨at Berlin, 14195 Berlin, Germany\nWe bridge an essential knowledge gap on the understanding of all-optical ultrafast switching in ferrimagnets;\nnamely, the connection between atomistic spin dynamics methods and macroscopic phenomenological models.\nAll-optical switching of the magnetization occurs after the application of a single femtosecond laser pulse to\nspecific ferrimagnetic compounds. This strong excitation puts the involved degrees of freedom, electrons, lattice\nand spins out-of-equilibrium between each other. Atomistic spin models have quantitatively described all-\noptical switching in a wide range of experimental conditions, while having failed to provide a simple picture\nof the switching process. Phenomenological models are able to qualitatively describe the dynamics of the\nswitching process. However, a unified theoretical framework is missing that describes the element-specific spin\ndynamics as atomistic spin models with the simplicity of phenomenology. Here, we bridge this gap and present\nan element-specific macrospin dynamical model which fully agrees with atomistic spin dynamics simulations\nand symmetry considerations of the phenomenological models.\nI. INTRODUCTION\nSince its experimental discovery [1], the theoretical de-\nscription of laser induced all-optical switching (AOS) of the\nmagnetization in GdFeCo ferrimagnetic alloys has remained\na challenge. Despite intense experimental and theoretical re-\nsearch in the field [1–12], an established and unified picture\nof the process is still missing. Experimental findings are\nmostly compared or interpreted in terms of atomistic spin\ndynamics simulations [13–17], multisublattice spin dynam-\nics based on symmetry arguments [5, 18, 19], and based on\nthe Landau-Lifshitz-Bloch equation [20–22]. The main goal\nof the present work is the revision, extension and merging of\nthese approaches into a unified model.\nAtomistic spin dynamics (ASD) models have been used be-\nfore to quantitatively describe ultrafast dynamics in 3 dtransi-\ntion metals [23, 24] and 4 frare-earth ferromagnets [25, 26].\nThey have also been used in GdFeCo, to describe the equilib-\nrium thermal properties [13], the thermal character of AOS\n[4], the so-called transient ferromagnetic-like state [3], the\ndemonstration of spin-current-mediated rapid magnon local-\nisation and coalescence [27] and the possibility of AOS using\npicosecond-long laser pulses [16]. Results from atomistic spin\nmodels also compare qualitatively well to an analytical the-\nory based on the excitation of spin-wave exchange modes [8],\nprovide insights for optimal electron, phonon and magnetic\ncharacteristics for low energy switching [28] and predict max-\nimum repetition rate using two consecutive laser pulses [29].\nMore sophisticated, orbital-resolved atomistic models provide\ninsights on the role of the intra-exchange coupling between\n4fand 5 delectrons in the dynamics of GdFeCo alloys[14].\nAtomistic models can naturally describe switching in Gd/Fe\nmultilayers composed of very thin layers [30, 31]. Recent ob-\nservations [32, 33] of single pulse switching in Mn 2RuxGa\nalloys are also well-described by ASD methods [34]. De-\nspite the demonstrated success in modeling AOS, ASD sim-\nulation results are cumbersome to interpret without an ana-\nlytical model that unveils the role of the different processes\nand interactions during the switching process. This potential\nsemi-analytical model has to capture most of the features ofthe ASD simulations.\nSemi-phenomenological models describing switching al-\nready exist. A macroscopic theory for the description of the\ndynamics and relaxation of the macroscopic (sublattice) mag-\nnetization of ferromagnets and antiferromagnets was devel-\noped originally by Baryakhtar [9, 35]. An extension of such\nphenomenology to ferrimagnets in the context of ultrafast spin\ndynamics was introduced in Ref. [5]. At the ultrafast scale,\nmagnetization dynamics are dominated by atomic scale spin\nexcitations, these spin dynamics are driven by dissipative pro-\ncesses which in ferrimagnets are two-fold, relativistic and ex-\nchange driven. Relativistic processes allow for exchange of\nangular momentum between the spins and lattice degree of\nfreedom due to the presence of spin-orbit interaction connect-\ning them. Exchange processes can arise due to transport of\nspin angular momentum – spin and magnon transport – which\nis the only mean to exchange angular momentum in ferromag-\nnets. In multisublattice magnets another, different pathway\nopens, namely, local exchange of angular momentum. To ac-\ncount for such local exchange processes in ferrimagnets, the\nequation of motion for the magnetization dynamics proposed\nby Landau and Lifshitz [36] is enhanced by an exchange re-\nlaxation term [5, 9, 19, 37]. Within this macroscopic model,\nthe exchange relaxation dominates the dynamics when the\nmagnetic sublattices are driven into mutual non-equilibrium.\nQualitative agreement to experiments in two-sublattice mag-\nnets has been demonstrated [19], such as AOS in ferrimag-\nnetic GdFeCo using fs laser pulses [5] and ps laser pulses\n[38], AOS in Heusler semimetals Mn 2RuxGa [39], or element-\nspecific demagnetization of ferromagnetic NiFe alloys [18].\nQuantitative comparison of this model to neither experiments\nnor ASD simulations have been conducted so far. While the\narguments behind such phenomenology are robust, the range\nof applicability and the validity of the model parameters could\nbe questioned. For instance, the parameters defining the rel-\nativistic and exchange relaxation are assumed to be constant\nand of the same order. The magnetic free energy functional\nis calculated for near thermal equilibrium states. This implies\na relatively strong coupling to the heat-bath, while switching\nconditions are supposedly fulfilled when exchange relaxationarXiv:2206.13593v1  [cond-mat.mtrl-sci]  27 Jun 20222\nbetween sublattices dominates over the relaxation to the heat-\nbath.\nAn alternative macroscopic model directly derived from an\natomistic spin model has also been proposed. This model is\nbased in the Landau-Lifshitz-Bloch (LLB) equation of mo-\ntion [20, 40–43]. The LLB model for two-sublattice mag-\nnets [20, 42] has been used in the context of AOS in GdFeCo,\ne.g. the element-specific demagnetization rates compare well\nto experiment, and it predicts that near the magnetic phase\ntransition the otherwise slower Gd sublattice becomes faster\nthan Fe [22], as recently observed [44]. The LLB model has\nbeen demonstrated to provide accurate analytical expressions\nfor the temperature dependence of the relativistic relaxation\nparameter as well as for the non-equilibrium effective fields\nbelow and above the critical temperature [42]. Moreover, the\nLLB model also describes the transverse motion of the mag-\nnetization. This makes it the preferred model for computer\nsimulations of heat-assisted magnetic recording [45] and re-\nalistic description of all-optical switching [46], and ultrafast\nspintronics, such as domain wall motion [47, 48] or skyrmion\ncreation by ultrafast laser pulses [49]. So far the LLB model\nand Baryakhtar-like models have been considered as comple-\nmentary approaches. Here, we merge them into one unified\napproach.\nIn this work we address the issues discussed above by di-\nrectly comparing both phenomenological models to ASD sim-\nulations. We do so since ASD simulations have been al-\nready quantitatively compared to experiments in literature.\nWe find that quantitative comparison between ASD and both\nphenomenological models is partially possible for laser exci-\ntation producing small deviation from equilibrium. However,\nthose models hardly reproduce magnetic switching using the\nsame parameter values describing the relaxation of small per-\nturbations. Here, based upon those phenomenological mod-\nels, we propose a macroscopic model that compares precisely\nto the magnetization dynamics calculated using ASD simula-\ntions, including element-specific magnetization relaxation and\nswitching. This model bridges atomistic spin dynamics based\nmodels and previously proposed phenomenological models.\nNotably, it provides a deeper understanding to the parameters\nentering the phenomenological models and sheds some light\ninto the process of ultrafast switching in ferrimagnets.\nThe work is broken down in the following way: in Sec. II,\nwe present the atomistic spin model for the calculation of the\nmagnetic equilibrium properties and non-equilibrium dynam-\nics. The equilibrium properties are compared to a mean field\nmodel. We then provide atomistic calculations of the ultra-\nfast magnetization dynamics with input from the two temper-\nature model. These results are the basis for the comparison to\nthe phenomenological models presented in Sec. III. Firstly,\nwe present the Baryakhtar model and the Landau-Lifshitz-\nBloch model. Secondly, we compare the ultrafast magneti-\nzation dynamics calculated with those models to the atomistic\nspin dynamics results. Finally, in Sec. III C we present the\nunified phenomenological model, a hybrid model combining\nBaryakhtar and LLB models, and its comparison to atomistic\nspin dynamics.II. ATOMISTIC SPIN MODEL\nFerrimagnetic materials characterise by spontaneous mag-\nnetization as a resultant of two or more components of non-\nparallel magnetic moments [50]. Atomistic spin models based\non the Heisenberg Hamiltonian can be considered one of the\nsimplest microscopic models able to reproduce the equilib-\nrium properties of ferrimagnets. The spin system energy due\nto only the exchange interactions can be described by an ef-\nfective Heisenberg model:\nH=\u0000å\ni6=jJaSa;i\u0001Sa;j\u0000å\ni6=jJbSb;i\u0001Sb;j\u0000å\ni6=jJabSa;i\u0001Sb;j(1)\nwhere Ja(b)(ab)is the exchange constant between neighbor-\ning sites represented by two classical spin vectors SiandSj\n(jSj=1). Further, one can include magnetic anisotropy terms\nto Eq. (1) to set a preferential axis for the magnetization.\nHowever, since the anisotropy energy is relatively low it plays\na marginal role in the switching process. This makes for a\nsimpler Hamiltonian and a more direct comparison to the phe-\nnomenological models. To model a ferrimagnet, one needs to\nconsider two alternating sublattices of unequal and antiparal-\nlel moments, with three exchange coupling constants: ferro-\nmagnetic for each sublattice ( JaandJb) and a third for the an-\ntiferromagnetic interaction between them, Jab. For instance,\nGdFeCo alloys are composed of a transition metal FeCo and\na Gd rare-earth sublattices. We model the Fe and Co spins\nas only one magnetic sublattice, and we assume a common\natomic magnetic moment of mFeCo=1:94mB. In these alloys\nthe rare-earth impurities add localised 4 fspins to the sys-\ntem assumed to be, mGd=7:6mB. The amorphous nature of\nGdFeCo is modelled by using a simple cubic lattice model\nbut with random placements of Gd moments within the lattice\nto the desired concentration. The applicability of the Heisen-\nberg approximation relies on the stability of local moments\nunder rotation and at high temperature where Stoner excita-\ntions are generally weak [51]. It is assumed that the electronic\nproperties are temperature-independent in the range where the\nsystem is magnetically ordered.\nA. Atomistic spin dynamics\nEquilibrium and non-equilibrium element specific mag-\nnetic properties of a ferrimagnet are calculated using atomistic\nspin dynamics simulations which are based in the stochastic-\nLandau-Lifshitz-Gilbert equation (s-LLG) [52]\n(1+l2\ni)ms;i˙Si=\u0000gSi\u0002[Hi\u0000li(Si\u0002Hi)]; (2)\nwhere gis the gyromagnetic ratio, and liis the so-called\nphenomenological sublattice specific damping parameter. By\nincluding a Langevin thermostat the spin dynamics includ-\ning statistical – equilibrium and non-equilibrium thermo-\ndynamic properties can be obtained. An effective field-\nlike stochastic term ziis added to the effective field Hi=\nzi(t)\u0000¶H\n¶Si, with white noise properties [53]: hzi(t)i=\n0 andhzi(0)zj(t)i=2likBTms;idi jd(t)=g:The variance3\n−2−1.6−1.2−0.8−0.400.40.81.21.6\n0 100 200 300 400 500 600 700M(T)[µB]\ntemperature / KMnet\nFe\nGd\nFIG. 1. Equilibrium magnetization of a GdFeCo alloy for Gd concen-\ntration, xGd=25%. Element-specific normalized equilibrium mag-\nnetization and net equilibrium magnetization, M(T) =xGdmGdmGd\u0000\nxFemFemFe, where mGd(Fe)is the atomic magnetic moment of Gd(Fe).\nLines correspond to the mean-field approximation with renormalized\nexchange parameters. Symbols correspond to atomistic spin dynam-\nics simulations.\nof the Langevin noise is chosen such that the fluctuation-\ndissipation theorem is full filled.\nB. Mean-field approximation\nExact analytical expressions for the M(T)curve are cum-\nbersome to derive due to the many body character of the prob-\nlem. Here we resort the mean field approximation (MFA),\nalready used in previous works [8, 13, 54]. We note that to\nbe able to apply the MFA for the GdFeCo impurity model,\nand thus translation non-symmetric with respect to spin vari-\nables Si, we need to transform the Heisenberg Hamiltonian to\na symmetric one. We use the spin analogy of the virtual crys-\ntal approximation (VCA) to transform the disordered lattice\nHamiltonian Hto a symmetric VCA Hamiltonian HVCA.\nWithin the VCA we evaluate the effective sublattice exchange\nparameters, given by the sum of the exchange interactions of a\ngiven spin at a site riof sublattice iwith all other atoms of this\nsublattice. This involves weighting the exchange parameters\nby the relative composition, xi\u0011concentration species i[8],\nJi=å\nri;r0\niJ(ri;r0\ni)\u0011|{z}\nVCAxiJ(ri;r0\ni)intrasublattice (3)\nwhereas the intersublattice effective exchange reads\nJi j=å\nri;r0\nj=2AiJ(ri;r0\nj)\u0011|{z}\nVCAxiJ(ri;r0\nj)intersublattice (4)\nThus the VCA Hamiltonian reads\nHVCA=å\nj2AiJiSi\u0001Sj+å\nj=2AiJi jSi\u0001Sj (5)where Airepresent the magnetic sublattice of the spin Si. In\nthe exchange approximation we define the MFA field as\nmaHMFA\na=zaJaama+zabJabmb (6)\nThe element-specific equilibrium magnetization is calculated\nvia the self-consistent solution of ma=L(b maHMFA\na)and\nmb=L(b mbHMFA\nb).zaandzabcorrespond to the number\nof first nearest neighbours of type aandb, respectively. It\nis well-known that the MFA overestimates the value of the\ncritical temperature TC. However, a very good agreement be-\ntween ASD and MFA can be obtained by using a reduced\nvalue for the exchange parameters, even for multilattice mag-\nnets [54]. Figure 1 shows element-specific Ma=xamama(T)\nusing ASD simulations and renornalized MFA for xGd=25%.\nNet magnetization is also shown in Fig. 1, which is defined\nasM(T) =xGdmGdmGd\u0000xFemFemFe. The agreement between\nASD and MFA is good enough for all the temperature regions.\nWe observe the presence of compensation temperature TMat\nroom temperature for xGd=25% at which the thermally av-\nerage magnetization of both sublattices are equal but oppo-\nsite, so that the magnetization of the system is equal to zero\nM(TM) =0. The mapping of the atomistic spin model and the\ncorresponding mean-field approximation turns out to be nec-\nessary for a quantitative comparison to the phenomenological\nmodels, and thereby paramount for the unification of both pic-\ntures.\nC. Two Temperature Model\nSingle pulse all-optical switching has been demonstrated to\nbe a thermal process in ferrimagnetic GdFeCo alloys [4] and\nin Mn 2RuxGa Heusler semi-metals [32]. Ultrafast heating by\noptical or electric means are sufficient to achieve switching in\nspecific GdFeCo alloys [55]. Although the minimum achiev-\nable duration of the electric pulses are limited to picoseconds,\nthose are better suited for potential integration into applica-\ntions. Laser pulses can be as short as only a few femtoseconds,\nwhich permits to excite the electron system in timescales of\nthe order of the exchange interaction allowing for the inves-\ntigation of fundamental physics governing switching. In this\nwork, we center in excitation of the ferrimagnetic GdFeCo\nusing femtosecond laser pulses. When a metallic ferrimag-\nnetic thin film is subjected to a near infrared laser pulse, only\nthe electrons are accessible by the photon electric field. Ini-\ntially, the absorbed energy is barely transferred to the lattice\nand consequently the electron system heats up. The electron\nand phonon temperatures are decoupled for up to several pi-\ncoseconds until the electron-phonon interaction equilibrates\nthe two heat-baths. This phenomenology is well captured by\nthe so-called two-temperature model (2TM) [56, 57] which\ncan be written as two coupled differential equations:\nCel¶Tel\n¶t=\u0000gep\u0000\nTel\u0000Tph\u0001\n+Pl(t) (7)\nCph¶Tph\n¶t= +gep\u0000\nTel\u0000Tph\u0001\n: (8)4\nCel=gelTelwhere gel=6\u0002102J/m3K2, and Cph=3:8\u0002106\nJ/m3K represent the specific heat of the electron- and phonon\nsystem. The electron-phonon coupling is taken temperature\nindependent, Gep=7\u00021017J/m3K. Here, P(t)is a Gaussian\nshaped pulse with a duration of 55 fs. The exact values of the\nparameters entering the TTM in GdFeCo are still unknown.\nThe values we use here are close to the commonly used, e.g.\nRefs. [4, 8, 34].\nD. Ultrafast magnetization dynamics using ASD\nElement-specific magnetization dynamics induced by a\nfemtosecond laser pulse are calculated by combining the\natomistic s-LLG equation for the spin dynamics (Eq. (2)) and\nthe 2TM for the electron temperature (Eq. (7)). The electron\nsystem acts as heat-bath for the atomic spins. We consider a\nlattice with N=50\u000250\u000250 spins, and damping parameters,\nlGd=0:01=lFe. Figure (2) shows, for t<0, the dynam-\nics of the element-specific magnetization from an initial sat-\nurated state ( T=0 K), towards thermal equilibrium with the\nheat-bath which is set to T=300 K. The relaxation dynamics\nof Fe sublattice is faster than those of the Gd sublattice. This\ncomes out naturally as the element-specific dissipation of an-\ngular momentum scales as ˙ mz\u0018gl=ms, in Gd ( mGd=7:6mB)\nis slower than in Fe sublattice ( mGd=1:94mB). Once the mag-\nnetic system is in thermal equilibrium with the heat-bath, we\napply the laser pulse, t>0, which introduces energy into the\nelectron system and induces ultrafast magnetization dynam-\nics. To illustrate the switching and no switching dynamics we\nconsider two limiting cases, dynamics induced by low laser\npower, P0, and large laser power, 2 P0. The electron temper-\nature increases up and above the Curie temperature in time\nscales of a few hundreds of femtoseconds Fig. (2) (a). This re-\nflects in the magnetic system as a fast demagnetization of both\nFe and Gd sublattices. For relatively low laser power, P0, the\nmagnetization of both sublattices reduces while the electron\ntemperature remains relatively high. Once the electron tem-\nperature reduces and equalizes to the lattice temperature, the\nmagnetization recovers to the thermal state given by the heat-\nbath temperature, which is higher than initially ( T=300 K).\nThis is why the final magnetization value is smaller than the\ninitial one. For higher laser powers, 2 P0, the magnetization\nof both sublattices reduces quickly. The Fe sublattice faster\nthan the Gd one. Once the magnetization of the Fe sublattice\nhits zero, instead of remaining demagnetized, the magnetiza-\ntion starts to develop toward the opposite direction, while the\nmagnetization of the Gd sublattice is still in the process of de-\nmagnetization. During a couple of picoseconds, both sublat-\ntice magnetization are aligned along the same direction, simi-\nlar to a ferromagnet. Consequently, this non-equilibrium state\nhas been named the transient ferromagnetic-like state [3]. One\ncan observe in Fig. (2) (b) that the demagnetization rates of\nboth sublattices slow down when the Fe magnetization crosses\nzero. This change reveals the set in of a process driving the\nmagnetization dynamics different to the one driving the initial\ndemagnetization. It has been argued that at this point direct\nexchange of angular momentum between sublattices domi-\n30060090012001500\n−1−0.75−0.5−0.2500.250.50.751\n-10 -5 0 5 10 15relaxation tothermal stateswitchingno switchingT[K]electronlatticemztime [ps]FeCoGd(a)(b)\nFIG. 2. (a) Electron and lattice temperature dynamics for two laser\npulse power values, P0and 2 P0. Both electron and lattice tempera-\nture are kept constant, T=300K, for t<0. At t=0 a laser pulse is\napplied and the dynamics of the electron and lattice temperature heat\nup. The dynamics of those temperatures are theoretically described\nby the two-temperature model. (b) Element-specific magnetization\ndynamics induced by the heat profile at (a). The dynamics are calcu-\nlated using atomistic spin dynamics methods. For lower laser powers\nP0, the magnetization of both sublattices demagnetize rapidly and re-\nmagnetize towards the new equilibrium. For laser power 2 P0, the\nmagnetization of both sublattices demagnetizes and switches. After\nswitching they relax towards the thermal equilibrium state. GdFeCo\nalloys with xGd=25% are calculated.\nnates over processes of relativistic origin, which in turn dissi-\npate angular momentum into the heat-bath. Interestingly, soon\nafter switching, both sublattice magnetization rapidly relax to\nequilibrium indicating that relaxation into the heat-bath dom-\ninates the dynamics.\nIII. PHENOMENOLOGICAL MODELS\nDifferently to ASD simulations, phenomenological mod-\nels describe the element-specific magnetization dynamics by\nsolving two coupled equations of motion, one for each sub-\nlattice. In this work we aim at finding a phenomenological\nmodel that describes the same element-specific magnetization\ndynamics as those coming out from the ASD simulations (Fig.\n2). The starting point is the comparison of the ASD simula-\ntions to well-known phenomenological models. We show that\nthose models are unable to describe in a satisfactory way the\ndifferent element-specific magnetization dynamics studied in\nthe previous section and summarized in Fig. 2.5\nA. Baryakhtar model\nThe simplest model to describe element-specific magneti-\nzation dynamics and switching in ferrimagnets was proposed\nby Mentink and co-workers [5]. Longitudinal spin dynamics\nwas derived from Onsager’s relations\nma\ngadma\ndt=aB\namaHa+aB\ne(maHa\u0000mbHb) (9)\nmb\ngbdmb\ndt=aB\nbmbHb+aB\ne(mbHb\u0000maHa) (10)\nhere, aB\na;bstands for the relaxation parameter of relativistic\norigin, which dissipates angular momentum out of the spin\nsystem, and aB\nestands for the exchange relaxation parame-\nter and describes the rate of dissipation of angular momen-\ntum between sublattices. By construction exchange relaxation\nconserves the total angular momentum. We emphasize here\nthe difference in the notation between the atomic relaxation\nparameter, l, describing the dissipation of the atomic spins\nin ASD simulations and the macrospin relaxation parameter,\na, describing the dissipation of the whole magnetic sample.\nWithin this model, the values for aB\na;bandaB\neare unknown\nbut used as fitting parameters when compared to experiments.\nThe internal effective field Ha(b), acting on sublattice a(b)are\nderived from a non-equilibrium mean-field approximation,\nmaHa=\u0000b\u00001L\u00001(ma)+maHMFA\na (11)\nwhere, L\u00001(x)is the inverse Langevin function, b=1=kBT,\nwhere Trepresents the temperature of the heat-bath to which\nthe spin system is coupled to. At equilibrium, the effective\nfield is Ha=0, as ma=L(b maHMFA\na). The same arguments\napply for sublattice b. It turns out that by solving Eqs. (9)\nand (10) together with the 2TM, described in Eqs. (7) and\n(8), one obtains similar ultrafast magnetization dynamics as\nthose using ASD simulations (Fig. (2)). Element-specific\ndemagnetization [18] and switching dynamics [19] based on\nthis approach have been discussed thoughtfully before. On\nthose works, the values for the relaxation parameters, rela-\ntivistic and exchange, are taken constant and of the same or-\nder,aB\nFe\u0019aB\nGd\u0019aB\ne. We note that here aB\nadefines the rate\nof change of angular momentum ( mm=g). It differs from the\ndefinition of intrinsic damping parameters in ASD, which are\nrelated to the rate of change of the magnetization ( m). Sim-\nilarly to ASD methods though, within the Baryakhtar model\nthe observed fast dynamics of the Fe sublattice is related to a\nsmaller value of atomic magnetic moment.\nThe switching process within the Baryakhtar-like model\nis explained in the following manner. Since the Fe sublat-\ntice reacts faster than Gd to heating it is expected to remain\ncloser to thermal equilibrium with the heat-bath. This trans-\nlates into a smaller non-equilibrium effective field acting on\nFe than in Gd, HFe\u001cHGd, during the action of the laser pulse.\nFor strong enough pulses, the Fe magnetization rapidly re-\nduces, mFe\u00190, still HFeis small in comparison to HGd, in a\nway that the dynamics of Fe can be fairly approximated by\n˙mFe\u0019aB\neHGd. This drives the magnetization of Fe towards\n(a)\n(b)\n-1-0.500.51\n-10 -5 0 5 10laser powerP0laser power 2P0-1-0.500.51laser powerP0laser power 2P0mztime [ps]αBe/αBa=0αBe/αBa=0.3αBe/αBa=3mzFeGdFIG. 3. Element-specific magnetization dynamics of GdFeCo cal-\nculated using atomistic spin dynamics (symbols) and macroscopic\nBaryakhtar-like equation (solid lines) for two laser pulse power val-\nues, (a) P0and (b) 2 P0. Both electron and lattice temperature are\nkept constant, T=300 K, for t<0. At t=0 a laser pulse is ap-\nplied. In the Baryakhtar-like model the relativistic relaxation pa-\nrameters aBahave a value different to the Gilbert damping in ASD\nsimulations, (g=mFe)aB\nFe=0:005 and (g=mGd)aB\nGd=0:01. The ex-\nchange relaxation parameter is varied, aBe=aB\nFe=0;0:3 and 3. The\nrelaxation to thermal state ( t<0) is only well described for the Fe\nsublattice. (a) For P0, the laser induced dynamics is well described\nbyaBe=aB\nFe=0:1. (b) For 2 P0the demagnetization phase of both\nsublattices is relatively well described in comparison to ASD sim-\nulations. Switching is also possible, here one instance, for a value\naBe=aB\nFe=3.\nthe opposite direction. The field, HGdis defined by the en-\nergy of the system, HMFA\nGd(Eq. (6)) and aB\nefrom the cou-\npling between the Gd and the Fe sublattices. After switching,\nHFe\u0019HGdand relativistic relaxation processes dominate the\ndynamics and drive magnetization to complete the switching.\nThe question here is to what extent the non-equilibrium fields\nas given by Eq. (11) are accurate, and how are the relaxation\nparameters related to atomic damping parameters in ASD.\nSo far the connection between the relaxation parameters in\nthe ASD and Baryakhtar-like model is unknown. In ASD sim-\nulations shown in Fig. 2 we have used lFe=lGd=0:01 as\natomistic relaxation parameter. One would expect that the re-\nlaxation parameters in the atomistic and macroscopic models\nare related as la\u0019aB\na(ga=ma). In an attempt to find this cor-\nrespondence, we directly compare results from ASD simula-\ntions and Baryakhtar-like models for different values of aB\na\nandaB\nein Eqs. (9) and (10). We numerically solve Eqs.\n(9),(10), and (11) coupled to the 2TM with exactly the same\nparameters as for the ASD simulations. After exploring the\nresults of the Baryakhtar model for a range of values for aB\na\nandae, we find that for some values the agreement is good,\nas one observes in Fig. 3, however, it is not possible to find a\ngood match for all scenarios.6\nIn order to illustrate this, we first focus on the dynamics in-\nduced by the laser pulse with power P0(Fig. (3)(a)). We find\na good match for the laser induced magnetization dynamics\n(t>0 for (g=mFe)aFe=0:005 and (g=mGd)aGd=0:01, and\nfor values of exchange relaxation of up to aB\ne=aB\nFe=0:3. For\nvalues aB\ne=aB\nFe<0:3, thermal relaxation ( t<0) of the Fe is\nalso well described, however the relaxation of the Gd sublat-\ntice is significantly faster. For larger values of the exchange\nrelaxation aB\ne=aB\nFe=3, the dynamics of both sublatttices are\nsubstantially speed up and strongly disagree with ASD simu-\nlations.\nFor larger laser pulse power 2 P0the magnetization switches\nusing ASD simulations. We keep the same values for the re-\nlaxation parameters in Baryakhtar-like model as for P0, and\ncompare to the ASD simulations. For small values of aB\ne\n(Fig. (3)(b)), differently to the P0case (Fig. (3)(a)), the dy-\nnamics described by the Baryakhtar-like model is not only\nslower than those of ASD simulations but it hardly reproduces\nmagnetization switching. In order to reproduce switching, we\nneed to use larger values of the exchange relaxation parameter,\naB\ne=aB\nFe=3. These findings are in agreement with previous\nworks using Baryakhtar-like model where switching was re-\nproduced for comparable values of aB\ne. However, as we have\ndiscussed before, for those values of aB\ne, thermal relaxation\ndynamics ( t<0) is much faster than in ASD simulations.\nThis brings us to the question of how much understanding\nabout switching can we gain by using this bare Baryakhtar-\nlike model, are we missing something?\nB. The Landau-Lifshitz-Bloch model\nSince the Baryakhtar-like model is based on symmetry ar-\nguments, the macroscopic magnetization dynamics coming\nout from ASD simulations should also be described by that\nmodel with adequate expression for the relaxation parameters\nand non-equilibrium effective fields. The magnetization dy-\nnamics coming out from ASD simulations is well described\nby the LLB equation of motion.\ndma\ndt=Gk;a(ma\u0000m0;a); (12)\nwhere\nGk;a=2lag\nmakBT1\nxaL(xa)\nL0(xa); (13)\nwith xa=b maHMFA\na, where HMFA\na is given in Eq. (6), and\nm0;a=L(xa). The same equation applies to the second sublat-\nticeb. Here, the relaxation rate Gk;adepends non-linearly on\nthe non-equilibrium sublattice magnetization, ma(b), through\nthe parameter xa. We note that Eq. (12) can be expanded\naround equilibrium for small perturbations of the magnetiza-\ntion. By doing so, the relaxation rates and effective fields are\nexpressed in terms of equilibrium properties such as equilib-\nrium magnetization and zero-field susceptibilities [20]. In the\npresent work, however, we use the version in Eq. (12). Direct\ncomparison between ASD simulations and the LLB model\n-1-0.500.51\n-10 -5 0 5 10laser powerP0laser power 2P0-1-0.500.51laser powerP0laser power 2P0mztime [ps]αe/αa=0αe/αa=0.1αe/αa=1mzFeGd(a)\n(b)FIG. 4. Element-specific magnetization dynamics of GdFeCo calcu-\nlated using atomistic spin dynamics (symbols) and macroscopic LLB\nequation (solid lines) for two laser pulse power values, (a) P0and (b)\n2P0. For t<0, electron and lattice temperature are T=300K, and at\nt=0 a laser pulse is applied. The exchange relaxation parameter is\nvaried, ae=aa=0;0:1 and 1, where aa=0:01, and a=FeCo or Gd.\nThe initial relaxation dynamics is well described by ae=aa=0. (a)\nFor laser power P0, the element-specific dynamics is well-described\nforae=aa=0:1. (a) For ae=aa=1, exchange relaxation dominates\nand the element-specific dynamics are similar. (b) For laser power\n2P0, the switching dynamics is not described by the LLB model.\nof element-specific magnetization dynamics is possible and\nwith relatively good agreement. Importantly, since the LLB\nmodel is derived directly from the ASD microscopic model,\nthe damping parameters, la(b)in Eqs. (13) and (2) stand for\nthe same physics, the rate of angular momentum dissipation of\nthe atomic spins. Differently to the Baraykhtar model where\naB\na(b)is taken as a fitting parameter, within the LLB model the\nvalue of la(b)in Eq. (13) is the same as in the ASD simu-\nlations. A key difference between the Baryakhtar-like model\nand the LLB model is that in the latter an exchange relaxation\nterm is missing. In order to find a meeting point between these\nphenomenological models, we rewrite Eq. (12) in terms of a\ndamping term multiplied by an effective field,\ndma\ndt=2laL(xa)\nxag\nmama\u0000m0;a\nbL0(xa)=gaaHa; (14)\nwhere\naa=2laL(xa)\nxa: (15)\nDifferently to Baryakhtar-like model, in the LLB model, the\nrelaxation parameter strongly depends on temperature and\nnon-equilibrium sublattice magnetization through the thermal\nfield, xa=b maHMFA\na. At the same time, the non-equilibrium\nfields maHawithin the LLB and Baryakhtar-like models differ.7\nThe effective field in the LLB model is defined as\nmaHa=(ma\u0000m0;a)\nbL0(xa): (16)\nEquation (16) provides a microscopic description of the effec-\ntive field driving the magnetization dynamics in ferrimagnets,\nbased on the Heisenberg spin model (Eq. (1)). Under the as-\nsumption of small perturbations around the equilibrium both,\nLLB and Baryakhtar-like effective fields, simplify to Landau-\nlike expressions [19]. Equation (14) describes with a very\ngood degree of accuracy the relaxation of the angular mo-\nmentum via dissipation to the heat-bath, which corresponds\nto the relativistic term in Eqs. (9) and (10). Previously, it has\nbeen found that ASD simulations compare well to Eq. (14) for\ncoupling parameters of around la\u00190:1\u00001 [20, 42]. These\nvalues can be considered to correspond to the intermediate-to-\nhigh coupling regime. Direct comparison between ASD sim-\nulations and experiments of single pulse switching in GdFeCo\nhas suggested values of lFe\u00190:06 and lGd\u00190:01 [16]. In\nthe context of the present work we find that Eq. (14) describes\nrelatively well the thermal relaxation dynamics in direct com-\nparison to ASD simulations (Fig. (4)).\nIn order to account for the exchange relaxation in the LLB\nmodel, we follow the Baryakhtar-like model ((9) and (10)),\nand add an exchange relaxation term to Eq. (14),\ndma\ndt=gaaHa+gae\nma(maHa\u0000mbHb) (17)\nwhere aeis a phenomenological exchange relaxation param-\neter to be determined by comparison to ASD dynamics. The\ninclusion of the exchange relaxation (second term in r.h.s) in\nthe LLB improves the agreement to ASD simulations. With\nthis addition, the LLB model describes well thermal relax-\nation for small values of the ratio ae=aaas demonstrated in\nFig. 4. For large values of aethe LLB model is unable to\ndescribe thermal relaxation dynamics ( t<0 in Fig. 4(a) and\n(b)). For laser power P0(Fig. 4(a) ( t>0)) the magnetization\ndynamics is slightly slower using the LLB model than those\ngained by ASD simulations for ae=aa=0. For ae=aa=0:1,\nthe agreement is even better than without exchange relax-\nation. The agreement vanishes when the exchange relax-\nation is increased to ae=aa=1. Critically, when the laser\npower is increased from P0to 2P0, for which ASD simula-\ntions show ultrafast switching, the LLB model only shows\ndemagnetization-remagnetization of both sublattices. We find\nsome agreement on the demagnetization time scales when a\nquite large exchange relaxation is used, ae=aa=1. These\ndynamics are similar to those observed using the Baryakhtar-\nlike model for intermediate values of the exchange relaxation\nparameter (Fig. (3)). It has been demonstrated previously\nthat by including the transverse components of the equation\nof motion, switching is possible via a precessional path when\na canting between the magnetization of each sublattice exists\n[21]. Here, we restrict to purely longitudinal switching within\nthe LLB model.\n02468 1 0-1-0.500.51−10−50 5 1 0laser powerP0laser power 2P0time [ps]mztime [ps]FeGd(a)(b)FIG. 5. Element-specific magnetization dynamics of GdFeCo calcu-\nlated using atomistic spin dynamics (symbols) and the unified phe-\nnomenological model derived here, following Eq. (17) (solid lines)\nfor two laser pulse power values, (a) P0and (b) 2 P0. Both electron\nand lattice temperature are kept constant, T=300 K, for t<0. At\nt=0 a laser pulse is applied, the same as in Figure (2). GdFeCo\nalloys with xGd=25% are calculated.\nC. Unified phenomenological model\nSo far we have constructed a phenomenological model\nbased on the LLB and Baryakhtar-like models, the dynam-\nics is given by Eq. (17), the effective field by Eq. (16)\nand the relativistic relaxation parameter Eq. (15). We still\nneed an expression for the exchange relaxation parameter. We\nconstruct this expression starting with single species ferro-\nmagnets, where sublattices aandbrepresent the same spin\nlattice, hence exchange of angular momentum is non-local.\nTherefore, maHa\u0000mbHb=maHexa2\n0Dma, with a0represent-\ning the lattice constant. Hence, the rate of non-local angu-\nlar momentum transfer reads Gnon\u0000loc:\nex =aex(maHa\u0000mbHb) =\naa(A=Ma(T))Dma, where Ais the so-called micromagnetic\nexchange stiffness [58]. Ma(T) = (ma=ua)mais the magne-\ntization density at temperature T, where uais the unit cell\nvolume. Therefore, we find that aex=aa=(zma). By con-\nsidering that the exchange relaxation rate should conserve the\nsymmetry under the exchange of lattice index, aex(M1;M2) =\naex(M2;M1), we find that\naex=1\n2\u0012aa\nzabma+ab\nzbamb\u0013\n: (18)\nThis expression is the extension of the non-local exchange re-\nlaxation in ferromagnets to local exchange relaxation in ferri-\nmagnets. This explicit expression for the exchange relaxation\nparameter in Eq. (18) completes our unified model, which\nbridges the atomistic spin dynamics model and the Baryakhtar\nand LLB macroscopic models.\nWe find that the agreement between our unified phe-\nnomenological model and ASD simulations is excellent, see\nFig. (5)(a) and (b). Figure 5(a) shows that for t<0, the sub-\nlattice magnetization relaxation towards thermal equilibrium\nvalue is described with a high level of accuracy by our model.\nFort>0 and a relatively low laser power P0, the agreement\nis also excellent for the demagnetization and remagnetization\ndynamics. Figure 5(b) shows the comparison between the uni-\nfied model and ASD simulations of the switching dynamics.\nWe conclude that Eq. (17) for the sublattice magnetization8\ndynamics together with the Eq. (16) for the effective field\nand Eqs. (15) and (18) for the relaxation parameters, unify\nthe Barayakhtar and the LLB phenomenological models for\nsingle-pulse all-optical switching in ferrimagnets.\nIV . DISCUSSION AND CONCLUSION\nThe macroscopic model presented in this work solves some\nopen questions in the field of ultrafast magnetization dynam-\nics in ferrimagnets. For example, it answers the question of\nthe range of applicability and the validity of the parameters of\nthe Barayakhtar and LLB phenomenological models. In the\none hand, within our model, the relativistic relaxation param-\neters ( aa) are element-specific and strongly depend on both\nthe temperature and the non-equilibrium sublattice magneti-\nzation. The temperature and magnetization dependence of\nthe relativistic relaxation parameters are well described by the\nLLB model. In the other hand, the exchange relaxation pa-\nrameter ( aex) is cast in terms of the element specific relativis-\ntic relaxation parameters and sublattice magnetization. We\nhave demonstrated that in order to reproduce the ASD sim-\nulations results, the relaxation parameters in the Barayakhtar\nmodel have to be both temperature and magnetization depen-\ndent. The explicit expression of the exchange relaxation pa-\nrameter is the main result of the present work since it allows\nus to unify the Barayakhtar and LLB models. While for the\nBarayakhtar model aeis unconnected to aa, within our pro-\nposed model they are proportional to each other, ae\u0018aa=ma.\nThis relation is the key to bridge both ASD simulations and\nBarayakhtar and LLB models together. Additionally, we have\nalso demonstrated the validity of the non-equilibrium effective\nfields given in Eq. (16) as derived in the LLB model instead\nof the Barayakhtar model.\nSingle-pulse switching in ferrimagnets has been described\nbefore by the Baryakhtar model. A necessary condition for\nswitching is that the system transits from the relativistic relax-\nation regime to the so-called exchange-dominated relaxation\nregime. Although details of switching in such a regime have\nbeen already discussed in detail [5, 19], so far it has remained\nunknown how this transition could be described theoretically.\nOur model resolves this question. When the system is at equi-\nlibrium or weakly excited, the exchange-relaxation parameter\nfulfills, ae\u001caa. For strong excitation, such that the mag-\nnetic order of one sublattice reduces significantly, close to\nzero ma!0, the exchange relaxation will dominate the dy-namics since ae\u0018aa=ma\u001daa. From our model, one can\nderive universal criteria for switching in ferrimagnets, includ-\ning GdFeCo and Mn 2RuxGa [59].\nThe provided understanding is paramount for further re-\nsearch on material engineering, for example, to find alter-\nnative material classes showing all-optical switching. No-\ntably, our model predicts that the exchange relaxation term\nis enhanced as the number of neighbours reduces. This de-\npendence suggests that magnetic systems of lower dimen-\nsion, e.g. 2D magnets [60], could show a faster, more ef-\nficient switching than bulk materials. Further, the extension\nof our model to the micromagnetic level will allow to opti-\nmize switching conditions. The use of micromagnetic com-\nputational solvers permits for a realistic description of ultra-\nfast AOS processes in ferrimagnetic alloys, such as helicity-\nindependent and helicity-dependent AOS, where multidomain\nstates and thermal gradients play an important role in the pro-\ncess [46].\nTo summarize, in the present work we have presented a\nunified model for single-pulse all-optical switching in ferri-\nmagnets. Our model merges and improves previous semi-\nphenomenological models, the Landau-Lifshitz-Bloch model\nand Barayakhtar-like models. To verify the accuracy of the\nproposed model, we directly compare the laser induced mag-\nnetization dynamics to atomistic spin dynamics computer\nsimulations. Differently to previous models, our model has\nthe advantage that it can be directly compared to ASD simu-\nlations. Further, we have established the connection between\nASD and macroscopic equations of motion. Importantly, we\nprovide here the stepping stone for the construction of a mi-\ncromagnetic model valid for ferrimagnets including exchange\nrelaxation between sublattices. This is paramount for a ro-\nbust construction of a multiscale scheme of the switching pro-\ncess in which not only local magnetization dynamics is de-\nscribed but also magnetic domain nucleation and motion un-\nder strong non-equilibrium. Multiscale-based micromagnetic\nmodels will allow for the description of realistic sample sizes\nand describe recent spintronics phenomena using laser pulses,\ne.g. magnetic skyrmion creation/deletion with fs laser pulses,\nor domain-wall motion under dynamics thermal gradients.\nACKNOWLEDGMENTS\nThe authors acknowledge support from the Deutsche\nForschungsgemeinschaft through SFB/TRR 227 ”Ultrafast\nSpin Dynamics”, Project A08.\n[1] C. Stanciu, F. Hansteen, A. Kimel, A. Kirilyuk, A. Tsukamoto,\nA. Itoh, and T. Rasing, All-Optical Magnetic Recording with\nCircularly Polarized Light, Physical Review Letters 99, 047601\n(2007).\n[2] K. Vahaplar, A. Kalashnikova, A. V . Kimel, D. Hinzke,\nU. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk,\nand T. Rasing, Ultrafast Path for Optical Magnetization Rever-\nsal via a Strongly Nonequilibrium State, Physical Review Let-ters103, 117201 (2009).\n[3] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A.\nD¨urr, T. A. Ostler, J. 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Berntsen1, Youguo Shi3, Tao Xiang3, Oscar Tjernberg1\u0003\n1Department of Applied Physics, KTH Royal\nInstitute of Technology, Stockholm 11419, Sweden\n2Department of Applied Physics, Stanford University, Stanford, CA 94305, USA\n3Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n4Helmholtz-Zentrum Berlin f ur Materialien und Energie,\nElektronenspeicherring BESSY II, Albert-Einstein-Stra\u0019e 15, 12489 Berlin, Germany\n5Synchrotron SOLEIL, L'Orme des Merisiers,\nD\u0013 epartementale 128, 91190 Saint-Aubin, France\n6Department of Physics and Beijing Key Laboratory of\nOpto-electronic Functional Materials &Micro-nano Devices,\nRenmin University of China, Beijing 100872, China\n7MAX IV Laboratory, Lund University, 22100 Lund, Sweden\n]These people contributed equally to the present work.\n\u0003Corresponding authors: conli@kth.se, oscar@kth.se\n1arXiv:2303.17302v1  [cond-mat.str-el]  30 Mar 2023The study of interaction between electromagnetism and elementary particles\nis a long-standing topic in physics. Likewise, the connection between particle\nphysics and emergent phenomena in condensed matter physics is a recurring\ntheme and condensed matter physics has often provided a platform for inves-\ntigating the interplay between particles and \felds in cases that have not been\nobserved in high-energy physics, so far. Here, using angle-resolved photoemis-\nsion spectroscopy, we provide a new example of this by visualizing the electronic\nstructure of a noncentrosymmetric magnetic Weyl semimetal candidate NdAlSi\nin both the paramagnetic and ferrimagnetic states. We observe surface Fermi\narcs and bulk Weyl fermion dispersion as well as the regulation of Weyl fermions\nby ferrimagnetism. Our results establish NdAlSi as a magnetic Weyl semimetal\nand provide the \frst experimental observation of ferrimagnetic regulation of\nWeyl fermions in condensed matter.\nIn recent decades, considerable e\u000borts have been made to identify topological quasi-\nparticles in condensed matter physics that follow the same physical laws as elementary\nparticles[1{7]. The discovery of topological semimetals has made this possible[8{20]. Weyl\nsemimetals is an important class of topological semimetals, which hosts emergent relativistic\nWeyl fermions in the bulk, and Fermi arc surface states connect two Weyl points with op-\nposite chirality on the boundary of a bulk sample[1, 6, 7, 13{15]. These emergent particles\nare the result of inversion symmetry (IS) [6, 7, 13{15, 21, 22] or time-reversal symmetry\n(TRS)[1, 2, 23{32] breaking. The TRS-breaking Weyl semimetals are usually derived from\nmagnetic materials[1, 2, 23{32], compared to the Weyl semimetals with IS-breaking, which\nprovide a platform for the study of the interplay between magnetism, electron correlation\nand topological orders. In the magnetic Weyl semimetals, a nonvanishing Berry curvature\ninduced by TRS-breaking can give rise to rich novel phenomena, such as a anomalous[33{35]\nand spin[36, 37] Hall e\u000bect, and chiral anomalous charge[17{20]. These exotic physical prop-\nerties make magnetic Weyl semimetals potential candidates for a wide range of applications\nin spintronics.\nRecently, a rare case of magnetic Weyl semimetal candidates, the RAlX (R: Rare earth; X:\nSi or Ge) family, has attracted attention. This family displays both IS and TRS breaking[38{\n63]. From the perspective of crystal structure, RAlX is a non-centrosymmetric crystal with\nmagnetism derived from R (Rare earth) atoms[38]. The magnetic structure of the RAlX\n2family is easy to regulate and presents diverse magnetic ordering. For example, the magnetic\nstructure of RAlX can be nonmagnetic[64{66], ferromagnetic[38{49], antiferromagnetic[49{\n56], ferrimagnetic[57{60] and even spiral magnetic[57, 61] by rare earth element substitution.\nSuch a rich magnetic structure makes the RAlX family an appropriate candidate to study\nthe interaction between magnetism and Weyl fermions. The interaction between magnetism\nand Weyl fermions includes two parts: the \frst part is the mediating e\u000bect of Weyl fermions\non magnetism[57], and the second part is magnetism regulation of the Weyl fermions and\nfurther e\u000bects on the topological ordering. To data, the mediating e\u000bect of Weyl fermions\non magnetism has been con\frmed by neutron di\u000braction measurements[57]. However, un-\nambiguous and direct experimental con\frmation of the regulation of Weyl fermions by mag-\nnetism remains unobserved until now.\nHere, we present angle-resolved photoemission spectroscopy (ARPES) measurements and\nband structure calculations to systematically investigate the electronic structure and topo-\nlogical properties of NdAlSi and how they are regulated by ferrimagnetism. We observe\nFermi arcs and bulk Weyl fermion dispersion in the paramagnetic state, after determining\na surface state associated with a Nd terminated surface cleaved at the Nd-Al plane. In ad-\ndition, we also observe that a new Weyl fermion dispersion corresponding to the new Weyl\nfermion generated in the ferrimagnetic state. These observations are in good agreement\nwith the prediction of theoretical calculations and con\frm the existence of Weyl fermions\nregulated by ferrimagnetism in NdAlSi. The results provide key insights into the interplay\nbetween magnetism and topological orders.\nNdAlSi was predicted to be a magnetic Weyl semimetal that crystallizes in the tetragonal\nstructure with the space group I41md(no. 109)[57], as shown in Fig. 1a. The crystal struc-\nture of NdAlSi has two vertical mirror planes, mxandmy, as well as two vertical glide mirror\nplanes,mxyandmxy, but lacks the mirror symmetry of the horizontal plane which induces\nthe inversion symmetry breaking. The corresponding three-dimensional (3D) Brillouin zone\n(BZ) of NdAlSi is shown in Fig. 1b. In order to determine the magnetic structure of NdAlSi\nat low temperature, magnetic susceptibility measurements were performed (for details see\nFig. S1 in the Supplemental Material). Based on the magnetic susceptibility (Fig. S1)\nand previous neutron scattering measurements[57], it can be inferred that the spontaneous\nmagnetization of NdAlSi appears as a up-up-down ( \"\"#, UUD) type ferrimagnetism at low\ntemperature (T <T com\u00183.4 K), as shown in Fig. 1c. The corresponding 3D BZ of NdAlSi in\n3the UUD ferrimagnetic state is shown in Fig. 1d. In order to study the electronic structure\nand topological properties of NdAlSi and its regulation by ferrimagnetic order, we performed\ndensity functional theory (DFT) calculations in the paramagnetic and the UUD ferrimag-\nnetic states, respectively. In the paramagnetic state, we \fnd a total of 40 Weyl nodes whose\nlocations within the \frst 3D BZ are shown in Fig. 1e-1f. These Weyl points are mainly\ndistributed in the k z\u00180\u0019=c(Fig. 1g) and k z\u0018\u00060.67\u0019=c(Fig. 1h) planes. In the UUD\nferrimagnetic state, the BZ is reconstructed relative to the paramagnetic state, resulting in\nthe folding of the electronic structure and Weyl points. Fig. 1i shows the distributions of all\nWeyl fermions in the ferrimagnetic state, to facilitate direct comparison with Weyl fermions\nin the paramagnetic state, we fold the Weyl fermions back into the \frst BZ in the paramag-\nnetic state (gray red and blue dots in Fig. 1i). It is found that the folded back Weyl nodes\nare mainly distributed in the k z\u00180\u0019=c(Fig. 1j), k z\u0018\u00060.33\u0019=c(Fig. 1k) and k z\u0018\u00060.67\n\u0019=c(Fig. 1l) planes of the \frst BZ of the paramagnetic state. Fig. 1m and 1n show the DFT\nband structure calculations along \u0000-X-Z 2-M-\u0000-Z directions in the paramagnetic (Fig. 1m)\nand ferrimagnetic (Fig. 1n) states. This elucidates the in\ruence of the UUD ferrimagnetic\norder on the electronic structure and topological properties of NdAlSi.\nAccording to the above DFT calculations, the UUD ferrimagnetic order not only regulates\nthe position of the existing Weyl nodes, but also generates new Weyl nodes in the 3D BZ.\nARPES measurements provide a tool to con\frm or refute this prediction. In order to do\nthis, we \frst study the electronic structure and topological properties of the paramagnetic\nstate in detail, and then observe the in\ruence of the UUD ferrimagnetic order induced by\nlowering the temperature.\nFigure 2 displays the overall constant energy contours and band structures of NdAlSi\nmeasured by ARPES in the paramagnetic state. The evolution of constant energy contours\nof the electronic bands at di\u000berent binding energies (Fig. 2a) exhibit sophisticated structures.\nA simple comparison with the DFT calculations of the bulk Fermi surface (Fig. S2 in the\nSupplemental Material) shows that the ARPES measurements (Fig. 2b) display a more\ncomplex structure, which may be related to the presence of surface states. In order to\ndetermine the contribution from surface states, we performed surface projected DFT band\ncalculations on six di\u000berent possible (001) termination surfaces of NdAlSi (for the detailed\nband structure calculations see Fig. S3 in the Supplemental Material). The results show\nthat the band structure calculations along the Y-\u0000-Y(Fig. 2g) and M-\u0000-M(Fig. 2h)\n4directions on the Nd atom terminated surface of the Nd-Al cleavage plane can capture most\nof the measured band features in the corresponding direction (Fig. 2e and Fig. 2f) except\nthe M shaped band at 0.8 eV binding energy and some very small details in Fig. 2e. When\nthe two orthogonal domain structures leading to superposition of bands along the X-\u0000-X\nandY-\u0000-Ydirections are taken into account (Fig. 2i and 2j), all the band features can be\nunderstood. The surface projected DFT Fermi surface calculations on the Nd terminated\nsurface of the Nd-Al cleavage plane is shown in Fig. 2c and agrees well with the measured\nconstant energy contours (Fig. 2b) and when the two orthogonal domain structures are\ntaken into account (Fig. 2d) it com\frms that the measured Fermi surface (Fig. 2b) is\nmainly from the surface states (for detailed analysis see the Supplemental Material).\nWe now proceed to explore Fermi arc candidates. By surface Green function calculations,\nwe \fnd two Fermi arc surface states connecting two pairs of Weyl nodes at the energy of 38\nmeV (Fig. 2k) and 56 meV (Fig. 2l) above the Fermi level (for details see the Supplemental\nMaterial). Similar arc features can also be found in the Fermi surface mapping in the\nsame position, as shown in Fig. 2b. To con\frm that the observed arc features in Fig. 2b\nare the Fermi arc surface states. We carried out photon energy dependent band structure\nmeasurements (Fig. 2m-2p) at 15 K along the direction of Cut3 ( ky=0.34 \u0017A\u00001) in Fig.\n2b. For further quantitative analysis, the extracted photon energy dependent momentum\ndistribution curves (MDCs) at a binding energy of 10 meV from Fig. 2m-2p are plotted in\nFig. 2q. The MDCs exhibit negligible photon energy dependence, suggesting that it is a\nsurface state. To further con\frm that the arc features observed in Fig. 2b are indeed the\nWeyl Fermi arcs, we examine the signatures of chiral charge in NdAlSi based on the bulk-\nboundary correspondence between the bulk Weyl fermions and surface Fermi arcs[1, 67]. In\norder to do so, band dispersion was measured along the straight Cut3 in Fig. 2b. Along\nthis cut (Fig. 2m-2p), a left-moving (Chern number n l=-1) and right-moving (Chern numbe\nnr=+1) edge mode related by a mirror plane kx=0 is clearly observed and associated with\na 2D momentum-space slice carrying a total Chern number n tot=0. In addition, we also\nexamine the signatures of chiral charge in the close loop cuts (Fig. 2b). For a closed loop\nin the surface BZ where the bulk band structure is everywhere gaped we add up the signs\nof the Fermi velocities of all surface states along this loop, with Chern number n=+1 for\nright movers and n=-1 for left movers[67]. Fig. 2r and 2s show the unrolling of the closed\nloop cuts along Loop1 (Fig. 2r, L1 in Fig. 2b) and Loop2 (Fig. 2s, L2 in Fig. 2b). Loop\n5L1 shows a left-moving chiral mode while a right-moving chiral mode is observed for loop\nL2. These results unambiguously show that the arc features in Fig. 2b are the topological\nFermi arcs.\nAfter having identi\fed the topological Fermi arc, we next search for the characteristic\nbulk Weyl fermion dispersion. Looking at the results from DFT calculations, we \fnd a total\nof 40 Weyl nodes in the paramagnetic state, with 24 of them (W2, W3 and W4) distributed\nin the k z\u00180\u0019=cplane (Fig. 3c) and the others (W1) are situated in the k z\u0018\u00060.67\u0019=c\nplane (Fig. 3b). In order to precisely identify the k zmomentum locations of the Weyl\nnodes, we performed broad-range (30 to 90 eV) photon energy dependent ARPES measure-\nments on a cleaved sample with a poor surface so that the surface states are completely\nsuppressed (for details see Supplemental Material). Due to the absence of interference from\nsurface states, bulk bands can be clearly distinguished and characterized. These results were\nfurther con\frmed using bulk sensitive soft X-ray ARPES (SX-ARPES) measurements (for\ndetails see Supplemental Material). Fig. 3a shows the photon energy dependent ARPES\nspectral intensity map (k x-kzFermi surface) at a binding energy of 0.2 eV along the X-\u0000-X\ndirection, measured at 30 K. By analyzing the periodic structure along the k zdirection, the\ncorrespondence between the high symmetry points of the BZ along the k zdirection and the\nphoton energy is determined as shown in Fig. 3a. Based on this, the k z\u0018\u00060.67\u0019=cplane\nhosting W1 type of Weyl fermions corresponds to a photon energy of \u001834 eV and the k z\u00180\n\u0019=cplane hosting W2, W3 and W4 types of Weyl fermion corresponds to a photon energy\nof\u001841 eV.\nFigure 3d and 3h show the Fermi surface mapping of NdAlSi measured at 15 K with\nthe photon energy of 34 eV (Fig. 3d) and 41 eV (Fig. 3h) under Positive Circular (PC)\npolarization on a high quality cleaved surface. The experimental data and the calculated bulk\nFermi surface (Fig. 3e and 3l) are distinctively di\u000berent due to the admixture of the surface\nstates. The Fermi surface mapping of the low quality surface (Fig. 3p), on the other hand,\nshows good overall agreement with the bulk calculations (Fig. 3l) due to the suppression\nof the surface states. In order to expose the Weyl fermion dispersion, we conducted band\ndispersion measurements along 4 cuts (Fig. 3b-c). The corresponding dispersion data is\ndisplayed for cuts 1-4 in Fig. 3f,i-k for the high quality surface and for cuts 2-4 in Fig.\n3q-s for the low quality surface. The corresponding surface projected DFT calculations are\nplotted in Fig. 3g and 3m-3o for comparison with the corresponding DFT bulk calculations\n6overlayed. It is observed that the measured dispersions on the low quality surface (Fig. 3q-\n3s) show good overall agreement with DFT bulk calculations and the measured dispersions\non the high quality surface (Fig. 3f and 3i-3k) are also consistent with the surface projected\nDFT calculations (Fig. 3g and 3m-3o). The observation of the topological Fermi arc and\nbulk Weyl fermions with linear dispersions, as well as the consistency between theoretical\ncalculations and experimental observations, establishes NdAlSi as a Weyl semimetal.\nWith the topological properties of the paramagnetic state well established, we turn to\nthe question of how these topological properties are regulated by the ferrimagnetism. The\nresults from the DFT bulk calculations show that the distribution of Weyl fermions in the\nUUD ferrimagnetic state (Fig. 1i) represent a folding into 1/3 of the paramagnetic state\nalong the in-plane diagonal direction. Based on the previous analysis, if the Weyl fermions\nare folded back into the \frst BZ of the paramagnetic state, the back folded Weyl nodes are\nmainly distributed in the k z\u00180\u0019=c(Fig. 1j), k z\u0018\u00060.33\u0019=c(Fig. 1k) and k z\u0018\u00060.67\u0019=c\n(Fig. 1l) planes of the \frst BZ of the paramagnetic state. By comparing the distribution of\nWeyl fermions to the paramagnetic state, it can be found that the folding of the BZ caused\nby ferrimagnetism generates 4 pairs of W2, 2 pairs of W3 and 2 pairs of W4 type Weyl\nfermions in the k z\u00180\u0019=cplane (Fig. 1j, Fig. 4a) as well as 4 pairs of W1 type of Weyl\nfermions in the k z\u0018\u00060.67\u0019=cplane (Fig. 1l) of the \frst BZ of the UUD ferrimagnetic\nstate. Furthermore, 4 pairs of W1 type of Weyl fermions are generated in the k z\u0018\u00060.33\n\u0019=cplane (Fig. 1k) of the \frst BZ of the paramagnetic state. Therefore, we can use the\ngeneration of the Weyl fermions mentioned above as a criterion to judge the presence of\nWeyl fermions regulated by the UUD ferrimagnetic order.\nFigure 4b and 4c show the constant energy contours at the binding energy of 70 meV\nmeasured with a photon energy of 41 eV at 15 K (Fig. 4b) and 2K (T <T com\u00183.4 K) (Fig.\n4c). As the temperature is lowered below Tcom, a new feature (marked by a green arrow\nin Fig. 4c) can be observed. This feature is not present at 15 K (Fig. 4b). In terms of\nshape, it is mirror-symmetric to the features marked by the red arrow and can be related to\nthe folding of the electronic structure caused by the UUD ferrimagnetic order (green lines\nin Fig. 4f). To further con\frm this conjecture, we conducted temperature dependent band\ndispersion measurements (Fig. 4e-4h) along Cut1 in Fig. 4a, and compared them to the\ncorresponding surface projected DFT band calculations in the paramagnetic state (Fig.4d).\nAll the band features measured at 15 K (Fig. 4e) are captured by the band calculations in\n7the paramagnetic state (Fig. 4d). However, the features marked by a green arrow in Fig.\n2h can not be found in Fig. 4d. It appears to be a duplicate of the band marked by the red\narrow in Fig. 2h (for detailed MDCs analysis see Fig. S10 in the Supplemental Material).\nThe noteworthy feature is that they are mirror-symmetric exactly around the boundary of\nthe BZ of the ferrimagnetic state and break at 15 K (for more comparison of band dispersion\ncuts see Fig. S11 in the Supplemental Material). Therefore, we infer that the new features\ngenerated at 2 K are the folded electronic structures caused by ferrimagnetism (green lines\nin Fig. 4f). It also implies that new Weyl fermions are generated due to the regulation of\nthe UUD ferrimagnetism. A cartoon picture of this process is shown in Fig. 4i.\nThe present observations establish NdAlSi as a ferrimagnetic Weyl semimetal and pro-\nvide strong evidence for the interaction between ferrimagnetism and Weyl fermions. This\nhelps us understand why there is an incommensurate ferrimagnetic order mediated by Weyl\nfermions between 3.4 K and 7.3 K[57]. In the ferrimagnetic state, ferrimagnetism not only\ndrives Weyl fermion shifting in the 3D BZ due to the breaking of TRS, but also generate\nnew Weyl fermions. This, in turn, results in an increase of the density of states near the\nFermi level, leading to higher conductivity. The expected increase in conductivity is consis-\ntent with transport measurements[57, 58]. In addition, new Weyl fermions generated in the\nferrimagnetic state can also impact the thermal conductivity of ferrimagnetic Weyl semimet-\nals. The increasing number of the Weyl nodes and Fermi arcs in the 3D BZ maybe lead\nto a reduction in the thermal conductivity due to the increased scattering of heat-carrying\nphonons. This reduction in thermal conductivity can result in improved thermoelectric per-\nformance, making ferrimagnetic Weyl semimetals potentially useful for energy conversion\napplications. Future heat transport measurements are needed to con\frm this.\nIn summary, ARPES measurements combined with band structure calculations paint a\nclear physical picture of the ferrimagnetic regulation of Weyl fermions. This interplay is\nkey to understanding the electrical and thermal transport properties of ferrimagnetic Weyl\nsemimetals and could turn to be important for the development of new technologies based\non these materials. Our study provides the \frst spectral evidence for the interplay between\nmagnetism and Weyl fermions in condensed matter physics and opens up a new avenue to\nstudy the magnetic regulation of topological ordering, a topic of great interest in topological\nphysics.\n8Methods\nSample Single crystals of NdAlSi were grown from Al as \rux. Nd, Al, Si elements were\nsealed in an alumina crucible with the molar ratio of 1: 10: 1. The crucible was \fnally\nsealed in a highly evacuated quartz tube. The tube was heated up to 1273 K, maintained\nfor 12 hours and then cooled down to 973 K at a rate of 3 K per hour. Single crystals were\nseparated from the \rux by centrifuging. The Al \rux attached to the single crystals were\nremoved by dilute NaOH solution.\nARPES Measurements High-resolution ARPES measurements were performed at the\nbeamline UE112 PGM-2b-13at BESSY II (Berlin Electron Storage Ring Society for Syn-\nchrotron Radiation) synchrotron, equipped with a SCIENTA R8000+DA30L electron ana-\nlyzer and at the CASSIOPEE beamline of the SOLEIL synchrotron, equipped with a SCI-\nENTA R4000 electron analyzer. The total energy resolution (analyzer and beamline) was\nset to 5\u001820 meV for the measurements. The angular resolution of the analyser was \u00180.1 de-\ngree. The samples were cleaved in situ and measured at di\u000berent temperatures in ultrahigh\nvacuum with a base pressure better than 1.0 \u000210\u000010mbar. Soft X-ray (SX) ARPES mea-\nsurements were performed at the VERITAS beamline of the MaxIV synchrotron equipped\nwith a R4000 electron analyzer.\nDFT calculations The electronic structure calculations for NdAlSi were performed based\non the density functional theory (DFT)[68, 69] as implemented in the VASP package[70,\n71]. The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE)\ntype[72] was chosen for the exchange-correlation functional. The projector augmented wave\n(PAW) method[73, 74] was adopted to describe the interactions between valence electrons\nand nuclei. In the calculation of the high-temperature paramagnetic phase of NdAlSi, the\nNd pseudopotential was chosen without the 4 felectrons. The kinetic energy cuto\u000b of the\nplane-wave basis was set to be 350 eV. A 16 \u000216\u000216 Monkhorst-Pack grids[75] was used\nfor the BZ sampling. For describing the Fermi-Dirac distribution function, a Gaussian\nsmearing of 0.05 eV was used. For structure optimization, both lattice parameters and\ninternal atomic positions were fully relaxed until the forces on all atoms were smaller than\n0.01 eV/ \u0017A. The relaxed lattice constants are: a0= 4.22 \u0017A and c0= 14.59 \u0017A, which agree well\nwith the experimental measurements[58]. The spin-orbit coupling e\u000bect was also included for\nstudying the non-trivial band topological. The positions and chirality of the Weyl points were\nstudied by using WannierTools package[76], which interfaces the Wannier90 package[77]. For\n9further studying the surface states of NdAlSi with di\u000berent terminal surfaces, we employed\na 36 atomic layers slab system with 20 \u0017A vacuum. The orbital weights of the top three\natomic layers were calculated as the surface states with respect to the corresponding terminal\nsurfaces.\nAs for the calculation of the low-temperature ferrimagnetic phase, we adopted the pseu-\ndopotential of Nd including the 4 felectrons. 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Pack, Special points for Brillonin-zone integrations. Phys. Rev. B\n13, 5188 (1976).\n[76] Q. Wu et al., WannierTools: An open-source software package for novel topological materials.\nComput. Phys. Commun. 224, 405 (2018).\n[77] A. A. Mosto\f et al., An updated version of wannier90: A tool for obtaining maximally-localised\nWannier functions. Comput. Phys. Commun. 185, 2309 (2014).\n[78] S. L. Dudarev et al., Electron-energy-loss spectra and the structural stability of nickel oxide:\nAn LSDA+U study. Phys. Rev. B 57, 1505 (1998).\n[79] V. Wang et al., VASPKIT: A user-friendly interface facilitating high-throughput computing\nand analysis using VASP code. Comput. Phys. Commun. 267, 108033 (2021).\nAcknowledgement\nThe work presented here was \fnancially supported by the Swedish Research council (2019-\n00701) and the Knut and Alice Wallenberg foundation (2018.0104). M.D. acknowledges\n\fnancial support from the G oran Gustafsson Foundation and Swedish Research Council\n(2022-03813). Y.G.S. acknowledges the National Natural Science Foundation of China\n(Grants No. U2032204), and the Informatization Plan of Chinese Academy of Sciences\n(CAS-WX2021SF-0102).\nAuthor Contributions\nC.L. proposed and conceived the project. C.L. carried out the ARPES experiments with the\nassistance from Y.W., Q.D.G., W.Y.C. and D.P.. J.F.Z., H.C.Y. and T.X. contributed to\nthe band structure calculations and theoretical discussion. H.X.L. and Y.G.S. contributed\nto NdAlSi crystal growth. C.L. characterized the single crystals. C.L. contributed to soft-\nware development for data analysis and analyzed the data with the assistance from Y.W..\nC.L. wrote the paper. E.R., B.F., H.F. and B.T. provided the beamline support. M.D.,\nM.H.B. and O.T. contributed to the scienti\fc discussions. All authors participated in and\ncommented on the paper.\n15FIG. 1. Crystal structure and band structure calculations for NdAlSi. (a) The crystal structure\nof NdAlSi with the space group I41md(no. 109). (b) The 3D BZ of the original unit cell of NdAlSi, and the\ncorresponding two-dimensional BZ projected on the (001) plane (green lines) in the pristine phase in (a).\n(c) A three dimensional lattice view of the NdAlSi spin structure. (d) The corresponding 3D BZ of NdAlSi\nwith the UUD ferrimagnetic order which exhibit the BZ folding relative to the paramagnetic state. (e) The\ndistribution of Weyl fermions in the 3D BZ in the paramagnetic state, while the red dots representing nodes\nwith chiralities +1 and blue dots representing -1. (f) The top view of (e). (g-h) The top view of (e) at the\nkz\u00180\u0019=cplane (g) and\u0018\u00060.67\u0019=cplane (h). (i) Distributions of all Weyl fermions from the top view of\nthe BZ in ferrimagnetic state. The gray lines show the boundary of the BZ in the ferrimagnetic state. The\nbright blue and red dots represent the Weyl fermions in the folded BZ due to the ferrimagnetic order. The\nshaded blue and red dots are the Weyl fermions in the ferrimagnetic state after folding back into the \frst BZ\nof the paramagnetic state (Weyl nodes folded back that are distributed in the \u0018\u0006\u0019=cplane are not shown\nhere). (j-l) The top view of (i) at the k z\u00180\u0019=cplane (j),\u0018\u00060.33\u0019=c(k) and\u0018\u00060.67\u0019=cplane (l). (m-n)\nCalculated band structures of NdAlSi along high-symmetry directions across the BZ in paramagnetic (m)\nand UUD type ferrimagnetic (n) state. The high symmetry points are de\fned in (b), while Z 2is the Z point\nin the second BZ in the k z=0\u0019=cplane. The blue lines are the bands corresponding to the paramagnetic\nstate, and the red lines are the folded bands due to the UUD type ferrimagnetic order.\n16FIG. 2. Determination of the surface termination and observation of topological Fermi arcs\nand chiral charges in NdAlSi. (a) Stacking plots of constant energy contours at di\u000berent binding energies\n(Eb), obtained from ARPES, display complex band structure evolution as a function of energy. (b) Fermi\nsurface and constant energy contours at di\u000berent binding energies measured with a photon energy of 41\neV at 15 K under positive circular (PC) polarization. (c) Surface projected DFT calculations of the (001)\nFermi surface and constant energy contours on the Nd atom terminated surface. (d) Surface projected DFT\ncalculations for two domain structures derived from (c) and their superposition after rotation of 90 degrees.\n(e-f) Band dispersions along Y-\u0000-Y[Cut1, (e)] and M-\u0000-M[Cut2, (f)] directions. The momentum directions\nof the cuts are show in the leftmost panel of (b). (g-h) The corresponding surface projected DFT calculations\nof the bands along Y-\u0000-Y(g) andM-\u0000-M(h) directions. (i-j) The surface projected DFT calculations of\nthe bands along Y-\u0000-Y(i) andM-\u0000-M(j) directions, considering the two domain structures. The spectral\nintensity is illustrated by the size and transparency of the markers. (k-l) Surface Green function calculation\nof constant energy contours, taking into account the double domain structures, at an energy of 38 meV (k)\nand 56 meV (l) above the Fermi level. (m-p) Photon energy dependent band dispersions along Cut3 in (b).\n(q) The momentum distribution curves (MDCs) extracted from (m-p) at a binding energy of 10 meV. The\nred and green diamond shaped markers indicate the peaks positions corresponding to two di\u000berent Fermi\narcs. (r-s) Band dispersions along Loop 1 [L1, (r)] and Loop 2 [L2, (s)].\n17FIG. 3. Weyl fermions in NdAlSi. (a) Photon energy dependent ARPES spectral intensity map at\nthe binding energy of 0.2 eV along the X-\u0000-Xdirection measured at 30 K. For the sake of comparison, the\nspectral intensities are normalized. (b-c) The distribution of Weyl fermions in the BZ at the k z\u00180.67\u0019=c\nplane (b) and\u00180\u0019=cplane (c) in which the paths of Cut1 to Cut4 and the types of Weyl fermion W1,\nW2, W3 and W4 are de\fned. (d) The Fermi surface map measured at 15 K with photon energy of 34 eV\nunder PC polarization. (e) Calculated bulk Fermi surface at the k z\u00180.67\u0019=cplane. The BZ is de\fned\nby orange solid lines. (f) Band dispersion along Cut1 measured with a photon energy of 34 eV at 15 K\nrelated to Weyl fermions W1, with DFT bulk calculations overlaid. (g) Corresponding surface projected\nDFT calculations for (f). The spectral intensity is denoted by the size and transparency of the markers.\n(h) The Fermi surface map measured at 15 K with a photon energy of 41 eV under PC polarization. (i-k)\nBand dispersion measured along Cut2 to Cut4 with a photon energy of 41 eV under PC polarization at 15\nK related to Weyl fermions W2 (i), W3 (j) and W4 (k), with DFT bulk calculations overlaid. (l) Calculated\nbulk Fermi surface at the k z\u00180\u0019=cplane. The BZ is de\fned by orange solid lines. (m-o) Corresponding\nsurface projected DFT calculations for (i-k). (p) The Fermi surface map measured on a low quality surface\nat 30 K with a photon energy of 41 eV under LH polarization. (q-s) Similar measurements as (i-k) but\nmeasured on the low quality surface sample at 30 K with photon energy of 41 eV under LH polarization.\n18FIG. 4. Ferrimagnetic regulation of Weyl fermions in NdAlSi. (a) The distribution of Weyl fermions\nin the BZ at the k z\u00180\u0019=cplane in the ferrimagnetic state, in which the path of Cut1 is de\fned. (b-c) The\nconstant energy contours at the binding energy of 70 meV measured with a photon energy of 41 eV at 15 K\n(b) and 2 K (c). (d) The surface projected DFT calculations of the bands in the paramagnetic state along\nthe path of Cut1 in (a). The spectral intensity is denoted by the size and transparency of the markers. (e-f)\nThe band dispersions (after symmetrization) along the path of Cut1 in (a) measured with photon energy of\n41 eV under PC polarization at 15 K (e) and 2 K (f), with DFT bulk calculations overlaid. (g-h) Zoom in\non the areas in (e-f) enclosed by the pink lines. The blue, green and red dashed lines are guides to the eye.\n(i) Cartoon picture of the regulation of Weyl fermions by ferrimagnetism in NdAlSi in the k z\u00180\u0019=cplane,\ncorresponding to the area enclosed by dashed pink lines in (a).\n19"    },    {        "title": "1711.00630v2.A_Comparative_Study_on_Spin_Orbit_Torque_Efficiencies_from_W_ferromagnetic_and_W_ferrimagnetic_Heterostructures.pdf",        "content": "1 \n A Comparative Study on Spin-Orbit Torque Efficiencies from W/ferromagnetic and \nW/ferrimagnetic Heterostructures \n \nTing-Chien Wang, Tian-Yue Chen†, Chun-Te Wu, Hung-Wei Yen, and Chi-Feng Pai∗ \n \nDepartment of Materials Science and Engineering, National Taiwan University, Taipei 10617, Taiwan  \n \nIt has been shown that W in its resistive form poss esses the largest spin-Hall ratio among all heavy transition \nmetals, which makes it a good candidate for generating efficient dampinglike spin-orbit torque (DL-SOT) acting \nupon adjacent ferromagnetic or ferrimagnetic (FM) layer. Here we provide a systematic study on the spin \ntransport properties of W/FM magnetic heterostru ctures with the FM layer being ferromagnetic Co 20Fe60B20 or \nferrimagnetic Co 63Tb37 with perpendicular magnetic anis otropy. The DL-SOT efficiency DLξ , which is \ncharacterized by a current-induced hysteresis loop shift method, is found to be correlated to the microstructure \nof W buffer layer in both W/Co 20Fe60B20 and W/Co 63Tb37 systems. Maximum values of 0.144DLξ≈  and \n0.116DLξ≈  are achieved when the W layer is partially amorphous in the W/Co 20Fe60B20 and W/Co 63Tb37 \nheterostructures, respectively. Our results suggest that  the spin Hall effect from resistive phase of W can be \nutilized to effectively control both ferromagnetic and ferrimagnetic layers through a DL-SOT mechanism. \n \n                                                 \n† Email: r05527067@ntu.edu.tw \n∗ Email: cfpai@ntu.edu.tw 2 \n I. INTRODUCTION \nThe spin Hall effect (SHE) [1-3] is the phenome non of conduction electrons with different spin \norientations being deflected to differ ent transverse directions due to sp in-orbit interactions in materials. \nAmong all classical materials (excluding emergent mate rials systems such as t opological insulators), \n5d transition metals (TM) like Pt [4], Ta [5], and W [6] are reported to have significant SHE. In order \nto utilize the SHE from these materials, transition metal/ferromagnetic metal (TM/FM) bilayer heterostructures are typically employed to obse rve SHE-induced magnetiza tion switchi ng [5,7], \ndomain-wall motion [8,9], and dynamics [10,11]. In  a TM/FM bilayer structure, the SHE-induced \ntransverse spin current \nsJ that quantifies the spins being absorbed by the FM layer can be expressed \nas ()TM/FM TM TM\nint /2sS H eJ eT J θ =h , where TM\nSHθ  is the internal spin Hall ratio of the TM layer and TM\neJ \nrepresents the longitudinal charge curre nt density flowing in the TM layer. TM/FM\nintT  represents the spin \ntransparency at the TM/FM interface [12,13]  (note that TM/FM\nint 1 T = for a perfect tr ansmission), which \nis related to the spin-mixing conductance [14,15] . The transmitted spin angular momentum from sJ \ncan be transferred to the magnetic moments in FM through a spin-tra nsfer torque mechanism [16]. The \nfinal effect is therefore a damp inglike spin-orbit torque  (DL-SOT) acting upon the FM layer, and the \nDL-SOT efficiency of a TM/FM bilayer heterostructure can be expressed as \n()TM TM/FM TM\nint 2/ /DLs e S H eJ J T ξ θ ≡=h  [12].  \nFor TMs, W has been reported to possess the la rgest internal spin Hall ratio and DL-SOT \nefficiency at room temperature, 0.30DLξ≈  [17]. The efficacy of the SHE from W has been 3 \n demonstrated and characterized through DL-SOT  switching measurements [6,18,19], harmonic voltage \nmethods [20], spin-pumping measurements [21], sp in-Hall magnetoresistance measurements [22-24], \nand optical approaches [25,26]. The strength of the SHE in W is also known to be strongly phase \ndependent: The resistive β-phase or amorphous phase has been e xperimentally shown to have greater \nSHE and DL-SOT efficiency while compare to the conductive α-phase or crystall ine phase [6,22,26]. \nIn this work, we provide a detailed systematic study on the DL-SOT efficiencies DLξ from two \nseries of W/FM magnetic heterost ructures using a newly-developed hysteresis loop shift measurement \nmethod [27]. The FM layers that we employe d are perpendicularly-magnetized ferromagnetic \nCo20Fe60B20 and ferrimagnetic Co 63Tb37. Ferromagnetic Co-Fe-B is a classical FM layer for magnetic \ntunnel junctions [28,29] and SHE three-terminal de vice applications [5,30], whereas ferrimagnetic \nCo-Tb and Co-Gd alloys have been adopted to study SOT behavior near the co mpensation point of the \nFM layer [31-33]. By characterizing DLξ from both W/Co 20Fe60B20 and W/Co 63Tb37 structures, the \ncomparative study allows us to examine differen t possible spin transport scenarios (different W/FM\nintT ) \nacross two different W/FM interfaces. We find that DLξ  depends on the thickness and the \nmicrostructure of buffer layer W in both systems. DLξ  can reach ~0.144 and ~0.116 for \nW/Co 20Fe60B20 and W/Co 63Tb37 heterostructures, respectively, when the W buffer layer is thin ( ≤4 \nnm), resistive, and partia lly amorphous. Our results suggest that re sistive W can serve as an efficient \nDL-SOT source and the corresponding W/FM\nintT  is comparable for FM being either ferromagnetic or \nferrimagnetic. 4 \n  \nII. HETEROSTRUCTURES PREPARATION \n Magnetic heterostructures of this work are de posited in a high vacuum sputtering chamber with \nbase pressure of 831 0 T o r r−× . We use dc (rf) magnetron sputteri ng with 3 mTorr (10 mTorr) of Ar \nworking pressure for depositions of metallic ( oxide) layers. Multilayer stack heterostructures \nW(Wt)/Co 20Fe60B20(1.4)/Hf(0.5)/MgO(2)/Ta(2) (numbers in th e parenthesis are in nanometers) and \nW(Wt)/Co 63Tb37(10)/Ta(2) are deposited onto thermally -oxidized silicon substrates, with Wt ranges \nfrom 1 nm to 16 nm. Co 20Fe60B20 (Co-Fe-B) is directly sputtere d from a single target while Co 63Tb37 \n(Co-Tb) is prepared by co-sputtering of a Co and a Tb target. Samples from the W/Co-Fe-B series are \nannealed at 300 \u001f for 1 hour in high vacuum to induce perpendicular magnetic anisotropy (PMA) of \nthe Co-Fe-B layer. The Hf(0.5) inse rtion layer serves as a PMA enha ncing layer and should be mostly \noxidized therefore having minimal effects on transpor t properties [34]. No fu rther heat treatment is \nneeded to obtain PMA for the W/Co-Tb series since ferrimagnet Co-Tb possesses bulk PMA [35,36]. \nThe Ta(2) capping layers protect th e heterostructures beneath from oxidation. To perform transport \nmeasurements, heterostructures are made into micron- sized Hall-bar devices with  lateral dimensions of \n5μm6 0μm ×  and 10μm6 0μm ×  by photolithography. \n \nIII. RESULTS FROM W/Co-Fe-B HETEROSTRUCTURES \nA. Magnetic properties and microstructures 5 \n  We first characterize magnetic anisotropy of W/ Co-Fe-B heterostructures with magneto-optical \nKerr effect (MOKE). As shown in Fig. 1(a) and (b ), the out-of-plane hystere sis loops of W/Co-Fe-B \nwith thin (4 nm) W buffer layer and thick (14 nm) W buffer layer i ndicate both hetero structures have \nPMA, with out-of-plane coercive fields of 35 OecH≈  and 175 OecH≈ , respectively. To unravel \nthe cause of this different magnitudes in coerci ve field, we perform cros s-section high resolution \ntransmission electron mi croscopy (HR-TEM) imaging on the W(4)/Co-Fe-B(1.4) and \nW(14)/Co-Fe-B(1.4) heterostructures. As presented in Fig. 1( c) and (d), the HR-TEM  results show that \nthe W buffer layer is partially amorphous for the thin W case, whereas it is crystalline \n(body-centered-cubic) for the thick W case. Therefor e, microstructure anal ysis suggests that the \nenhanced out-of-plane coercive field for the thick W heterostructure is correlated to the change of W buffer layer texture. This correlation might origin ate from the change of domain nucleation sites or \ndomain wall pinning sites during the phase transition of  W buffer layer. Note that the MgO layers in \nboth samples are not crystalline, which indicates th at the templating of Co-F e-B crystallization during \nannealing process is mainly affect ed by the W buffer layer, in cont rast to the prev iously-studied \nCo-Fe-B/MgO templating effect [37,38] . Nevertheless, all annealed W(\nWt)/Co-Fe-B(1.4) \nheterostructures with W 1n m 1 6n m t≤≤  show PMA and allow us to perform DL-SOT \ncharacterization using tec hniques that are suitable for devices with PMA. \n \nB. Hysteresis loop shift measurements 6 \n To systematically characterize the DL-SOT effi ciency from W/Co-Fe-B heterostructures, we \nperform the current-induced hysteresis loop shif t measurements [27,31,36] on patterned Hall-bar \ndevices, as shown in Fig. 2(a). In this type of me asurement, we sweep out-of-plane field zH to obtain \nhysteresis loops from anomalous Hall voltage under th e influence of an extra in-plane bias field xH. \nThe purpose of xH is to overcome the effective fi eld originating from interfacial \nDzyaloshinskii-Moriya interaction (DMI) DMIH  and to realign domain wall moments [39,40]. When \nDMI xHH≥ , the current-induced DL-SOT acting on the do main wall moments can be observed as an \nout-of-plane effective field effzH, which will lead to a shift in the out-of-plane hysteresis loop. \nTherefore, by obtaining the hysteresis loop shifts under different applied currents (DCI) as shown in \nFig. 2(b), the DL-SOT strength can  be quantified. We summarize the DCI dependence of switching \nfields and effzH for W(4)/Co-Fe-B(1.4) and W(15)/Co-Fe- B(1.4) samples in Fig. 2(c) and (d), \nrespectively. It is obvious that for both cases effzH depends on the applied DCI linearly, which is \nconsistent with prev ious studies [27,31,36]. Ho wever, the slope of eff-to- z\nDC HI  is greater in \nW(4)/Co-Fe-B(1.4) than in W(15)/C o-Fe-B(1.4), suggesting a more si gnificant DL-SOT effect in the \nthin W case. The DL-SOT efficiency DLξ can be quantitatively estim ated from the ratio between effzH \nand the current density eJ in the spin-Hall buffer layer by [27,41] \n eff eff\n0F M22z\nDL s\neH eMtJξμπ⎛⎞ ⎛⎞= ⎜⎟ ⎜⎟⎝⎠ ⎝⎠ h, (1) \n 7 \n where sM and eff\nFMt represent saturation magnetization and ef fective thickness of the FM layer, \nrespectively. eff\nFMt is the nominal FM thickness subtracted by the magnetically de ad layer thickness. \nThese parameters for Co-Fe-B are characterized by vibrating sample magnetometer (VSM) to be \n31500emu/cmsM≈ (61.5 10 A/m×  in SI units) and eff\nCo-Fe-B 0.7 nm t ≈ . Since the applied charge current \nwill be flowing in both W and Co-Fe-B layers, eJ in the W layer is calculated by \n()Co-Fe-B W Co-Fe-B W W Co-Fe-B /eD CJJ t t t ρρ ρ =⋅ +⎡⎤⎣⎦, where ()W /DC DCJIw t=⋅  is the nominal current \ndensity with 5μm w=  (width of the device). Co-Fe-Bρ  and Wρ are the resistivities of Co-Fe-B and \nW layers, respectively. Therefore, the resulting equation to estimate DLξ from measured eff/z\nDC HI  \ncan be expressed as \n \n eff Co-Fe-B W W Co-Fe-B eff\n0F M W\nCo-Fe-B W22z\nDL s\nDCtt H eMt w ttIρρξμπρ⎛⎞ ⎛ ⎞ + ⎛⎞= ⎜⎟ ⎜ ⎟ ⎜⎟⎝⎠ ⎝⎠ ⎝ ⎠ h. (2) \n \n The estimated magnitudes of DL-SOT efficiency DLξ as well as relevant magnetic and \nelectrical properties as f unctions of W buffer layer thickness ar e summarized in Fig. 3. The W buffer \nlayer thickness dependence of out-of-plane coercive field cH (Fig. 3(a)) again verifies the \nmicrostructure evolution from amor phous to crystalline pha se mentioned earlier. Resistance (Fig. 3(b)) \nand inverse of resistance (Fig. 3(c)) of the devices al so indicate the existence of two regimes: resistive \nthin W (Wt≤4 nm) and conductive thick W (Wt≥4 nm). The corresponding resistivities of W buffer 8 \n layer in these two regimes are estimated to be thin (amorphous)\nW 185.7 cm ρμ ≈Ω  and \nthick (crystalline)\nW 90.9 cm ρμ ≈Ω . The resulting DL-SOT efficiency DLξ is strongly phase dependent, as \nshown in Fig. 3(d). For Wt≤4 nm, DLξ increases while increasing W thickness and can be \nwell-fitted to a spin diffusion model () ()thin(amorphous) W\nWW 1 sechDL DL sttξξ λ ⎡ ⎤ =−⎣ ⎦[4] with \nthin(amorphous)0.144DLξ ≈  and spin diffusion length thin(amorphous) W0.9nmsλ ≈ . In contrast, for all Wt≥4 nm \nsamples 0.03DLξ≈ , which suggests that thick(crystalline)0.03DLξ ≈  and thick(crystalline) W1.5nmsλ ≤ . This \nphase dependence of DLξ in W/Co-Fe-B heterostructures is consistent with previous reports \n[6,22,26]. However, in this work we further qua ntitatively estimate the DL-SOT efficiency from \ncrystalline W buffer layers to be thick(crystalline)0.03DLξ ≈  in W/Co-Fe-B heterostructures. Also note that \nfor all W/Co-Fe-B heterostructures, the magnitude  of DMI effective field is estimated to be \nDMI 100Oe H ≤ . \n \nC. Demonstration of spin-orbit torque switching \nIn order to demonstrate current-induced DL-SOT sw itching in these heterost ructures, as shown in \nFig. 4(a), we apply an in-plane bias field 80OexH=  to realign magnetic domain wall moments [39] \nto facilitate domain expansion. A puls e of charge current with amplitude swI and pulse width 50 ms is \nsent into the device to generate a pulsed spin current from the SHE of W buffer layer. The resulting \nSHE-induced DL-SOT acting on the adjacent Co-F e-B layer will switch the magnetization when swI \nreaches a critical value. As shown in Fig. 4(b), a representative W(4)/Co- Fe-B(1.4) device can be 9 \n reversibly switched between tw o magnetization states by cri tical switching current of ~ 0.6 mA , which \ncorresponds to critical switching current density 10 22.5 10 A/mcJ≈× . Although this value is much \nlower than the previously reported 11 21.6 10 A/mcJ≈×  in a similar heterostructure [18], the result is \nnot unexpected: Since DL-SOT sw itching in micron-sized samples is mainly governed by domain \nnucleation and domain wall propagation processes, a lower coercive field sample (in our case \n30OecH≈ ) will give rise to a lower cJ [42]. The feasibility of switching W/Co-Fe-B devices \nthrough DL-SOT mechanism can also be predicted fr om Fig. 3(c) and (d), by viewing them as \nswitching phase diagrams. Based on this concept, we can predict that within the range of swI we send \ninto devices ( 10 mAswI≤ ), no DL-SOT switching will be observed from the W(15)/Co-Fe-B(1.4) \nsample. Indeed, we only observe successful curren t-induced DL-SOT switching in thin (amorphous) W \nsamples but not in thick (crystalline) samples.  \nIV . RESULTS FROM W/FERRIMAGNETIC BILAYERS \nA. DL-SOT characterization \nRecently, it has also been shown that SOT from 5d transition metals (Ta and Pt) or topological \ninsulators (Bi 2Se3) can be utilized to contro l magnetic moments in a ferrimagnetic layer such as Co-Tb \nand Co-Gd [31-33,43]. However, a detailed study on DL-SOT from W/ferrimagnet heterostructures \nhas yet to be reported. We employ th e same characterization procedure for W(Wt)/Co-Fe-B(1.4) \nheterostructures on our W(Wt)/Co-Tb(10) heterostructures. As  shown in Fig. 5(a), the Co 63Tb37(10) 10 \n sputtered onto W(3) buffer layer shows bulk PMA in its as-deposited state, with 350 OecH≈ , which \nis greater than the cH’s of W/Co-Fe-B films. Representative HR -TEM result, as shown in Fig. 5(b), \nfurther indicates that both W(3) and Co-Tb(10) la yers are amorphous. Representative hysteresis loop \nshift measurement results from W(3)/Co-Tb(10) a nd W(10)/Co-Tb(10) Hall-ba r devices are shown in \nFig. 5(c) and (d), respectively. The eff/z\nDC HI  ratios are estimated to be -3.3 Oe/mA for \nW(3)/Co-Tb(10) and -0.7 Oe /mA for W(10)/Co-Tb(10). The eff/z\nDC HI  difference between these two \nsamples again suggests W phase dependence of the SHE.  We further estimate the DL-SOT efficiencies \nof these two samples with different W thickness: ()W3 nm 0.10DLt ξ =≈  and \n()W10 nm 0.026DLt ξ =≈ . To calculate DLξ  for W/Co-Tb samples from Eqn. (2), \n3265emu/cmsM≈ and Co-Tb 200  cm ρμ =Ω of the Co-Tb layer are separately characterized by VSM \nand four-point measurements. No obvious magnetic dead layer is found in the as-deposited W/Co-Tb \nheterostructures.  \nWe summarize the W thickness de pendence of out-of-plane field cH and DL-SOT efficiency \nDLξ for W(Wt)/Co-Tb(10) in Fig. 6(a) and (b), re spectively. While no obvious W thickness \ndependent trend is found for cH (since the PMA for Co-Tb has a bulk origin), the behavior of DLξ \nfor W/Co-Tb resembles that found in the W/ Co-Fe-B system. The thickness dependence of DLξ \nagain can be fit into a spin-diffusi on model and we are able to extract thin(amorphous)0.116DLξ ≈  and \nthin(amorphous) W1.1nmsλ ≈  for thin W regime. For thick W regime, thick(crystalline)0.026DLξ ≈  and \nthick(crystalline) W2n msλ ≤ . Note that the estimated DLξ’s for W/Co-Tb are all slightly smaller but fairly 11 \n close to the estimated DLξ’s for W/Co-Fe-B heterostructures, whic h means that the spin transparency \nfactor W/Co-Tb W/Co-Fe-B\nint intTT ≤ , with the assumption that W\nSHθ have the same phase dependent trend in \nboth cases. It is possible that th e amorphous nature of W/Co-Tb in terface leads to a smaller spin \ntransparency factor while compar e to the (partially) crystalline W/ Co-Fe-B interface. A similar result \nof smaller spin transparency fact or has also been reported by Finl ey and Liu for Ta/Co-Tb system \n(while compare to Ta/Co-Fe-B) [31]. In contrast, th e magnitude of DMI effective fields are estimated \nto be DMI 250Oe H ≈  for W/Co-Tb samples, which is greater than those in W/Co-Fe-B \nheterostructures. More detailed studies on Co -Tb domain wall structure and domain expansion \ndynamics might elucidate the difference of  DMI strength in these two systems. \n \nB. DL-SOT switching \nNote that current-induced DL-SOT switching cannot be realized in W(Wt)/Co-Tb(10) devices, as \ncan be predicted by the switching phase diag rams Fig. 5(c) and (d). For example, 10 mAswI>  is \nrequired to see possible cu rrent-induced switching in the W(3)/Co- Tb(10) device, but a current this \nlarge will typically destroy our samples. To achieve DL-SOT switching, we further reduce the \nthickness of deposited Co-Tb from 10 nm to 3.5 nm. In this thin Co-Tb case, the coercive field of \nCo-Tb is reduced to 10OecH≈ , which is beneficial for observi ng magnetization switching due to a \nlower depinning field. In Fig. 6(d), we show  a representative current-induced DL-SOT switching curve \nfrom a W(4)/Co-Tb(3.5) Hall-bar de vice with lateral dimensions of 10μm6 0μm × . The critical 12 \n switching current is of ~ 1m A , from which the critical switching current density is estimated to be \n10 21.4 10 A/mcJ≈× . This number is even smaller than the case of W/Co-Fe-B, mainly due to the \nsmaller cH and magnetization eff\nFMsMt of thinner Co-Tb layer. Over all, our results indicate that \ncurrent-induced DL-SOT in W/Co-Tb heterostru cture is an efficient mechanism to induce \nmagnetization dynamics therein. Although previous st udy had shown that the field-like component of \nSOT could also possibly exist in a TM/Co-Tb system  [32], the magnitude is much smaller than its \ndampinglike counterpart. Therefore, we conclude that the magnetizatio n switching we observe here is \nmainly due to current-induced DL-S OT from the SHE of amorphous W. \n \nV . CONCLUSION \n To summarize, we show that current-induced DL-SOT efficiencies DLξ from both W/Co-Fe-B \n(TM/ferromagnetic) and W/Co-Tb (TM/ferrimagnetic) heterostructures depend on the microstructure \nof W buffer layer. Thr ough hysteresis loop shift measurements, we estimate \nthin(amorphous)0.116 0.144DLξ ≈−  for magnetic heterostructures  with thin, amorphous W, while \nthick(crystalline)0.026 0.030DLξ ≈−  for heterostructures with thick, crystalline W. By comparing results from \nboth systems, we find the spin transparency factor W/Co-Tb W/Co-Fe-B\nint intTT ≤ . We further demonstrate \ncurrent-induced DL-SOT switching in both W/Co-Fe-B and W/Co-Tb heterostructure systems. 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Rohart, E. Jue, V . Cros, and A. Fert, Europhys. Lett. 100, 57002 (2012). \n[42] T. Y . Chen, C. T. Wu, H. W. Yen, and C. F. Pai, Phys. Rev. B 96, 104434 (2017). 15 \n [43] J. H. Han, A. Richardella, S. A. Siddiqui, J. Fi nley, N. Samarth, and L. Q. Liu, Phys. Rev. Lett. \n119, 077702 (2017). \n \n  16 \n  \nFigure 1. Out-of-plane hysteresis loops of (a ) W(4)/Co-Fe-B(1.4)/Hf(0.5)/MgO(2) and (b) \nW(14)/Co-Fe-B(1.4)/Hf(0.5)/MgO(2) magnetic hete rostructures. Cross s ection HR-TEM imaging \nresults from (c) W(4)/Co-Fe-B(1.4)/Hf(0.5)/MgO( 2) and (d) W(14)/Co-Fe-B(1.4)/Hf(0.5)/MgO(2) \nmagnetic heterostructures. The subpanels are the diffractograms derived by reduced fast Fourier \ntransformation (FFT) from the regi ons of interests (white boxes). \n \n  \n17 \n  \nFigure 2. (a) Schematic illustration of anomalous Ha ll voltage measurement. (b) Representative shifted \nHall voltage loops from a W(4)/Co-Fe-B(1 .4) sample with different DC currents DCI and an in-plane \nbias field xH=600 Oe. (c,d) Switching fields swH of W(4)/Co-Fe-B(1.4) and W(15)/Co-Fe-B(1.4) \nsamples for down-to-up (red circles)  and up-to-down (blue triangles) switching pr ocesses as functions \nof DCI, with xH=600 Oe and 1500 Oe, respectively. effzH (black squares) represent the center of \nHall voltage loops. The solid li nes represent lin ear fits to effzH data. \n18 \n  \nFigure 3. (a) Out-of-plane coercive field cH of the Co-Fe-B layer, (b) Hall-bar device resistance, (c) \ninverse of the Hall-bar device resistance, and (d) the magnitude of DL-SOT efficiency DLξ of \nW/Co-Fe-B magnetic heterostructures  as functions of W thickness (Wt). L and w in (c) stand for length \nand width of the Hall-bar device channel, respectivel y. The solid line and dashed line in (c) represent \nlinear fits to W4nm t≤  and W4nm t>  data, respectively. The red solid  line and blue dashed line in \n(d) represent fits to a spin diffusion model for W4nm t≤  and W4nm t>  data, respectively.  \n \n19 \n  \nFigure 4. (a) Schematic illustration of cu rrent-induced SOT switching measurement. swI represents \nthe amplitude of injected current pulse. The applied current pulse duration is 50 ms. (b) A \nrepresentative current-induced SO T switching result from a W(4)/Co-Fe-B(1.4) Hall-bar sample under \nin-plane bias field 80OexH= . The black arrows represent the sweeping directions of applied current \npulse swI. \n  \n20 \n  \nFigure 5. (a) Representative out-of-plane hysteresis loop of a W(3)/Co-Tb(10)  heterostructure. (b) \nCross section HR-TEM imaging result from a W( 3)/Co-Tb(10) sample. (c,d) Switching fields swH of \nW(3)/Co-Tb(10) and W(10)/Co-Tb( 10) samples for down-to-up (red circles) and up-to-down (blue \ntriangles) switching processes as functions of DCI, both with xH=1000 Oe. effzH (black squares) \nrepresent the cente r of Hall voltage loops. The solid lines represent linear fits to effzH data. \n  \n21 \n  \nFigure 6. (a) Out-of-p lane coercive field cH of the Co-Tb layer and (b) the magnitude of DL-SOT \nefficiency DLξ of W(Wt)/Co-Tb(10) magnetic heterostructur es as functions of W thickness (Wt). \nThe red solid line and blue dashed line in (b) represent fits to a spin diffusion model for W4nm t≤  \nand W4nm t>  data, respectively. (c) Out-of-plane  hysteresis loop of a W(4)/Co-Tb(3.5) \nheterostructure. (d) Current-indu ced SOT switching curve of a W(4)/Co-Tb(3.5) Hall-bar sample under \nin-plane bias field 800OexH= . The black arrows represent the sweeping directions of applied \ncurrent pulse swI. The dashed lines serve as guide to the eye.  \n"    },    {        "title": "0811.2118v1.Selection_rules_for_Single_Chain_Magnet_behavior_in_non_collinear_Ising_systems.pdf",        "content": "arXiv:0811.2118v1  [cond-mat.mtrl-sci]  13 Nov 2008Selection rules for Single-Chain-Magnet behavior in non-c ollinear Ising systems\nAlessandro Vindigni1∗and Maria Gloria Pini2\n1Laboratorium f¨ ur Festk¨ orperphysik, Eidgen¨ ossische Te chnische Hochschule Z¨ urich, CH-8093 Z¨ urich, Switzerlan d\n2Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche,\nVia Madonna del Piano 10, I-50019 Sesto Fiorentino (FI), Ita ly\n(Dated: November 2, 2018)\nThe magnetic behavior of molecular Single-Chain Magnets is investigated in the framework of a\none-dimensional Ising model with single spin-flip Glauber d ynamics. Opportune modifications to\nthe original theory are required in order to account for reci procal non-collinearity of local anisotropy\naxes and the crystallographic (laboratory) frame. The exte nsion of Glauber’s theory to the case of\na collinear Ising ferrimagnetic chain is also discussed. Wi thin this formalism, both the dynamics\nof magnetization reversal in zero field and the response of th e system to a weak magnetic field,\noscillating in time, are studied. Depending on the geometry , selection rules are found for the\noccurrence of slow relaxation of the magnetization at low te mperatures, as well as for resonant\nbehavior of the a.c.susceptibility as a function of temperature at low frequenc ies. The present\ntheoryapplies successfully tosome real systems, namely Mn -, Dy-, andCo-based molecular magnetic\nchains, showing that Single-Chain-Magnet behavior is not o nly a feature of collinear ferro- and\nferrimagnetic, but also of canted antiferromagnetic chain s.\nPACS numbers: 75.10.-b, 75.10.Pq, 75.50.Xx, 75.60.Jk\nI. INTRODUCTION\nSlow dynamics of the magnetization reversal is a cru-\ncial requirement for potential applications of Single-\nChain Magnets (SCM’s)1,2,3, and nanowires in general,\nin magnetic-memory manufacture. For nanowires with\na biaxial anisotropy, provided that their length is much\ngreater than the cross section diameter but smaller than\nexchange length, this phenomenon is governed by ther-\nmal nucleation and propagation of soliton-antisoliton\npairs; the associated characteristic time is expected to\nfollow an Arrhenius law4,5. For genuine one dimensional\n(1D) Ising systems with single spin-flip stochastic dy-\nnamics, a slow relaxation of the magnetization was first\npredicted by Glauber6in 1963. Through Glauber’s ap-\nproach,manyphysicalsystemswereinvestigated,ranging\nfrom dielectrics7,8,9to polymers7,10,11. More fundamen-\ntally, this model has been employed to justify the use\noftheKohlrauch-Williams-Wattsfunction10,12(stretched\nexponential) to fit the relaxation of generalized 1D spin\nsystems. Also the universality issue of the dynamic crit-\nical exponent13,14,15,16,17of the Ising model18, as well as\nstrongly out-of-equilibrium processes (magnetization re-\nversal19, facilitated dynamics19, etc.) have been studied\nmoving from the basic ideas proposed by Glauber.\nIn this paper, single spin-flip Glauber dynamics is\nused to investigate theoretically the slow dynamics of\nthe magnetization reversal in molecular magnetic sys-\ntems. In particular, we extend Glauber’s theory6to\nthe Ising collinear ferrimagnetic chain, as well as to the\ncase of a chain in which reciprocal non-collinearity of lo-\ncal anisotropy axes and the crystallographic(laboratory)\nframe is encountered. Such extensions are motivated by\nthefact that(i) inmolecular-basedrealizationsofSCM’s,\nantiferromagnetic coupling typically has a larger inten-\nsity than the ferromagnetic one; in fact, the overlappingof magnetic orbitals, which implies antiferromagnetic ex-\nchange interaction between neighboring spins, can be\nmore easily obtained than the orthogonality condition,\nleadingtoferromagnetism20,21,22; (ii)non-collinearitybe-\ntweenlocalanisotropyaxesandthecrystallographic(lab-\noratory) frame takes place quite often in molecular spin\nchains. Besides magnetization reversal, the dynamic re-\nsponse of the system to a weak magnetic field, oscillating\nin time at frequency ω, is also studied. Depending on the\nspecific experimental geometry, selection rules are found\nfor the occurrence of resonant behavior of the a.c.sus-\nceptibility as a function of temperature (stochastic reso-\nnance) at low frequencies, as well as for slow relaxation\nof the magnetization in zero field at low temperatures.\nThe paper is organized as follows. In Sect. II we ex-\ntend Glauber’s theory6, originally formulated for a chain\nofidenticalandcollinearspins, tothemoregeneralmodel\nof a chain with non-collinear spins, possibly with Land´ e\nfactors that vary from site to site. In Sect. III we use\ntwo different theoretical methods (the Generating Func-\ntions approach and the Fourier Transform approach) to\ninvestigate the relaxation of the magnetization after re-\nmoval of an external static magnetic field, starting from\ntwo different initial conditions: fully saturated or par-\ntially saturated. In Sect. IV we calculate, in a linear ap-\nproximation, the magnetic response of the system to an\noscillatingmagnetic field. Fora chain of Nspins, the a.c.\nsusceptibility is expressed as the superposition of Ncon-\ntributions, eachcharacterizedby itstime scale; througha\nfew simple examples, we show that, depending on the ge-\nometryof the system ( i.e., the relativeorientationsofthe\nlocal easy anisotropy axes and of the applied field), dif-\nferent time scalescan be selected, possibly giving rise, for\nlow frequencies, to a resonant peak in the temperature-\ndependence of the complex magnetic susceptibility. In\nSect. V we show that the theory provides a satisfactory2\naccount for the SCM behavior experimentally observed\nin some magnetic molecular chain compounds, charac-\nterized by dominant antiferromagnetic exchange interac-\ntionsandnon-collinearitybetween spins. Finally, inSect.\nVI, the conclusions are drawn and possible forthcoming\napplications are also discussed.\nII. THE NON-COLLINEAR ISING-GLAUBER\nMODEL\nIn a celebrated paper6, Glauber introduced, in the\nusual 1D Ising model18, a stochastic dependence on the\ntime variable t:i.e., the state of a spin lying on the k-\nth lattice site was represented by a two-valued stochastic\nfunction σk(t)\nHI=−N/summationdisplay\nk=1/parenleftbig\nJIσkσk+1+gµBHe−iωtσk/parenrightbig\n, σk(t) =±1.\n(1)\nJIis the exchange coupling constant, that favors nearest\nneighboring spins to lie parallel ( JI>0, ferromagnetic\nexchange) or antiparallel ( JI<0, antiferromagnetic ex-\nchange); gis the Land´ e factor of each spin, and µBthe\nBohr magneton. In the original paper6a 1D lattice of\nequivalent and collinear spins was studied; there the re-\nsponse to a time-dependent magnetic field H( t), applied\nparallel to the axis of spin quantization and oscillating\nwith frequency ω, as in typical a.c.susceptibility exper-\niments, was also considered.\nIn order to investigate the phenomena of slow relax-\nation (for H=0) and resonant behavior of the a.c.sus-\nceptibility (for H ∝ne}ationslash= 0) in molecular SCM’s, we are going\nto generalize the Glauber model (1) accounting for non-\ncollinearity of local anisotropy axes and crystallographic\n(laboratory) frame. To this aim, we adopt the following\nmodel Hamiltonian\nH=−N/summationdisplay\nk=1/parenleftbig\nJIσkσk+1+GkµBHe−iωtσk/parenrightbig\n, σk(t) =±1.\n(2)\nJIis an effective Ising exchange coupling that can ap-\nproximately be related to the Hamiltonian parameters\nof a real SCM23,24: see later on the discussion in Sec-\ntion V. Like in the usual Ising-Glauber collinear model\n(1), the spins in Eq. (2) are described by classical, one-\ncomponent vectors that are allowed to take two integer\nvaluesσk(t) =±1, but now the magnetic moments may\nbe oriented along different directions, ˆzk, varying from\nsite to site. Within this scheme, the Land´ e tensor of a\nspin on the k-th lattice site has just a non-zero compo-\nnent,g/bardbl\nk, along the local easy anisotropy direction ˆzk.\nDenoting by ˆeHthe direction of the oscilla.ting magnetic\nfield,H(t)=He−iωtˆeH, we define the generalized Land´ e\nfactorGkappearing in Eq. (2) as\nGk=g/bardbl\nkˆzk·ˆeH (3)i.e., a scalar quantity that varies from site to site. Fol-\nlowing Glauber6, we define the single spin expectation\nvaluesk(t) =∝an}bracketle{tσk∝an}bracketri}htt, where brackets denote a proper en-\nsemble average, and the stochastic magnetization along\nthe direction of the applied field\n∝an}bracketle{tM∝an}bracketri}htt=µBN/summationdisplay\nk=1Gk∝an}bracketle{tσk∝an}bracketri}htt=µBN/summationdisplay\nk=1Gksk(t).(4)\nThe basic equation of motion of the Glauber model12,14\nreads\nd\ndtsk(t) =−2∝an}bracketle{tσkwσk→−σk∝an}bracketri}ht, (5)\nin which wσk→−σkrepresents the probability per unit\ntime to reverse the k-th spin, through the flip + σk→\n−σk. For a system of Ncoupled spins, this probability\nis affected by the interaction with the other spins, with\nthe thermal bath and, possibly, with an external mag-\nnetic field. Among all possible assumptions for the tran-\nsition probability wσk→−σkas a function of the N+ 1\nvariables11,19,25,26{σ1,...,σk,...,σN,t}, again following\nGlauber6we require wσk→−σkto be independent of time\nand to depend only on the configuration of the two near-\nest neighbors of the k-th spin. In zero field, these re-\nquirements are fulfilled by\nwH=0\nσk→−σk=1\n2α/bracketleftbigg\n1−1\n2γσk(σk−1+σk+1)/bracketrightbigg\n(6)\nwhile in the presence of an external field\nwH\nσk→−σk=wH=0\nσk→−σk(1−δkσk), (7)\nis usually chosen; the attempt frequency1\n2α(i.e., the\nprobability per unit time to reverse an isolated spin) re-\nmains an undetermined parameter of the model; γac-\ncounts for the effect of the nearest neighbors; the param-\netersδkhave the role of stabilizing the configuration in\nwhich the k-th spin is parallel to the field, and destabiliz-\ning the antiparallel configuration. Thanks to the partic-\nular choices (6) and (7) for the transition probability, by\nimposing the Detailed Balance conditions6it is possible\nto express γandδkas functions of the parameters in the\nspin Hamiltonian (2)\nγ= tanh(2 βJI), δk= tanh(βGkµBH).(8)\nwhereβ=1\nkBTis the inverse temperature in units of\nBoltzmann’s constant. Another advantage of Glauber’s\nchoices (6) and (7) is that the equation of motion (5)\ntakes a simple form. In particular, for H = 0, Eq. (5)\nwith the choice (6) becomes\nds(t)\ndt=−αAs(t), (9)\nwheres(t) denotes the vector of single-spin expectation\nvalues{s1(t), s2(t),···, sN(t)}andAis a square N×N3\nsymmetric matrix, whose non-zero elements are A k,k=1\nand A k,k−1=Ak,k+1=−γ\n2, with A 1,N=AN,1=−γ\n2if peri-\nodic boundary conditions are assumed for the N-spin\nchain. A closed solution of this set of first-order differen-\ntial equationscan be obtained expressingthe expectation\nvalue of each spin, sk(t), in terms of its spatial Fourier\nTransform (FT) /tildewidesq\nsk(t) =/summationdisplay\nq/tildewidesqeiqke−λqt. (10)\nSubstituting Eq. (10) into Eq. (9), one readily obtains\nthe dispersion relation\nλq=α(1−γcosq), q=2π\nNn (11)\nwithn= 0,1,..., N−127. For ferromagnetic cou-\npling (JI>0, hence γ >0) the smallest eigenvalue\nλq=0=α(1−γ) occurs for n=0, independently of the\nnumber of spins Nin the chain. For antiferromagnetic\ncoupling ( JI<0, hence γ <0) andNeven, the smallest\neigenvalue λq=π=α(1− |γ|) occurs for n=N\n2; while in\nthe case of Nodd, the smallest eigenvalue corresponds\ntoα/bracketleftbig\n1−|γ|cos/parenleftbigπ\nN/parenrightbig/bracketrightbig\n, thus depending on the number of\nspins in the antiferromagnetic chain28. The characteris-\ntic time scales of the system, τq, are given by\nτq=1\nλq=1\nα(1−γcosq). (12)\nAt finite temperatures the characteristic times τqare fi-\nnite because |γ|<1; forT→0 one has that 1 −|γ|van-\nishes irrespectively of the sign of JI, because γ→JI\n|JI|=\n±1. Thus, for H= 0, there is one diverging time scale\nin theT→0 limit:τq=0for ferromagnetic coupling and\nτq=πfor antiferromagneticcoupling (and even N). In the\npresence of a non-zero, oscillating field H( t) = He−iωt,\nthe equation of motion (5) with the choice (7) takes a\nform (see Eq. 24 in Section IV later on) which can still\nbe solved, though in an approximate way6, for a suffi-\nciently weak intensity of the applied magnetic field.\nIII. RELAXATION OF THE MAGNETIZATION\nIN ZERO FIELD\nThe originalGlauber model was formulated for a chain\nof collinear spins with the same Land´ e factors: i.e., in\nEq. (2) one has Gk=g,∀k= 1,···,N. Assuming\nthat the system has been fully magnetized by means of\na strong external field, one can study how the system\nevolves if the field is removed abruptly. This corresponds\nto take a fully saturated initial condition\nsk(0) = 1,∀k. (13)\nIn ferrimagnetic chains, on the other hand, a “par-\ntial” saturation can be reached, provided the antiferro-\nmagnetic coupling ( JI<0) between nearest neighbors is“strongenough”, in a sense that will be clarified later on.\nIn fact, if the Land´ efactorsfor odd and even sites arenot\nequal (go∝ne}ationslash=ge), through the application of an opportune\nfield the sample can be prepared in a configuration with\n/braceleftBigg\nsk(0) = +1 ,fork= 2r+1 (kodd)\nsk(0) =−1,fork= 2r(keven).(14)\nWith respect to the case considered by Glauber, it is\nconvenient to separate explicitly the expectation values\nof the odd sites, s2r+1(t), from those of the even sites,\ns2r(t). Thus, for H = 0, the set of Nequations of motion\n(9) can be rewritten as\n/braceleftBigg\nd\ndts2r=−α/bracketleftbig\ns2r+1\n2γ(s2r+1+s2r−1)/bracketrightbig\nd\ndts2r+1=−α/bracketleftbig\ns2r+1+1\n2γ(s2r+s2r−2)/bracketrightbig(15)\nIn the following, the solutions of (15) will be found using\ntwo different approaches that yield identical results.\nA. The Generating Function approach\nThe Generating Function approach, which closely fol-\nlows the original Glauber’s paper, is exposed in detail in\nAppendix. Here, in order to distinguish between the fer-\nromagnetic and ferrimagnetic relaxations, we specialize\nthe general solution, Eq. (A5) and Eq. (A6), to the two\ndifferentkindsofinitialconditions, Eq.(13)andEq.(14).\nIn both cases, we will assume that the exchange cou-\nplingJIis negative. The “partially saturated” configu-\nration, Eq. (14), reflects a typical experimental situation,\nin which the antiferromagnetic coupling is much bigger\n(JI≈100÷1000 K) than the Zeeman energy associated\nwith accessible magnetic fields. On the other hand, the\ninitial configurationwith allthe spins alignedin the same\ndirection, Eq. (13), clearly reflects the experimental situ-\nation of a fully saturated sample. This condition is easily\nobtained for ferromagnetic coupling ( JI>0), while it\nmay require very strong fields (eventually unaccessible)\nfor antiferromagnetic coupling ( JI<0).\nLet us startfrom the saturatedconfiguration, Eq.(13).\nSubstituting the initial condition sk(0) = 1 for all kin\nboth (A5) and (A6), we obtain\n\n\ns2r(t) =e−αt+∞/summationtext\nm=−∞/bracketleftbig\nI2(r−m)(γαt)+I2(r−m)−1(γαt)/bracketrightbig\ns2r+1(t) =e−αt+∞/summationtext\nm=−∞/bracketleftbig\nI2(r−m)(γαt)+I2(r−m)+1(γαt)/bracketrightbig\n.\nHence, exploiting the property (A4) of the Bessel func-\ntions (taking y= 1), and redefining the sums by a unique\nindexj, we get\n\n\ns2r(t) =e−αt+∞/summationtext\nj=−∞Ij(γαt) =e−α(1−γ)t\ns2r+1(t) =e−αt+∞/summationtext\nj=−∞Ij(γαt) =e−α(1−γ)t.(16)4\nThis meansthat, startingwith all the spinsalignedin the\nsamedirection,eachspinexpectationvalue(bothoneven\nand odd sites) decays obeying a mono-exponential law\nwith relaxation time τq=0= [α(1−γ)]−1, which is just\nthe characteristic time scale obtained as the inverse of\nthe dispersion relation λqwith zero wave number q= 0,\nseeEq. (11). Noticethat τq=0candivergefor T→0only\nin the case of ferromagnetic coupling, JI>0 (γ >0).\nLet us now consider the partially saturated configu-\nration, Eq. (14), in which sk(0) = 1 for kodd and\nsk(0) =−1 forkeven. Substituting these initial con-\nditions in both (A5) and (A6), we obtain\n\n\ns2r(t) =−e−αt+∞/summationtext\nm=−∞/bracketleftbig\nI2(r−m)(γαt)−I2(r−m)−1(γαt)/bracketrightbig\ns2r+1(t) =e−αt+∞/summationtext\nm=−∞/bracketleftbig\nI2(r−m)(γαt)−I2(r−m)+1(γαt)/bracketrightbig\nand, still exploiting the property (A4) (but now for y=\n−1), we get\n\n\ns2r(t) =−e−αt+∞/summationtext\nj=−∞(−1)jIj(γαt) =−e−α(1+γ)t\ns2r+1(t) =e−αt+∞/summationtext\nj=−∞(−1)jIj(γαt) =e−α(1+γ)t.\n(17)\nAlso in this case all the spins of the system relax with\na mono-exponential law, but now the relaxation time is\nτq=π= [α(1+γ)]−1, which corresponds to the inverse of\nthe eigenvalue λqwith wave number q=π, see Eq. (11).\nNotice that τq=πcan diverge for T→0 only in the case\nof antiferromagnetic coupling, JI<0 (γ <0).\nSummarizing, according to the sign of the exchange\nconstant, both time scales τq=0(forJI>0) andτq=π\n(forJI<0) diverge in the low temperature limit T→0,\nfollowing an Arrhenius law\nτ=1\n2αe4β|JI|(18)\nwith energy barrier 4 |JI|(slow relaxing mode). It is\nworth noting that the remaining relaxation times, given\nby the inverse of the eigenvalues in Eq. (11) with q∝ne}ationslash= 0\nandq∝ne}ationslash=π, always remain of the same order of mag-\nnitude as α−1(fast relaxing modes). This time scale is\ntypically very small ( ∼ps) in real systems, and negligible\nwith respect to the characteristic times involved in any\nexperimental measurement we refer to.\nB. The Fourier Transform approach\nThe solutions, (16) and (17), to the set of equations\n(15) can alternatively be deduced within the Fourier\nTransform (FT) formalism, which has already been ex-\nploited to obtain the dispersion relation (11). Recalling\nthe definition (10) of sk(t) and its spatial FT\n/tildewidesq=1\nN/summationdisplay\nksk(t)e−iqkeλqt, (19)we evaluate /tildewidesqat time t= 0,/tildewidesq=1\nN/summationtext\nksk(0)e−iqk,\nfor the two initial conditions of interest, (13) and (14).\nStarting from the all-spin-up configuration, Eq. (13), we\nhave\n/tildewidesq=1\nN/summationdisplay\nke−iqk=δq,0. (20)\nHence the solution for the expectation value of a spin\nlocalized on the klattice site at time tis\nsk(t) =/summationdisplay\nqδq,0eiqke−λqt=e−λq=0t,(21)\nwhich is identical to (16) since λq=0=α(1−γ).\nStarting from the partially saturated configuration,\nEq. (14), it is useful to rewrite it as sk(0) =−eiπk, so\nthat the FT at t= 0 is\n/tildewidesq=−1\nN/summationdisplay\nkeiπke−iqk=−δq,π.(22)\nHence the solution is readily obtained\nsk(t) =−/summationdisplay\nqδq,πeiqke−λqt=−eiπke−λq=πt,(23)\nwhich is identical to (17) since λq=π=α(1+γ).\nFinally, we observe that Eqs. (21) and (23) hold even\nfor a ring with a finitenumberNof spins, while Eqs.\n(16) and (17) were obtained in the infinite-chain limit.\nC. Slow versus fast relaxation of the spontaneous\nmagnetization\nThe expectation values, sk(t), of spins localized on\nthe even and odd sites of a linear lattice at time t,\ncomputed either with the Generating Functions or the\nFourier Transform approach, have been shown to dis-\nplay a mono-exponential relaxation, see Eqs. (16) and\n(17), with different time scales, τq=0= [α(1−γ)]−1and\nτq=π[α(1 +γ)]−1respectively, depending on the differ-\nent initial conditions, Eqs. (13) and (14). As a conse-\nquence, also the macroscopic magnetization, expressed\nby Eq. (4), displays the same mono-exponential relax-\nation as the single site quantities sk(t).\nA chain in which all the magnetic moments are equal\ncan be prepared only in the saturated initial configu-\nration, Eq. (13), with all the spin aligned in the same\ndirection, through the application of an external field.\nThus, when the field is abruptly removed, such a sys-\ntem will relax slowly at low temperature only if the ex-\nchange coupling is ferromagnetic ( JI>0). In contrast, if\nthe exchange coupling is antiferromagnetic ( JI<0) and\nthe chain is “forced” in the saturated state by a strong\napplied magnetic field, the system will relax very fast\n(in a typical time of the order of α−1) when the field\nis removed. Let us now discuss how these results, first\nobtained by Glauber6, are generalized to the case of a5\nchain in which the magnetic moments are collinear, but\nnotequal on each site.\nAs pointed out in the introduction, a model with an-\ntiferromagnetic coupling ( JI<0) but non-compensated\nmagnetic moments on the two sublattices is more akin\nto real SCM’s1,3,29. Yet it is very interesting since, de-\npending on the intensity of the applied magnetic field,\nthe system can be prepared either in the saturated ini-\ntial configuration, Eq. (13), where all spins are parallel\nto each other, or in the partially saturated one Eq. (14),\nwhere nearest neighbors are antiparallel. In the former\ncase, a very strong field is required in order to overcome\nthe antiferromagnetic coupling between nearest neigh-\nbors; once the field is removed, the relaxation of the\nmagnetizationis expected to be fast at lowtemperatures,\non the basis of the solution (16). In the latter case, the\npartiallysaturatedinitial configuration(14) can easilybe\nobtainedthroughtheapplicationofasmaller,experimen-\ntally accessible magnetic field; when the field is abruptly\nremoved, the relaxation is expected to be slow accord-\ning to the solution (17). The solution (17) justifies the\nobservation of SCM behavior in ferrimagnetic quasi-1D\ncompounds like CoPhOMe29(see Sect. V).\nSummarizing, we have found that when a collinear\nferrimagnetic chain is prepared in an initial state –\nfully or partially saturated depending on the intensity\nof the applied magnetic field – once the field is re-\nmoved abruptly, the spin system can show fast or slow\nrelaxation, respectively. Fast relaxation corresponds to\nstrongerfields; unfortunatelyforthe quasi-1Dchaincom-\npound CoPhOMe29,30, the antiferromagnetic exchange\nconstant is so large ( |JI| ∼100K) that the realization\nof a fully saturated initial configuration would require a\nvery high, almost unaccessible field ( ∼1000 kOe). Thus\nthis compound is not a good candidate for such a kind\nof experiments31.\nIV. MAGNETIC RESPONSE TO AN\nOSCILLATING MAGNETIC FIELD\nIn the presence of a magnetic field H, the transition\nprobability to be put in the equation of motion (5) is\nwH\nσk→−σk, defined in Eq. (7). One obtains\ndsk(t)\ndt=−α/braceleftBig\nsk(t)−γ\n2[sk+1(t)+sk−1(t)]\n+γδk\n2[∝an}bracketle{tσkσk+1∝an}bracketri}htt+∝an}bracketle{tσk−1σk∝an}bracketri}htt]−δk/bracerightBig\n(24)\nthat differs from Eq. (15), considered earlier for H= 0, in\nthe presence of both a non-homogeneous term, δk, and\nthe time-dependent pair-correlation functions ∝an}bracketle{tσkσk±1∝an}bracketri}htt.\nThe latter ones, assuming that the field is so weak to\ninduce just small departures from equilibrium, can be\napproximated by their time-independent counterparts32\n∝an}bracketle{tσkσk+1∝an}bracketri}htt=∝an}bracketle{tσk−1σk∝an}bracketri}htt≈tanh(βJI)≡η.(25)As it is usual in a.c.susceptibility measurements,\nwe also assume the time-dependent magnetic field\nH(t)=He−iωtˆeH, oscillating at frequency ω, to be weak\nso that the δkparameters can be linearized\nδk= tanh(βµBGkH(t))≈βµBGkH(t).(26)\nThe system of equations of motion (24) then takes the\nform\ndsk(t)\nd(αt)=−sk(t)+γ\n2[sk+1(t)+sk−1(t)]\n+βf(βJI)µBGkH(t), (27)\nwhere\nf(βJI) = 1−γη=1−η2\n1+η2(28)\nis a function of the reduced coupling constant βJIand\nwe have taken into account that γ= 2η/(1+η2). After\na brief transient period, the system will reach the sta-\ntionary condition in which the magnetic moment of each\nspin oscillates coherently with the forcing term at the\nfrequency ω. Expressing the expectation value of a spin\non thek-th lattice site, sk(t), through its spatial FT, /tildewidesq,\nthe trial solution is\nsk(t) =/summationdisplay\nq/tildewidesqeiqke−iωt. (29)\nSubstituting the latter in the system (27) we get\n/tildewidesq=βf(βJI)µBHα/tildewideGq\nα(1−γcosq)−iω,(30)\nwhere/tildewideGqis the FT of Gk:\n/tildewideGq=1\nNN/summationdisplay\nk=1e−iqkGk. (31)\nThe average of stochastic magnetization can readily be\nobtained from (4) as\n∝an}bracketle{tM∝an}bracketri}htt=µBe−iωtN/summationdisplay\nk=1/summationdisplay\nqq′/tildewideGq/tildewidesq′eiqkeiq′k,(32)\nwhich accounts for non-collinearity of local anisotropy\naxes with respect to the field direction. Performing the\nsum over all the lattice sites ( kindices) yields a factor\nNδq,−q′in Eq. (32); substituting the expression (30) for\n/tildewidesq, one obtains\n∝an}bracketle{tM∝an}bracketri}htt=Nµ2\nBβf(βJI)He−iωt/summationdisplay\nqq′α/tildewideGq/tildewideGq′δq,−q′\nα(1−γcosq)−iω.\n(33)\nThen, considering that /tildewideGq/tildewideG−q=|/tildewideGq|2, thea.c.suscep-\ntibility is finally obtained dividing (33) by H e−iωt\nχ(ω,T) =Nµ2\nBβf(βJI)/summationdisplay\nqα|/tildewideGq|2\nα(1−γcosq)−iω.(34)6\nFIG. 1: Temperature dependence of the imaginary part of the c omplex susceptibility, Eq. (38), of a collinear one-dimens ional\nIsing model with alternating spins. Resonant behavior in re sponse to an oscillating magnetic field is possible, at low fr equency,\nonly when magnetic moments are uncompensated (a,c,d), whil e a broad peak is found when the net magnetization is zero (b).\n(The curves refer to reduced frequency ω/α= 0.001)\nIn principle, the a.c.susceptibility of a chain with N\nspinsadmits Npoles, correspondingtothe Neigenvalues\nλqin Eq. (11). Each mode is related to a different time\nscaleτq= 1/λq. Inpractice, notallthetimescaleswillbe\ninvolved in the complex susceptibility χ(ω,T), butonly\nthe ones selected by /tildewideGq. A result similar, at first glance,\nto Eq. (34) was deduced by Suzuki and Kubo27, but in\ntheir case the relationship was between the time scale τq\nand the wave-vector-dependent susceptibility χ(q,ω). In\ncontrast, in an a.c.susceptibility experiment only the\nzero-wave-vector susceptibility χ(q= 0,ω) is accessible;\nthe peculiarity of Eq. (34) is that other time scales, dif-\nferent from τq=0, can be selected thanks to the depen-\ndence of the gyromagnetic factors Gkand of the local\nanisotropy axes on the site position k. This is the main\nresult of our study and will be clarified hereafter through\na few examples.\nA. The a.c.susceptibility of a collinear Ising\nferrimagnetic chain\nLet us start considering the case of a one-dimensional\nIsing model with two kinds of spins (aligned parallel on\nantiparallel to the chain axis) alternating on the odd\nand even magnetic sites of the lattice with Land´ e fac-torsG2r+1=goandG2r=ge(integerr), respectively.\nStrictly speaking, a collinear Ising ferrimagnet is char-\nacterized by an antiferromagnetic coupling JI<0, but\nalso the case JI>0 can be treated through Eq. (34).\nIn fact, since the local axis of anisotropy has the same\ndirection for all the spins, the FT of the site-dependent\nLand´ e factor is\n/tildewideGq=1\nNN/2/summationdisplay\nr=1[gee−iq2r+goe−iq(2r−1)]\n= (ge+eiqgo)1\nNN/2/summationdisplay\nr=1e−iq2r. (35)\nTaking into account that, in the presence of periodic\nboundary conditions, one has\nN/2/summationdisplay\nr=1e−iq2r=N\n2(δq,0+δq,π), (36)\nitfollowsthatthe onlynon-zerovaluesof /tildewideGqareforq= 0\nandq=π\n/tildewideG/bardbl\nq=1\n2[(ge+go)δq,0+(ge−go)δq,π].(37)\nThus, according to Eq. (34), the parallel a.c.suscepti-\nbility (∝bardbl=zz) of a collinear Ising chain with alternating7\nspins is\nχ/bardbl(ω,T) =Nµ2\nBβ f(βJI)\n×1\n4/bracketleftbigg(ge+go)2\n(1−γ)−i(ω\nα)+(ge−go)2\n(1+γ)−i(ω\nα)/bracketrightbigg\n(38)\nIt appears that both the relaxation times obtained by\nSuzuki and Kubo27for the ordinary and the staggered\nsusceptibility of the usual Ising model, namely τq=0=\n[α(1−γ)]−1andτq=π= [α(1 +γ)]−1respectively, do\ncoexist in the a.c.susceptibility (38). Notice that, in\ntheω→0 limit, the static susceptibility of the Ising\nferrimagnet in zero field17is recovered from Eq. (38),\nsince one hasf(βJI)\n1∓γ=1−η2\n1+η21\n1∓γ=e±2βJI.\nAs regards the dynamic response of the system to an\noscillating magnetic field applied along the chain axis,\ndepending on the sign of the effective exchange coupling\nconstant JI, the ferromagnetic ( ge+go) or the antifer-\nromagnetic ( ge−go) branch of the parallel susceptibility\n(38) are characterized by a diverging time scale at low\ntemperature. In particular, for a collinear Ising ferri-\nmagnet one has JI<0, so that τq=πis diverging, while\nτq=0is short (of the order of α−1, the attempt frequency\nof an isolated spin). Thus, for JI<0, a resonant be-\nhavior of the a.c.susceptibility versus temperature (at\nlow frequencies ω/α≪1) can only be observed in the\ncasege∝ne}ationslash=go(see Fig. 1d) when magnetic moments are\nuncompensated , while a broad peak is found in the case\nge=gowhen the net magnetization is zero (see Fig. 1b).\nClearly, for JI>0, a resonant peak is found in both\ncases (see Fig. 1a and 1c), because a net magnetization\nis always present in the system.\nSuch a resonant behavior of the a.c.susceptibility ver-\nsusT, in ferromagnetic33as well as in ferrimagnetic17\nIsing chains with single spin-flip Glauber dynamics, is a\nmanifestation of the stochastic resonancephenomenon34:\ni.e., the response of a set of coupled bistable systems to a\nperiodic drive is enhanced in the presence of a stochastic\nnoise when a matching occurs between the fluctuation-\ninduced switching rate of the system and the forcing fre-\nquency. In a magnetic chain, the role of stochastic noise\nis played by thermal fluctuations and the resonant peak\nin the temperature-dependence of the a.c.susceptibil-\nity occurs when the statistical time scale, associated to\nthe slow decay of the magnetization, matches with the\ndeterministic time scale of the applied magnetic field\nτq(Tpeak)≈1\nω. (39)\nB. The a.c.susceptibility of an n-fold helix\nNext, as an example of a non-collinear spin arrange-\nment, we consider a system of spins with the local axes\nof anisotropy arranged on an n-fold helix (see Fig. 2); θ\nis the angle that the local axes form with z, the unique\naxis of the helix ( i.e., the chain axis). In this case the\nLand´ e factors are equal on all lattice sites, but differentFIG. 2: Thick arrows denote the projections on the xyplane,\nperpendicular to the chain (helix) axis z, of magnetic mo-\nments in a one-dimensional Ising helimagnet, for different f old\nsymmetries ( n= 2,3,4,6). Dashed lines are the projections\nof the local axes of anisotropy, ˆzk.\nspins experience different fields because of the geometri-\ncal arrangement of magnetic moments. In the following\nwe will make the approximation that the Land´ e tensor\nof a spin on the k-th lattice site has just a non-zero com-\nponentgalong the easy anisotropy direction ˆzk, so that\nGk=gˆzk·ˆeH(see Eq. (3)). In the crystallographicframe\n(x,y,z), the directors ˆzkread (integer k)\nˆzk= sinθ/bracketleftbigg\ncos/parenleftbigg2πk\nn/parenrightbigg\nˆex+sin/parenleftbigg2πk\nn/parenrightbigg\nˆey/bracketrightbigg\n+cosθˆez.\n(40)\nLet us first consider the case of an oscillating magnetic\nfield H applied parallel to z, the helix axis. All the spins\nactually undergo the same field, and since Gk=gcosθ\nindependently of the lattice site k, the only peak in the\nFT/tildewideGqoccurs at q= 0\n/tildewideGz\nq=gcosθ δq,0(∀n). (41)\nFollowing the same procedure as in the previous para-\ngraph, the parallel a.c.susceptibility ( ∝bardbl=zz) takes an\nexpression (valid for any value of the fold index nof the\nhelix)\nχ/bardbl(ω,T) =Nµ2\nBβ f(βJI)g2cos2θ\n(1−γ)−i(ω\nα)(∀n) (42)\nthat differs from Glauber’s result for the collinear Ising\nchain6only by the geometrical factor cos2θ. For fer-\nromagnetic coupling, JI>0, the relaxation time τ0=\n[α(1−γ)]−1diverges as T→0, and a resonant behavior\nof thea.c.parallel susceptibility versus temperature is\nfound, at low frequency, when the oscillating field is ap-\nplied parallel to the helix axis, z, along which spins are\nuncompensated : see Fig. 3a, which refers to the case of a\ntwo-fold helix ( n= 2) .8\nFIG. 3: (color online) Temperature dependence of the imagin ary part of the parallel (42) and perpendicular (45) complex\nsusceptibility of an Ising chain with two-fold helical spin arrangement. The local axes ˆz1andˆz2were assumed to form an angle\nθ=π\n3withz, the chain axis (unique axis of the helix). Different curves r efer to different values of ω/α: 0.0001 (continuous, red\nline); 0.0002 (dashed, green line); 0.0005 (dashed single- dotted, blue line); 0.0010 (dashed double-dotted, violet l ine). Resonant\nbehavior in response to an oscillating magnetic field is poss ible, at low frequency, only for field applied in a direction w here\nmagnetic moments are uncompensated (a,c), while a broad pea k is found (b,d) when there is no net magnetization along the\nfield direction.\nLet us now consider the case of an oscillating magnetic\nfield H applied perpendicularly to the chain axis. In this\nconfiguration, it is useful to distinguish the case n= 2\nfrom the general case n >2.\n•n= 2\nIn this case, it is worth noticing that for H parallel\ntoy, one has identically Gr≡0 for any lattice\nsiter. Thus, /tildewideGy\nq≡0 and the corresponding a.c.\nsusceptibility is identically zero\nχyy(ω,T)≡0 (43)\n(not shown). In contrast, for H parallel to x, one\nhasGr=−gsinθon odd sites and Gr= +gsinθ\non even sites. The FT is\n/tildewideGx\nq=1\nNN/2/summationdisplay\nr=1gsinθ(−e−iq(2r−1)+e−iq2r)\n=gsinθ1\n2(δq,0+δq,π)(1−eiq)\n=gsinθ δq,π (44)\nwhere we have taken into account Eq. (36). Thus,\nfor ferromagnetic coupling, JI>0, the relaxationtimeτπ= [α(1+γ)]−1does not diverge as T→0,\nand the perpendicular a.c.susceptibility\nχxx(ω,T) =Nµ2\nBβf(βJI)g2sin2θ\n(1+γ)−i(ω\nα)(45)\ndoes not present a resonant behavior as a function\nof temperature; rather, it presents a broad max-\nimum (see Fig. 3b). Clearly, in the case of an-\ntiferromagnetic coupling, JI<0, the behavior of\nthe susceptibility components is reversed: a broad\nmaximum is found for the temperature dependence\noftheparallelsusceptibility χzz(ω,T)(seeFig. 3d),\nwhile a resonant behavior is found for the perpen-\ndicular susceptibility χxx(ω,T) (see Fig. 3c).\n•n >2\nIn this case, denoting by ˆex·ˆeHandˆey·ˆeHthe\ndirectors of the in-plane field, the FT’s of Gkare\ngiven by\n\n\n/tildewideGx\nq=1\nN(ˆex·ˆeH)gsinθ/summationtextN\nk=1cos/parenleftbig2πk\nn/parenrightbig\ne−iqk\n= (ˆex·ˆeH)gsinθ1\n2/parenleftBig\nδq,2π\nn+δq,−2π\nn/parenrightBig\n/tildewideGy\nq=1\nN(ˆey·ˆeH)gsinθ/summationtextN\nk=1sin/parenleftbig2πk\nn/parenrightbig\ne−iqk\n= (ˆey·ˆeH)gsinθ1\n2i/parenleftBig\nδq,2π\nn−δq,−2π\nn/parenrightBig.(46)9\nRemarkably, as just |/tildewideGq|2appears in Eq. (34), the\ngeneral result for the in-plane susceptibility turns\nout to be independent of the field direction. Thus,\nforn >2, the perpendicular ( ⊥)a.c.susceptibility\nof then-fold helix is given by\nχ⊥(ω,T) =Nµ2\nBβ f(βJI)\n×1\n2sin2θg2\n[1−γcos/parenleftbig2π\nn/parenrightbig\n]−i(ω\nα),(47)\nwhere we have exploited the fact that cos/parenleftbig\n−2π\nn/parenrightbig\n=\ncos/parenleftbig2π\nn/parenrightbig\nfor the term appearingat the denominator\nof Eq. (34).\nSummarizing, in the general case of an Ising chain\nwith an n-fold helical spin arrangement ( n≥2), we\nhave explicitly shown that a resonant behavior of the\na.c.susceptibility versus temperature, similar to the one\ndisplayed by ferromagnetic6,33and ferrimagnetic17Ising\nchains with collinear spins, is possible only for field ap-\nplied in a direction where magnetic moments are uncom-\npensated. In contrast, a broad peak is found when there\nis no net magnetization along the field direction.\nV. APPLICATION TO REAL SINGLE CHAIN\nMAGNETS\nIn this Section we will apply the developed formalism\nto some real compounds as representative realizations of\nSCM’s; for the three selected systems – we know this\nrestriction is far from being exhaustive1,3–a.c.suscep-\ntibility data on single crystal are available, which is a\nfundamentalrequirementforcheckingtheproposedselec-\ntion rules. The considered systems29,35,36are character-\nized by the alternation of two types of magnetic centers\nalong the chain axis, so that at least two spins per cell\nhave to be considered; moreover, the magnetic moments\nare not collinear, the dominant exchange interactions are\nantiferromagnetic and a strong single-ion anisotropy is\npresent, which favors magnetization alignment along cer-\ntain crystallographic directions ˆzk. The static properties\nof these compounds, like magnetization and static sus-\nceptibility, are generally well described using a classical\nHeisenberg model with an isotropic exchange coupling J\nand a single-ion anisotropy D. Thus, in order to describe\nthe dynamic behavior in response to a weak, oscillating\nmagnetic field by means of the previously developed the-\nory, it is necessary to relate the Hamiltonian parame-\nters of such a classical spin model to the exchange con-\nstantJIof the effective Ising model (2). In the following\nwe will show, through a few examples on real systems,\nthat indeed, depending on the geometry, selection rules\nare obeyed for the occurrence of slow relaxation of the\nmagnetization at low temperatures ( β|JI| ≫1), as well\nas for resonant behavior of the a.c.susceptibility as a\nfunction of temperature at low frequencies. As regards\nthe frequencies involved in an a.c.susceptibility exper-\niment on real SCM’s, generally1,3they lie in the range10−1÷104Hz, while the attempt frequency αis of the\norder of 1010÷1013Hz. Thus, for a typical experiment,\na resonant peak in the a.c.susceptibility can safely be\nobserved provided that at least one of the characteristic\ntime scales τqinvolved in (34) diverges at low T, in order\nfor the condition (39) to be satisfied.\nA. The MnIII-based Single Chain Magnet\nIn the one-dimensional molecular magnetic compound\nof formula [Mn(TPP)O 2PPhH]·H2O, obtained by re-\nacting Mn(III) acetate mesotetraphenylporphyrin with\nphenylphosphinic acid35, hereafter denoted by MnIII-\nbased SCM, the phenylphosphinate anion transmits a\nsizeable antiferromagnetic exchange interaction that,\ncombined with the easy axis magnetic anisotropy of the\nMnIIIsites, gives rise to a canted antiferromagnetic ar-\nrangement of the spins. The static single-crystal mag-\nnetic propertieswereanalyzedin the frameworkofaclas-\nsical spins Hamiltonian\nH=−N/2/summationdisplay\nr=1{JS2r−1·S2r+D/bracketleftbig\n(Sz1\n2r−1)2+(Sz2\n2r)2/bracketrightbig\n+e−iωtµBHαgαβ[Sβ\n2r−1+Sβ\n2r]} (48)\nwhereJ <0 is the antiferromagnetic nearest neighbor\nexchange interaction between S= 2 spins. D >0 is\nthe uniaxial anisotropy favoring two different local axes,\nalternating along odd and even sites respectively; both\naxes form an angle θ= 21.01owith the crystallographic\ncaxis, while they form opposite angles of modulus φ=\n56.55owith the aaxis (see Fig. 4). Thus we can write\nˆz2r−1= sinθcosφˆex−sinθsinφˆey+ cosθˆezandˆz2r=\nsinθcosφˆex+sinθsinφˆey+cosθˆez.\nA best fit of the static single-crystal magnetic suscep-\ntibilities, calculated via a Monte Carlo simulation35pro-\nvidesJ=−1.34 K and D= 4.7 K; the gyromagnetic\ntensorGαβis diagonal and isotropic with g/bardbl= 1.97.\nEquivalent results can be obtained calculating the static\npropertiesofmodel(48)viaatransfermatrixapproach37.\nSince the uniaxial anisotropy Dis rather strong with re-\nspect to the exchange coupling |J|, as a first approxima-\ntion one can assume the two sublattice magnetizations\nto be directed just along the two easy axes, ˆz2r−1and\nˆz2r, so that the chain system (48) can be described by a\nnon-collinear Ising model formally identical to Eq. (2),\nwith an effective38Ising exchange coupling JIand a gen-\neralized Land´ e factor Gkdefined as, respectively\nJI=JS(S+1)cos( ˆz2r−1·ˆz2r)\nGr=g/bardbl\nr/radicalbig\nS(S+1) (ˆzr·ˆeH). (49)\nDepending on the orientation of the oscillating mag-\nnetic field with respect to the crystallographic axes, the\nFT of the generalized Land´ e factor takes the following\nforms\n/tildewideGq=g/bardbl/radicalbig\nS(S+1)10\nFIG. 4: (color online) Disposition of local axes ( ˆz2r−1and\nˆz2r) and magnetic moments (red arrows) in the MnIII-based\nreal SCM, discussed in Sect. V.A, with antiferromagnetic\neffective Ising exchange coupling JI<0 . Right: Schematic\nview of the chain structure ( zis the chain axis) along the\ncrystallographic xaxis. Left: projections oflocal axes(dashed\nlines) and of magnetic moments (red arrows) in the xyplane,\nperpendicular to the chain axis.\n×1\nNN/2/summationdisplay\nr=1e−iq2r[eiq(ˆz2r−1·ˆeH)+(ˆz2r·ˆeH)]\n=g/bardbl/radicalbig\nS(S+1)\n\nsinθ1cosφ1δq,0,H∝bardblx\nsinθ1sinφ1δq,π,H∝bardbly\ncosθ1δq,0,H∝bardblz(chain axis)(50)\nThe corresponding a.c.susceptibility takes the expres-\nsion\nχ(ω,T) =Nµ2\nBβf(βJI)(g/bardbl)2[S(S+1)]\n×\n\nsin2θcos2φ1\n(1−γI)−i(ω\nα),H∝bardblx\nsin2θsin2φ1\n(1+γI)−i(ω\nα),H∝bardbly\ncos2θ1\n(1−γI)−i(ω\nα),H∝bardblz(chain axis)(51)\nTakingintoaccountthat, fortheMnIIISCMunderstudy,\nthe “true” exchange coupling, Jin Eq. (48), is antifer-\nromagnetic, and that the angle between the two easy\nanisotropy axes ˆz1andˆz2isδ= 34.6o<90o(see Fig.\n4, right), from Eq. (49) it follows that also the effective\nIsing exchange coupling is antiferromagnetic, JI<0. As\na consequence, in the low temperature limit β|JI| → ∞,\nthe relaxation time τq=πdiverges, while τq=0does not.\nThus, for low frequencies ω/α≪1, thea.c.susceptibil-\nity presents a resonant behavior only when the oscillat-\ning magnetic field is applied along the crystallographic yaxis,i.e.the direction, perpendicular to the chain axis,\nalong which the magnetizations of the two sublattices\nareuncompensated (see Fig. 4). In contrast, when H\nis applied parallel to z(the chain axis) or to x, namely\ntwo directions along which the magnetizations of the two\nsublattices are exactly compensated, no resonant behav-\nior is expected. These theoretical predictions turn out\nto be in excellent agreement with experimental a.c.sus-\nceptibility data35obtained in a single crystal sample of\n[Mn(TPP)O 2PPhH]·H2O, thus confirming that such a\nMnIII-based canted antiferromagnet is a bona fide SCM.\nB. The DyIII-based Single Chain Magnet\nThe molecular magnetic compound of formula\n[Dy(hfac) 3(NITPhOPh)], hereafter denoted by DyIII-\nbased SCM, belongs to a family of quasi one-dimensional\nmagnets in which rare earth ions (with spin S) and or-\nganic radical ions (with spin s= 1/2) alternate them-\nselvesalongthe chainaxis, z, which in this compound co-\nincides with the crystallographic baxis. Static measure-\nments in single crystal samples suggest36that there is an\nantiferromagnetic exchange interaction between neigh-\nboring DyIIIions, whose easy anisotropy axes are canted\nwithrespecttothechainaxisinsuchawaytogeneratean\nuncompensated moment along b, while the components\nin theacplane are compensated. Thus, as far as the\ndominant exchange interaction J <0 between DyIIIions\nis taken into account, the spin Hamiltonian of the sys-\ntem is quite similar to Eq.(48). However, with respect to\nthe MnIII-based chain, the crystal structure of the DyIII-\nbased SCM is more complicated, not only owing to the\npresence of two kinds of magnetic centers (the DyIIIions\nand the organic radical ions), but mainly because the\nsystem is formed by two different families of chains, with\ntwo almost orthogonal projections of the easy axes in\ntheacplane, perpendicular to the chain axis: this “ac-\ncidental” (in the sense that it is not imposed by symme-\ntry) orthogonality is the reason for the nearly isotropic\nmagnetic behavior displayed by the system within such\na plane36.\nWe adoptasimplified model formallyequivalentto Eq.\n(48). Taking into account only the dominant antiferro-\nmagnetic exchange interaction ( J <0) between neigh-\nboring DyIIIions (which indeed are next nearest neigh-\nbors in the real system) and their uniaxial anisotropy\n(D >0), the system can approximately be described by\nthe classical spins Hamiltonian (48), where now |Sk|= 1.\nBy means of a classical Transfer Matrix calculation, the\nstatic properties of the DyIII-based SCM turn out to be\nsatisfactorily fitted36byJ=−6 K,D= 40 K, g/bardbl= 10,\nwith the two easy anisotropy axes ˆz2r−1,ˆz2rforming\nequal angles θ≈75owith the chain axis z. (Notice that\nthe latter propertyholds true forboth families ofchains.)\nAlso in the case of the DyIII-based SCM, the uniaxial\nanisotropy Dturns out to be sufficiently strong with re-\nspect to the exchange coupling |J|in order to assume,11\nFIG. 5: (color online) Disposition of odd and even local axes\n(ˆz2r−1andˆz2r) and magnetic moments (thick arrows) in the\nDyIII-based real SCM, discussed in Sect. V.B, with ferro-\nmagnetic effective Ising exchange coupling JI>0 . Top:\nSchematic view of the chain structure ( zis the chain axis),\ndisplaying the two families of chains (A, with red magnetic\nmoments, and B, with green magnetic moments). Bottom:\nprojections of magnetic moments in the xyplane, perpendic-\nular to the chain axis.\nas a first approximation38, the two sublattice magneti-\nzations of DyIIIto be directed just along the two easy\naxes. Thus one can define an equivalent non-collinear\nIsing model (2), where the effective Ising exchange cou-\nplingJIand the generalized Land´ e factor Grare now\ndefined as\nJI=Jcos(ˆz2r−1·ˆz2r)\nGr=g/bardbl\nr(ˆzr·ˆeH). (52)\nDepending on the orientation of the oscillating magnetic\nfield with respect to the crystallographic axes, the FT of\nthe generalized Land´ e factor takes the form\n/tildewideGq=g/bardbl1\nNN/2/summationdisplay\nr=1e−iq2r[eiq(ˆz2r−1·ˆeH)+(ˆz2r·ˆeH)]\n∝g/bardbl/braceleftBigg\ncosθ δq,0,H∝bardblz(chain axis)\nsinθ δq,π,H⊥z.(53)\nIt is important to notice that this result holds true for\nbothfamilies(A,B)ofchains. Next, weobservethatsince\nin the DyIII-basedSCM, the spinsonopposite sublattices\narecoplanarwith the chain axis, the angle between ˆz2r−1\nandˆz2risjust2θ≈150o>90o. Takingintoaccountthat\nthe “true”exchangeconstantinEq. (48) isantiferromag-\nnetic,J <0, from Eq. (52) it follows that the effective\nIsingexchangecouplingisnowferromagnetic, JI>0(see\nFig. 5, top). As a consequence, in the low temperaturelimitβJI→ ∞, the relaxation time τq=0diverges, while\nτq=πdoes not. Thus, the a.c.susceptibility\nχ(ω,T)∝Nµ2\nBβf(βJI)(g/bardbl)2\n×/braceleftBigg\ncos2θ1\n(1−γI)−i(ω\nα),H∝bardblz(chain axis)\nsin2θ1\n(1+γI)−i(ω\nα),H⊥z(54)\nis expected to have a resonant behavior, for low frequen-\nciesω/α≪1, only when the oscillating magnetic field is\nappliedparalleltothechainaxis, z,alongwhichthemag-\nnetizations of the two sublattices are uncompensated (see\nFig. 5, top). Such a theoretical prediction turns out to\nbe in excellent agreementwith the experimental a.c.sus-\nceptibility data36obtained in a single crystal sample of\n[Dy(hfac) 3(NITPhOPh)] ∞, thus confirming that also the\nDyIII-based canted antiferromagnet is a bona fide SCM.\nTheonlyqualitativedifference, withrespecttotheMnIII-\nbased chain is that, due to the different geometry of the\nspin arrangement and of the local anisotropy axes with\nrespecttothechainaxis,theresonantbehaviorofthe a.c.\nsusceptibility is now observed for field applied parallel to\nthe chain axis, rather than perpendicular to it.\nC. The CoPhOMe (CoII-based) Single Chain\nMagnet\nIn the molecular magnetic compound of for-\nmula [Co(hfac) 2NITPhOMe], hereafter denoted by\nCoPhOMe29,30, the magnetic contribution is given by\nCobalt ions, with an Ising character and effective S=\n1/2, and by NITPhOMe organic radical ions, magnet-\nically isotropic and with s= 1/2. The spins are ar-\nranged on a helical structure, schematically depicted in\nFig. 6, right, whose projections in a plane perpendic-\nular to the helix axis z(coincident with the crystallo-\ngraphiccaxis), are represented in Fig. 6, left. The prim-\nitive magnetic cell is made up of three Cobalts (black\narrows) and three organic radicals (red arrows). Al-\nthough the effective spins of the two types of magnetic\ncenters have the same value, the gyromagnetic factors\nare different: gCo∝ne}ationslash=gR; thus, since the nearest neighbor\n(Cobalt-radical) exchange interaction is negative (and\nstrong,|J| ≈100 K)30, the sublattice magnetizations are\nnot compensated along z, whereas they are compensated\nwithin the xyplane perpendicular to the chain axis z.\nFor this compound, which was the first to display SCM\nbehavior29,30, static measurements on single-crystalsam-\nples has not been interpreted in terms of a simple model\nyet, due to the complexity of the system itself. Thus,\na relationship such as (49) and (52), which associate the\nIsing Hamiltonian (2) parameters( JIandGk) with those\nofa more realistic Hamiltonian, is still missing. However,\nthe dynamic behavior has been thoroughly investigated\ntreating - for the sake of simplicity - both the CoIIand\nthe organic radical spins as Ising variables, with σ=±1.12\nFIG. 6: (color online) Disposition of even and odd local axes\n(dashed lines) and magnetic moments (thick arrows) in the\nCoPhOMe real SCM, discussed in Sect. V.C, with antiferro-\nmagnetic effective Ising exchange coupling JI<0 . Right:\nSchematic view of the chain structure ( zis the chain axis)\nalong the crystallographic yaxis. Left: projections of local\naxes (dashed lines) and of magnetic moments (thick arrows)\nin thexyplane, perpendicular to the chain axis.\nThe effective Ising Hamiltonian reads\nH=−N/6/summationdisplay\nl=13/summationdisplay\nm=1{JIσl,2m[σl,2m−1+σl,2m+1]+e−iωtµBH\n[gRσl,2m−1(ˆz2m−1·ˆeH)+gCoσl,2m(ˆz2m·ˆeH)]}(55)\nwithlmagneticcellindexand msitelabelwithboundary\nconditions σl,7=σl+1,1. Since all the local axes ˆzk(k=\n1,···,6) form the same angle θ≈55owith the zaxis,\nwhen a magnetic field is applied along z, the FT of the\ngeneralized Land´ e factor is simply given by\n/tildewideG/bardbl\nq= cosθ1\nNN/2/summationdisplay\nr=1(gCoe−iq(2r−1)+gRe−iq2r)\n=cosθ\n2[(gCo+gR)δq,0+(gCo−gR)δq,π] (56)\nwhich, except for the prefactor cos θ, is quite similar to\nEq. (37) for the collinear Ising chain with alternating\nspins. Thus, the parallel a.c.susceptibility is\nχ/bardbl(ω,T) =Nµ2\nBβ f(βJI)cos2θ\n4\n×[(gCo+gR)2\n(1−γI)−i(ω\nα)+(gCo−gR)2\n(1+γI)−i(ω\nα)].(57)\nTaking into account that the effective exchange coupling\nof CoPhOMe is negative ( JI<0), the antiferromagnetic\nbranch of the parallel susceptibility is characterized by\na diverging time scale τq=π= [α(1+γI)]−1at low tem-\nperature, so that, for low frequencies ω/α≪1,χ/bardbl(ω,T)\ndisplays a resonant behavior.Let us now consider the case of a field applied in the\nplane perpendicular to the chain axis. For H ∝bardblx(see Fig.\n6, left) one has, letting k0=π\n3\nGx\n2r−1=gRsinθcos[k0(2r−1)]\nGx\n2r=gCosinθcos[k02r] (58)\nso that the FT takes the form\n/tildewideGx\nq= sinθ1\nNN/2/summationdisplay\nr=1e−iq2r/parenleftBig\ngCocos(k02r)\n+gReiq[cos(k0)cos(k02r)+sin(k0)sin(k02r)]/parenrightBig\n=1\n4sinθ[(gCo+gRei(q−k0))(δq,k0+δq,π+k0)\n+ (gCo+gRei(q+k0))(δq,−k0+δq,π−k0)] (59)\nwhere, as usual, we have exploited Eq. (36). Thus it\nfollows that\n/tildewideGq=±π\n3=sinθ\n4(gCo+gR)\n/tildewideGq=π±π\n3=sinθ\n4(gCo−gR).(60)\nThe corresponding relaxation times are τq=±π\n3=α\n1−1\n2γ\nandτq=±2π\n3=α\n1+1\n2γso that, summing the four contribu-\ntions we obtain the perpendicular a.c.susceptibility\nχ⊥(ω,T) =Nµ2\nBβ f(βJI)sin2θ\n8\n×/bracketleftbigg(gCo+gR)2\n(1−1\n2γ)−i(ω\nα)+(gCo−gR)2\n(1+1\n2γ)−i(ω\nα)/bracketrightbigg\n.(61)\nIn conclusion, for the six-fold helix model with alter-\nnating spins and Ising exchange coupling in Eq. (55),\nthe parallel and perpendicular components of the a.c.\nsusceptibility, χ/bardbl(ω,T) andχ⊥(ω,T), display a behavior\nsimilar to that of a ferrimagnetic chain with alternating\nspins (see (38)) and of an n-fold helical spin arrangement\nwith equivalent spins (see (47)), respectively. In spite of\nthe approximations involved in model (55) to describe\nthe real CoPhOMe molecular magnetic chain, the two\ncalculated susceptibilities (57) and (61), qualitatively re-\nproduce the dynamic behavior of this compound29,30. In\nfact, no out-of-phase a.c.susceptibility (imaginary part)\nis observed when the field is applied in the plane per-\npendicular to the chain axis, z, for the experimental fre-\nquencies (1 ÷105Hz)30. In contrast, when the oscillating\nfield is parallel to z, a resonant behavior is observed as\na function of temperature. Even though our theoretical\ntreatment holds only for small deviations from equilib-\nrium, it is worth mentioning that the absence of slow\nrelaxation for fields applied in the perpendicular plane is\nevidenced in the low temperature magnetization curve as\nwell: at low enough temperatures, a finite-area hysteresis\nloop is present only when a static field is applied parallel\nto the chain axis, while no hysteresis is observed in the\nin-plane magnetization curve29,30.13\nVI. CONCLUSIONS\nIn conclusion, in the framework of a one-dimensional\nIsing model with single spin-flip Glauber dynamics,\ntaking into account reciprocal non-collinearity of local\nanisotropy axes and the crystallographic (laboratory)\nframe, we have investigated: (i) the dynamics of mag-\nnetization reversal in zero field, and (ii) the response of\nthe system to a weak magnetic field, oscillating in time.\nWe have shown that SCM behavior is not only a feature\nof collinear ferro- and ferrimagnetic, but also of canted\nantiferromagnetic chains. In particular, we have found\nthat resonant behavior of the a.c.susceptibility versus\ntemperature in response to an oscillating magnetic field\nis possible, at low frequency, only for fields applied in a\ndirection where magnetic moments are uncompensated.\nIn contrast, a broad peak is expected when there is no\nnet magnetization along the field direction.\nThe role played by geometry in selecting the time\nscales involved in a process is an important and pecu-\nliarresult, typicalofmagneto-molecularapproachtolow-\ndimensional magnetism. In fact, magnetic centers with\nuniaxial anisotropyusually correspondto building blocks\nwith low symmetry39,40, which – in turn – often crystal-\nlize in more symmetric space groups, realizing a recip-\nrocal non-collinearity between local anisotropy axes as a\nnatural consequence1,2. Thus the family of real SCM’s,\nto which our model applies, does not restrict to ad-hoc\nsynthesized compounds but, instead, is expected to grow\nlarger in the future3. As a validity check of our selec-\ntion rules (as well as a tutorial exemplification), we have\nshown how our theory applies successfully to three differ-\nent molecular-based spin chains; when possible, we have\nput the parameters of our model Hamiltonian (2) in re-\nlationship with those of more general models, typically\nused tofit the staticpropertiesofthe correspondingcom-\npounds. Needless to say that the possibility of schema-\ntizing the chosen three compounds with Hamiltonian (2)\nrelies on the fact that at low enough temperatures they\nbehave as chains consisting of two-level units coupled by\na fully anisotropic exchange interaction. The latter as-\nsumption is expected to hold also for spin larger than\none-half in the presence of strong single ion anisotropy,\nprovided that domain walls still remain sharp24,38. In\nthis case each single magnetic center follows a thermally\nactivated dynamics, with an energy barrier ∆ 0, and well\nestablished heuristic arguments41suggest to replace the\nattempt frequency by α=α0e−β∆0.\nA naive application of our 3-fold-helix results (42)\nand (47) to the recently synthesized non-collinear Dy 3\ncluster would prevent the observation of slowrelaxation,\nwhile Single-Molecule-Magnet dynamics is indeed there\nobserved even in the presence of compensated magnetic\nmoments42. However, such a behavior in the classical\nregime,i.e.far from level crossings where underbarrier\nprocesses of quantum origin are important, is observed\nfor Dy 3in non-zero field and the resonant behavior is\ndue to a change of the relative population between thelowest and the first excited Kramers doublets of each Dy\nion: For sure this mechanism cannot be accounted for\nwhen dealing with two-level elementary variables, like σk\nin Hamiltonian (2). An extension of our model to mul-\ntivalued σk’s definitely deserve to be considered in the\nnext future.\nBeyondmolecularspinchains, ourapproachmightalso\nbe used to model monatomic nanowires showing slow re-\nlaxation ofthe magnetization at low temperatures43and,\npossibly, one-dimensional spin glasses44(provided that\nquenched disorder is somehow taken into account). In\nthis regard, the question of distinguishing between SCM\nand spin-glass behavior in quasi-1D systems is still a hot\ntopic of discussion45,46,47,48.\nAfter the successful organization of Single-Molecule\nMagnets onto surfaces49,50,51, the grafting of properly\nfunctionalized SCM’s on substrates represents a foresee-\nable goal as well as a fundamental step for their possi-\nble use as magnetic-memory units3. Technologies em-\nploying more traditional materials but based on alter-\nnative geometrical arrangement of magnetic anisotropy\naxes with respect to the switching field, such as in per-\npendicular recording52or processional switching53, are\nalready at the stage of forthcoming implementation in\ndevices. Were SCM’s to be considered as a possible route\nto tackle the main issues of high-density magnetic stor-\nage –i.e.optimization of the signal-noise ratio, thermal\nstability and writability52– the proposed selection rules\nfor slow relaxation, and related bistability, might find an\napplication in magnetic-memory manufacture as well.\nAcknowledgments\nWe wish to thank R. Sessoli and A. Rettori for stim-\nulating discussions, and J. Villain for interest and fruit-\nful suggestions in the early stages of this research work.\nFinancial support from ETHZ, the Swiss National Foun-\ndation, and Italian National Research Council is also ac-\nknowledged.\nAPPENDIX A: THE GENERATING FUNCTIONS\nAPPROACH\nAssuming periodic boundary conditions and defining\nthe two generating functions\nL(y,t) =+∞/summationdisplay\nr=−∞y2r+1s2r+1(t)\nG(y,t) =+∞/summationdisplay\nr=−∞y2rs2r(t), (A1)\nEqs.(15) taketheformoftwodifferentialequations(with\nthe dot indicating the first derivative with respect to the14\nadimensional variable αt)\n/braceleftBigg˙L(y,t) =−L(y,t)+1\n2γ(y+y−1)G(y,t)\n˙G(y,t) =−G(y,t)+1\n2γ(y+y−1)L(y,t).(A2)\nThe system (A2) can be decoupled through the substi-\ntution\nU(y,t) =L(y,t)+G(y,t)\nW(y,t) =L(y,t)−G(y,t), (A3)\nfrom which we directly get\n/braceleftBigg˙U(y,t) =−(1−ν)U(y,t)\n˙W(y,t) =−(1+ν)W(y,t),\nwithν=1\n2γ(y+y−1). The solutions of these two\nequations are U(y,t) =U(y,0)e−(1−ν)αtandW(y,t) =\nW(y,0)e−(1+ν)αtthat, exploiting the property\nexp/bracketleftbigg1\n2(y+y−1)x/bracketrightbigg\n=+∞/summationdisplay\nk=−∞ykIk(x) (A4)\nof the Bessel functions of imaginary argument Ik(x), can\nbe rewritten as\n\n\nU(y,t) =U(y,0)e−αt+∞/summationtext\nk=−∞ykIk(γαt)\nW(y,t) =W(y,0)e−αt+∞/summationtext\nk=−∞(−1)kykIk(γαt)\nPerforming the inverse transformation of (A3), the solu-\ntions to the system (A2) can be obtained\nL(y,t) =1\n2e−αt×\n+∞/summationdisplay\nk=−∞yk/bracketleftbig\nU(y,0)Ik(γαt)+(−1)kW(y,0)Ik(γαt)/bracketrightbig\nG(y,t) =1\n2e−αt×\n+∞/summationdisplay\nk=−∞yk/bracketleftbig\nU(y,0)Ik(γαt)−(−1)kW(y,0)Ik(γαt)/bracketrightbig\n.\nThen, separatingthe k-oddfromthe k-eventermsinboth\nsums, we get\nL(y,t) =e−αt+∞/summationdisplay\nr=−∞[y2rL(y,0)I2r(γαt)+y2r+1G(y,0)I2r+1(γαt)]\nG(y,t) =e−αt+∞/summationdisplay\nr=−∞[y2rG(y,0)I2r(γαt)\n+y2r−1L(y,0)I2r−1(γαt)].\nAccording to (A1), now L(y,0) andG(y,0) can be ex-\npressed again in terms of the initial single spin expecta-\ntion values s2r(0) ands2r+1(0) respectively\nL(y,t) =e−αt+∞/summationdisplay\nr=−∞\n×[y2r+∞/summationdisplay\nm=−∞y2m+1s2m+1(0)I2r(γαt)\n+y2r+1+∞/summationdisplay\nm=−∞y2ms2m(0)I2r+1(γαt)]\nPuttingk′=k+mwe have\nL(y,t) =e−αt+∞/summationdisplay\nk′=−∞y2k′+1+∞/summationdisplay\nm=−∞\n[s2m+1(0)I2(k′−m)(γαt)+s2m(0)I2(k′−m)+1(γαt)].\nComparing this latter result with the definition of L(y,t)\n(A1) and requiring for the terms corresponding to the\nsame power of yto be equal, an explicit function for the\nodd spin expectation values is readily obtained\ns2r+1(t) =e−αt+∞/summationdisplay\nm=−∞[s2m+1(0)I2(r−m)(γαt)\n+s2m(0)I2(r−m)+1(γαt)]. 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Back et al., Science 285, 864 (1999)."    },    {        "title": "1507.06847v1.Unveiling_hidden_ferrimagnetism_and_giant_magnetoelectricity_in_polar_magnet_Fe2Mo3O8.pdf",        "content": "1 \n Unveiling hidden ferrimagnetism and giant m agnetoelectricity in polar magnet Fe 2Mo 3O8 \nYazhong Wang1, Gheorghe L. Pascut1, Bin Gao1, Trevor A. Tyson1,2, Kristjan Haule1, Valery \nKiryukhin1, and Sang-Wook Cheong1,* \n1Rutgers Center for Emergent Materials and De partment of Physics and Astronomy, Rutgers \nUniversity, Piscataway, New Jersey 08854, USA \n2Department of Physics, New Jersey Institut e of Technology, Newark, New Jersey 07102, USA \n*Corresponding author, sangc@physics.rutgers.edu \n  \nMagnetoelectric (ME) effect is recognized for its utility for low-power electronic devices. \nLargest ME coefficients are often associated with phase transitions in which ferroelectricity is induced by magnetic order. Unfort unately, in these systems, larg e ME response is revealed only \nupon elaborate poling procedures. These procedur es may become unnecessary in single-polar-\ndomain crystals of polar magnets. Here we report giant ME effects in a polar magnet Fe\n2Mo 3O8 \nat temperatures as high as 60 K. Polarization jumps of 0.3 μC/cm2, and repeated mutual control \nof ferroelectric and magnetic mo ments with differential ME co efficients on the order of 104 ps/m \nare achieved. Importantly, no electri c or magnetic poling is needed, as necessary fo r applications. \nThe sign of the ME coefficients can be switche d by changing the applied “bias” magnetic field. \nThe observed effects are associated with a hidde n ferrimagnetic order unve iled by application of \na magnetic field. \n \n \n  \n 2 \n Introduction  \n    A significant effort has been invested into finding new material s in which macroscopic \nproperties, such as the magnetizat ion and the electric polarization,  are coupled and controlled by \nexternal parameters like temperature and electric or magnetic fields1-5. Materials where these \nquantities are interconnected are highly desired due to their importance in developing devices \nwith new functionalities6-8. Examples of materials falling into  this category are the pyroelectric \nand multiferroic materials9-12. The practical aspect of pyroelect ric materials is the capacity to \ngenerate a current when they are subjected to a temporal temperature gradient through heating or cooling. Due to the efficient conversion of ther mal energy into electrical energy, pyroelectric \nmaterials have offered numerous device applic ations, for example for temperature-sensing\n13,14 \nand for thermoelectric applications15. The practical aspect of multiferroic materials is the ability \nto mutual control the magnetization (polarizati on) by the use of external electric (magnetic) \nfields, the effect known as magnetoelectric (ME) effect16-18. The ME effect can be linear or/and \nnon-linear with respect  to the external fields and it is characterized by the appropriate ME \ncoefficients19,20. At the present time, materials with la rge ME coefficients are exploited for \ndeveloping low-power magnetoelectronic-based devices and new multiple state memory elements\n21,22. Recently, materials with significant ME re sponse associated with ferroelectricity \ninduced by magnetic orde r have been identified23-25. Unfortunately, elaborate poling procedures, \nsuch as cooling in applied electric and magnetic fields, are needed to reveal the largest ME \ncoefficients in these systems. Finding new materi als with colossal ME coefficients lacking this \ndrawback is of primary importan ce for prospective applications. \n    Materials belonging to the polar crys tallographic symmetry groups  lack the inversion \nsymmetry at all temperatures. Ma ny of these materials contain magnetic ions, and they often \nexhibit long-range magnetic order. We call these materials “polar magnets”. The prerequisite for \nnon-trivial magnetoelectricity is simultaneous br eaking of time reversal symmetry and space \ninversion symmetry. Thus, all polar magnets should exhibit non-trivial ME effects below \nmagnetic ordering temperatures. Importantly, m onodomain polar single crystals can often be \ngrown, potentially eliminating the need for any poling procedures to reveal the largest possible \nME response. While the polar magnets are numerous, the investigation of their \nmagnetoelectricity has been extremely limited. The few examples of polar magnets whose 3 \n magnetoelectricity has been  studied include GaFeO 3 (ref. 26) and Ni 3TeO 6 (ref. 24). Clearly, a \ntargeted search for enhanced ME effects am ong polar magnets holds significant promise. \n    For the ME device applications, ferro- or ferrimagnetic polar magnets are some of the best \ncandidates, as macroscopic magnetic moment is needed for their functio nality. Such compounds \nare rare. However, in some cases macroscopic ma gnetic moment is “hidden” within a nominally \nantiferromagnetic state, and can be easily reveal ed in a modest applied magnetic field, thereby \nleading to a potentially large ME response. A well-known example of such  a hidden moment is \nrealized in La 2CuO 4, the parent compound of high- TC cuprate superconductors27. Each Cu-O \nplane exhibits a weak ferromagnetic moment due to  canting of the spins of the otherwise regular \nNeel order. The canting results from Dzyaloshin sky-Moria interaction. Weak ferromagnetism is \nmasked in zero magnetic field because of the antiferromagnetic interplane coupling. However, \nspin canting is responsible for the many distinct magnetic propert ies of this compound, including \nthe unusual shape of the magnetic sus ceptibility in the vicinity of TN and in an applied field, and \nthe atomic-scale giant magnetoresistence in th e field-induced weakly ferromagnetic phase. \nAnother layered magnet in which a small ferrimagne tic moment of each layer is hidden at zero \nfield is multiferroic (but nonpolar at high T) LuFe 2O4 (ref. 28). The giant magnetic coercivity \nand the unusual ME relaxation properties of LuFe 2O4 are related to the ferrimagnetism in its Fe-\nO layers. Similar to these compounds, a hidden ma gnetic moment in a polar magnet could result \nin a strongly enhanced magnetic  response, which should lead to large ME effects when the \nmagnetic moment and crysta l structure are coupled. \n \nHerein, we report giant ME effect s in a monodomain polar magnet Fe 2Mo 3O8, that possesses \nboth multiferroic and pyroelectric characteristics. Below TN ≈60 K, it exhibits a layered collinear \nmagnetic structure with a small fe rrimagnetic moment in each layer29. As in La 2CuO 4, this \nmoment is “hidden”, but can be reveal ed in a modest a pplied magnetic field28. As a result of \nfield- and temperature-induced magnetic transitions in Fe 2Mo 3O8, the electric polarization \nexhibits changes as large as 0.3 μC/cm2. As the hidden ferrimagnetism is converted to a bulk \nmoment by an applied magnetic field, giant differential ME coefficients approaching 104 ps/m \nare achieved. The observed effects are significantly larger than those previ ously reported in polar \nmagnets, such as Ni 3TeO 6 (ref. 24). The ME control is mutual, as both the magnetization and \nelectric polarization can be tune d by the electric and magnetic fi eld, respectively. Importantly, no 4 \n electric or magnetic poling is n eeded, and the sign of the differe ntial ME coefficients can be \nswitched by simply changing the applied “bia s” magnetic field. Us ing first principles \ncalculations, we show that exchange striction is the leading mechanism responsible for the \nobserved ME effects. Our results demonstrate the promise of polar magnets as ME systems, and \nindicate that their functional properties could be further enha nced by presence of a local \n(“hidden”) magnetic moment that can be easily converted to m acroscopic magnetization by an \napplied field.                                                                                                                                                              \nResults  \n    Fe 2Mo 3O8, known as the mineral kamiokite30,31, consists of honeycomb-like Fe-O layers \nseparated by sheets of Mo4+ ions, See Fig 1(a). The la yers are stacked along the c axis. The Fe-O \nlayer is formed in the ab plane by corner-sharing FeO 4 tetrahedra and FeO 6 octahedra, as shown \nin Fig 1(b). In this layer, the tetrahedral (Fe t) and octahedral (Fe O) triangular sublattices are \nshifted along the c axis by 0.614  Å with respect to each other31, leading to short and long \ninterlayer Fe-Fe distances, see Fig 1(a). The vertices of the FeO 4 tetrahedra point along the \npositive c axis, reflecting the polar structure of Fe 2Mo 3O8 (ref. 31). The Mo kagome-like layer is \ntrimerized. The Mo trimers are in the singlet  state, and do not contribute to magnetism32. Below \nTN≈60 K, the Fe2+ moments exhibit the antif erromagnetic (AFM) order in the honeycomb layers, \nsee Fig 1(c). As discussed below, Fe O has larger spin than Fe t, and therefore each of the Fe-O \nlayers is ferrimagnetic33. Along the c axis, the nearest Fe spins are aligned in the same direction, \nimplying ferromagnetic interlayer coupling. The resulting stacking of the ferrimagnetic Fe-O \nlayers along the c axis leads to vanishing macroscopic magnetic moment, and we call this state \nAFM.  \n    The temperature variation of DC magnetic susceptibility χ in zero field-cooled (ZFC) and \nfield-cooled (FC) processes is shown in Fig 1(e)  for the magnetic field both parallel and normal \nto the c axis. The shapes of the curves are consiste nt with the transition to the AFM order shown \nin Fig 1(c) at TN=61 K, with Fe2+ spins pointing along the c axis. The large difference between \nthe c-axis and in-pane susceptibilities in the paramagnetic state demonstrates appreciable \nanisotropy of the Fe2+ spins. No thermal hysteresis is obs erved, see Supplementary Fig 1. A large \nspecific heat ( CP) anomaly is present at the magnetic transition, see Fig 1(f). To account for the \nphonon part, the specific heat was fi t to a double Debye model for T>TN (90 to 200 K). The best 5 \n fit, shown in Fig 1(f), was obtai ned for the Debye temperatures θD1=174 K and θD2=834 K. It \nfails for T<TN as it implies a negative magnetic contribution for T<50 K. This indicates an \nadditional lattice contribution for these temperatures , suggesting a structural  transition associated \nwith the magnetic order. This suggestion is co rroborated by the temperature dependence of the \ndielectric constant ε(T) and the variation of the electric polarization ΔP(T)≡P(T)-P(T=120 K), \nboth along the c axis, shown in Figs 2(a,b). In particular, the jump of ΔP at TN clearly indicates \nsimultaneous magnetic and structural transitions. The magnitude of this jump, ~0.3 μC/cm2, is \nlarger than the values typically observed in mult iferroics, and is the largest measured value in \npolar magnets, to our knowledge. Im portantly, no poling is needed in an alrea dy polar material to \nobserve the changes shown in Figs 2(a,b). In particular, ΔP was measured by integrating the \npyroelectric current on warming after cooling down to T=5 K in zero electric field (see \nSupplementary Fig 2 for details). In our  measurements, the direction of the ΔP vector (along or \nopposite to the positiv e direction of the c axis defined above) is unde termined. First principles \ncalculations described below indicate that ΔP points in the positive c direction, hence we adopt \nthis convention here. \n    Magnetic field ( H) induces a metamagnetic transition signaled by sharp magnetization ( M) \njumps, see Fig 2(c). It is accompanied by a stru ctural transition indicated by the corresponding \njumps in the electric polarization, as shown in Fig 2(d). A small hysteresis is observed in the \nlatter transition. The ΔP≡P(H)-P(H=0 T) vector is in the negative c axis directi on, and its value \nat T=50 K is roughly twice as small as the ΔP induced at TN for H=0 T. No poling of any kind is \nneeded. Replacement of Fe with Mn, as well as  Zn doping on the Fe site are known to convert \nthe AFM state observed in Fe 2Mo 3O8 into a ferrimagnetic (FRM) state32,33, in which the AFM \norder in the Fe-O layers is preserved, but the sp ins in every second layer are flipped, see Fig. 1(c). \nIn the FRM state, the ferrimagnetic moments of the Fe-O layers are co-aligned, giving rise to a \nmacroscopic magnetization. The extr apolation of the high-field M(H) data of Fig 2(c) to zero \nfield gives a positive intercept of ~0.5 μB/f.u. at T=50 K, indicating the ferrimagnetic character of \nthe high-field state, which we assume to have th e same FRM structure as shown in Fig 1(c). This \nassumption is corroborated by the Fe-O  net ferrimagnetic moment of 0.6 μB/f.u. for a single layer, \nexpected from the Moessbauer measurements of the Fe O and Fe t moments33 (4.83 μB and 4.21 μB, \nrespectively), as well as by the results of the fi rst principles calcula tions described below. 6 \n Discussion \n    To understand the microscopic origin of th e observed ME effects, we have carried out ab-\ninitio calculations in the fram ework of density functional theory adding an on-site Coulomb self-\ninteracting potential U (DFT+U). For the DFT part, the generalized gradient approximation \nPerdew-Burke-Ernzerhof (GGA-PB E) functional was used. For U=0, the ground state is metallic \nwith the FRM structure, but m oderate correlation strength ( U=4 eV) leads to an AFM insulating \nground state. While U of the order of 4 eV is required to  obtain the correct ground state, its exact \nvalue was found to be unimportant for the magnetic exchange energies releva nt to this work. The \ndetails of the DFT calculations can be found in the Methods section. The ionic positions were optimized for two imposed magnetic structures , the AFM and FRM. The FRM structure was \nfound marginally higher in total energy (less than 10 meV/f.u.), i ndicating that this phase is \nexpected to be induced in modest magnetic fi elds, consistent with our experimental data. \n    The calculated ionic shifts for the transiti ons from the paramagnetic (PARA) to the AFM state, \nand from AFM to FRM, are shown in Figs 1(c), a nd 3(c,d). The ionic shifts for every ion in the \nunit cell are given in Supplementary Table I. Th e experimental paramagnetic structure, and the \ncalculated AFM and FRM structures were used. The ionic shifts can be utilized for an estimate of the magnetically-induced electric polarization change ΔP. While the total polarization is a \nmultivalued quantity, the difference ΔP between two structures is a well-defined quantity\n34. For a \nqualitative comparison with experiment, it is sufficient to use the ionic-like formula for ΔP given \nby ଵ\n∑(ݖ\n−ݖ)ܳ, where ݖ and ݖ are the c-axis ionic coordinate s for the initial and the final \nstructures, respectively, ܳ are the formal ionic charges, V is the unit cell volume, and the sum is \ntaken over the unit cell. For the PARA to AF M, and AFM to FRM transitions, we obtain ΔP \nvalues of 0.60(11) μC/cm2 and -0.55(11) μC/cm2, respectively. The calculated magnitudes and \nthe relative signs of ΔP are in good qualitative agreement with our experiments. The positive \nsign of ΔP for the PARA to AFM transition indicates that the ΔP vector points al ong the positive \nc axis, justifying the conv ention used in our work. \n    The calculated ionic shifts also allow to ge t an insight into the mech anism of the ME effect. \nThe atoms shift to maximize the magnetic ener gy gains in the AFM and FRM states. Oxygen \nions exhibit the largest shifts, and therefore the ME energy gains should be associated with the 7 \n modifications of the superexchange  paths between the interacting Fe2+ spins. Lattice structure, as \nwell as preservation of the in-p lane magnetic order in applied magnetic field imply that the \nlargest magnetic coupling ( J) is between the nearest Fe2+ ions, see Fig 1(b). The calculations \nshow that upon the transition from the paramagnetic to the AFM state, the Fe-O-Fe angle ( θ) \nbetween the nearest Fe2+ increases from 109º to ~111º, mos tly due to the oxygen shifts, see Fig \n1(d). The in-plane antiferromagnetic J increases with increasing θ due to the more favorable Fe-\nO-Fe orbital overlap, resulting in the magnetic ener gy gain. Thus, we ascribe the ionic shifts, as \nwell as the accompanying ΔP, to the exchange striction in the AFM state.  \n    The FRM state can be induced both by a positive and a negative magnetic field along the c \naxis. The two states differ only by 180º rotation of every spin in  the system. While the field-\ninduced magnetizations should be  opposite for the opposite fields, ΔP induced by exchange \nstriction should be identical. This prediction is clearly confirmed by the data of Figs 3(a,b). The \ncalculated ionic shifts for the AFM to FRM tr ansition, shown in Fig 3( d), are opposite (but \nsmaller) to those occurring at the PARA to AFM transition, see Fig 3(c) . In other words, the \nlattice partially relaxes towards the paramagnetic st ructure in the FRM state. This is consistent \nwith the magnetic energy loss due to the inte rlayer interactions in the FRM phase, and \ncorresponding relaxation of the lattice distorti on realized in the AF M state. As a result, ΔP is \nnegative in the AFM to FRM transi tion. Thus, the data of Fig 3(b), in combination with our first \nprinciples calculations, show that exchange stric tion underlies the ME effect in the transition to \nthe FRM state, as in the AF M transition discussed above. \n    The sharpness of the field-induced transitions shown in Figs 2(d) and 3( a,b) gives ri se to giant \nvalues of the differential ME coefficient d P/dH in the vicinity of th e transition field, reaching \nalmost -104 ps/m for T=55 K. (Consult Supplementary Fig 2(d) for the field-dependent d P/dH for \ndifferent temperatures). Combined with absenc e of poling requirements and the small hysteresis \n(0.02 T at 55 K, 0.007 T at 58 K), it leads to gian t, reproducible, and almo st linear variation of P \nwith H, as shown in Fig 4(a) for T=55 K. In the range shown, ΔP oscillates, varying by 0.08 \nµC/cm2 as H goes from 3.25 to 3.5 T and back. The inverse effect, in which an applied electric \nfield ( E) changes the magnetization is al so giant, reproducible, and linear, as shown in Fig 4(b). \nAt T=55 K and H=3.345 T, the magnetization varies by 0.35 μB/f.u. in the field oscillating \nbetween ±16.6 kV/cm, resulting in the d M/dE of -5700 ps/m. Similarly large differential ME 8 \n coefficients d P/dH and d M/dE are observed at other points  on the AFM-FRM transition \nboundary shown in Fig 4(c). These co efficients are more than an order of magnitude larger than \nthose reported for the polar magnet Ni 3TeO 6 (ref. 24), see Fig 4(d).  \n    When both external fields H and E are collinear and their direc tion coincides with the positive \nc axis of the crystal, both ΔP and ΔM are negative in applied positive H and E, respectively. \nThus, both d P/dH and d M/dE are negative. The data of Figs 3( a,b) show that, consistent with \nexchange striction mechanism, ΔM changes sign in negative H, while ΔP does not. As a result, \nboth d P/dH and d M/dE change their sign and become positive in negative H and E, while \nretaining the same magnitudes. This sign revers al is illustrated in Supplementary Fig 4. \n    In conclusion, polar magnets clearly possess a great potential as ME materials. The absence of \npoling requirements makes possible utilization of giant ME coefficients associated with sharp \nmetamagnetic transitions practical , because reproducible, hysteresis -free linear responses can be \nachieved, as necessary for applications. In Fe 2Mo 3O8, hidden ferrimagnetism of the Fe-O layers \nstrongly enhances the magnetic response at the transition fiel d, providing explanation for the \nobserved giant differential ME coefficients. Excha nge striction mechanism of the ME effect in \nFe2Mo 3O8 provides an additional functional capabil ity of controlling the sign of these \ncoefficients by the direction of the applied “bias” magnetic fiel d. Therefore, studies of other \npolar magnets, especially with exchange strict ion ME mechanism and local ferrimagnetism, are, \nin our opinion, of significant promise. \nMethods  \nSingle crystal preparation and structure analysis \nFe2Mo 3O8 single crystals were grown using a ch emical vapor transp ort method at 1000 °C for 10 \ndays, followed by furnace cooling. They are black hexagonal plates with typical size ~1 × 1 × 0.5 \nmm3. Powder X-ray diffraction measurement was performed on crushed powders of Fe 2Mo 3O8 \nsingle crystals. Refinement shows that the room-temperature space group is P6 3mc. a- and c- \nlattice constants ar e 5.773(3) and 10.054(3) Å, respectively. \nMeasurements  9 \n All measurements of magnetic properties M(H), χ(T) and M(E) were performed in a Quantum \nDesign MPMS-XL7. The dielectric constant ε(T), specific heat CP(T), electric polarization P(T) \nand P(H) properties were performed using Quantum Design PPMS-9. ε(T) was measured with \n1V a.c. electric field applied along the c axis using a Quadtech 7600 LCR meter at 44 kHZ. \nSpecific heat measurements were conduc ted using the standard relaxation method. P(T) and P(H) \nwere obtained by integrating the pyroelectric current J(T) and magnetoelectric current J(H), \nwhich were measured using Ke ithely 617 programmable electromet er at 5 K/min warming rate \nand ramping magnetic field with 200 Oe/s.  \nFirst principles calculations \nAb-initio calculations were pe rformed using the full-potential li nearized augmented plane wave \n(FP-LAPW) method as implemented in the WIEN2k code35 within the framework of density \nfunctional theory36,37. The electronic, magnetic and structural properties of Fe 2Mo 3O8 were \ncalculated using the generalized  gradient approximation (GGA) for the exchange-correlation \npotential, in the form of Perdew, Burke and Ernzerhof38,39 (PBE) plus an on-site Coulomb self-\ninteraction correction potential ( U) treated by DFT+U, and the double-counting in the fully \nlocalized limit40. Since the symmetry of low temperature crystal structur e is not know n, the point \ngroup symmetry of the hexagonal paramagnetic space group P6 3mc (ref. 31) was artificially \nreduced for the purpose of optimizations of internal  parameters (OIP). All the calculations were \ndone in the triclinic  space group P1, with  the lattice parameters kept fixed to a = b = 5.773 Å, \nc=10.054 Å, α = 90°, β = 90°, γ = 120°. OIP were performed with imposed AFM and FRM \nmagnetic configurations, using as the initial guess the experimentally determined internal \nparameters of the paramagnetic phase31. The search for equilibrium ionic positions was carried \nout by means of the PORT method41 with a force tolerance ≤ 0.5 mRy/Bohr. The calculations \nwere performed with more than 200 k-points in the irreducible wedge of the Brillouin zone (10 × \n10 × 4 mesh). The total energy, charge a nd force convergence criteria were ~10-4 Ry, ~10-4 \nelectrons and 0.25 mRy/Bohr, resp ectively. The muffin-tin radii RMT were chosen as 1.90, 1.93 \nand 1.66 bohr for Mo, Fe and O, respectively. To en sure that no charge l eaks outside the atomic \nspheres, we have chosen the energy which sepa rates the core and the valence states to be −10 Ry, \nthus treating the Mo(4s, 4p, 4d, 5s),  Fe(3s, 3p, 3d, 4s) and O(2s, 2p)  electrons as valence states. \nAll other input parameters were used with their default values.  10 \n 1 Choi, Y. J., Zhang, C. L., Lee, N. & Cheong, S. W. 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Étude par effet Mössbauer de l'ion \nFe2+ en symétrie trigonale dans les composés du type (Fe, M) 2Mo 3O8 (M = Mg, Zn, Mn, Co, Ni) et \npropriétés magnétiques de (Fe, Zn) 2Mo 3O8. J. Phys. France  33, 549-564 (1972). \n34 King-Smith, R. D. & Vanderbilt, D. Theory  of polarization of crystalline solids. Phys. Rev. B  47, \n1651-1654 (1993). \n35 Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, D. & Luitz, J. WIEN2K, An Augmented Plane \nWave + Local Orbitals Program for Calculating Crystal Properties .  (Karlheinz Schwarz, Techn. \nUniversität Wien, Austria, 2001). \n36 Hohenberg, P. & Kohn, W. Inhomogeneous Electron Gas. Phys. Rev.  136, B864-B871 (1964). \n37 Kohn, W. & Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. \nPhys. Rev. 140, A1133-A1138 (1965). \n38 Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. \nRev. Lett.  77, 3865-3868 (1996). \n39 Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple [Phys. \nRev. Lett. 77, 3865 (1996)]. Phys. Rev. Lett.  78, 1396-1396 (1997). \n40 Anisimov, V. I., Solovyev, I. V., Korotin, M. A., Czy żyk, M. T. & Sawatzky, G.  A. Density-functional \ntheory and NiO photoemission spectra. Phys. Rev. B  48, 16929-16934 (1993). \n41 PORT is a method implemented in the WIEN2K code for finding the equilibrium positions by \nminimization of the total energy; for furt her details see the WIEN2K User Guide. \n \n \n  \n \n 12 \n Acknowledgements \nThis work was supported by DOE under Gran t No. DE-FG02-07ER46382. K.H. was supported \nby NSF-DMR 1405303. Heat capacity measurements at NJIT (T.A.T.) were supported by DOE \nunder Grant DE-FG02-07ER46402. The PPMS (used for heat capacity measurements) was \nacquired under NSF MRI Grant DMR-0923032 (ARRA aw ard). Discussions with Karin M. Rabe \nare gratefully acknowledged.  \nAuthor contributions  \nY.Z.W. carried out magnetoelectric, magnetic a nd dielectric measurements. T.A.T. performed \nthe heat capacity measurement. B.G. synthesize d single crystals and performed XRD. G.L.P. \ndeveloped the theoretical model under the supervis ion of K.H. Y.Z.W., G. L.P., V.K. and S.-W.C. \nwrote the manuscript. S.-W.C. initiated and supervised the research. \nCompeting financial interests \nThe authors declare no compe ting financial interests. \n \n \n \n \n \n \n \n \n \n 13 \n Figure 1| Magnetic transition in Fe 2Mo 3O8. (a) Crystal structure of Fe 2Mo 3O8. Vertical lines connect \nthe nearest Fe ions along the c axis (blue lines are longer than the red ones).  (b) The Fe-O layer in the ab \ncrystallographic plane. Thick line depicts the largest Fe-Fe magnetic coupling J. (c) Schematic view of \nthe AFM and FRM orders. Pink arrows represent the fe rrimagnetic moments of the individual Fe-O layers. \n(d) The AFM order, together with the calculated largest ionic  shifts associated with the paramagnetic to \nAFM transition. The direction of the magnetically-induced ΔP is shown with a thick arrow. (e) \nTemperature dependence of DC magnetic susceptibility χDC in zero field-cooled (ZFC) and field-cooled \n(FC) processes along two crystallographic dir ections, parallel and perpendicular to the c axis, in μ0H=0.2 \nT. (f) Specific heat anomaly at the Neel temperatur e. Red line represents the double Debye model fit \ndiscussed in the text.  Insert: the image of as-grown Fe 2Mo 3O8 single crystal.  \nFigure 2| Magnetically-induced electric polariza tion, and the metemagnetic transition. (a)  \nTemperature dependence of the c-axis dielectric constant ϵ(T), f=44 kHZ. (b) Variation of the c-axis \nelectric polarization ΔP with temperature. (c,d) Magnetic field dependence of magnetization M(H) and \npolarization ΔP(H) at various temperatures. In (d), solid (open) circles depict the data obtained upon \nsweeping the magnetic field up (down). \nFigure 3| Magnetoelectric effect, and the associated ionic  shifts. (a,b ) Magnetic field dependence of \nmagnetization M(H) and polarization P(H) at T=55 K. Numbers and arrows indicate the measurement \nsequence. The insert in (b) shows the magnetic orders and the ferrimagnetic mome nts of the Fe-O layers \nfor the phases involved. (c) The calculated ionic shifts for the paramagnetic to AFM transition. Thick \narrow represents the corresponding change of the electric polarization, ΔP. (d) the same as (c), but for the \nAFM to FRM transition.  \nFigure 4| Reproducible magnetoelectric control of the electric polarization and magnetization with \ngiant ME coefficients. (a)  Periodic modulation of electric polariza tion (blue) induced by a magnetic field \nlinearly varying between 3.25 T and 3.5 T (black) at 55 K. (b) Periodic modulation of magnetization \n(green) induced by an electric field (red) linearly varying between ±16.6 kV/cm, for T=55 K and \nߤ=ܪ3.345 T. (c) Phase diagram of Fe 2Mo 3O8. Black dots determined from M(H), and red diamonds – \nfrom  χ(T) curves. (d) Electric field dependence of magnetization for Fe 2Mo 3O8 (from panel (b), \naveraged), and for Ni 3TeO 6 (×10). The insert illustrates the experimental setup, with directions of the \napplied fields shown. In all figures, the magnetizatio n, polarization, and the applied fields are along the c \naxis. \n "    },    {        "title": "1811.06717v1.Ferrimagnetic_organelles_in_multicellular_organisms.pdf",        "content": "Ferrimagnetic organelles in multicellular organisms  \n \nGorobets S. V.1, Gorobets О. Yu.1, 2, Gorobets Yu.  I.2,1, Bulaievska М. О.1 \n \n1National Technical University of Ukraine \"Igor Sikors ky Kyiv Polytechnic Institute\", 03056, Kyiv, \nUkraine  \n2Institute of Magnetism NAS of Ukraine and MESYS of Ukraine, 03142, Kyiv, Ukraine  \n \nAbstract  \n \nIntroduction  \nBiogenic ferrimagnetism has been intensively studied for more than four decades in connection with \nthe discovery of biogenic magn etic nanoparticles (BMN) in a wide variety of organisms1-40. In the \nnarrow sense, BMN s are  nanoparticles of strong natural magnets (ferrites – magnetite, maghemite, \ngreigit) in organisms that create stray magnetic fields  in their vicinity41,42. In a broad sense, BMNs are \nnanoparticles of iron -containing materials that have sufficient magnetic susceptibility for movement in \nthe magnetic field of laboratory magnets10. These nanoparticles arecalled biogenic because their \nbiosynthesis is programmed at the genet ic level41-43. The main impetus for the study of biogenic \nferrimagnetism  was the ability of m agnetotactic bacteria to move along the lines of force of the Earth’s \nmagnetic field (i.e. magnetotaxis)1,2. Subsequently, the idea of magnetotaxis was developed  into the \nidea of magnetoreception of animals (i.e. their ability to orient in the geomagnetic field) due to the \npresence of BMN9,33. \nBut when examining organs and tissues for the presence of BMN, these magnetic nanoparticles \nwere found not only in those  organs and tissues that may be responsible for the orientation of animals \nin the Earth’s external magnetic field (brain, beak of migratory birds, lateral line and ethmoid bone of \nmigratory fish), but also in a number of other organs, both migrat ory and non-migratory \norganisms8,14,19,22,24 ,25,28-30,33,37 -39,44-50 (Fig. 1, Table 1) . \n \n \nFig. 1. The organs of migratory fish, in which the presence of BMNs is experimentally shown: \nethmoid bone22,45, 46,51, brain19,22,45, muscle22,45,46, skin22,45,46,51, eye22,45,46,51, gill s45, heart45, liver45, \nintestine45, lateral line22,45. The organs of non -migratory fish, in which the presence of BMNs is \nexperimentally shown: ethmoid bone, brain, lateral line24,25. \n \nAll these numerous experimental data on the detection of BMN s in analogous and homologous \norgans  of organisms of different phylogenetic groups confirm the existence of the theoretically \npredicted single genetically programmed mechanism of biosynthesis of BMN s in all organisms42,52,53. \nAt the same time, the theoretically predicted presence of BMNs in a number of human organs and \ntissues by the methods of comparative genomics50is confirmed by experimental studies on the presence \nof BMNs in human organs34,37,38, 54-56, as well as analogous and homologous organs and tissues of \nanimals8,14,19, 22,25,33, 44-47,49,50 (Table 1).  \n \nTable 1. Human organs, as well as analogous and homologous organs of animals, in which BMNs are \nfound.  \n \nHuman organs \nwith theoretically \npredicted BMNs \npresence  (+) Experimentall\ny confirmed \nBMNs  \npresence in \nhuman organs  Experimentally confirmed BMNs presence in relevant \nor analogous fish organs  \nBrain50 (+) \n Brain34,54,55 Brain of the  Oncorhynchus nerka19, brain of the whale \nMegaptera novaeangliae44, \nhead  Pachycondyla  marginata14, brain of the silver carp  \n(Fig. 9) \nHeart50 (+) Heart37 Heart of the  Sus domestica49, fish heart45 \nLiver50 (+) \n Liver37 Liver of the  Sus domestica49, fish liver45, liver of the \nhouse mouse Mus musculus  (Fig. 2, 3) \nSpleen50 (+) Spleen37 Spleen of the  Sus domestica50 \nEthmoid bone50 (+) \n Ethmoid \nbone56 Ethmoid bone of the  Salmo salar , ethmoid bone of the \nCyprinus  carpio , ethmoid bone of the Esox  lucius25, \nethmoid bone of the Oncorhynchus nerka8, ethmoid bone \nof the  Delphinus delphis33, ethmoid bone of the \nThunnus  albacares46, antennas of the  \nPachycondyla  marginata14 \nAdrenal glands50(+) Adrenal \nglands38  \nLung50(+)  Lung of the  Sus domestica49, fish gills45 \nKidney50 (+)  Kidney of the  Sus domestica50 \nIntestine50 (+)  Intestine of the fish45, abdominal cavity of the ants14, \nintestine of the house mouse Mus musculus  (Fig. 5) \nMuscle tissue50(+)  Muscle of the fish22,46  \nSkin50 (+)  Skin of the  Salmo salar22, skin of the  Oncorhynchus \nnerka47, skin of the  Thunnus  albacares46 \nEyes   Eyes of the fish22,45 \nJoints   Joints of the  Pachycondyla  marginata14 \n \nIn this work, the localization of BMN s was investigated using atomic force (AFM) and magnetic \nforce microscopy (MFM) in a number of organs of various organisms (animals, plants, fungi) in order \nto identify which systems of multicellular organisms include BMNs . \n \nExperiment and discussion  \nIn this paper, the following samples were examined using AFM and MFM , namely, animal samples : \nliver, intestine, pancreas  of mouse , lungs, kidneys, spleen  of pig , brain  of carp ; plant samples: leaves \nand root  of tobacco , stem and tuber  of potato ; and fungi samples: fruit  bodies of the common and \nshiitake  mushroom s. The cells and other structural components (shown in the images by arrows) in the \nsamples of the listed organs were id entified and the localization of BMNs in the indicated organs was \nshown.  \nThe results of studies of samples of the liver of a mouse with the use of AFM and MFM showed \nthat BMNs  in the liver of a mouse are located in the wall of sinusoids (capillaries of the liver) (Fig. 2 -3 \nand Table 2). Sinusoids57, sinusoidal endothelial cells of the liver58, nuclei of sinusoidal endothelial \ncells of the liver57,59, hepatocytes, nuclei of hepa tocytes57, fenestra e58-61presented in Fig. 2 -3 in this \npaper have typical morphology and dimensions, characterized in the works57-61. \n \n \n \n \n \n Table 2. Dimensional characteristics of the structura l components of the mouse liver  \n \nStructural components  \n Calculated dimensions \n(see Fig. 2 -3), µm  Dimensions in \nliterature , µm  References  \nCapillary density  400-600/µm2 400-500/µm2 62 \nSinusoids  5-20 5-15 63-65 \nSinusoidal endothelial cells  15-25 20-25 58 \nHepatocytes  15-25 20-30 58 \nHepatocyte nuclei  5-8 6-8 58 \nFenestrae  200-500 50-500 58-61 \n \n \n \nFig. 2. AFM (left) and MFM (right) images of a sample of the liver of the house mouse Mus musculus : \n1 hepatocyte, 2 nuclei of hepatocyte s and 3 sinusoids.  \n \n \n \nFig. 3. AFM (left) and M FM (right) images of a sample of the liver of the house mouse Mus musculus : \n1 sinusoidal endothelial cells of the liver, 2 nuclei of sinusoidal endothelial cells of the liver, 3 \nfenestra e and 4 sinusoids.  \n \nSimilar results were obtained on the localization  of BMNs in the samples of the pancreas and \nintestines of a mouse. So, BMNs in the pancreas and the intestines of the mouse are located in the wall \nof the capillaries  as can be seen from Figures 4 -5 and data on the size and dist ance between the \ncapillaries in Tables 3 -4. The capillaries of the pancreas are shown in Fig. 4 in this paper and have \ntypical morphology and size66-68. \n \n \n \n \n \n \n1 2 1 \n3 \n1 \n2 3 4 Table 3. Dimensional characteristics of the structural components of the mouse pancreas  \n \nStructural components  \n Calculated dimensions \n(see Fig. 4), µm  Dimensions in literature , \nµm \n References  \nDistance between the \ncapillaries  10-30 20-50 66 \nCapillaries  4-10 3-7 66-68 \n \n \n \nFig. 4. AFM (left) and MF M (right) image s of a sample of the pancreas of the house mouse Mus \nmusculus : arrows indicate the capillaries.  \n \n Capillaries of the intestine are presented in Fig. 5 in this work and have also typical \nmorphology a nd size, described in the work69,70. \n \nTable 4.  Dimensional characteristics of the structural components of the mouse intestine  \n \nStructural components  \n Calculated dimensions \n(see Fig. 5), µm  Dimensions in literature , \nµm References  \nDistance between the \ncapillaries  5-30 30-40 71,72 \nCapillaries  3-6 3-5 69,70  \n \n \n \nFig. 5. AFM (left) and MFM (right) images of a sample of the intestine of the house mouse Mus \nmusculus : arrows indicate the capillaries.  \n \n AFM and MFM examinations of porcine specimens gave similar results, namely, BMNs in the \nlungs (Fig. 5 and Table 6), kidneys (Fig. 6 and Table 7) and spleen (Fig. 7 and Table 8) of the pig are \nlocated in the capillary wall. So, capillaries of lungs, are pre sented in fig. 6 in this paper and have \ntypical morphology and size, described in the works73,74. \n \nTable 5. Dimensional characteristics of the structural components of the pig lung  \n \nStructural \ncomponents  Calculated dimensions \n(see Fig. 6), µm  Dimensions in  literature , \nµm References  \nDistance between the \ncapillaries  5-25 10-20 73 \nCapillary density  600-900/µm2 700-900/µm2 75-78 \nCapillaries  5-10 5-20 73,75  \n \n \n \nFig. 6. AFM (left) and MFM (right) images of a sample of the lungs of a domestic pig Sus domestica : \narrows indicate the capillaries.  \n \n The capillaries of the kidneys are presented in Fig. 7 in this paper  and have typical morphology \nand size, described in79. \n \nTable 6. Dimensional characteristics of the structural components of the pig kidney  \n \nStructural components  \n Calculated dimensions \n(see Fig. 7), µm  Dimensions in literature , \nµm \n References  \nCapillary density  100-300/µm2  25-220/mm2 80,81  \nCapillaries  7-10 3-20 82,83  \n \n \nFig. 7. AFM (left) and M FM (right) image s of a sample of  thekidney  of the domestic pig Sus \ndomestica : arrows indicate the capillaries.  \n \n The c apillaries of a spleen, are presented in fig. 8 in this work, have typical morphology and \nsize, described in84,85. \n \nTable 7. Dimensional characteristics of the structural component s of the pig spleen  \n \nStructural components  \n Calculated dimensions \n(see Fig. 8), µm  Dimensions in literature , \nµm References  \nDistance between the \ncapillaries  5-10 10-30 86 \nCapillaries  4-7 4-10 87-90 \n \n \n \nFig. 8.  AFM (left) and MFM (right) images of a sample of the spleen of the domestic pig Sus \ndomestica : arrows indicate the capillaries.  \n \nAs noted above, BMNs are most thoroughly investigated in the organs and tissues of fishes. \nHowever, only transmission electron  microscopy (TEM) and MFM provide information on the \nlocation of BMNs  among all experimental methods that are used today to study BMNs34,35,91 -114. This \npaper presents the results of AFM and MFM studies of samples of the silver carp brain, and it is shown \nthat BMNs are located in the capillary wall (Fig. 9 and Table 8)  in the brain of the silver carp, as in the \norgans of mice and pigs . The c apillaries of  brain are p resented in fig. 9 in this work and  have typical \nmorphology and dimension s, characterized in115. \n \nTable 8. Dimensional characteristics of the structural components of the silver carp brain  \n \nStructural components  \n Calculated dimensions \n(see Fig. 9), µm  Dimensions in literature , \nµm References  \nDistance between the \ncapillaries  10-30 15,3 116 \nCapillaries  3-9 3-8 116,117  \n \n \n \nFig. 9. AFM (left) and MFM (right) images of a sample of the silver carp brain of Cyprinus carpio : \narrows indicate the capillaries.  \n \nThe BMNs in the human ethmoid bone are located in the sinuses of the bone56, through which \nthe capillaries and nerves pass118-120. The location of BMN s in the ethmoid bone of fish es is similar25. \nThus, a common feature of the location of BMNs in the ethmoid bone and in other organs of various \nanimals (Fig. 2 -9, Table 1) is their localization in the vicinity of the capillaries.  \nDue to the fact that for many years, BMN s was investigated mainly in connection with the \nideas about magnetotaxis and magnetoreception, BMN s in plants and fungi has been little studied. For \nexample, the paper11presents experimental data on the detection of magnetically sensitive \nnanoparticles in plants, but the authors of this work consider it more likely that they have detected \nphytofer ritin and its aggregates, and not plant BMNs. As for fungi, BMN s is experimentally detected \nonly in the microfungi Fusarium oxysporum  and Verticillium spp .121. \nThe study of BMNs in plants was carried out on the example of tobacco, as the most studied \nmodel organism among plants, as well as potatoes. The BMNs in the leaf and  the roots of tobacco (Fig. \n10-11) are located on the membrane of phloem sieve tubes. Phloem is a conducting  plant tissue that \nforms a network of sieve tubes through which organic substances , synthesized by leaves during \nphotosynthesis , are transported  to all plant organs122-126, unlike xylem, which transports water and \nminerals from the soil127-130. Sieve tubes of tobacco leaf are presented in Fig. 10 in this paper and  have \nthe typical morphology and dimensions described in131. \n \n \n \nFig. 10.  AFM (left) and M FM (right) image s of a sample of a leaf of the cultivated tobacco Nicotiana \ntabacum . Elements of sieve tubes, including pores, are visible in the AFM image. White arrows in the \nAFM image indicate the membrane of  the sieve tubes, red arrows  indicate the pores of the sieve tubes.  \n \n \n \nFig. 11. AFM (left) and MFM (right) images of a sample of a root of the cultivated tobacco Nicotiana \ntabacum . Elements of sieve tubes, including pores, are visible in the AFM image. White arrows in the \nAFM image indicate the membrane of the sieve tubes, red arrows indicate the pores of the sieve tubes.  \n \nSimilar data on the location of BMNs were obtained on the s amples of potatoes. The BMNs in \nthe potato stalk are located in the walls of the conducting  tissue (phloem) (as can be seen from Fig. 12 -\n13), wh ich is characterized in132. BMNs in a potato tuber are located at the boundary between starch \ngrains133 and conducting  tissue (phloem).  \n \n \n \nFig. 12. AFM (left) and MFM (right) images of a sample of the stem of a potato Solanum tuberosum : \nconducting  tissue/ sieve tubes (white arrows) phloem, cell wall (red arrows).  \n \n \n \nFig. 13.  AFM (left) and MFM (right) images of a sample of the tuber of a potato Solanum tuberosum : \nconducting  tissue/sieve tubes (white arrows) phloem, cell wall (red arrows).  \n \nThe study of BMNs in mushrooms was carried out using the example of higher mushrooms, \nsuch as mushroom s Agaricus bisporus  and Lentinula edodes , which are the most common edible \nmushrooms. As can be seen from Fig. 14 -15 BMN s in the mushroom are located in the walls of the \nvascular hyphae, which are characterized  in the works134-136. \n \n \n \nFig. 14. AFM (left) and MFM (right) images of a sample of the common mushroom Agaricus \nbisporus : arrows indicate the hyphae.  \n \n \n \nFig. 1 5. AFM (left) and MFM (r ight) images of a sample of the shiitake  mushroom Lentinula edodes : \narrows indicate the hyphae.  \n \nThus, studies of BMNs in samples of organs and tissues of animals, plants and fungi, conducted \nin this work showed that:  \n• BMN in the organs of multicellular organisms form chains;  \n• BMN in multicellular organisms are  part of the transport system. Thus, BMNs in animals are located \nin the walls of capillaries (all examined organs and tissues except the ethmoid bone) or in the vicinity \nof the capillaries (the ethmoid bone). BMNs in plants are located in the wall of the conducting  tissue, \nnamely in the wall of sieve tubes of the phloem. BMNs in fungi are located in the wall of the \nconducting tissue, namely in the walls of the vascular hyphae134-136. \n \nConclusions  \nThe chains of BMN are components of the cells that form the w alls of the capillaries of animals and \nthe walls of the conducting tissue of plants and fungi. At the same time, the functions of the capillaries \nof animals and the functions of the conducting tissue of the phloem of plants, as well as the vascular \nhyphae of fungi, are similar122,137,138. In particular, capillaries (exchange vessels) exchange nutrients, \ngases, liquids, metabolites, signal substances (hormones), immune cells, etc. between the blood and \nbody tissues137,139 -141. Conducting  tissue of the phloem of plants serves to transfer organic substances, \nhormones, etc. throu gh the body122-126. The same functions have vascular hyphae of fungi138,142 -144. The \nlocation of BMNs  in various organs and tissues of animals, plants and fungi as p art of systems with \nsimilar functions cannot be accidental, taking on account that the genetically programmed mechanism \nof biosynthesis of BMNs  arose at the beginning of evolution42,43,145 -147. Such localization of BMNs  \nargues in favour  of the idea that BMNs chains are directly involved in metabolic processes25,42,43,148 -150 \nand perform vital functions. In addition, bioinformatics analysis showed that BMNs in multicellular \norganisms, as well as in magnetotactic bacteria151-154, form inside the cell and are associated with the \ncell membrane8,9,15,34,39. This means that the chains of BMNs  in multicellular organisms are part of a \nnew type of organelles with the  ferrimagne tic properties – the ferrimagnetic  organelles of a specific \npurpose. Such ferrimagnetic  organelles create scattering magnetic fields of the order of several \nfractions of T and gradients of magnetic field induction of the order of 105-106 T/m in their vicinity, \nwhich can significantly affect the mass transfer processes near the cell membrane of vesicles, \ngranules41,42,150, organelles, structural elements of the membrane and others.  \nIn connection with the foregoing, it is clear that an extremely important task is to search for the \ngeneral functions of BMNs in various organs and tissues of multi cellular organisms. To solve this \nproblem, it is necessary to identify the types and functions of cells containing BMNs in various organs \nof multicellular organisms.  \n \nMaterials and Methods  \nAtomic force microscopy and magnetic force microscopy.  Determinatio n of the presence of BMNs \nand the study of their localization in the investigated samples was carried out using the «Solver PRO -\nM» scanning probe microscope by atomic force microscopy (AFM) and magnetic force microscopy \n(MFM).  \nThe magnetic probe MFM_LM series with chip size 3.4x1.6x0.3mm, coated by CoCr was used. \nThis probe was used both for AFM and MFM imaging. The non -contact AFM (NC -AFM) mode was \napplied. The MFM scanning was carried out at the constant distance from the samp le surface after \nAFM scanning. The probe “lift” height was 100 nm. The cantilever was calibrated using the test \nsamples. Calibration of the probe was carried out immediately before the measurements.  \nThe biological material was prepared before AFM and MFM s canning. Fixation of biological \nmaterial was carried out in a 10% formalin solution. The duration of fixation was 24 hours. 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"    },    {        "title": "2001.00707v1.Neutron_diffraction_and_ab_initio_studies_on_the_fully_compensated_ferrimagnetic_characteristics_of_Mn2V1_xCoxGa_Heusler_alloys.pdf",        "content": "1 \n Neutron diffraction and ab initio  studies  on the      \nfully compensated f errimagnetic characteristics of  \nMn 2V1-xCo xGa Heusler alloys  \nP V Midhunlal1, C Venkatesh2, J Arout  Chel vane3, P D Babu4 and N Harish Kumar1 \n  \n1. Department of Physics, Indian Institute of Technology Madras, Chennai -600036, India.  \n2. Saha Institute of nuclear physics,  Bidhannaga r, Kolkata, West Bengal -700064 . \n3. Defense  Metallurgical  Research  Laboratory,  Kanchanbagh  (PO),  Hyderabad -500058,  India.  \n4. UGC-DAE Consortium for Scientific Research, Mumbai Center, R -5 shed, BARC, Trombay,   \n    Mumbai -400085, India . \n \n \nKeywords :  Compensated ferrimagnets, high Curie temperature, Neutron diffraction,  \n                    ab initio  studies . \n \nAbstract:  Neut ron diffraction and ab initio  studies were  carried out on Mn 2V1-xCoxGa        \n(x=0, 0.25, 0.5, 0.75, 1) Heusler alloys which exhibits high T C fully compensated ferrimagnet ic \ncharacteristic  for x=0.5 . A combined analysis of neutron diffraction and ab initio  calculations \nrevealed the crystal structure and magnetic configuration  which could not be determined f rom \nthe X-ray diffraction and magnetic measurements . As reported earlier, R ietveld refinement of \nneutron diffraction data confirmed L21 structure for Mn 2VGa and Xa structure  for Mn 2CoGa . The \nalloy s with x=0.25  and 0.5  possess L21 structure with Mn (C)-Co disorder.  As the Co \nconcentration reaches 0. 75, a structural transition  has been observed from disordered L21 to \ndisord ered Xa. Detailed ab initio  studies also confirm ed this structural transition. The reason for \nthe magnetic moment compensation in Mn 2(V1-xCox)Ga was identified  to be different from that \nof the earlier reported fully compensated ferrimagnet (MnCo )VGa . With the help of neutron \ndiffraction and ab initio  studies, it is identified that t he disordered L21 structure with antiparallel  \ncoupling between the ferromagnetically  align ed magnetic moments of  (Mn (A)-Mn (C)) and  (V-Co) \natom pairs enables the compensation in  Mn 2V1-xCoxGa. 2 \n 1. Introduction  \n \n     Half-metallic fully compensated ferrimagnets  (HMFCF i) with high Curie temperatures are  \npotential candidates for spintronic application s especially for the spin-transfer torque based \nrandom access memory (STT -RAM)  [1]. They posse ss 100 % spin polarization at the Fermi \nlevel along with zero net magnetization and produces no  stray magnetic field which  increases  the \nefficiency of spintronic devices . Even though HMFCF i posses ses zero magnetic moment  \nmacroscopically , they significantly differ from conventional antiferromagnets . Unlike the \nconventional antiferromagnets , zero magnetization occurs in HMFCFi due to the magnetic \nmome nt compensation  arising from the different  sub-lattices which are chemically or \ncrystallographically inequivalent [2]. There are reports which discuss  the possibility of having \nzero moment half -metallicity in  different kinds of materials such as Heusler alloys  [3], dilute \nmagnetic semiconductors  [4], double perovskites  [5] and transition metal pnictides  [6]. Among \nthese, the most favourable  candidate is Heusler alloys  which are ternary intermetallics having \ngeneral formula X 2YZ  for full -Heusler alloys and XYZ for half -Heusler alloys  (where  X, Y are \ntransition metal and Z is an sp valent atom) . The  band structure and thereby the physical \nproperties of these alloys can be tuned by modifying the chemical composition through the \nsubstitution of  certain elements at specific sites.  The Slater -Pauling rule (S-P rule) , i.e.,  Mt=Zt-24 \n(for full -Heusler  alloys) , which relates the total magnetic moment per formula unit (M t) to the \ntotal number of valence electrons  (Zt) of half -metallic Heusler alloys can be utilized as an \nimportant tool to engineer new materials with improved physical properties like zero moment  \nhalf-metallicity.  The rule predicts hal f-metallic zero magnetic moment state  in full-Heusler \nalloys having 24 valence electrons  [7]. Galanakis I et al . have shown that half-metallic fully \ncompe nsated fe rrimagnetism  is achievable in  well-known  half-metallic Mn 2VZ (Z=Al ,Si) \nHeusler alloys  by substituting  Co at the Mn site  [8]. The alloys (MnCo) VAl and (Mn 1.5Co0.5)VSi \nhaving 24 valence electrons are predicted to be HMFCFi . Based on the ir theoretical prediction , \nexperimental investigations have been reported on (Mn 2-xCox)VZ (Z=Ga ,Al) alloy s [9,10] . This \nis interesting since the parent alloys Mn 2VZ (Z=Ga ,Al) are reported to have half -metallic \ncharacteristics  [11,12] . Even though the results show ed that the magnetic moment is \ncompensating  in these Co substituted systems , the Curie temperature decrease s below room \ntemperature with increase in Co concentration for both the alloy  series  which is not favourable  3 \n for devise applications. On the o ther hand , our recent  experimental  investigation show ed that the \nCo substitution at the V site has resulted in total magnetic moment compensation without \nsignificant change in Curie temperatures in the ferrimagnetic Mn 2VZ (Z=Ga ,Al) Heusler alloys  \n[13]. The observed Curie temperature s were  well above the room temperature  (> 650 K)  for \nMn 2V1-xCoxZ (Z=Ga ,Al; x=0,0.25,0.5,0.75,1)  alloys which were in contrast to the earlier reports \non Co substitution at the Mn site. The observed magnetic moment  and T C were 0.10 f.u.and \n706 K for Mn 2V0.5Co0.5Ga and  0.06 f.u.and 659 K for Mn 2V0.5Co0.5Al respectively . The \npossible reason for the magnetic moment compensation  was attributed to the  antiparallel  \ncoupling between the ferromagnetically  aligned  moments of  (Mn (A)-Mn (C)) and  (V-Co) atom \npairs . But there was no experimental or theoretical evidence for t his proposed magnetic \nconfiguratio n. Thus in order to determine the crystal  structure  and magnetic configuration  of the \ninteresting alloys Mn 2V1-xCoxGa (x=0,0.25,0.5,0.75,1) , neutron diffraction  measurement and \ndensi ty functional theory (DFT) based ab initio calcu lations  were carried out . The c rystal  \nstructure and the magnetic  configuration  were  identified  by comparing the neutron diffraction  \n(ND)  results with that of the ab initio  calculation s.  \n2. Experimental  & Computational  details  \n     The details on  sample preparation, discussions on X-ray diffraction and magnetic \nmeasurements  can be found in our earlier report  [13]. Room temperature  neutron diffraction \n(ND) measurements were carried out at a wavelength of 1.48 Å using the position -sensitive \ndetector based focusing crystal diffractometer PD -3 installed by the UGC -DAE CSR Mumbai \nCentre at the Dhruva reactor, Trombay  [14]. Spin-polarized  band structure calculations were \ncarried out for the Mn 2V1-xCoxGa (x= 0, 0.25, 0.5, 0.75, 1)  Heusler  alloys using full -potential \nlinearized augmented plane wave method (FP -LAPW) implemented in WIEN2k code [15]. \nPedrew -Burke -Ernzerhof  parameterization scheme was used for the  generalized gradient \napproximation (GGA)  exchange -correlation potential  [16]. The c alculation s were  carried out for \nthe 16 atom s supercell . The k -space integration was carried out by the tetrahedron method [17] \nby taking  150 k points in the irreducible Brillouin zone. The plane wave cut off parameter \nRMT×Kmax was taken as 9. RMT×Kmax is the product of the  smallest  atomic sphere radius  times the \nlargest K -vector of the plane wave expansion of the wave function which determines the size of \nthe basis set and thereby the accuracy  of the calculation.  The angular momentum truncation 4 \n parameter lmax was set to 9.  The energy convergence ( -ec) and charge convergence ( -cc) criterion \nwas set to 10-4 Ry and 10-4 e respectively.  \n       X2YZ Heusler alloys  (X, Y are  transition metal atom and Z is sp valent atom)  crystallize in \nthe cubic  Cu2MnAl type structure (Space group: 225, L21, Fm𝟑̅m) having four interpenetrating \nfcc sub -lattices  namely A, B, C and D  with coordinates (0,0,0), (1\n4 ,1\n4,1\n4), (2\n4,2\n4 ,2\n4) and (3\n4 ,3\n4 ,3\n4) \nrespectively . X atoms  are equally occupied in the A and C sub-lattices . The  Y and Z atoms \noccupy the B and D sub-lattices respectively.  Apart from the usual Cu2MnAl structure , Heusler \nalloys are found to crystalli ze in Hg 2CuTi type structure also, commonly known as inverse \nHeusler structure (space group: 216, Xa, F4̅3m). Unlike the  L21 alloys, the valance of X \ntransition -metal atom is smaller than that of the Y transition -metal atom for the Xa alloys and the \natoms in the B and C sub-lattices will be swapped as shown  in figure 1(a)&( b). In the case of  \nMn 2V1-xCoxGa, the atomic ordering along cube diagonal would be  Mn (A)- (V/Co )(B)-Mn (C)-Ga(D) \nin the L21 structure and Mn (A)-Mn (B)-(V/Co )(C)-Ga(D) in the Xa structure . So t he structural \nrefinement of ND pattern and ab initio  calculations  were carried out by considering both  L21 and \nXa crystal structure s.  \n \n \n \n \n \n \n \n \n X Y Z \nL21 Xa (a) (b) \nA B C D A \nFigure 1 : (a) L21 crystal structure with X (A)-Y(B)-X(C)-Z(D)  atomic configuration along \nthe cube diagonal (b) Xa crystal structure with X (A)-X(B)-Y(C)-Z(D)  atomic \nconfiguration. A, B, C and D represents the four interpenetrating fcc lattices.  5 \n 3. Results & discussion  \n3.1 Neutron diffraction studies on Mn 2V1-xCo xGa \n       Earlier neutron diffraction studies showed that Mn 2VGa alloy  were  in the cubic L21 structure  \n[18]. So i n our previous report on the structural and magnetic properties of the Mn 2V1-xCoxGa \n(x=0,0.25,0.5,0.75,1)  alloys, Rietveld refinement of X-ray diffraction  (XRD)  patterns was carried \nout by assuming the L21 structure . At the same time, the report by Liu et al. show s that the other \nend member  Mn 2CoGa crystallize in the inverse  (Xa) crystal structure  [19]. So the crystal \nstructure of substituted alloys could be either L21 or Xa. This was not taken into consideration in \nour previous report.  So in order to remove the uncertainty in the crystal structure , we have \ncarried out the refinement by considering the Xa structure also. The Rietveld refinement  carried \nout on Mn 2V1-xCoxGa alloys  showed that the X-ray diffraction patterns  were  fitting well for both \nL21 and Xa structural models  resulting in the similar  lattice parameter and χ2 value s. This makes it \ndifficult to distinguish the crystal structure of the se Co substituted alloys  through  XRD analysis . \nTo illustrate this problem  in a  better  manner , XRD  patterns  of Mn 2V1-xCoxGa \n(x=0,0.25,0.5,0.75,1)  alloys in both L21 and Xa crystal structures have been simulated using \nFullProf software [20] as displayed in figure 2 (a)&(b). It is clear  that there are no noticeable \ndifferences in the simulated L21 and Xa X-ray diffraction pattern s. Our earlier experimental report  \nshowed similar X -ray diffraction pattern s without having any order dependent  superlattice  \nreflections  (111) and (200). The absence of superlattice  reflections was  attributed to the similar \natomic scattering factors of X-ray for the elements Mn , V, Ga and Co . The presence of atomic \nanti-site disorder was also unavoidable . The ambiguity in the crystal structure  could be resolved \nby doing neutron diffraction  (ND)  measurements  since the coherent scattering length for \nindividual elements differs  significantly  (Mn=  -3.73 fm; V=  -0.38 fm; Ga=  7.288 fm; Co=  2.49 \nfm). In order to check this, the neutron diffraction patterns of the alloys were simulated as shown \nin figure 3 (a)&(b). For simulating the X -ray and neutron diffraction patterns, earlier reported \nexperimental lattice parameter was used.  Unlike the X-ray diffraction patterns, the simulated  ND \npatterns show  that e xcept the (333) peak , all other peaks are paired in both L21 and Xa crystal \nstructure . L21 and Xa crystal structures are easily distinguishable  by looking at  the relative \nintensities of these paired peaks  which were not the case  with the X-ray diffraction pattern s. For \nthe L21 structure, the second peak in each pair is more intense than the first peak and for the Xa 6 \n structure , the intensity of the first peak is higher than that of the second one.  Another noticeable \ndifference  between the two simulated ND pattern  is that the (333) peak in Xa structure is more \nintense than that in the L21 structure . Here a striking difference between the simulated XRD and \nthe ND patterns has to be mentioned. The visible peaks such as (220), (400), (422), (440) and \n(620) in the XRD patterns (indexed in green)  are having negligible intensity in the ND patterns. \nAt the same time, the low intense peak s such as (111), (200), (311), (222), (331), (420) etc…  in \nthe XRD patterns (indexed in black ) are pronounced in the ND patterns.  Thus the simulated ND \npatterns clear ly show the difference between L21 and Xa crystal structure .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 2 : Simulated XRD patterns of Mn 2V1-xCoxGa alloys in the (a) L21 and (b) Xa \ncrystal structure . \n15 30 45 60 75 90105 12002040600204060021426302346690214263\n(333)\n(620)(440)(422)(400)(220)x=0 \n \n (111)\n(200)\n(311)\n(222)\n(331)\n(420)\n(531)\n(442)x=0.25 \n x=0.5 \n x=0.75 \n x=1  \n Intensity (103 counts )\n2 (degree)\n(a)L21\n15 30 45 60 75 90105 1200275402754029580306003060\n  \n (333)\n(620)(440)(422)(400)(220)(111)\n(200)\n(311)\n(222)\n(331)\n(420)\n(531)\n(442) \n  \n  \n Intensity (103 counts )  \n x=1\nx=0.75\nx=0.5\nx=0.25\nx=0\n2 (degree)\n(b)Xa7 \n  \n \n \n \n \n \n \n \n \n \n \n \nAs the ND patterns would distinguish the L21 and Xa crystal structure , we have recorded the ND \npatterns of Mn 2V1-xCoxGa alloys at room temperature  which is shown in figure 4 (a). Before \ndiscussing the Rietveld refined patters as shown in figure 4 (b), some distinguishable features  can \nbe observed from the unrefined ND patterns.  Comparison of  the simulated L21 and Xa ND \npatterns  with experimental data reveals that Mn 2VGa  crystallizes  in L21 structure  and Mn 2CoGa \ncrystallizes in  Xa structure . The relative intensity of the nearby peaks is in good agreement with \nthe simulated patterns.  It is evident that the order dependent superlattice reflections (111) and \n(200) are present with a huge  intensity which was absent in the XRD patterns. So the reported \nabsence of superlattice reflections in the XRD patterns of Mn 2V1-xCoxGa alloys can be attributed Figure 3 : Simulated neutron diffraction patterns of Mn 2V1-xCoxGa alloys in the (a) \nL21 and (b) Xa crystal structure . \n15 30 45 60 75 90105 120081609170101901122091726\n(620)(440)(422)(400)(220)\n(622)(533)(442)(531)(333)(420)(331)(222)(311)(200) \n \n2 (degree)x=0Intensity (count)\nL21(111) x=0.25\n  x=0.5\n  x=0.75\n \n(a) x=1 \n \n15 30 45 60 75 90105 120091701019032640102001122 \n Intensity (count)x=0.25\nx=0\n2 (degree) Xa\n(b)(111)\n(200)\n(311)\n(222)\n(331)\n(420)\n(333)\n(531)\n(442)\n(533)\n(622)(220)\n(400)\n(422)\n(440)\n(620) \n  x=0.5\n  x=0.75\n  x=1 \n 8 \n to the similar atomic scattering factors of X-ray for the elements Mn , V, Ga and Co  which are in \nthe same period and not due to any atomic antisite disorder .  \n \n \n \n \n \n \n \nFor the Co substitution with x=0.25, some changes in the ND pattern can be observed  compared \nto the parent Mn 2VGa . The relative intensity of the (111) peak has been increased and the \nintensity of the (311) peak is more compared to the (222) peak.  Apart from these intensity \nchanges the overall pattern resembles that of the L21. The changes in intensity suggest the \npossibility of having an atomic disorder which will be discus sed later.  When the Co \nconcentration is increased to x=0.5, more changes in  the peak intensities were observed . The \nrelative intensity of peak s in the  pairs (111) & (200),  (311) & (222), (531) & (442)  and the \nintensity of the single peak (333) resembles the Xa structure and the pairs (331) & (420), ( 533) & \n(622)  resembles the L21 structure . Another interesting point to note is that the intensity mismatch  Figure 4 : (a) Neutron diffraction patterns of Mn 2V1-xCoxGa alloys recorded at room \ntemperature. (b) Nuclear refinement of the data for 2θ>40.  \n20 40 60 80 100061301020303690112201428\nx=0 \n (111)\n(200)\n(311)\n(222)\n(331)\n(420)\n(333)\n(531)\n(442)\n(533)\n(622) \n x=1\nx=0.75\nx=0.5\nx=0.25 \n  \n   \n Intensity (103 Counts )\n2 (degree)\n(a)\n0918\n01020\n047\n0918\n40 60 80 1000611\n  \n  \n  \n  \n  \n  \n  \n  \n(b) Yobs\n Ycalc\n Yobs-Ycalc\n Bragg position\n  \n  \n(311)\n(222)\n(331)\n(420)\n(333)\n(531)\n(442)\n(533)\n(622)\nIntensity (103 Counts )\n2 (degree)x=0x=0.25x=0.5x=0.75x=19 \n between the peaks in each pair is less compare to  the simulated pattern s. The above points \nindicate  that the crystal structure of Mn 2V0.5Co0.5Ga alloy could be  either  Xa or L21 with atomic \nantisite disorder.  For the Co concentration with x=0.75, the ND pattern is similar to that of \nMn 2CoGa which is Xa. This implies that there could be a structural transition from L21 to Xa as \nthe Co concentration increases  which sho uld be confirmed by doing the R ietveld refinement . By \nkeeping these things in mind, we tried to do the Rietveld refinement for  the ND pattern  by using \nthe FullPro f software . Since Mn 2V1-xCoxGa alloys are  ferrimagnetic at room temperature         \n(TC > 650 K) , nuclear refinement was carried out by considering the scattering angle  (2θ) above \n40˚at the beginning of the refinement . The refined ND patterns are shown in figure 4 (b). The \nrefinement showed that the Mn 2VGa alloy cry stallizes in the L21 structure with 6 % atomic \nantisite disorder between V and Ga . For x=0.25 and 0.5,  different structural models  such as  L21, \nXa, L21 + (various disorders), Xa + (various disorders)  have been considered.  For x=0.25, the best \nfit was obtained only for L21 + (Mn (C)-Co disorder).  For the alloy with x=0.5 , the structural \nmodels  L21 + (Mn (C)-Co disorder) and  Xa + (Mn (B)-Co disorder)  have resulted in identical fit to \nthe pattern . The ambiguity in crystal structure for this alloy  has been removed with the help of  \nab initio  calculations  which is di scussed in the next section. This calculation  identified the \ncorrect structure to be L21+(Mn (C)-Co) disorder . For the alloy with x=0.75, Xa + (Mn (B)-Co \ndisorder)  structural model has given the best fit.  Mn 2CoGa was found to crystallize in the Xa \nstructure with 5 % atomic antisite disorder between Mn (B) and Co . The considered structural \nmodels, the observed disorders, lattice parameter and the respective χ2 values are shown in Table \n1. It was evident from the  refinement that  the majority of the substituted Co has go ne to the Mn \nsite and the corresponding Mn atoms have gone to the V site  for the alloys with x=0.25 and 0.5 . \nThe antisite Co content was reduced from 89 % to 5 % as the Co concentration increases from \nx=0.25  to 1. The obtained crystal structure after ND pattern refinement is in agreement with the \npreliminary observations. Interesting point is that there is a gradual structural transition from  L21 \nto Xa as the Co concentration increases from x=0 to 1 which was not evident from the XRD \nanalysis.  \n  10 \n  \n \n \n \n \n \n \n x Fitted structural models  Co d isorder  \n(%)  \nLattice \nparameter  \n(Å) χ2 \n0 L21 + (V-Ga disorder)  6  5.895  11 \n0.25 L21+ (Mn (C)-Co disorder)  89  5.878  4.6 \n0.5 L21 + (Mn (C)-Co disorder)  \nor \nXa + (Mn (B)-Co disorder)  86  \n \n71  5.862  \n \n5.866  5.7 \n \n4.6 \n0.75 Xa + (Mn (B)-Co disorder)  25  5.844  8.1 \n1 Xa + (Mn (B)-Co disorder)  5  5.862  4.1 \nTable 1 : The considered structural models, the observed disorders, lattice parameter and the \nrespective χ2 values for neutron diffraction refinement of Mn 2V1-xCoxGa alloys.                                 \n \n \n 11 \n Having identified the crystal structure of Mn 2V1-xCoxGa, the next step is to do the refinement by \nconsidering the magnetic contribution to the neutron diffraction pattern by taking the whole \nscattering  angle 5 -120˚. The magnetic refinement showed that the ferromagnetically aligned Mn \natoms in the A and  C sublattices couples ferrimagnetically with V atoms in Mn 2VGa. The \nobserved magnetic moments were -1.28 μB for Mn (A) & Mn (C) and 0.71  μB for V which sums up \nto -1.85 μB/f.u. For Mn 2CoGa, refinement showed that the Mn atoms in the A and B sublattices  \nare aligned ferrimagnetically with a moment of -1.82 μB for Mn (A), 3.20  μB for Mn (B) and 0.52 μB \nfor Co which sums up to 1.9 μB/f.u. For the substituted alloys (x=0.25, 0.5, 0.75), the magnetic \nphase refinement of ND data was not possible d ue to certain constraints. Since the disordered  L21 \nand Xa structure s have 6 magnetic atoms per formula unit (Mn (A), Mn (C/B), V (B/C), Co (B/C), \ninterchanged Mn and Co),  fixing the individual moment became difficult. For example, in the \ncase of x=0.25, As the ‘ C’ sublattice is occupied by Mn and Co atoms and the ‘ B’ sublattice is \noccupied by V, Co and Mn atoms, it is difficult to figure out the individual atomic contribution \nto the total neutron scattering amplitude from these sub -lattices. Another difficulty is th at, due to \nthe ferrimagnetic  nature, magnetic reflections were overlapped with the nuclear reflection in the  \ndiffraction patterns and very small magnetic contribution was observed in the low angle (111) \nand (200) superlattice reflections only. This adds to the difficulty of magnetic phase  refinement \nas the goodness of the fit has to be monitored by observing these two peaks alone.  So in order to \nidentify the magnetic configurations and to confirm the crystal structure, density functional \ntheory based (DFT) ab initio  calculations were carried out.  The following section discusses  the \nab initio  calculations on  Mn 2V1-xCoxGa alloys.  Conclusions on the crystal structure and the \nmagnetic configurations  are given on the basis of a  combined analysis of ND and ab initio  \ncalculations .                                             \n3.2 Ab initio  studies  on Mn 2V1-xCo xGa (x=0, 0.25, 0.5 , 0.75 and 1)  \n          Spin-polarized  ab initio  calculations were carried out for L21 and Xa crystal structures for \neach x value. For each structure , energy minimization  with respect to lattice parameter  (volume) \nhas been carried  out and the lattice parameter corresponding to the minimum energy was used \nfor the self -consistent field (scf) calculations. Experimental lattice parameter  for Mn 2V1-xCoxGa \nwas used for the energy minimization.  The Figure 5(a)-(e) shows the energy minimization with \nrespect to the volume for Mn2V1-xCoxGa alloys in L21, Xa and disordered structures.   12 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 5 (a) -(d): Energy minimization with respect to the volume for Mn2V1-xCoxGa \nalloys in L21, Xa and disordered structures.  \n \n \n \n \n1.20 1.28 1.36 1.44 1.52 1.60-0.4-0.3-0.2-0.1\n(a) Volume (1000 ) [a.u]3\n  Energy (- 41686 ) (Ry)\n L21\n Xa x=0\n1.20 1.28 1.36 1.44 1.52-0.6-0.5-0.4\nEnergy (- 42574 ) (Ry) x=0.25\n Xa\n L21\n L21+ disorder\n  \n(b) Volume (1000 ) [a.u]3\n1.20 1.28 1.36 1.44 1.52-0.90-0.84-0.78-0.72-0.66 x=0.5\n  Energy (- 43462 ) (Ry)\nVolume (1000 ) [a.u]3 (c) L21\n L21+disorder\n Xa\n Xa+disorder\n1.26 1.33 1.40 1.47-0.18-0.16-0.14-0.12-0.10-0.08 \nEnergy (- 44351 ) (Ry) \n x=0.75\n L21\n Xa\nVolume (1000 ) [a.u]3 (d)\n1.20 1.28 1.36 1.44 1.52 1.60-0.52-0.48-0.44-0.40-0.36\n L21\n Xa\n  Energy (- 44351 ) (Ry) x=1\n(e) Volume (1000 ) [a.u]313 \n From the energy minimization , it is clear that the alloy with x=0 ha s minimum energy in the L21 \ncrystal structure . ND analysis indicate d that for the alloy with x=0.25, most of the substituted Co \natoms have gone  to the Mn (C) site and equivalent Mn (C) atoms have gone to the V site  in the L21 \nstructure . So along with  the L21 and Xa structures , L21+(Mn (C)-Co) disordered structure  is also \nconsidered  for the energy minimization . The stoichiometry used for the calculation is \n(M1.75Co0.25)(V 0.75Mn 0.25)Ga which still ha s the total valence electron number of 23 and thereby a \nmoment of -1 μB/ f.u. as per the S-P rule.  As shown in figure 5(b) , L21+(Mn (C)-Co) disordered \nstructure has exhibited low er energy  which is in agreement with the ND result . Similarly , ND \nrefinement suggested either a L21+(Mn (C)-Co) disordered structure or Xa+(Mn (B)-Co) disordered \nstructure for x=0.5 . So the  energy minimization was carried out by considering these disordered \nstructure s also. The minimization curve s clearly show that the L21+(Mn (C)-Co) disordered \nstructure has the minimum energy compared to other structures.  The stoichiometry used for the \ncalculation is (M 1.5Co0.5)(V 0.5Mn 0.5)Ga which still has the total valence electron number of 2 4 \nand thereby a moment of 0 μB/ f.u. as per the S-P rule . For x=0.75 and 1, Xa crystal structur e has \nthe minimum energy compared to  the L21 structure. Thi s clearly shows a crossover of crystal \nstructure from L21 to Xa as the Co concentration increases  which are in agreement with the ND \nresults . Table 2 shows the  calculated lattice parameters which are compared to that of the values \nreported in other experimental and ab initio  studies.  Table 3  shows the total and atom -resolved \nmagnetic moments  and spin polarization values of Mn 2V1-xCoxGa. The expected magnetic \nmoments from S -P rule are -2, -1, 0, 1, and 2 μ B/f.u. and the reported experimental values in our \nearlier report  are 1 .80, 0.97, 0.10, 1.11 and 2.05 μB/f.u for x=0, 0.25, 0.5, 0.75 and 1 respectively. \nThe calculated total moments also follows the experimental trend of magnetic moment \ncompensation as the Co concentration reaches x=0.5. For the x=0  alloy  (Mn 2VGa), the \ncalculated total magnetic moment  in the low er energy L21 structure is  -1.98 μB/f.u. which is close \nto the S -P value.  \n \n \n \n 14 \n  \n \n \n \n \n \n \n  \nx Crystal structure  Lattice parameter (Å)  \nPresent \nCalculation  Other Reports  \nCal. Exp. \n0 L21 5.824 5.820 [21] \n5.685 [22] \n5.808 [23] 5.914 [13] \n5.905 [11] \n6.095 [24] \n0.25 L21+disorder  5.805  - 5.874 [13] \n0.5 L21+disorder  5.796  - 5.878 [13] \n0.75 Xa 5.824 - 5.881 [13] \n1 Xa 5.789 5.78 [21] \n5.61 [25] 5.877 [13] \n5.862 [19] \n5.869 [26] \nTable 2 : Crystal structure and the corresponding calculated lattice parameters of Mn 2V1-\nxCoxGa alloys. Calculated (Cal) and experimental (exp) lattice parameters reported by other \nresearchers are also shown for the comparison.  \n                                \n \n \n 15 \n  \n \nThe obtained moment value is in agreement with the earlier report  [18]. The Mn moments in the \nA and C sublattices are found to be in ferromagnetic alignment which is common in the case of \nMn-based Heusler  alloys in the L21 structure.  For Mn 2CoGa ( x=1), Mn moments in the A and B \nsublattices are found to be in antiparallel  alignment which is observed  in the case of Mn -based \nHeusler  alloys in the Xa structure . This is also in agreement with the earlier report [27]. For \nx=0.25  and 0.5 , L21 structure  with Mn (C)-Co disorder is the minimum energy state  having  a total \nmoment of -0.99 and 0 μB/f.u. respectively. This is close to the experimental and S -P value  \n(shown in  Table 3 ). For x=0.75 also, the calculated moment  is close to the experimental value.  \nThe obtained magnetic configuration from the ab initio  calculation for the alloy with x=0.5  is \nsimilar  to the assumption made for explain ing the magnetic moment compensation  in our \nprevious report.  L21 structure with dduu magnetic configuration was assumed for Mn (A), Mn (C),  \nx \n Crystal \nstructure  Mag.  \nConfig.  atom resolved magnetic moments  \n(μB) Total \nmoment \n(Cal)  \nμB/f.u. Total \nmoment  \n(S-P rule)  \nμB/f.u. Total \nmoment    \n(Exp)[13] \nμB/f.u. P \n(%) \nMn(A) Mn(B/C) V Co \n L21 ddu -1.45 -1.45 0.82 - -1.98 -2 1.80 94 \n0.25 L21+disorder  dduu  -1.32 -1.53 0.73 \n2.90 (Mn)  0.48 -0.99 -1 0.97 99 \n0.5 L21+disorder  dduu  -1.35 -1.31 0.65 \n2.82 (Mn)  0.66 0 0 0.10 99 \n0.75 Xa dudu  -1.91 2.69 -1.43 0.94 0.99 1 1.11 98 \n1 Xa duu -1.82 2.85 - 0.99 2.00 2 2.05 99 \nTable 3: Identified crystal structure, calculated (cal) atom -resolved and total magnetic moments \nand spin polarization values of Mn 2V1-xCoxGa. Experimental (exp) magnetic moments are also \nshown for the camparison. The antisite Mn moment is shown near to the V mo ment in the case \nof x=0.25 and 0.5.  For representing the magnetic configuration  of Mn(A) Mn(B/C)V Co , upward \nand downward directions are denoted as ‘u’ and ‘d’ respectively.  \n                                \n \n \n 16 \n V(B), Co (B) atoms (here u implies upward moment direction and d implies downward moment \ndirection).  The present calculation also showed dduu  configuration  irrespective of the disorder . \nIn addition to this, the disordered Mn atom at the V site has aligned  in the upward direction.  This \nimplies t hat the  L21 structure and ferromagnetic coupling  of Mn atoms in the A and C sublattices  \nof Mn 2VGa ha ve not altered with Co substitution up to x=0.5 . Also , the substituted Co and V \natoms couples  ferro magnetically for x up to 0.5 . In short , the disordered L21 structure with the \nferrimagnetic  coupling between the ferromagnetically  aligned (Mn (A)-Mn (C)) and  (V-Co) atom \npairs enables the magnetic moment compensation in  Mn 2V1-xCoxGa. Figure  6 shows the \nvariation of atomic moments with the Co concentration.  For the  Co-substituted alloys, t he \nMn (B/C) and Co  moment increases and the Mn (A) and V moment decrease gradually  as x  increases \nfrom 0.25  to 1.  \n \n \n \n \n \n \n \n \n \n \n \nL.Wollm an et al . have carried out a detailed theoretical investigation on the structural and \nmagnetic properties of the cubic Mn 2Y(3d/4d)Z (Z=Ga,  Al) alloys  [21]. The results show  that for \nthese Mn -based alloys with Z t ≤ 24, L21 structure is favoured  (eg: Mn 2VGa) and when Z t >24, Xa \nstructure is favoured  (eg: Mn 2CoGa).  Another interesting observation is that due to the \ndifferences in the nearest neighbour  atoms, L21 Mn-based alloys have ferromagnetic coupling for Figure 6 : Variation of atomic moments with the Co concentration \nfor Mn 2V1-xCoxGa.                                \n \n \n \n0.00 0.25 0.50 0.75 1.00-2-10123\n  \n   Mn(A)\n Mn(B/C)\n V\n CoAtomic moment (B)\nCo concentration x17 \n the Mn atoms at the A and C sub -lattices  and in the case of Xa Mn-based alloys, the coupling \nbetween the Mn atoms in the A and B sub -lattices is ferrimagnetic . This is one of the \ncharacteristic features of Xa Mn-based alloys.  This observation  is comparable to our present \ninvestigation. In our investigation, the valence electron number (Z t) of Mn 2VGa  is varied from \n22 to 26 by substituting Co at the V site. Our calculation also shows a ferro magnetic  coupling \nbetween the Mn atoms at different sub -lattices for the alloys with x= 0, 0.25 and 0.5 which \ncrystallizes in  the disordered L21 structure .  The alloys with x=0.75 and 1 crystallize in the Xa \nstructure  and the Mn moments are coupled ferrimagnetically . \n       Since  the Mn 2V0.5Co0.5Ga alloy exhibits a fully compensated ferrimagnetic state with high \nTC, it is interesting to investigate the half -metallic characteristic of Mn 2V1-xCoxGa. Figure  7(a)-\n(e) shows the spin-resolved  total density of states (TDOS) and partial density of states versus \nenergy diagram for the Mn 2V1-xCoxGa alloys.  Only  the dominant d band  contribution of Mn, V \nand Co is considered for the partial density of states.  As shown in figure 7 (a), Mn 2VGa  (x=0) \nexhibit  a gap at the Fermi level in the spin-down  subband . The non -zero density of states at the \nFermi level in the spin -up subband  is mostly contributed by the Mn (A,C) - 3d states . For Mn 2VGa,  \nthe calculated  spin polarization is 92 %  which indicates that it is nearly a half -metal . This is in \nagreement with the earlier reported values which  is around 94 %  [21,23] . For the Co substituted \nalloys also, the spin polarization values are more than 90 %. The calculated spin polarization \nvalues for the alloys with x=0.25, 0.5, 0.75 and 1 are 99, 99, 98 and 99 % respectively.  As \nexpected the V density of states decreases ( shown in pink colour) and the Co density of states \nincreases ( shown in green colour) with the increase in Co concentration. The earlier ab initio  \nresults on the Mn 2-xCoxVGa  alloys  (Co substitution at the Mn site) showed that the spin \npolarization  decreases to 34 %  for the compensation point x= 1 [23]. The experimental \ninvestigation carried out by Ramesh Kumar et al. also indicates the absence of ha lf-metallicity \nwhen Co is substituted at the Mn site  [9]. In contrast to this, our present investigation shows that \nthe spin polarization for Mn 2V0.5Co0.5Ga alloy is 99 %.  The difference in the density of states \ncurve for the spin up and spin down  subbands  indicates that the exchange potential for the \nmagnetic atoms are having opposite signs resulting in the ferrimagnetic  state which is evident \nfrom the atomic moments shown in Table 3  [28].  18 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 7(a) -(e): Spin-resolved  total and partial density of states versus \nenergy diagram for the Mn 2V1-xCoxGa (x=0, 0.25, 0.5, 0.75, 1)  alloys.  \n-6 -4 -2 0 2 4 6-6-30369\nx=0\n(a) Total DOS\n Mn(A,C) - d\n V(B) - d\n   \n E (eV)DOS  (states/eV)\nF\n-6 -4 -2 0 2 4 6-6-4-2024 x=0.25\n Total DOS\n Mn(A) - d\n Mn(C) - d\n Co(C)  - d\n V(B)    - d\n Mn(B) - d\nF\n   \n \nDOS  (states/eV)\n(b) E (eV)\n-6 -4 -2 0 2 4 6-4-2024 Total DOS\n Mn(A) - d\n Mn(C) - d\n Co(C)  - d\n V(B)    - d\n Mn(B) - d\nx=0.5\n   \n DOS  (states/eV)\nE (eV) (c)F\n-6 -4 -2 0 2 4 6-6-4-20246\nx=0.75\n(d)  Total DOS\n  Mn(A) - d\n  Mn(B) - d\n  V(C) - d\n  Co(C) - d\n   \n \nDOS  (states/eV)\nE (eV)F\n-6 -4 -2 0 2 4 6-6-4-20246\n(e) Total DOS\n Mn(A) - d\n Mn(B) - d\n Co(C) - d\n   \n DOS  (states/eV)\nE (eV)Fx=119 \n 4. Conclusions  \n     The c rystal structure and the magnetic configuration of Mn 2V1-xCoxGa alloys which exhibit \nfully compensated ferrimagnetic property  are identified through combined  neutron diffraction \nand ab initio  studies . Our investigation showed that the L21 and Xa structure of the Mn 2V1-\nxCoxGa alloys are well distinguishable from the neutron diffraction pattern which could not be \nidentifiable  from the X -ray diffraction  studies . Neutron diffraction studies on Mn 2V1-xCoxGa \nclearly indicated a structural transition from L21 to Xa as the Co concentration increases. Alloy \nwith x=0 crystallize s in L21, x=0.25 and 0.5 crystallize s in disordered L21, x=0.75  crystallizes in \ndisordered Xa and  x=1  alloy crystallize s in Xa structure. When  a slight  V-Ga disorder was present \nin Mn 2VGa, Mn -Co atomic antisite disorder was observed in the Co -substituted alloys. Ab initio  \nstudies  have confirmed the structural transition observed in the neutron diffraction patterns.  The \nidentified magnetic configuration from the ab initio  studies for x=0, 0.25, 0.5. 0.75 and 1 are \nddu, dduu , dduu , dudu  and duu respectively . It was evident that the disordered L21 structure \nwith the antiparallel  coupling between the ferromagnetically  aligned (Mn (A)-Mn (C)) and  (V-Co) \natom pairs enables the magnetic moment compensation in  Mn 2V1-xCoxGa. A high spin \npolarization value of more  than 90 %  was observed for all the alloys including the high T C fully \ncompensated ferrimagnet Mn 2V0.5Co0.5Ga.  \n \nAcknowledgement  \nP V Midhunlal acknowledge IIT Madras for all the funding  and providing VIRGO supercluster \nfor the High -Performance  Computing Environment (HPCE) . P V Midhunlal and N Harish \nKumar  acknowledg e the UGC-DAE Consortium for Scientific Research, Mumbai Center, BARC  \nfor the neutron diffraction measurement s. \n \n \n \n \n 20 \n References  \n[1] Balke B, Fecher G H, Winterlik J and Felser C 2007 Mn 3Ga, a compensated ferrimagnet \nwith high Curie temperature and low magnetic moment for spin torque transfer \napplications Appl. Phys. Lett.  90 1–4. \n[2] Hakimi M, Venkatesan M, Rode K, Ackland K and Coey J M D 2013 The zero -\nmagnetization Heusler ferrimagnet J. Appl. Phys.  113 2013 –6. \n[3] Galanakis I, Özdoǧan K, Şaşinoǧlu E and Aktaş B 2008 Ferrimagnetism and antiferro -\nmagnetism in half -metallic Heusler alloys Phys. Status Solidi Appl. Mater. 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B  60 \n13006 –10 \n \n "    },    {        "title": "1510.05087v1.Ferrimagnetic_nanostructures_for_magnetic_memory_bits.pdf",        "content": "1 \n Ferrimagnetic nanostructures for magnetic memory bits  \nAUTHOR NAMES  \nA. A. Ünal ,1 S. Valencia,1 D. Marchenko,1 K. J. Merazzo,2 F. Radu,1 M. Vázquez,2 and J. \nSánchez -Barriga1,* \nAUTHOR ADDRESS  \n1Helmholtz -Zentrum Berlin für Materiali en und Energie, Albert -Einstein -Str. 15, 12489 \nBerlin, Germany  \n2Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Madrid, Spain \nKEYWORDS  Heat -assisted magnetic recording, bit- patterned media, magnetic data storage, \nferrimagnetic antidots  \n \n \n     2 \n ABSTRACT   \nIncreasing the magnetic data recording density requires reducing the size of the individual \nmemory elements of a recording layer as well as employing magn etic materials with \ntemperature- dependent functionalities. Therefore, it is predicted that t he near future of \nmagnetic data storage technology involves a combination of energy -assisted recording on \nnanometer -scale magnetic media. We present the potential of heat -assisted magnetic \nrecording on a patterned sample; a ferrimagnetic alloy composed of a rare earth and a transition metal, DyCo\n5, which is grown on a hexagonal -ordered nanohole array membrane. \nThe magnetization of the antidot array sample is out- of-plane oriented at room temperature \nand rotates towards in- plane upon heating above its spin- reorientation temp erature (TR) of \n~350 K, just above  room temperature.  Upon cooling back to room temperature (below T R), we \nobserve a well -defined and unexpected in- plane  magnetic domain configuration modulating \nwith ~45 nm. We discuss the underlying mechanisms giving rise to this behavior by \ncomparing  the magnetic properties of the patterned sample with the ones of its extended thin \nfilm counterpart .  Our results pave  the way for  novel applications of ferrimagnetic  antidot \narrays  of superior functionality  in magnetic nano -devices near room temperature.   \nINTRODUCTION  \nThe research on artificially engineered spintronic materials in the for m of layered thin films \nand low -dimensional structures has experienced a tremendous boost due to the versatility of \ntheir applications in computing technology. This is reflected by the development of magnetic \ntunneling based read- out technologies [ 1], magnetic sensors [ 2,3], and non- volatile \nmagnetoresistive random -access memory devices [ 4]. Because conventional magnetic data \nstorage relies on two -dimensional arrays of magnetic bits, developing higher density, cheaper \nand faster devices relies on reducing the lateral size of individual memory elements or data 3 \n storage bits [ 5]. Ultimately controlling the magnetic anisotropy and  shape of nanostructures, \nas well as understanding their magnetic properties in reduced dimensions are key aspects that \nhave received a sustained interest over the past decades [ 6]. For instance, in antiferromagnetic \n(AF) materials, which have crucial impo rtance in exchange- bias systems, finite -size effects \nlead to a scaling of the magnetic properties such as the ordering temperature or the magnetic anisotropy [ 7], while in ferromagnetic (FM) materials control of the shape anisotropy can be \nachieved dependi ng on the geometry of the objects [ 8]. \nNanostructured magnetic materials often exhibit novel  properties over their bulk \ncounterparts as a consequence of reduced atomic coordinates and modified density of states. Magnetic systems of low dimensionality like artificially grown or self -organized FM \nnanostructures [ 9,10], arrays of nanostructures [ 11,12], nanowires [ 13,14,15], and antidot \narrays [ 16] have been intensively investigated, mainly triggered by the discovery of \nspontaneous self -organization of magnetic FePt  nanoparticles on a surface [ 11]. In all these \nfields, there is a rapid decrease of the relevant length scales of the magnetic structures down to the sub- 100 nm range. To illustrate this , a storage density of 1 Tbi ts/in\n2 would require  a bit \nlength of around 25 nm. The underlying idea is to replace the comparatively large  randomly -\noriented magnetic grains in conventional media with a nanoscopic region of a single magnetic \ndomain [17]. As the magnetic stability of individual particles scales with the  material  \nanisotropy constant and the particle volume , the use of FM FePt alloyed nanoparticles with \nvery high magnetic anisotropy (only in their chemically ordered face -centered tetragonal L1 0 \nphase) has been  considered  one of the promising routes towards  future ultrahigh- density \nrecording -media applications [ 18,19]. In this case, the suppression of superparamagnetic  \neffects at very small volumes  might prove crucial to achieve  stable and enhanced magnetic \nproperties at the smallest length scales.  On the other hand, the use of ferrimagnetic  \nnanoparticles with controlled coercivity near room temperature might be  a more desirable  \ncompromise  to additionally  achieve full  control ove r the magnetic properties.  4 \n Recently, ferrimagnetic (FI) alloy systems  with relatively low coercive fields , such as  the \ncobalt (Co) -  dysprosium (Dy) alloy  DyCo 5, have been proposed as ideal candidates for  \nmagnetic data storage and spintronic  applications  [20] owing to their flexibility in controlling \na tunable perpendicular exchange -bias effect at room temperature [ 21]. Here the Dy-Co alloy \nacts as a hard FI material with  relatively low coercive field and well-defined perpendicular \nmagnetic anisotropy at room temperature . The system  exhibits a spin -reorientation \ntemperature (TR) of ~350 K above which the uniaxial magnetic anisotropy rotates from out -\nof-plane to in -plane . Such  a remarkable change of the anisotropy axis within a narrow \ntemperature window nea r room temperature  renders  this type of F I alloys  one of the  \ntechnologically relevant and  promising  materials  for future spintronic  applications. Therefore , \nparticularly taking into account that the magnetic properties of DyCo 5 nanostructures has \nremained unexplored, patterning  of this type of  FI alloys  down to the nanoscale  as well as \nunderstanding their magnetic properties  at ultimate length scales  are both of scientific and \ntechnological interest.  \nAmong the different growth methods of  magnetic nanostructure s, the anodization of \nalumina templates and the subsequent deposition of  magnetic antidot array s have  certain \nadvantages that overcome  the unwanted effects of  superparamagnetism.  This technique \nemploys ultrahigh -density substrates th at are patterned down to the nanoscale  as temp lates to \ngrow thin films on top  [16,22]. In the case of nanoporous alumina membranes, nanoholes are \narranged in a fully -controllable hexagonal  symmetry [ 23,24], and growing a thin film on top \nresults in an array of antidots  [25]. The antidots act as well -ordered nonmagnetic inclusions \nwithin the magnetic thin film, promoting the creation of single nano- domain structures whose \nsizes can be manipulated depending on the anodization conditions. This is a  much  less \nexpensive technique compared to lithographic patterning methods. These nanostructures can \nbe promising candidates for a new generation of ultrahig h-density magnetic -storage media 5 \n mainly due to the absence of a superparamagnetic limit, as there are  no isolated small \nmagnetic entities . This is due to the fact that the nanoholes introduce a locally distributed \nshape anisotropy [ 26], act themselves as pi nning centers for magnetic  wall displacements, and \ntheir periodic distribution determines the whole magnetization process as well as  the \nmagnetoresistance behavior [ 16]. Investigations of FI antidot arrays with tunable and \ncontrollable magnetic anisotropies  might in fact  provide superior functionalities that have not \nbeen addressed  so far .  \nIn this work, we present a nanoscale -patterned, hexagonally -ordered FI DyCo 5 antidot array \nwhere the local nanoscopic distribution of the magnetization  at room temperature  can be \ncontrolled via heati ng the sample slightly above  its spin -reorientation temperature . We report \non controlling the magnetic anisotropy as a function of temperature and we compare the nano-\npatterned sample with its plain film counterpart. U sing x -ray magnetic circular dichroism \n(XMCD) in combination with photoelec tron emission microscopy (PEEM), we reveal that the \ntemperature- dependent changes of the magnetic anisotropy in the presence of antidots results \nin well- defined and stable magnetic nano -domain configurations modulating within ~45 nm at \nroom temperature . We discuss the underlying mechanisms giving rise to this behavior . Our \nresults pave the way for future applications of FI nanostructures in heat-assisted magnetic \nnano- devices . \nMETHODS   \n   We prepared  antidot arrays of DyCo 5 by magnetron sputtering  on top of  hexagonally -\nordered alumina me mbranes , which were f abricated u sing a two -step anodization process  [20, \n25]. For the present experiments, we produced  membranes containing nanopores  with a \ntypical diameter of ~68 nm and  center -to-center  inter-pore distances of ~105 nm  [see Fig. \n1(a)] . A DyCo 5 extended film was  grown  on top of an un- patterned alumina substrate for 6 \n comparison purposes . The growth of the antidot arrays  and the extended film  was perform ed \nsimultaneously within the sputtering chamber, and their thickness was ~25 nm. The samples \nwere  obtained by  co-deposition from Co and Dy targets keeping the substrate temperature at \n~200°C. The co-deposition was done with a relative composition of 1:5 to ensure  the spin-\nreorientation transition to be closest to room temperature. The samples were capped with a 2  \nnm thick Al layer to avoid oxidation upon transport in air. The magnetic properties of the  \nextended films , their compensation and spin reorientati on temperatures, coercive fields, and \nmagnetic anisotropy axes were determined by element specific hysteresis loops at the Dy and \nCo absorption [ 21] edges u sing XMCD in transmission and scattering geometries  [27]. \n   High -resolution magnetic domain configurations around the nanoholes and across the spin-\nreorientation transition were  obtained by XMCD -PEEM imaging [ 28]. Experiments were \nperformed  at the UE49 -PGM a beamline of the synchrotron BESSY II . XMCD element -\nspecific images  were obtained by tuning the synchrotron photon energy  to the L 3 and M 5 \nresonances of  Co (779.2 eV) and Dy  (1293.9 eV), respectively .  Each of the XMCD images \nwas calculated from a sequence of images taken with circular polarization (90% of circular \nphoton polarization) and alternating the light helicity. After normalization to a bright- field \nimage, the sequence was  drift-corrected, and frames recorded at the same photon energy and \npolarization were averaged. T he difference of the two resulting  images  with opposite helicity, \ndivided by their sum , showed Co and Dy magnetic -domain contrasts, which represent the \nmagnetization vector pointing along the incidence direction of the x -ray be am. Due to the 16º  \nshallow incidence angle of the x -rays on the sample, our XMCD- PEEM measurements are  \nmainly  sensitive to the in -plane magnetization.  The samples were priorly  magnetized  by \napplying a magnetic field of ~0.5 T perpendicular to the surface, so that the XMCD images in the initial remanent  magnetic state at room temperature exhibited nearly -zero magnetic \ncontrast.  7 \n RESULTS AND DISCUSSION  \nWe investigated the formation of magn etic domains in  FI DyCo 5 grown on extended \nsubstrates and on nanoporous membranes, controlling the sample tempe rature above and \nbelow the spin- reorientation transition temperature of ~ 350 K. Scanning electron microscopy \n(SEM) [Fig. 1(a)] and XPEEM images [ Fig. 1(b)] from the nanohole array show the \nhexagonal  arrangement of  antidots. Circles and lines on the SEM image mark several \npunctual defects and structural domains that occur during the self -assembling anodization \nprocess. The distance between  the borde rs of two nanoholes, where  the metallic alloy is \nsitting, is around 37 nm. The XPEEM image shows  photoelectrons emitted from the material \ndeposited on the surroundings of the nanoholes as the bright intensity contrast, and dark \ncontrast corresponds to the nanohole locations. By varying the x -ray photon energy and \nsimultaneously rec ording XPEEM images, we obtain x -ray absorption spectra  (XAS)  for Co \n[Fig. 1(c)] and Dy  [Fig. 1(d)], showing L 2,3 and M 4,5 absorption edges, respectively.  \nAt a temperature of  385 K, just above T R = 350 K, the XMCD -PEEM magnetic contrast  \nshown in Fig. 2 reveals  clear  in-plane  ferrimagnetic ordering of Co and Dy elements, both \nbeing anti -ferromagnetically aligned with respect to each other in the imaged x -y surface \nplane.  Very c learly, t he red (blue) regions of the Co XMCD images in Fig. 2 correspond to \nblue (red) regions of the Dy XMCD  signal . Figure s 2(a -b) show  the magnetic configuration \nfrom the antidot array sample and Fig s. 2(c -d) from its  extended thin film counterpa rt. Re d \nand blue -color contrast correspond to magnetic domains pointing in the surface plane and \nprojected along the incident x -ray beam direction, with a maximum of ~10 % XMCD \nasymmetry. White (or faint) color contrast indicates eith er out -of-plane magnetization , \nmagnetization perpendicular to the incident beam direction or the zero net magnetization as  it \nwould be expected from the nanohole positions. Comparing  the XMCD -PEEM images of the \nantidot array to  the ones of the extended film sample reveals clear  differ ences in the size of  8 \n the magnetic domains; the  antidot sample exhibits nanometer -sized domains separated by \nwhite color contrast, whereas the extended thin film sample exhibits domains in the order of \nseveral micrometers. This observation suggests that the  antidots act as p inning centers for the \nmagnetization  stabilizing  magnetic nano -domains that are  naturally separat ed from each other. \nBecau se the extended thin film lacks of such pinning centers, small nucleation domains  which \ninitiate  the magneti zation reversal have  collapsed to  form larger ones . \nIn the following, we present Co XMCD -PEEM  images for three selected temperatures \nfrom a heating -and-cooling cycle; initial state at 300 K, high -temperature state at 470 K, and \nthe final state  reached  after cooling back to 300 K again, for both the extended thin film and \nantidot array samples. Figures 3(a) and 3(d) show the initial states for both samples, where nearly -zero local magnetization  signal  along the in- plane orientation is observed. The reason \nfor the vanishing magnetic contrast  is the out -of-plane spin confi guration  below  T\nR. Upon \nheating  to 470 K, which is well above the transition temperature, we clearly observe the \nappearance of  in-plane oriented magnetic domains , as shown by the  XMCD images in Fig. \n3(b) and 3(e). When  we cool the samples  back to room temperature  and re -examine the \nresulting magnetic  properties , the extended thin film sample re -establishes its original out-of-\nplane  spin configuration,  as expected [Fig. 3(c)] ; however, the antidot array exhibits  a non-\nzero local magnetization as if the magnetic domains are mostly pinned to the in-plane  spin \norientation even in the absence of any  magneti c fields upon cooling [Fig. 3(f)]. To take a \ncloser look at the remanent local  magnetization of the antidot array, we extract line profiles \nalong red and blue bits of magnetic domains as well as the real str ucture as seen by SEM and \nXPEEM. The results are summarized in Fig s. 3(g) and 3(h), where we show l ine profiles  \nextracted  along magnetic im ages for opposite magnetization directions  (blue and red lines) . \nWe clearly observe magnetic information repeating every 45- 50 nm (between M ↑ and M=0 \nstates, and between M ↓ and M=0 states) , which correlates well with the difference between 9 \n the center -to-center inter -pore distance and the nanohole diameter . One bit of magnetic \ninformation being ~ 45 nm, one can estimate a data storage density of around 75 Gbits/inch2, \nwhich is in the same order of magnitude with the stora ge density of conventiona l hard -disk \ndrives.  \nFigure 4 (a) summarizes  key experimental results together with a schematic representation \nof the magnetic anisotropy Ku of the samples as a function of temperature. It shows various \nchannels of writing and storing data using  a medium of  DyCo 5 antidot array s. The easy axis \nof the magnetization rotates from out -of-plane to in- plane as the samples are heated above T R, \nand spins of the two sublattices of Co and Dy align in the sample plane. The change of the \neasy axis  is outlined by the  Co X MCD images from an antidot hexagonal unit cell and fr om \nthe extended thin film sample  on the right -hand side of  the schematic representation depicting \nthe in -plane anisotropy (K||) at high t emperatures.  C ooling the extended thin film sample back \nto room t emperature leads to  either one of the out -of-plane anisotropy configurations in the \nabsence of an external field , as seen in Fig. 3(c) . However, the antidot array sample has more \noptions upon cooling: it can decay into two in- plane and two  out-of-plane ani sotropy \nconfigurations , as shown by the Co XMCD -PEEM images on the left -hand side  of Fig. 4(a) . \nRegions within  the antidot sample with out -of-plane magnetization show a faint color \ncontrast;  therefore not much is seen in these images. Two regions of the sa mple with two \nhexagonal unit cells of antidots showing in- plane magnetization are also shown. We attribute \nthe occurrence of  this magnetic multi- domain state  to the presence of nanoholes , which act as \npinning centers for  the magnetic anisotropy of the samp le. To illustrate this in more detail, in \nFigs. 4(b -d) we show the results of micromagnetic simulations within about one hexagonal \nunit cell obtained using the OOMMF package [ 29]. In Fig. 4(b) we show a SEM image from \nthe antidot array superimposed with blue circles around the nanoholes. The circles are used as \na mask to construct the input for the calculations. For simplicity, we simulate a Co layer with 10 \n the magnetic anisotropy oriented along the + y direction and an initial out- of-plane \nmagnetization.  In Figs. 4(c) and 4(d) we show the calculated magnetization state for the \nantidot array and the extended film in the fully -relaxed state, respectively. Very clearly, the \nantidot ar ray in Fig. 4(c) exhibits a magnetic multi- domain configuration, while the extended \nfilm remains in a single -domain state in qualitative agreement with the experiments. The \ncalculations clearly reveal that the nanoholes pin the magnetization so that it follows  circular \ncontours around them , leading to local changes in the magnetic anisotropy.  These changes are \ncaused by the fact that every nanohole gives rise to six constrictions with its first neighbors, so that magnetic domains and domain walls are pinned or trapped by these constrictions. In \nthis way, every domain wall experiences a spatially -dependent dipolar interaction originating \nfrom a landscape of pinning potentials within the whole structure, resulting in local rotations of the magnetization and thus in magnetic multi- domain configurations .  \nHaving observed these variety of magnetic domains  in the experiment , one can consider \ncontrolling their occurrence by applying either in -plane or out -of-plane magnetic field pulses \n(H\n‖ or H┴) to selectively force the system to choose one of the four possibilities  mentioned \nabove . The experimental results  summarized in Fig. 4 (a) are therefore quite analogous to the \ncase where one applies a local magnetic field pulse during the cooling process to manipulate the mag netism of the nano- domains . It would be equally possible to employ this method for \nboth longitudinal and perpendicular recording as they follow the same basic principle. When \nthe energized write head passes by the medium, it leaves behind a magnetization p attern. For \nperpendicular recording, this magnetization pattern is ‘up’ and ‘down’ rather than ‘left’ and ‘right’ for longitudinal recording.  \nContrary to the high anisotropy HAMR materials (among others: FePt, CoPt, SmCo\n5), FI \nantidots such as the ones st udied here  would be a promising candidate that facilitates writin g \ndata by means of manipulating the magnetic easy and hard axes as a function of mild-11 \n temperature treatment. T he critical temperatu re of  our magnetic  system is not the Curie \ntemperature TC but the  spin reorientation temperature  TR, which is very close to room  \ntemperature.  Note that  differently from this, high anisotropy HAMR materials need to be \nheated to temperatures close to their T C, which are around 750 -1000 K [ 30], to sufficiently \nreduce their high coercivity so that a moderate write field can induce a new magnetic state. In \nthis respect, there have been also attempts to reduce the high T C of the recording layers with \ndoping. In one example, Thiele et al.  used Ni doping to reduce the T C of FePt; however, at the \nsame time the anisotropy decreases [ 31]. Assuming that in the near future  plasmonics  and \nnear-field optics  will focus and transmit optical energy to spot sizes of around (25- 50 nm)2 \nand demagnetize the sample [ 19,30 ], lower  phase -transition  temperatures  would still be more \nfeasible considering  issues su ch as power consumption, excess -heat dissipation and risk of \ndamaging medium materials.  \nSummarizing , we fabricated ferrimagnetic  DyCo 5 antidot arrays for heat -assisted and bit -\npatterned magnetic recording. Our work de monstrates a novel functionality  of ferrimagnetic  \nbit-patterned materials that can be used to write and store  magnetic data  near room \ntemperature. The size of the nanoholes used in this work was around ~ 68 nm and their \nhexagonal lattice constant  ~105 nm. W e have shown that the magnetization of the antidot \narray sample can be tuned from an out -of-plane to an in -plane spin configurat ion upon heating \nabove its spin- reorientation temperature of ~ 350 K. We have compared the magnetic \nproperties of the patterned s ample with the ones of its extended thin film counterpart and \nobserved remarkable differences in the size of the magnetic domains.  Our results reveal  the \nformation of small magnetic nano -domains in between antidots with well -defined and stable \nmagnetic dom ain configurations. Magnetic modulations of the nano- domains contrast are \nobserved to occur within ~45 nm for opposite magnetizations . We have attributed this \nbehavior to the change in the magnetic anisotropy caused by the nonmagnetic inclusions, 12 \n which at the same time act as pinning centers upon formation of  multiple  magnetic domains  \nin agreement with our micromagnetic simulations . Finally, w e have discussed the \nimplications of our findings for heat -assisted  magnetic recording using local magnetic probes.  \nOur work demonstrates that ferrimagnetic antidot arrays exhibit tunable and controllable \nmagnetic traits with the potential to provide superior functionality for energy -assisted bit -\npatterned media near room temperature.  \n \n   \n  13 \n FIGURES  \n \n \n \n \nFigure 1.  (a) SEM and (b) XPEEM images of the hexagonal lattice antidot array of 68 nm pore size \nand 105 nm separation between pore centers . XAS spectra of (c) Co and (d) Dy elements at their L 2,3 \nand M 4,5 edges, as obtained from XPEEM energy scan s using linear horizont al polarized x -rays from \nthe field of view seen in (b).  \n  \n14 \n  \n \n \n \nFigure 2.  XMCD -PEEM images measured at 385 K at the Co L 3 and Dy M 5 edges from the antidot \narray (a- b) and the extended thin film sample (c -d) showing the ferrimagnetic ordering of the Co and \nDy magnetic moments via the corresponding blue -red XMCD contrast. Antidots stabilize nanometer -\nsized domains  separate d from each other.  \n \n   \n15 \n  \n \nFigure 3.  XMCD -PEEM images measured at Co L 3 edge from the extended thin film sample (a -c) \nand from the antidot array sample (d -f) at the labeled temperatures. (g) Line profile along the structure \nof five neighbor ing antidots, XPEEM (black line) and SEM (dashed line). (h) Line profiles along \nXPEEM magnetic images for opposite magnetizations (blue and red lines) show  magnetic information \nrepeating every 45 -50 nm (between M ↑ and M=0 states, and between M ↓ and M=0 states).  \n16 \n  \n \nFigure 4. (a) S chematic representation of the heat -assisted magnetic recording process and the spin \nreorientation transition as observed by XPEEM. A selection of images at the Co  L3 edge summarizes  \nthe experimental results.  On the left, the room -temperature results correspond to different XMCD \nimages of the antidot array.  (b-d) Results of micromagnetic simulations. In (b ), a SEM image from the \nantidot array superimposed with blue circles around the nanoholes is shown. The circles are used as a \nmask to construct the input for the calculations. The magnetic anisotropy is oriented along the + y \ndirection and the initial magnetization is out -of-plane. (c ) Calculate d multi -domain configuration of  \nthe antidot array in the relaxed state. Small arrows depict the magnetization directions, blue and red \ncolors the projection of the magnetization along the y  direction. (d) Same calculations as in (c ) for the  \nextended film, revealing a single- domain configuration.  \n17 \n AUTHOR INFORMATION \nCorresponding Author  \n*E-mail: jaime.sanchez -barriga@helmholtz -berlin.de  \nACKNOWLEDGMENT  \nK. J. M and M. V gratefully acknowledge the support from MINECO under project \nMAT2013- 48054- C2-1-R. \n \nREFERENCES  \n                                                           \n[1]  S. S. P. Parkin , C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S.- H. \nYang,  Nature Mater.  3, 862 ( 2004).  \n[2]  M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne, G. \nCreuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).  \n[3]  G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989).  \n[4]  S. S. P. Parkin, K. P. Roche, M. G. Samant, P. M. 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Appl. \nPhys. 91, 6595 (2002).  "    },    {        "title": "1010.4872v1.High_spin_polarization_in_epitaxial_films_of_ferrimagnetic_Mn3Ga.pdf",        "content": "High spin polarization in epitaxial films of ferrim agnetic Mn 3Ga \n \nH. Kurt,* K. Rode. M. Venkatesan, P. S. Stamenov an d J. M. D. Coey \n \nSchool of Physics and CRANN, Trinity College, Dubli n 2, Ireland \n \n* kurth@tcd.ie \n \nPACS number(s): 75.76.+j, 75.70.-i, 75.60.Ej, 75.50 .-y  \n \nAbstract \n \nFerrimagnetic Mn 3Ga exhibits a unique combination of low saturation magnetization \n(Ms = 0.11 MA m -1) and high perpendicular anisotropy with a uniaxial  anisotropy \nconstant of Ku = 0.89 MJ m -3. Epitaxial c-axis films exhibit spin polarization as high \nas 58%, measured using point contact Andreev reflec tion. These epitaxial films will \nbe able to support thermally stable sub-10 nm bits for spin transfer torque memories.  \n \n \n \n \n \n \n \n There is a revival of interest in Heusler alloys, m otivated by the extraordinary \nvariety of electronic ground states that can be ach ieved by varying the number of \nvalence electrons 1, 2, composition 3-5 and atomic order in these materials. The Heusler \nfamily include semiconductors, metals and half meta ls 6, as well as ferromagnets, \nantiferromagnets and superconductors, and even perh aps compensated ferrimagnetic \nhalf metals 7 and topological insulators 8, 9. Thin films of half-metallic Heuslers with a \nhigh Curie temperature, such as Co 2MnSi, have been successfully used in magnetic \ntunnel junctions 10-13  and spin valves 14 . The magnetization of these films lies in-plane, \nbut tetragonally-distorted Heuslers could offer hig h perpendicular anisotropy, \nnecessary for thermally-stable sub-10 nm tunnel jun ctions and spin valves. Here we \npresent epitaxially grown tetragonal Mn 3Ga films, that exhibit a spin polarization of \nup to 58%, together with uniaxial anisotropy ( K1 = 0.89 MJ m -3) and low \nmagnetization ( Ms = 0.11 MA m -1), a combination of properties that may be ideal fo r \ntiny perpendicular spin-torque switchable elements that will allow for scalable \nmagnetic memory and logic.  \n \nThe general composition of the L21 cubic Heusler alloys is X 2YZ. In the \nperfectly-ordered state, illustrated in Fig. 1a, th e Y and Z atoms occupy two \ninterpenetrating face-centred cubic lattices, where  each is octahedrally coordinated by \nthe other, and the X atoms form a simple cubic latt ice, where they are tetrahedrally \ncoordinated by both Y and Z atoms, at the corners o f a cube. The X–Y , X–X, Y–Y \nbond lengths are √3a0/4, a0/2 and a0/√2, respectively, where a0 is the cubic lattice \nparameter. The material we discuss here, Mn 3Ga, forms two stable crystal structures. \nThe high temperature hexagonal D019  phase is a triangular antiferromagnet, easily \nobtained by arc melting 15, 16 . The tetragonal D022  phase is a ferrimagnet, usually obtained by annealing the hexagonal material at 350 -400 °C for 1-2 weeks 15, 17, 18 . The \nspin polarization at the Fermi level has been calcu lated to be 88 % for the tetragonal \nphase, which has been suggested as a potential mate rial for spin torque applications 19 . \nTo this end, thin films with appropriate magnetic p roperties were needed. \n \nThe D022  structure is a highly-distorted tetragonal variant  of the L21 Heusler \nunit cell, which has been stretched by 27% along c axis. The unit cell is outlined on \nFig. 1a by the red line; lattice parameters of the unit cell (space group I4/ mmm ), \nwhich contains two Mn 3Ga formula units, are a = 394 pm and c = 710 pm. As a result \nof the tetragonal structure, both 4 d tetrahedral X-sites and 2b octahedral Y-sites are \nsubject to strong uniaxial ligand fields, which lea d to uniaxial anisotropy at both Mn \npositions. Imperfect atomic order results in some M n population of the 2 a Z sites, \nwhich are similarly distorted.  \n \nGenerally, the Mn moment and Mn-Mn exchange in meta llic alloys are very \nsensitive to the inter-atomic distances. Widely-spa ced Mn atoms with a bond length ≥ \n290 pm tend to have a large moment, of up to 4 µB, and couple ferromagnetically 20 . \nNearest-neighbour Mn atoms have much smaller moment s, and couple \nantiferromagnetically when the bond length is the r ange 250 – 280 pm. As a result, the \nmagnetic order is often complex. αMn, for example has a large cubic cell ( a = 890 \npm) with four different manganese sites having mome nts ranging from 0.5 to 2.6 µB, \nand a complex noncollinear antiferromagnetic struct ure with TN = 95 K. The \nmagnetism of tetragonal Mn 3Ga is simpler. The Y sublattice is ferromagneticall y \ncoupled, as expected from the long Y-Y bonds (391, 450 pm), but there is a strong \nantiferromagnetic X-Y intersublattice interaction ( bond length 264 pm) which leads to ferrimagnetic order. This overcomes the X-X interac tions, which are \nantiferromagnetic in-plane (bond length 278 pm) and  ferromagnetic along the c-axis \n(bond length 355 pm). The antiferromagnetic X-X int eractions are frustrated by the \nstrong antiferromagnetic X-Y coupling, and the X su blattice remains ferromagnetic, \nwith its moment aligned antiparallel to that of the  Y sublattice. Different occupancies \nand magnetic moments in X and Y sites leads to a co mpensated ferrimagnetic \nstructure with alternating spins on atomic planes a long c axis (Fig. 1b). Imperfect \natomic ordering of Mn and Ga may, however, produce local deviations from collinear \nferrimagnetism. The Curie temperature of Mn 3Ga is high. It would be much greater \nthan 770 K were it not for the transition to the an tiferromagnetically ordered \nhexagonal D019 structure at this temperature 15, 19 .  \n \nFig. 1 (a) The cubic unit cell of the L21 Heusler alloys. The tetragonal unit cell \nof the D022  structure is indicated by the red line. (b) In Mn 3Ga, it is stretched along \nthe c-axis. The X and Y positions are occupied by Mn, an d the Z position is occupied \nby Ga. The magnetic couplings in X and Y sublattice s are antiferromagnetic and \nferromagnetic respectively, but the stronger antife rromagnetic intersublattice coupling \nbetween X and Y sites frustrates the X site moments  leading to a ferrimagnetic structure. \n \nWe have grown oriented films of Mn 3Ga on MgO (001) substrates with various buffer \nlayers by dc magnetron sputtering from a Mn 3Ga target in a chamber with 2.0 × 10 -8 \nTorr base pressure. The Mn 3Ga growth rate was approximately 1nm/min. The \nsubstrate temperature was varied from 250-375 °C. Buffer layers were 10-30 nm of \noriented Pt (001), Cr (001) or MgO (001). Pt seed l ayers were prepared by pulsed \nlaser deposition at 500 °C substrate temperature under 40 µbar oxygen partia l \npressure. The films grow with (001) orientation ( c-axis normal to plane) on all three. \nMost of our stacks were made by dc-magnetron sputte ring; Mn 3Ga films grown by \npulsed laser deposition turned out to have the hexa gonal D019  structure. X-ray \ndiffraction was carried out using a Bruker D8 Disco very diffractometer with a \nmonochromated Cu K α1 source.  Magnetization measurements were made usin g a \nQuantum Design MPMS SQUID and PPMS vibrating sample  magnetometers. \n \nStructural, magnetic and spin polarization data for  representative films are \nsummarized in Table 1, and some X-ray diffraction d ata are shown in Fig. 2a-c. \nSample 345 was a perpendicular spin valve structure  with two Mn 3Ga layers, \nseparated by a 10 nm Cr spacer. The relatively high  magnetization and low spin \npolarization of the Mn 3Ga layer grown on Cr indicate that it has a high de gree of \ndisorder. \n  \nFig. 2. a) X-ray diffraction patterns of Mn 3Ga layers grown on Pt, Cr and MgO seed \nlayers. b) Substrate temperature ( Ts) dependence of Mn 3Ga layers grown on Pt (001); \nthe inset shows the order parameter S vs. Ts, at Ts = 375 °C the sample is not single \nphase. c) φ scans showing the four-fold symmetry of the films a nd the MgO (001) \nsubstrate. d) Atomic force microscope images of fil ms grown on MgO, e) Pt, f) Cr, all \ngrown at Ts = 350 °C. \n \n \n \nTable 1. Structural, magnetic and spin polarization  data on Mn 3Ga films. Films grown \non well oriented Pt and MgO exhibit lower magnetiza tion accompanied with a small \nin-plane canted moment and higher spin polarization , whereas the films grown on Cr \nhave higher moment and reduced polarization. Sample  345a and b represent a bilayer \nMn 3Ga structure; MgO/Pt/Mn 3Ga(60)/Cr(10)/Mn 3Ga(55). \n \nSample Buffer Ts t c M // M  ⊥ m µ0Hc P  (300K)  (300K)  (300K)  (2.2K)  \n#  °C nm  pm  MA m -1 MA m -1 µB fu -1 T % \n343  MgO/Pt  250  90  707  0.022  0.11  0.65  1.80  45  \n346  MgO/Pt  250  90  712  0.017  0.10  0.59  1.90  55  \n388  MgO  350  60  707  0.015  0.14  0.81  1.80  n.a.  \n408  MgO/Cr  350  40  696  0 0.14  0.81  1.68  n.a.  \n345a  MgO/Pt  250  60 708  n.a.  0.15  0.89  1.50  58  \n345b  Cr  250  55  700  n.a.  0.2 2 1.3 0 1.02  40  \n* 1 MA m -1 corresponds to 5.93 µB fu -1 \n \n \n \nThe composition of one of the films grown on MgO wa s determined by ICP \nmass spectroscopy; the Mn:Ga atomic ratio was 3.0:1 .0. The films were metallic, and \nthe resistivity of those grown on Pt was estimated as 1.6 ×10 -6 Ω m after correcting \nfor  the resistance of the underlying Pt. Resistivi ty of a film grown directly on MgO \ncould not be measured because of the discontinuous island growth mode in this case, \nillustrated in Fig. 2d.  \n \nAn order parameter S is defined by the square root of the intensity rat io of \n(101) to (204) reflections, divided by the theoreti cal ratio. The variation of S with \nsubstrate temperature is shown in the inset of Fig.  2b. The highest value of 0.8 is \nmeasured for the samples grown on Pt (001) at 350 °C. Platinum has the smallest \nlattice mismatch with tetragonal Mn 3Ga (0.4%) and almost strain-free epitaxial \ngrowth takes place easily. The surfaces of Mn 3Ga films exhibit variations depending on the substrate type and growth temperature as sho wn in Fig. 2d-f. The pinhole-free \nareas of films grown on Pt seed layers at 350 °C have an rms surface roughness of 0.8 \nnm, whereas the samples grown on thin Cr seed layer s were pinhole free, but much \nrougher (4-5 nm rms). In addition, there are large,  randomly distributed islands ~ 50 \nnm high on the surface (Fig. 2f).  Moreover, the Mn 3Ga films grown on Cr have a \nshorter c-axis, which is probably caused by the large lattic e mismatch with Cr (4.2 \n%). The D022  unit cell has to expand in the ab - plane for epitaxial growth. The films \non Pt and MgO grow in cube on cube mode (Fig. 2c), whereas the films on Cr are \nrotated in-plane by 45 degrees to facilitate epitax ial growth. The reason for the island \ngrowth mode on MgO is thought to be the large latti ce mismatch (7.7%), as islands \ncan relax the lattice strain. The top surface of th ese films is smooth, but continuous \nfilm growth was not possible for thicknesses up to 60-70 nm. The pinholes seen on \nthe surface of Mn 3Ga grown on Pt simply reflect pinholes formed on th e surface of \nthe thin Pt (001) seed layer, which arise because o f the lattice mismatch between Pt \nand MgO. Pinhole free surfaces can be obtained on t hick Pt (001) seedlayers (> 200 \nnm) grown on MgO (001) substrates, which  show feat ureless surfaces 21 .  \n \nThe relatively high surface roughness of epitaxial Mn 3Ga may limit its use to \ngiant magnetoresistance (GMR) spin valve devices. T his may not be a disadvantage. \nAs the dimensions of the memory bits decreases belo w 50 nm the large impedance of \ntunnel junctions can limit their use in high-speed circuits, whereas the impedance of \nGMR devices based on Mn 3Ga could be engineered to 50 Ω in these dimensions, \nwhich is compatible with high-speed operation. \n \nA typical room-temperature hysteresis loop is shown  in Fig. 3a. Data are corrected for substrate diamagnetism. The substrate  also exhibits weak Curie-law \nparamagnetism at low temperatures, which is attribu ted to 50 ppm of Fe 2+  impurities \npresent in the MgO. The broad hysteresis loops show  a spontaneous magnetization of \n130 ± 20 kA m -1 and coercivity of up to 1.9 T at 300 K. Epitaxial Mn 3Ga films grown \non MgO and Pt also exhibit a small canted magnetic moment which appears in the \nmagnetization measurements with field parallel to t he plane. There is considerable \nvariation in coercivity, but similar values of magn etization are found in all samples, \ncorresponding to a moment of 0.59 – 0.89 µB fu -1. Values reported in bulk samples \ndepend on the stoichiometry of the D022  compound, increasing from about 1.0 µB fu -1 \nfor a material with a 3:1 Mn:Ga ratio to 1.4 µB fu -1 for a 2:1 ratio 16-19 . The \nmagnetization in thin films is similar, and it has been shown to depend on the degree \nof atomic ordering 22, 23 . The magnetization is consistent with an essential ly \nantiparallel arrangement of moments on the 4 d and 2 b sites, determined by neutron \ndiffraction for a sample of composition Mn 2.85 Ga 1.15  to be 1.6 ± 0.2 and -2.8 ± 0.3 µB, \nrespectively 15 . A higher moment in the 2 b site is consistent with the calculations of \nKübler et al .24 . Electronic structure calculations  predict larger  moments on both sites, \nand indicate that the compound is nearly half-metal lic with a spin polarization of 88 \n% at the Fermi level 19 .  \n \nThe strength of the uniaxial anisotropy can best be  determined from the \nperpendicular and parallel magnetization curves. Th e anisotropy field determined \nfrom the extrapolation in Fig. 3a is 16 T, which co rresponds to a uniaxial anisotropy \nconstant K1 = 0.89 MJ m -3. A somewhat larger value was reported by Wu et al . in \nMn 2.5 Ga, which is mainly due to the higher magnetization  on account of non-\nstoichiometric composition 23 . The inverse correlation of coercivity and magneti zation indicated in Table 1 reflects the variation of the anisotropy field Ha = 2 K1 /Ms. The \nsmallest volume V for which the stability condition KuV/kBT  ≥  60 is satisfied at room \ntemperature is 280 nm 3. The corresponding nanopillar with height equal to  diameter \nhas dimensions 7 nm. \n \nIn uniformly magnetized, homogeneous uniaxial magne ts, the anisotropy field \nsets an upper limit on the coercivity as pointed ou t by Stoner  and Wohlfarth 25 . In \nreality, the coercivity is always much lower, a res ult known as Brown’s paradox 26 . \nThe explanation is that material is never perfectly  homogeneous; reversal begins in a \nlocalized nucleation volume of δB3, where δB = π √(A/Ku) is the Bloch wall width, and \nA is the micromagnetic exchange constant. Assuming A ≈ 10 pJ m -1, we find δB = 10 \nnm.  The expected activation volume is of order 100 0 nm 3. We anticipate that \nnanostructured Mn 3Ga elements will exhibit stable coherent reversal a t dimensions \nwhich are less than 10 nm. It therefore promises sc aling to these sizes.  \n \nMultilayer spin valve stacks were realized by growi ng the second layer of Mn 3Ga \non a Cr spacer. The magnetization measurements reve al a two step switching curve \nfor the trilayer structure as shown in Fig. 3b. The  magnetization reversal of the film \ngrown on Cr is sharper with a coercivity of 1T, whe reas the film grown on Pt switches \nat 1.5 T. The highest measured spin polarization of  the Mn 3Ga films is 58 %, which \ncompares with the calculated value of 88% 19 . The polarization decreases with order, \nstrain as well as Mn deficiencies, which increases magnetization.  \n  \n  \nFig.  3. (a) Magnetization curves for a 90 nm thick  film of tetragonal Mn 3Ga measured at room \ntemperature with the field applied perpendicular an d parallel to the plane of the film. Anisotropy fie ld \nat 300 K was determined by extrapolating in plane d ata through origin and is 16 T. (b) Magnetization \nof Mn 3Ga/Cr/Mn 3Ga bilayer structure. (c) Point-contact Andreev ref lection spectrum of a Mn 3Ga film \ngrown on Pt.  \n \n \nThe tetragonal D022 -Mn 3-xGa (x = 0 - 1) offers a wide variety of magnetic \nproperties that can be engineered to suit a specifi c application. In the case of \nstoichiometric Mn 3Ga, presented here, the magnetization is reduced du e a higher \ndegree of compensated spins and high anisotropy wou ld allow thermally-stable sub-\n10 nm spin torque devices. For magnetic random acce ss memories, a low \nmagnetization has the advantage that it requires a lower current to switch by STT. A \nparticular advantage is that the Mn 3Ga nanopillars would be switchable by STT but \nimmune to magnetic field of even NdFeB permanent ma gnets due to the high \ncoercivity of the films, and therefore would not re quire any magnetic shielding.   \nIn conclusion, tetragonal Mn 3Ga thin films look promising for nanoscale spin-\ntransfer torque memory and logic applications. Ther e is sufficient anisotropy to ensure \nthermal stability to sub-10 nm dimensions, and the small magnetization is \nadvantageous for spin torque switching. There is a degree of flexibility in the D022  \nstructure, in terms of composition and degree of at omic order, which should enable \nthe magnetic properties to be optimized for a speci fic magnetic application.  The immediate challenge now is to observe spin torque s witching a Mn 3Ga nanopillar. \n \n \n \n \nAcknowledgements \nThis work was supported by SFI as part of the MANSE  project 2005/IN/1850, and \nwas conducted under the framework of the INSPIRE pr ogramme, funded by the Irish \nGovernment's Programme for Research in Third Level Institutions, Cycle 4, National \nDevelopment Plan 2007-2013. \n \n \n  \n \n \n1 I. Galanakis, P . Mavropoulos, and P . H. Dederichs, Journal of Physics D: Applied Physics 39 , \n765 (2006). \n2 I. Galanakis, P . H. Dederichs, and N. Papanikolaou,  Physical Review B 66 , 174429 (2002). \n3 P . J. Webster and K. R. A. Ziebeck, in Landolt-Börnstein, New Series, Group III , edited by W. H. \nR. J. (Springer, Berlin, 1988), Vol. 19c, p. 77  \n4 H. C. Kandpal and et al., Journal of Physics D: App lied Physics 40 , 1507 (2007). \n5 B. Hillebrands and C. Felser, Journal of Physics D:  Applied Physics 39  (2006). \n6 R. A. de Groot, F. M. Mueller, P . G. v. Engen, and K. H. J. Buschow, Physical Review Letters 50 , \n2024 (1983). \n7 S. Wurmehl, H. C. Kandpal, G. H. Fecher, and C. Fel ser, Journal of Physics Condensed Matter \n18 , 6171 (2006). \n8 S. Chadov, X. Qi, J. Kübler, G. H. Fecher, C. Felse r, and S. C. Zhang, Nat Mater 9, 541 (2010). \n9 H. Lin, L. A. Wray, Y . Xia, S. Xu, S. Jia, R. J. Ca va, A. Bansil, and M. Z. Hasan, Nat Mater 9, 546 \n(2010). \n10 W. Wang, H. Sukegawa, R. Shan, S. Mitani, and K. In omata, Applied Physics Letters 95 , 182502 \n(2009). \n11 S. Tsunegi, Y . Sakuraba, M. Oogane, K. Takanashi, a nd Y . Ando, Applied Physics Letters 93 , \n112506 (2008). \n12 N. Tezuka, N. Ikeda, S. Sugimoto, and K. Inomata, A pplied Physics Letters 89 , 252508 (2006). \n13 Y . Sakuraba, M. Hattori, M. Oogane, Y . Ando, H. Kat o, A. Sakuma, T. Miyazaki, and H. Kubota, \nApplied Physics Letters 88 , 192508 (2006). \n14 T. Iwase, Y . Sakuraba, S. Bosu, K. Saito, S. Mitani , and K. Takanashi, Applied Physics Express 2, \n063003 (2009). \n15 E. Krén and G. Kádár, Solid State Communications 8, 1653 (1970). \n16 H. Niida, T. Hori, and Y . Nakagawa, Journal of the Physical Society of Japan 52 , 1512 (1983). \n17 H. Niida, T. Hori, H. Onodera, Y . Yamaguchi, and Y .  Nakagawa, Journal of Applied Physics 79 , \n5946 (1996). \n18 B. Balke, G. H. Fecher, J. Winterlik, and C. Felser , Applied Physics Letters 90 , 152504 (2007). 19 J. Winterlik, B. Balke, G. H. Fecher, C. Felser, M.  C. M. Alves, F. Bernardi, and J. Morais, \nPhysical Review B 77 , 054406 (2008). \n20 N. Yamada, Journal of the Physical Society of Japan  59 , 273 (1990). \n21 G. Cui, V. Buskirk, J. Zhang, C. P . Beetz, J. Stein beck, Z. L. Wang, and J. Bentley, in Materials \nResearch Society Symposium  Proceedings 310 , 1993), p. 345. \n22 F. Wu, E. P . Sajitha, S. Mizukami, D. Watanabe, T. Miyazaki, H. Naganuma, M. Oogane, and Y . \nAndo, Applied Physics Letters 96 , 042505 (2010). \n23 F. Wu, S. Mizukami, D. Watanabe, H. Naganuma, M. Oo gane, Y . Ando, and T. Miyazaki, Applied \nPhysics Letters 94 , 122503 (2009). \n24 J. Kübler, A. R. William, and C. B. Sommers, Physic al Review B 28 , 1745 (1983). \n25 E. C. Stoner and E. P . Wohlfarth, Philosophical Tra nsactions of the Royal Society of London. \nSeries A, Mathematical and Physical Sciences 240 , 599 (1948). \n26 W. F. Brown, Reviews of Modern Physics 17 , 15 (1945). \n \n "    },    {        "title": "1811.05362v1.X_ray_magnetic_linear_dichroism_as_a_probe_for_non_collinear_magnetic_state_in_ferrimagnetic_single_layer_exchange_bias_systems.pdf",        "content": "X-ray magnetic linear dichroism as a probe for\nnon-collinear magnetic state in ferrimagnetic sin-\ngle layer exchange bias systems\nChen Luo1,2,3,*, Hanjo Ryll1, Christian H. Back2,3, and Florin Radu1,**\n1Helmholtz-Zentrum-Berlin f¨ ur Materialen und Energie, Albert-Einstein-Strasse 15, 12489 Berlin, Germany\n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, Germany\n3Institute of Experimental Physics of Functional Spin Systems, Technical University Munich, James-Franck-Str.\n1, 85748 Garching b. M¨ unchen, Germany\n*chen.luo@ur.de\n**florin.radu@helmholtz-berlin.de\nNovember 14, 2018\nAbstract\nFerrimagnetic alloys are extensively studied for their unique magnetic properties leading to possi-\nble applications in perpendicular magnetic recording, due to their deterministic ultrafast switching\nand heat assisted magnetic recording capabilities. On a prototype ferrimagnetic alloy we demon-\nstrate fascinating properties that occur close to a critical temperature where the magnetization is\nvanishing, just as in an antiferromagnet. From the X-ray magnetic circular dichroism measure-\nments, an anomalous ’wing shape’ hysteresis loop is observed slightly above the compensation\ntemperature. This bears the characteristics of an intrinsic exchange bias effect, referred to as\natomic exchange bias . We further exploit the X-ray magnetic linear dichroism (XMLD) contrast\nfor probing non-collinear states which allows us to discriminate between two main reversal mech-\nanisms, namely perpendicular domain wall formation versus spin-flop transition. Ultimately, we\nanalyze the elemental magnetic moments for the surface and the bulk parts, separately, which\nallows to identify in the phase diagram the temperature window where this effect takes place.\nMoreover, we suggests that this effect is a general phenomenon in ferrimagnetic thin films which\nmay also contribue to the understanding of the mechanism behind the all optical switching effect.\nNon-collinear magnetism is emerging as a crucially important trait of magnetic systems which\nare indispensable to antiferromagnetic spintronics [1]. Magnetic skyrmions, helical and conical\nstates, canted spins specific to frustrated systems, domain walls in ferromagnetic and antiferro-\nmagnetic materials are all attempted to be controlled through external stimuli (electric currents,\nvoltages, laser excitations, strain) towards functionalization for applications in modern devices.\nMoreover, non-collinear spin textures in ferrimagnets can give rise to anomalous Hall effect [2, 3],\nwhich enables readout in magnetic sensors. While important progress is made on the under-\nstanding of complex magnetic textures in single crystals, the miniaturization of devices requires\nnano-scaling of the materials, which leads to significant modifications of their bulk magnetic prop-\nerties.\nAmong magnetic materials, rare-earth-transition-metals (RE-TM) ferrimagnetic alloys have\nattracted great interest because they exhibit superior flexibility in designing desired properties for\n1arXiv:1811.05362v1  [cond-mat.mtrl-sci]  13 Nov 2018ultimate functionality and as model systems for basic research in the field of spintronics. They\ncan be easily engineered as two ferromagnetic oppositely oriented sub-lattices in form of thin\nfilms and nanostructures with controllable perpendicular anisotropy, variable net magnetization\nas a function of stoichiometry and tunable spin reorientation transition temperature [4]. They\ncan also be assembled as heterostructures in form of spin valves and tunnel junctions [5, 6].\nFor example, DyCo/Ta/FeGd has been demonstrated to exhibit interlayer exchange coupling\nand a tunable and robust perpendicular exchange bias at room temperature, which can be set\nwithout additional field cooling cycles [5, 4]. The DyCo 5material has also been proposed to be\nsuitable for heat-assisted magnetic recording near room temperature [7]. For the archetypical\nGdFeCo and other RE-TM ferrimagnets like TbFe, TbCo it has been demonstrated that their\nmagnetization can be controlled using femtosecond laser pulses at large lateral length scales and at\nthe nanoscale, without applying any external magnetic field [8, 9, 10, 11, 12]. RE-TM ferrimagnetic\nalloys can be tuned to behave as true antiferromagnets at a compensation temperature where the\nmagnetic moments of the RE and TM sub-lattices are equal in size but oppositely oriented,\nleading to a zero net magnetization. For some RE elements with low orbital magnetic moments,\ncomparable to the orbital magnetic moment of the TM element, the angular momentum may\nbe quenched for a certain temperature which may lead to an acceleration of the precessional\nspin dynamics [13, 14]. Ultrafast magnetization reversal across the compensation temperature of\nRE-TM alloys may deterministically provide the ultimate switching speeds that can be achieved\ntoday [15, 16, 17, 18, 19].\nThe complex physics near the compensation temperature is enriched by one more fascinating\neffect. Anomalous magnetic behavior in form of wing shape hysteresis loop has been reported\nin several RE-TM ferrimagnetic alloys that exhibit a perpendicular magnetic anisotropy such as,\nGdCo [20], HoCo [21], TbFe [22], GdFe [23],DyCo 4[24], and GdFeCo [25, 26, 27] thin films. When\napplying a magnetic field perpendicular to the sample, it is expected that the net magnetization\nwill naturally align with the external field. However, a counter-intuitive effect is observed: the\nmagnetization diminishes when the magnetic field overcomes a certain value, leading to a decrease\nor even a vanishing magnetization of the sample.\nOriginally, this intriguing effect was interpreted based on models assuming an alloy composi-\ntion gradient across the film thickness or even across lateral directions of the sample. According\nto these assumptions, a compensation temperatures range will occur and, as a result the mag-\nnetic hysteresis loop will reflect the relative weight of the corresponding ”below” and ”above”\ncompensation parts of the film [20, 25]. These early models have been addressed critically in\nrelation to similar observations in HoCo films [21], questioning the original proposals based on the\nstructural or magnetic inhomogeneities. Recently, the observation of similar anomalous loops in\nGdFeCo was observed to occur at faster time scales [23]. For this case the origin for the effect was\nsuggested to be caused by a transient temperature range which extends over the compensation\ntemperature of the film. Even more recently, similar anomalous magnetic behavior in the same\nGdFeCo films was reported in equilibrium with the suggestion that its origin actually may relate\nto a spin-flop mechanism [27]. This last attempt to resolve the debate, however, comes at odds\nwith previous observation of this effect in a DyCo 4film where it is suggested that the effect bears\nthe characteristics of an exchange bias effect [24]. As a result, although this effect is off paramount\nimportance for ultrafast magnetization research, its fundamental origin is still highly debated. We\ncenter our study on resolving the origin of the effect utilizing one of the most powerful modern\nexperimental tools to magnetism, namely soft x-ray spectroscopy. Moreover, we suggests the this\neffect may explain the origin of all optical switching in ferrimagnetic films (see Discussion section\nand Supplementary).\n20.02.04.06.0\n-2.0-1.00.0\n0.02.04.0\n1280 1290 1300 1310 1320 1330 1340 1350-0.20.00.20.4XAS & XMCD (arb.unit)\nXASXAS\nXMCDXAS & XMLD (arb.unit)\nPhoton energy (eV)XMLD\nX-rayTEY Photodiode\nSample𝐻𝑀 Circular X -ray\n𝑀 Linear X -ray(a)(b)\n(c)\n𝐸Figure 1: (a) Sketch of the XMCD and XMLD measurements, the fluorescence and TEY signals\nfrom the sample are recorded. (b) The XAS and XMCD spectra for the Dy M 4,5edges at 300 K.\n(c) The XAS and XMLD spectra for the Dy M 4,5edges at 300 K. The three peaks of Dy M 5edge\nare marked by the dashed lines at E= 1294.3, 1296.5 and 1298.4 eV. The data in panels (b) and\n(c) were both recorded in FY mode.\nResults\nWe make use of x-ray circular magnetic dichroism (XMCD) [28] and x-ray linear magnetic dichro-\nism (XMLD) assembled from magnetic field dependent absorption spectra (XAS) measured by\ntotal electron yield (TEY) and by fluorescence yield (FY) to resolve the origin of the magnetic\ntransition that occurs close to the compensation temperature of RE-TM ferrimagnetic alloys.\nXMCD is sensitive to the ensemble averaged orbital and spin contribution to the magnetic mo-\nments projected along the circular polarization direction which is set to be parallel to the x-ray\nbeam direction. Through the sign of XMCD one can distinguish the directional sense of the\nmagnetic moments. However, through its magnitude one cannot uniquely discriminate on their\neventual non-collinear arrangement with respect to magnetic domain formation. To achieve this\ncapability, the XMLD contrast will be involved.\nXMLD [29, 30, 31, 32, 33] is the difference in XAS cross section for the /vectorEvector of linear\npolarized X-rays oriented parallel and perpendicular to the magnetic moments. XMLD depends\non the square of the magnetic moment <M2>and on the magneto-crystalline anisotropy, which\nmakes it favorable for the study of antiferromagnetic systems. In spite of the key dependences to\nintrinsic magnetic properties, for 3d transition metal elements (Mn, Co, Fe) the size of the XMLD\neffect is extremely small, hindering its further development [31, 34]. Also, it requires a rather\nhigh precision of reproducibility of energy set since the size of its sign changing contrast at the L 3\nresonant edge occurs in a narrow energy range. We exploit here, as demonstrated further below,\nthe XMLD at the RE (Dy) edges which is expected to be much larger [29] and occurs as a sizable\nintensity change at the M 5edge for both TEY and FY detection modes.\nFirst we present the demonstration of XMCD and XMLD geometries and the novel character-\nistics of the XMLD effect at the Dy M edges through an experiment to probe the non-collinear\nstates in DyCo 5ferrimagnetic thin films at room temperature. Figure 1(a) shows the experimental\n3-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0\n-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0\n-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0\n-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0\n-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0\n-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0\n-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0\n-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.010 K 80 K 150 K 160 K\n165 KNormalized Bulk Dy hysteresis loops (arb. unit) 220 K\nm0H (T)180 K 300 KHebHcFigure 2: Temperature dependent hysteresis loops recorded by FY signal with circular polarized\nX-rays set to the Dy M 5edge (E= 1298.4 eV). The shift of the side hysteresis loop denoted\nasHeband the coercive field of the central loop denoted as Hcare depicted schematically in the\ntop-right panel.\ngeometry for the surface and bulk measurements (see also Supplementary). All the measurements\nwere performed in a perpendicular geometry, the bulk sensitive FY signal with a probing depth of\n∼100 nm [35] was recorded by a photodiode located 2 cm away from the sample surface. Fig. 1(b)\nshows the XAS and XMCD measurements for Dy at 300 K. The XMCD spectrum was obtained by\ntaking the difference of ( σ+−σ−), whereσ+andσ−represent the XAS spectra measured by FY\nand using the circular polarized X-rays with the magnetic field ( µ0H= 1 Tesla) parallel and anti-\nparallel to the beam direction. The Dy XMLD spectrum was obtained by taking the difference of\nthe XAS spectra measured by keeping the linear polarized X-rays /vectorEparallel to the synchrotron\nplane and perpendicular to the beam direction, and setting the direction of the magnetic moments\nperpendicular and parallel to /vectorE, respectively, as shown in Fig. 1(c). The magnetic field of 1 Tesla\nwas checked to saturate the magnetization of the sample whether in-plane or out-plane at room\ntemperature. The field was applied parallel to the beam direction and perpendicular to it. As\nshown in the figure, there are three peaks at the Dy M 5edge, which are marked by the dashed\nlines. The maximum XMCD signal appears enhanced at the third peak whereas the maximum\nXMLD signal is located at the middle peak. The intensity difference at the M 5edge for the XMLD\nspectra is sufficiently large to be exploited for intensity measurements as a function of the external\nfield. The sensitivity to the angular orientation between the magnetic moments and the direction\nof the polarization vector is clearly demonstrated: strong XMLD contrast appears by re-orienting\nthe magnetic moments only, from parallel to perpendicular directions with respect to /vectorE, using\nvectorial magnetic fields. To exclude further contributions to the XMLD contrast possible caused\nby crystalline electric field effect, further orthogonal field directions in plane of the sample were\nmeasured (Supplementary).\nTo approach the compensation temperature, we performed temperature-dependent XMCD and\nhysteresis loop measurements at the Dy M 4,5and Co L 2,3edges. Part of the bulk sensitive hys-\nteresis loops taken by measuring the FY intensity at the Dy M 5edge (E= 1298.4 eV) are shown\nin Fig. 2. The hysteresis loops signal reverses when the temperature crosses a critical tempera-\nture, called magnetic compensation temperature T comp. Also, the occurrence of the perpendicular\nmagnetic anisotropy is clearly distinguished as the full remanent magnetization and by the sharp-\nness of the magnetization reversal. Besides the main sharp reversal of the film, we observe an\n4”anomalous” behavior at higher fields which develops at temperatures higher than T comp. This\nintriguing response of magnetization at higher magnetic fields is counter-intuitive in nature. For\nmagnetic films which exhibits a net magnetization like ferromagnets, an external field will cause\nfull magnetization, aligning the spins as the field is increased. By contrast, the ”anomalous”\nhysteresis loop show that the magnetization decreases as the magnetic field is increased.\nIn Fig. 3(a) we plot the coercive field of the hysteresis loops and the shift of the side hysteresis\nloops as a function of temperature. The divergence of the coercive field, which occurs at the\nvanishing net magnetization of the film, reveals with good accuracy the absolute value of the\nintrinsic T compwhich is about 154 K, in close agreement with previous experiments and simulation\nresults [36, 37]. Also, we plot the field of the center of the wing hysteresis loops as a function\nof temperature, denoted as exchange bias field H eb. We observe that the shift of the side loop\nincreases as the temperature increases up to the highest measured value of about 7 T. Due to\nthe finite available external fields (up to 9 T) we could not further follow the shift of the side\nloop beyond the 7 T. This limitation is visible at 165 K, where the side loop begin to behave\nas a minor hysteresis loops. Instead, we can determine the limiting temperature where the side\nloop will vanish. To this end we show in Fig. 3(b) and (c) the inverse of the total net magnetic\nmoments characteristic of bulk and surface, respectively. They are extracted from XMCD spectra\nmeasured by TEY and by FY (Supplementary). We observe that both the inverse of the bulk\nand the surface net magnetic moments exhibit a divergent behaviour, at 154 K for the bulk part\nof magnetization and at 200 K for the surface part. As such, the system exhibits a different\ncompensation temperature for the probed surface, which is about 50 K higher as compared to the\nbulk magnetic compensation [24]. In-between these two compensation temperatures the system\nis in a frustrated state leading to the peculiar wing shape hysteresis loops. The occurrence of\nthese two compensation temperatures correlates very well with the temperature range where the\nanomalous magnetic behaviour occurs at high fields. Outside this region, we observe no side\nhysteresis loops because the surface and the bulk spins are in a stable configuration.\nTo characterize the reversal of the anomalous loops we make use of the XMCD and XMLD\neffects, focusing on the relevant temperature of 160 K. Fig. 4(a) shows the hysteresis loops at\nboth Dy M 5edge and Co L 3edge. They exhibit a similar behaviour with opposite signs. This\ndemonstrates that the magnetic moments of the Dy and Co sublattices are basically anti-parallelly\ncoupled to each other, even when they enter the anomalous spin state at higher fields. At this\nstage, we can resolute that a spin-flop transition does not occur. Such a spin-flop would appear in\nthe Fig. 4(a) as a significant difference between the shape of the magnetization curves of Dy and\nCo, which is not observed. However, a weak non-collinear state between Co and Dy net magnetic\nmoments can be distinguished by the difference of the relative magnetizations present at ±8 T.\nThis observation is further supported by the demonstration of the same effect in a FeGd film\n(Supplementary). We can consider that the Co and Dy sublattices are essentially anti-parallelly\noriented for all external magnetic fields. This, however, does not exclude a non-collinear behavior\nfor the elemental sublattices: comparing the surface and bulk hysteresis loops, see Fig. 4(b),\nclearly reveals that the surface signal is almost completely reversed at high fields while the bulk\nsignal is only half reversed, which strongly indicates that the mechanism is due to the reversing\nof the surface magnetic moments.\nBy taking advantage of the strong linear dichroism of the Dy element at the M 5absorption\nedge, XMLD measurements were applied to understand the anomalous magnetic behavior. Here,\nwe use linear polarized X-rays to perform hysteresis loop measurements, as shown in Fig. 4(c).\nThe XMLD hysteresis loops were recorded at the middle peak of the Dy M 5edges atE= 1296.5\neV, where we observed a maximum XMLD signal in Fig. 4(d) and Fig. 1(c). From Fig. 4(c) one\ncan see that there are two hysteresis-loop-like structures for both surface and bulk. The two loops\nappear at the same field of the XMCD ’wing shape’ hysteresis loops but end up at the same\nlevel of intensity at ±8 T. These results reveal that parts of the magnetic moments rotate from\nthe out-of-plane to the in-plane direction at high fields, which directly indicates the existence of\n50 50 100 150 200 250 3000246\nT[K]Hc[Tesla]\nHeb[Tesla]\n0 50 100 150 200 250 30002468\nT[K]1/Mtot\n0 50 100 150 200 250 3000.00.51.01.52.0\nT[K]1/MsurfaceFigure 3: Phase diagram of the magnetic states as a function of temperature: (a) Temperature\ndependence of the coercivity field µ0Hcand the exchange bias field Heb. The values of µ0Hc\nwere extracted from the rectangular hysteresis loops at the crossing point with respect to the\nmagnetization axis. The shift of the side wings, denoted as Heb, were extracted at the half height\nof the side loops alone. The maximum value of µ0Hcis about 4.8 T slightly below T comp, whereas\nthe highest measured shift side hysteresis loop is about 7 Tesla; (b) The inverse of the total net\nmagnetic moment extracted by analyzing the XMCD spectra characteristic for the bulk part of the\nfilm (FY data). This show a divergent behavior at the closely similar compensation temperature as\nin panel (a). (c) The inverse of the total net magnetic moment extracted by analyzing the XMCD\nspectra characteristic for the surface part of the film (TEY data). They also exhibit a divergent\nbehavior near 200 K, showing the probed surface has a higher compensation temperature. In\nbetween these two compensation temperatures, the side hysteresis loops occur.\n6-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0Normalized hysteresis loops \nH(T) Dy FY\n Dy TEY\n0.02.04.06.0\n1280 1290 1300 1310 1320 1330 1340 1350-0.20.00.20.4 8.0T\n 2.5TXAS & XMLD (arb.unit)\nPhoton energy (eV) XMLD\n-8 -6 -4 -2 0 2 4 6 8-1.0-0.50.00.51.0Normalized hysteresis loops \nH(T) Dy\n Co(a)\n(c)(b)\n(d)DyCo\nnon-collinear state\nSaturation𝐻Net moment\ncollinear state\nRemanence(e)\nDomain Wall\nSurface\nBulk & FYTEY\n-8 -6 -4 -2 0 2 4 6 80.80.91.0Dy XMLD intensity (arb.unit)\nH (T) TEY\n FYFigure 4: XMCD and XMLD measurements at 160K: (a) Hysteresis loops measured by recording\nthe FY signal with circular polarized X-rays at the third peak of the Dy M 5edges (E= 1298.4\neV) and the Co L 3edge (E= 777.0 eV). (b) Comparison of surface (TEY) and bulk (FY) hysteresis\nloops for Dy. (c) XMLD hysteresis loops measured by recording the FY and TEY signal with\nlinear polarized X-rays at the second peak of the Dy M 5edges (E= 1296.5 eV). (d) XAS and\nXMLD spectra taken at 8 T and 2.5 T for Dy with linear polarized X-rays. (e) Sketch of the\nmagnetic spin structure for the remanence and saturation state.\nnon-collinear spin structure between the surface and the bulk. One needs to point out that there\nare also significant differences between the bulk and surface signals. The surface XMLD hysteresis\nloop reaches a maximum value around ±6 T, indicating that the surface magnetic moments rotate\nto the in-plane direction around this field, then decreases and approaches flattening near ±8 T.\nThis further indicates that the surface magnetic moments have a higher rotation angle at high\nfields. This is in full agreement with a simple domain wall structure initiated at the surface of the\nfilm.\nFigure 4(d) shows the XAS spectra of Dy and their difference taken at 8 T and 2.5 T with linear\npolarized X-rays. By comparing the image with the standard XAS and XMLD measurements at\n300 K, one can see that the XAS at 8 T is similar to the σ/bardblXAS while the XAS at 2.5 T is close\nto theσ⊥XAS in Fig. 1(c). By defining the relative XMLD amplitude as ( σ/bardbl-σ⊥)/(σ/bardbl+σ⊥), we\nget a value of ∼4.4% here and ∼5.6% at 300 K. Note that the XMLD amplitude is believed to be\nproportional to the square of the total magnetic moments M2[29]. The magnetic moments at 160\nK are about m160K\nbulk =ms+ml= 6.8µB/atom and m160K\nsurface = 5.7µB/atom, which are about 1.45\ntimes of the magnetic moments at 300 K m300K\nbulk= 4.8µB/atom and m300K\nsurface = 3.8µB/atom. Thus\nthe relative XMLD amplitude of 5 .6%×1.452= 11.6% can be expected at 160 K for the situation\nthat all the magnetic moments align in-plane versus out-of-plane. Here the experimental value\nof 4.4% means that the in-plane contribution at 8 T is about/radicalBig\n4.4%/11.6%≈62% of the total\nmagnetic moment, which indicates a very thick domain wall probably throughout the whole film.\nDiscussion\nBased on the experimental facts, one can draw a sketch of the spin structure for the anomalous\nmagnetic behaviour, as shown in Fig. 4(e). The surface magnetic moments are always smaller\nthan the bulk magnetic moments for both Dy and Co. At the magnetic remanence, the spins of\n7Dy and Co are in a collinear state and anti-parallelly coupled to each other. Due to the strong\nexchange coupling between the surface and bulk, and due to the fact that the surface magnetism\nis dominated by Dy while the bulk magnetism is dominated by Co, the net moments of the surface\nwould prefer to align anti-parallel towards the bulk moments. At high fields, this frustrated state\nbecomes unstable, which forces the spin structure of the whole system to turn into a non-collinear\nstate with an out-of-plane partial domain wall. Within this wall, the exchange energy is stored and\nreleased, which resembles closely exchange bias interactions with the difference that the atomic\nexchange is not affected by additional interfaces in an otherwise generic ”two magnetic layers”\nsystem. By analyzing the net magnetic moments for the bulk and surface, separately, one can\nlocalize the temperature range where the occurrence of the side loop takes place. Within this\ntemperature range the shift of the side loop increases as a function of temperature. This can\nbe understood within the general theories for exchange bias [38], which postulates the shift of\nthe hysteresis loop is inverse proportional to the M×t, where M is the magnetization of the\nactive magnetic layer and t is its thickness. In our case the active layer is the surface which has\na compensation temperature of 200 K, therefore, when approaching this temperature the shift of\nthe hysteresis should increase, as observed experimentally in Fig. 3.\nTo suggest that this effect is a general phenomenon which occurs in thin ferrimagnetic films\nclose to the compensation temperature, we provide supplementary data (Supplementary) on yet\nanother system, namely FeGd ferrimagnetic film. There, the same effects are observed. Never-\ntheless, since FeGd has a lower magnetic anisotropy and stiffness (due to the nearly vanishing\nmoment of Gd), the temperature range where the shift of the hysteresis loop occurs is larger.\nAt a more general level, we speculate that our observations may have an impact towards deeper\nunderstanding of key aspects of ultrafast magnetic switching in ferrimagnetic films [8, 17]. The\nmere occurrence of two compensation temperatures comprising two magnetic states in a frustrated\narrangement should motivate further experiments which includes this peculiar phase diagram, also\nconsidering thickness [39] as a control parameter. As an alternative mechanism for the all optical\nswitching, measured from below compensation in an external field, we can assume an intersection\nbetween the intrinsic dynamical paths and the static ones. For instance, if the Hebwill scale down\nin the non-equilibrium state during the pump probe delay, one can transiently cross over the\nformation of the domain wall state leading to the so called transient-ferromagnetic state observed\nfor an FeCoGd film in [17]. In fact, strong evidence for this scenario is seen in the Figure S5(a)\n(Supplementary). There, we observe that the regions between the -6 T and -2 and between 2 T\nand 6 T are very similar to the so called transient ferromagnetic-like state observed in all optical\nswitching experiments on FeCoGd samples. The projection of the Gd magnetization on the field\ndirection is opposite (after a crossing field equal to 3.5 T) with respect to the magnetization at\nmagnetic remanence, whereas the projection of Fe sublattice magnetization is vanishing. As such,\nthe total net magnetic moment of the whole film is oppositely oriented at high fields (larger than\n3.5 T) with respect to the net magnetic moment in lower applied fields (smaller then 3.5 T). Thus,\nassuming that during the pump probe delay, the time-dependent non-equilibrium states causes\nthe system to cross the critical field for the formation of this ”spring” spin configuration, one\ncan understand the origin of the all optical switching in ferrimagnetic thin films by analogy to\nthe effect we reported here. Note, that this mechanism excludes the occurrence of the same all\noptical switching effect in ferromagnetic materials, because this magnetic spring does not occur in\nthese materials. Instead, the thermally assisted all-optical switching may (disjunctively from the\neffect we report) take place in ferromagnetic as well as in ferrimagnetic materials as considered\nin [40, 41].\nIn conclusion, DyCo 5ferrimagnetic thin films were investigated with XMLD and XMCD tech-\nniques. An anomalous ’wing shape’ hysteresis loop, referred to as atomic exchange bias effect\nwith a large exchange bias field of µ0HEBup to a maximum measurable value of 7 Tesla, was\nobserved slightly above the compensation temperature T comp≈154 K. The origin of this effect\nwhich is demonstrated to be mediated by the formation of an out-of-plane partial domain wall\n8during the hysteresis measurements, is directly confirmed via XMLD measurements. Such a huge\nperpendicular exchange bias effect in a single film may be a good candidate for future perpen-\ndicular magnetic recording applications. The technique of using the XMLD contrast at the rare\nearth M 4,5edges to probe the non-collinear states could be very useful for characterization of\nnon-collinear magnetism which is intimately related to the spintronics research field.\nMethods\nSample preparation\nThe 20 nm thick DyCo 5thin films were grown on sapphire substrates by magnetron sputtering\n(MAGSSY chamber at HZB) in an ultra-clean Argon atmosphere of 1 .5×10−3mbar with a base\npressure of <2×10−8mbar at room temperature. The stoichiometry of the ferrimagnetic alloy\nwas controlled by varying the deposition rate of Co and Dy targets in a co-evaporation scheme.\nA 3 nm Ta capping layer was grown on top of the samples to prevent surface oxidation. We\nhave characterized the lateral homogeneity using energy dispersive scanning X-Ray spectroscopy\ntechnique (EDS), but were unable to observe any phase separation at the sensitivity level of about\nfew hundred nanometers provided by this method.\nX-ray absorption spectroscopy\nThe XAS measurements were performed at the VEKMAG end-station [42] installed at the PM2\nbeamline of the synchrotron facility BESSY II. This end station offers unique capabilities for this\ntype of research, since it provides a vector magnetic field option with a maximum magnetic field\nup to 9 Tesla in the beam direction, 2 Tesla in the horizontal plane and 1 Tesla in all directions for\na temperature range of 2 K - 500 K. The XAS spectra were recorded by means of total electron\nyield (TEY) and by fluorescence yield (FY), and for each constituent element, separately.\nThe TEY is measured by recording the drain current as a function of the x-ray photon energy\nnormalized by a Pt grid x-ray monitor mounted in a magnetically shielded environment as the\nlast optical element before the sample. The TEY is known to be surface sensitive, providing\ninformation over the escape length of the electrons which exhibits a mean free path of about 3\nnm. As such, the surface magnetic properties are provided in a selective manner by this recording\nchannel.\nThe FY is measured by a magnetically insensitive x-ray detector, placed at 2 cm away from the\nsample surface. FY is a photon-in photon-out spectroscopic technique which provides information\nintegrated over the penetration depth of the x-ray, which can be of order of tens of nm. 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The mechanics of the vekmag experiment. In Volker RW Schaa, editor,\nProc. of MEDSI2016, Barcelona, Spain, September 11?16, 2016 , number Geneva, Switzerland\nin 9, pages 370–373. JACoW, 2017.\nAcknowledgments\nThe x-ray absorption measurements were carried out at the VEKMAG end-station installed at\nthe PM2-VEKMAG beamlines, of BESSY II, Helmholtz-Zentrum Berlin (HZB). We thank the\nHZB for the allocation of synchrotron radiation beamtime. The authors acknowledge the financial\nsupport for the VEKMAG project and for the PM2-VEKMAG beamline by the German Federal\nMinistry for Education and Research (BMBF 05K10PC2, 05K10WR1, 05K10KE1) and by HZB.\nSteffen Rudorff is acknowledged for technical support.\n12Suplementary information for:\nX-ray magnetic linear dichroism as a probe\nfor non-collinear magnetic state in ferrimagnetic\nsingle layer exchange bias systems\nChen Luo1,2,3,*, Hanjo Ryll1, Christian H. Back2,3, and Florin Radu1,**\n1Helmholtz-Zentrum-Berlin f¨ ur Materialen und Energie, Albert-Einstein-Strasse 15, 12489 Berlin, Germany\n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, Germany\n3Institute of Experimental Physics of Functional Spin Systems, Technical University Munich, James-Franck-Str.\n1, 85748 Garching b. M¨ unchen, Germany\n*chen.luo@ur.de\n**florin.radu@helmholtz-berlin.de\nNovember 14, 2018\nSupplementary Notes\nS1 XMLD: experimental geometries utilizing vectorial magnetic fields\nWe describe below the experimental geometries for the linear dichroism measurements using vector\nmagnetic fields on the Ta/DyCo 5/Al2O3sample of the main body of the paper. This demonstrates\nin a self-consistent manner the sensitivity of the XMLD contrast to the angle of magnetization\ndirection with respect to the direction of the linear polarization. There are two common ways\nto measure a XMLD spectra. One is to keep the magnetization along the easy direction and to\nrotate the polarization direction, the other one is to keep the linear polarization direction fixed\nand change the magnetization direction [1]. Here, by taking advantage of the 3D vector magnet\nof the VEKMAG end-station, the second method is being used for our measurements. We set the\nlinear polarization /vectorEoriented perpendicular to the beam direction and parallel to the storage ring\nplane. The sample is oriented perpendicular to the beam direction, as for transmission geometry.\nThe magnetization of the sample is set oriented by the external field in three orthogonal directions:\nH/bardbl\nIP,H⊥\nIP, andH⊥\nOP, as shown in Fig. S1(a). The OPandIPindexes refer to the direction of the\nmagnetization with respect to the sample, namely out-of-plane and in-plane, respectively. The /bardbl\nand⊥indexes refer to the orientation of the magnetization with respect to the linear polarization\ndirection of the X-rays, namely parallel or perpendicular to /vectorE, respectively.\nThe XAS spectra and their difference are shown in Fig. S1(b,c,d). The difference between the\nspectra measured for the orthogonal directions H/bardbl\nIPandH⊥\nOP, as well as for H/bardbl\nIPandH⊥\nIP, do\nshow a XMLD contrast (black line in Fig. S1(c,d)). By contrast, when we compare the spectra\nrecorded for H⊥\nIPandH⊥\nOP, we observe a vanishing XMLD contrast (black line in Fig. S1(b)). This\ndemonstrates that the XMLD contrast is sensitive only to the orientation between the direction of\nmagnetization with respect to the direction of the linear polarization. Given that the amplitude\nof the XMLD is large, reaching a ∼5.6% at room temperature, the intensity difference at the\nmiddle peak of the M5edge, allows for detecting non-collinearity between the magnetization and\nthe direction of the external field during hysteresis measurements, as described in the main body\nof the paper.\n1arXiv:1811.05362v1  [cond-mat.mtrl-sci]  13 Nov 2018X-ray𝐸𝐻𝐼𝑃\n∥𝐻𝑂𝑃(a) (b)\n(c) (d)⊥\n⊥Figure S1: (a) Sketch of the XMLD measurements at room temperature. Here H/bardbl\nIPrepresents\nthe field direction which lies in-plane and parallel to the /vectorEvector of the linear polarized X-rays,\nH⊥\nIPrepresents the field direction which lies in-plane and perpendicular to /vectorE, andH⊥\nOPis the\nout-of-plane direction (perpendicular to the /vectorE). (b,c,d) show the XAS spectra measured at H/bardbl\nIP,\nH⊥\nIPandH⊥\nOP, as well as their differences. The XAS spectra were measured by recording the FY\nas a function of x-ray energy across the M5andM4edges of Dy.\n2S2 XMCD: analysis of the magnetic moments\nDue to the existence of the self-absorption and radiative decay effects, the FY spectra may ex-\nhibit a nonlinear dependence with respect to the absorption cross section [ ?, 2, 3]. To compare\nthe difference between the XMCD measured in FY, TEY and transmission mode, we prepared a\ncalibration sample Ta(2.5 nm)/DyCo 5(20 nm)/Ta(5 nm) grown on a 150 nm thick Si 3N4mem-\nbrane substrate. This allows us to measure the XAS and XMCD using TEY, fluorescence and\ntransmission signals at the same time, as shown in Figure S2 (a,b,c). For a direct comparison, we\nhave re-plotted all three XMCD spectra in Figure S2 (d). Comparing the XMCD spectra mea-\nsured in transmission with the XMCD spectra measured in TEY mode, we observe that they have\nan identical line shape, with a clear difference in amplitude. This directly demonstrates that the\nmagnetic moments of the surface are smaller as compared to the magnetic moments of the bulk\npart. However, the magnetic moments of the bulk measured by transmission and FY should have\nthe same amplitude, which should be reflected in a similar amplitude and line-shape for the XMCD\nspectra. This is, however, not the case as directly demonstrated in Figure S2 (d): the amplitude\nof the XMCD spectra is similar for the transmission and FY modes, but the XMCD line-shape\nmeasured by FY exhibits an enhanced spectral feature at 1294.3 eV. This difference prevents an\naccurate determination of the spin and orbital moments through FY measurements. Nevertheless,\nwe have applied the sum rules to the FY and transmission XMCD spectra and obtained the fol-\nlowing magnetic moments at room temperature: mFY\ntotal= 4.8µB/atom and mTR\ntotal= 3.9µB/atom.\nOne can see that the magnetic moment obtained from the fluorescence spectra is about 23% larger\nas compared to the value from the transmission spectra.\nWe then further use this scaling factor to correct for the magnetic moments extracted by\napplying the sum rules to the temperature dependence of the Dy XMCD spectra measured by\nFY in the main paper. For XMCD measured by FY for Co (not shown), this factor turned\nout to close to 1. We observe that both Co and Dy exhibit lower moments on the surface for\nthe whole temperature range, with different percentages: the Co surface has about 67% of the\nbulk net moment value, whereas the Dy surface has about 84% of its bulk value. The fact that\nthe Co moment is reduced stronger as compare to the Dy moment may indicate a non-collinear\narrangement between the Co and the Dy spins at the top surface. This can be a result of a reduced\nsurface coordination. The results are plotted in Fig. S3 to serve as the basis for the observed\ndifference between the compensation temperatures of bulk and surface parts of the film.\n3(a) (b)\n(c) (d)\n0.00.30.50.81.01.3\n1280 1290 1300 1310 1320 1330 1340 1350-0.6-0.4-0.20.00.2XAS & XMCD (arb.unit) s+@TEY\n s-@TEY\nPhoton energy (eV) XMCD\n0.00.30.50.81.01.31.5\n1280 1290 1300 1310 1320 1330 1340 1350-0.9-0.6-0.30.0XAS & XMCD (arb.unit) s+@FY\n s-@FY\nPhoton energy (eV) XMCD\n0.00.30.50.81.01.31.5\n1280 1290 1300 1310 1320 1330 1340 1350-0.9-0.6-0.30.00.3XAS & XMCD (arb.unit) s+@TR\n s-@TR\nPhoton energy (eV) XMCD\n1280 1290 1300 1310 1320 1330 1340 1350-1.0-0.8-0.6-0.4-0.20.00.2XMCD (arb.unit)\nPhoton energy (eV) TEY\n FY\n TRFigure S2: The Dy XAS and XMCD spectra measured by recording the TEY (a), FY (b) and\ntransmission signals (c). (d) Comparison between the TEY, FY and transmission XMCD spectra.\n0 50 100 150 200 250 300-8.0-6.0-4.0-2.00.02.04.06.08.010.0ms+ml (mB/atom )\nTemperature (K) Dy FY\n Dy TEY\n Co FY\n Co TEY\nFigure S3: Temperature dependence of the total magnetic moments ms+ml. The spin msand\norbitalmlmagnetic moments were obtained by applying the XMCD sum rules [4, 5, 6] for the\nXAS and XMCD spectra measured at remanent magnetization. The sign of the magnetic moments\nreverses after crossing T comp.\n40 50 100 150 200 250 300\nT (K)0123456Hc (Tesla)\n0123456\nHeb (Tesla)Figure S4: Temperature dependence of the coercive field Hc(open black circles) and the exchange\nbias fieldHeb(filled blue circles) for an FeGd film. The values of Hcwere extracted from the\nrectangular hysteresis loops at the crossing point with respect to the magnetization axis. The\nshift of the side wings, denoted as Heb, were extracted at the half height of the side loops alone.\nThe maximum value of Hcis about 2 T slightly aside T comp, whereas the highest measured shift\nof the side hysteresis loop is about 6.2 Tesla. The lines are guides to the eyes.\nS3 Support study of the atomic exchange bias in FeGd film\nFor the purpose of supporting our results and method, and to demonstrate a general character of\nour observations, we show one more model system, namely a FeGd ferrimagnetic thin film. The\nsample has the following structure Ta(2.5 nm)/Fe 77Gd 23(20 nm)/Ta(5 nm)/Si 3N4. The Gd layer\nexhibits a nearly vanishing orbital magnetic moment, therefore the magnetic anisotropy and the\nstiffness of the film are much lower as compared to the Dy based alloys. As such it can be used as\nsoft magnetic element in ferrimagnetic spin valves and it is the preffered system for the all optical\nultrafast magnetic switching effect.\nIn Figure S4 we show the dependence of the coercive field and the shift of the side loop as a\nfunction of temperature. The coercive field exhibits a typical divergent behaviour at the compen-\nsation temperature which for this sample is about 25 K. Above the compensation temperature,\nthe system develops an additional hysteresis loop denoted as H ebwhich increases steadily up to a\nmaximum measured value of about 6.2 T. Comparing this phase diagram with the one of DyCo 5\nfilm we notice a similar character, but with some markedly differences: the temperature range\nwhere the atomic exchange bias occur is larger for FeGd (more 100 K) as compared to DyCo 5.\nAlso, the temperature dependence of the H ebshows deviations from a linear behaviour which is\nactually expected within the theories of exchange bias effect. Note that for the FeGd samples\nwe did not measure the magnetic moments, as such we cannot provide the exact compensation\ntemperature of the surface part.\nIn Figure S5 we provide an overview of the magnetic behaviour for a constant temperature\nequal to 35 K. The sample has been measured in transmission and TEY modes. The element\n56\n 4\n 2\n 0 2 4 6\n0H (T)\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00XMCD hysteresis loops (arb. unit)(a)Fe XMCD\nGd XMCD\n6\n 4\n 2\n 0 2 4 6\n0H (T)\n0.0000.0250.0500.0750.1000.1250.1500.175XMLD hysteresis loops (arb. unit)(b)\nGd TEY\nGd TR1170 1180 1190 1200 1210 1220 1230 1240 1250\nPhoton energy (eV)0.00.20.40.60.81.0TR XAS & XMLD (arb. unit)(c)XAS at 6T\nXAS at 0T\nXMLD\n1170 1180 1190 1200 1210 1220 1230 1240 1250\nPhoton energy (eV)0.00.20.40.60.81.0TEY XAS & XMLD (arb. unit)(d)XAS at 6T\nXAS at 0T\nXMLDFigure S5: (a) The out-of-plane XMCD hysteresis loops are measured by recording the trans-\nmission signals with circular polarized X-rays at the Gd M 5edgeE= 1184.9 eV and at Fe L 3\nedgeE= 707 eV. They clearly demonstrate that the magnetic moments of the Gd and Fe are\ncoupled anti-parallel one to each other. (b) The XMLD hysteresis loops are measured using the\nlinear polarized X-rays at the Gd M 5edgeE= 1183.7 eV. Note that, for the TEY loop the data\nat smaller fields are not shown due to a common artefact caused by the electrons near zero field\nduring field sweep measurements, leading to large distortions of the curve. (c,d) The XAS and\nXMLD spectra measured at 0T and 6T. Here we observe a ≈13.5% XMLD signal for transmission\n(c) and≈10.6% for TEY (d).\n6specific (for Fe L 3and Gd M 5edges) magnetic hysteresis loops (Figure S5(a)) were measured in\ntransmission mode (bulk part) with circular polarized beam (XMCD) impinging perpendicular\nto the sample surface. They display a central rectangular loop which is characteristic for the\nperpendicular anisotropy, and the side loops. These hysteresis loops clearly show that the Fe and\nGd sub-lattice magnetizations are anti-parallelly oriented with respect to one another. At the\nhighest fields, one notices that a weak non-collinearity occurs. Strikingly, the regions between the\n-6 T and -2 and between 2 T and 6 T are very similar to the so called transient ferromagnetic-like\nstate observed in all optical switching experiments on FeCoGd samples [7] (see also the Discussion\nsection in the main manuscript).\nThe hysteresis loops with linear polarization (XMLD) for both the surface (TEY) and the\nbulk (transmission) parts exhibit a clear imprint of a large rotation of the sub-latices from out-of-\nplane direction to an in-plane direction (Figure S5(b)). The differences between the surface and\nthe bulk are also clearly observed, suggesting that the surface spins are more susceptible to the\nexternal fields. Finally, the XMLD spectra are shown for both TEY and transmission modes in\nFigure S5(c, d). Similar to Dy, the Gd does also exhibit a large XMLD contrast which can be\nused for non-collinearity studies.\n7References\n[1] Gerrit van der Laan. Magnetic linear x-ray dichroism as a probe of the magnetocrystalline\nanisotropy. Phys. Rev. Lett. , 82:640–643, Jan 1999.\n[2] S. Eisebitt, T. B¨ oske, J.-E. Rubensson, and W. Eberhardt. Determination of absorption\ncoefficients for concentrated samples by fluorescence detection. Phys. Rev. B , 47:14103–14109,\nJun 1993.\n[3] M. Pompa, A. M. Flank, P. Lagarde, J. C. Rife, I. Stekhin, M. Nakazawa, H. Ogasawara, and\nA. Kotani. Experimental and theoretical comparison between absorption, total electron yield,\nand fluorescence spectra of rare-earth M5edges. Phys. Rev. B , 56:2267–2272, Jul 1997.\n[4] Paolo Carra, B. T. Thole, Massimo Altarelli, and Xindong Wang. X-ray circular dichroism\nand local magnetic fields. Phys. Rev. Lett. , 70:694–697, Feb 1993.\n[5] B. T. Thole, P. Carra, F. Sette, and G. van der Laan. X-ray circular dichroism as a probe of\norbital magnetization. Phys. Rev. Lett. , 68:1943–1946, Mar 1992.\n[6] C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs, E. Chaban, G. H. Ho, E. Pellegrin,\nand F. Sette. Experimental confirmation of the x-ray magnetic circular dichroism sum rules\nfor iron and cobalt. Phys. Rev. Lett. , 75:152–155, Jul 1995.\n[7] I Radu, K Vahaplar, C Stamm, T Kachel, N Pontius, HA D¨ urr, TA Ostler, J Barker, RFL\nEvans, RW Chantrell, et al. Transient ferromagnetic-like state mediating ultrafast reversal of\nantiferromagnetically coupled spins. Nature , 472(7342):205–208, 2011.\n8"    },    {        "title": "1903.04330v2.Exchange_enhanced_Ultrastrong_Magnon_Magnon_Coupling_in_a_Compensated_Ferrimagnet.pdf",        "content": "Exchange-enhanced Ultrastrong Magnon-Magnon Coupling in a\nCompensated Ferrimagnet\nLukas Liensberger,1, 2,\u0003Akashdeep Kamra,3,yHannes Maier-Flaig,1, 2\nStephan Gepr ags,1Andreas Erb,1Sebastian T. B. Goennenwein,4\nRudolf Gross,1, 2, 5, 6Wolfgang Belzig,7Hans Huebl,1, 2, 5, 6and Mathias Weiler1, 2,z\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie\nder Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, 7491 Trondheim, Norway\n4Institut f ur Festk orper- und Materialphysik,\nTechnische Universit at Dresden, 01062 Dresden, Germany\n5Nanosystems Initiative Munich, 80799 Munich, Germany\n6Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany\n7Department of Physics, University of Konstanz, 78457 Konstanz, Germany\n(Dated: September 24, 2019)\nAbstract\nWe experimentally study the spin dynamics in a gadolinium iron garnet single crystal using\nbroadband ferromagnetic resonance. Close to the ferrimagnetic compensation temperature, we\nobserve ultrastrong coupling of clockwise and counterclockwise magnon modes. The magnon-\nmagnon coupling strength reaches almost 40% of the mode frequency and can be tuned by varying\nthe direction of the external magnetic \feld. We theoretically explain the observed mode-coupling\nas arising from the broken rotational symmetry due to a weak magnetocrystalline anisotropy. The\ne\u000bect of this anisotropy is exchange-enhanced around the ferrimagnetic compensation point.\n1arXiv:1903.04330v2  [cond-mat.mtrl-sci]  23 Sep 2019The strong and ultrastrong interaction of light and matter is foundational for circuit\nquantum electrodynamics [1{3]. The realizations of strong spin-photon [4{6] and magnon-\nphoton [7{12] coupling have established magnetic systems as viable platforms for frequency\nup-conversion [13, 14] and quantum state storage [15]. Antiferromagnets and ferrimagnets\nfurther host multiple magnon modes. Their coupling allows for coherent control and en-\ngineering of spin dynamics for applications in magnonics [16, 17] and antiferromagnetic\nspintronics [18, 19].\nRecently, it has been shown [20{22] that the weak interlayer exchange interaction be-\ntween two magnetic materials can cause magnon-magnon coupling. However, the much\nstronger intrinsic exchange has not yet been leveraged for coupling phenomena. While the\nTHz-frequency dynamics in antiferromagnets is challenging to address experimentally [23],\nthe sublattice magnetizations in compensated ferrimagnets can be tuned to achieve GHz-\nfrequency quasi-antiferromagnetic dynamics. Here, we report the experimental observation\nof ultrastrong exchange-enhanced magnon-magnon coupling in a compensated ferrimagnet\nwith the coupling rate reaching up to 37% of the characteristic magnon frequency. We\nfurthermore demonstrate that the coupling strength can be continuously tuned from the\nultrastrong to the weak regime.\nWe investigate spin dynamics, or equivalently the magnon modes, in a compensated,\ne\u000bectively two-sublattice ferrimagnet in the collinear state. Around its compensation tem-\nperature, this system can be viewed as a \\quasi-antiferromagnet\" due to its nearly identical\nsublattice magnetizations MA&MB. Figure 1 schematically depicts the typical spatially\nuniform spin dynamics eigenmodes of the system [25]. Within the classical description, these\nbecome clockwise (cw) and counterclockwise (ccw) precessing modes which correspond to\nspin-down and spin-up magnons, respectively, in the quantum picture. The key physics\nunderlying our experiments is the tunable exchange-enhanced coupling, and the concomi-\ntant hybridization, between theses two modes. The essential ingredients - mode coupling\nand exchange-enhancement - are both intuitively understood within the quantum picture\nas follows. First, due to their opposite spins, a spin-up magnon can only be coupled to\nits spin-down counterpart by a mechanism that violates the conservation of spin along the\nsublattice magnetization, and thus magnon spin, direction [24]. Since angular momentum\nconservation is a consequence of rotational invariance or isotropy, an anisotropy about the\nmagnon spin axis provides such a coupling mechanism. Achieving the equilibrium sublattice\n2MA\nMBMA\nMBMA\nMB[       ±       ]1\n2\nClassical\nMA ≳ MBQuantum\nPolariza�onSpin-up/\nccwSpin-down/\ncwSpin-zero/\nHybrid\nf1f2FIG. 1. Classical and quantum representations of the magnetization dynamics in a two-sublattice\ncompensated ferrimagnet. The eigenmodes of the compensated ferrimagnet close to its compensa-\ntion temperature are similar to that of an antiferromagnet since the sublattice magnetizations are\nalmost identical (we choose MA&MB). In the quantum picture, the classical modes with counter-\nclockwise (ccw) and clockwise-precession (cw) are identi\fed as spin-up and spin-down magnons.\nThe hybridized modes with linear polarization correspond to spin-zero magnons [24]. The angles\nbetween the two sublattice magnetizations have been exaggerated for clarity.\nmagnetizations, or equivalently the magnon spin axis, to lie along directions with varying de-\ngrees of local axial anisotropy allows to correspondingly vary the resultant magnon-magnon\ncoupling. This explains the nonzero mode-coupling along with its tunability. However, the\ntypically weak magnetocrystalline anisotropy may not be expected to yield observable e\u000bects\nand, therefore, has typically been disregarded. This is where exchange-enhancement in a\nquasi-antiferromagnet makes the crucial di\u000berence. The antiferromagnetic magnons, despite\ntheir unit net spin, are formed by large, nearly equal and opposite spins on the two sublat-\ntices [26]. The anisotropy-mediated mode coupling results from, and is proportional to, this\nlarge sublattice spin instead of the unit net spin, and is therefore strongly ampli\fed. This\nampli\fcation e\u000bect is termed exchange-enhancement within the classical description [26{28].\nIn our corresponding experiments, we study the magnetization dynamics of a (111)-\noriented single crystal Gd 3Fe5O12(gadolinium iron garnet, GdIG) disk by broadband mag-\nnetic resonance (BMR) [29]. A schematic depiction of the setup is shown in Fig. 2(a).\nWe use a vector network analyzer to record the complex transmission S21as a function of\n3the microwave frequency fand the external magnetic \feld H0applied in the (111)-plane.\nOur experiments are performed at T= 282 K, slightly below the ferrimagnetic compensation\npointTcomp= 288 K, as determined by SQUID-magnetometry [30]. At this temperature, the\nresonance frequencies of the spin-up and spin-down modes are in the microwave frequency\nrange.\nIn Fig. 2(b), we show the normalized background-corrected \feld-derivative of S21[31]\nrecorded at \fxed magnetic \feld magnitude \u00160H0= 0:58 T applied at '= 90\u000e. As dis-\ncussed later, this is a situation in which the magneto-crystalline anisotropy energy has axial\nsymmetry about the magnetic \feld direction. We refer to this case as an e\u000bectively ax-\nially symmetric (e.a.s.) direction. By \ftting the data to Eq. (S7) [30], we extract the\nresonance frequencies f1andf2of the two observed resonances, their di\u000berence \u0001 fresand\ntheir linewidths \u00141and\u00142. In Fig. 2(c) we show corresponding data and \fts for '= 0\u000eand\n\u00160H0= 0:65 T, which corresponds to a situation in which the magneto-crystalline anisotropy\nenergy is anisotropic about the applied magnetic \feld direction, which we refer to as an axial\nsymmetry broken (a.s.b.) direction, as explained below. Again, two resonances are observed.\nIn contrast to the data in Fig. 2(b), the resonances are now clearly separated.\nWe repeat these experiments for a range of magnetic \feld magnitudes H0applied along\nthe two directions (e.a.s. and a.s.b.) of interest. The obtained resonance frequencies are\nshown as symbols in Figs. 2(d) and (e). In the e.a.s. case shown in Fig. 2(d), we clearly\nobserve two resonance modes. The \frst one follows @fres=@H 0>0 and is the spin-up\nmodef\"and the second resonance with @fres=@H 0<0 is the spin-down mode f#. The\nvertical dashed line corresponds to \u00160H0= 0:58 T where \u0001 fresis minimized and the data\nshown in Fig. 2(b) is obtained. The resonance frequencies are in excellent agreement with\nthose obtained from numerical (see Supplemental Material [30]) and analytical (see below)\nsolutions to the Landau-Lifshitz equation.\nWhen applying H0along the a.s.b. axis, we obtain the resonance frequencies shown in\nFig. 2(e). Here, we observe a more complex evolution of the resonance frequencies for two\nreasons. First, for \u00160H0/0:4 T, the equilibrium net magnetization is titled away from H0\nand varies with H0. Second, and crucially, f\"andf#exhibit a pronounced avoided crossing.\nThe dashed vertical line indicates the value of H0of minimal \u0001 fres(c.f. Fig. 2(e)).\nWe plot \u0001fresand the half-width-at-half-maximum (HWHM) linewidths \u0014\"and\u0014#as\na function of the magnetic \feld H0in Figs. 2(f) and (g) for the e.a.s. and a.s.b. cases,\n4hrf\nCPWP1\nP2w\ncc=250ʅm\nd=6.35mm\ntсϱϬϬʅŵ\n[111]\n[121]\n[101]GdIG\nʔH0ɾ=90°[111]\n[121], \ne.a.s.\n[101], a.s.b.(a)\n12 1518 2124-0.30.00.30.6=90°,e.a.s.∂DS21/∂H0(1/T)\nf(GHz)(b)\n∆fresRe\nIm\n0510152025-0.2-0.10.00.10.2=0°,a.s.b.\nf(GHz)(c)\n∆fres0.00.51.01.52.00510152025\n0.00.51.01.52.00510152025\n0.00.51.01.52.00123\n0.00.51.01.52.00246numerical\nanalyticalf(GHz)\n0H0(T)(d) =90°,e.a.s.\nf↓f↑\nf↑f↓\n0H0(T)(e)=0°,a.s.b.\nf↓f↑\ngc/2π/2π(GHz)\n0H0(T)↓/2π(f)\n↑/2πfres/2\ngc/2π\n0H0(T)↑/2π\n↓/2π(g)\nfres/2FIG. 2. (a) Schematic broadband ferromagnetic resonance (BMR) setup, with the GdIG disk\non the coplanar waveguide (CPW). The angle 'de\fnes the in-plane direction of the magnetic\n\feldH0. (b),(c) BMR spectra obtained for \fxed magnetic \feld strengths applied along the (b)\ne\u000bectively axially symmetric (e.a.s.) direction in the (111)-plane at '= 90\u000e(\u00160H0= 0:58 T) and\nalong the (c) axial symmetry broken (a.s.b.) axis '= 0\u000e(\u00160H0= 0:65 T) recorded at T= 282 K\n(Tcomp = 288 K). The solid lines are \fts to Eq. (S7) [30]. The resonance frequencies are indicated\nby the red arrows and their di\u000berence is denoted as \u0001 fres. (d),(e) Mode frequencies vs. applied\nmagnetic \feld strength measured at T= 282 K where MGd&MFe. Open circles and triangles\ndenote measured resonance frequencies. The red dotted curves depict results of our analytical\nmodel and the blue dashed lines are obtained by numerical simulation. Along the e.a.s. direction\n'= 90\u000e(d), weak coupling is observed, whereas along the a.s.b. direction '= 0\u000e(e), we \fnd\nultrastrong coupling (see text). The solid gray lines in (e) indicate the uncoupled case taken from\nthe analytical solution of panel (d). (f),(g) Linewidths \u0014=2\u0019of the spin-up \u0014\"and spin-down \u0014#\nmodes, and resonance frequency splitting \u0001 fres=2 as a function of H0. The coupling strength gc=2\u0019\nis given by the minimum of \u0001 fres=2.\n5respectively. We \fnd the mutual coupling strength gc=2\u0019= minj\u0001fres=2j= 0:92 GHz\nfor the e.a.s. case and gc=2\u0019= 6:38 GHz for the a.s.b. con\fguration. In the former case,\ngc.\u0014\";\u0014#(c.f. Fig. 2(f)). Thus, the system is in the weak to intermediate coupling regime.\nFor the a.s.b. case, the linewidths \u0014are at least three times smaller than gc. Hence the\ncondition for strong coupling gc> \u0014\";\u0014#is clearly satis\fed. Furthermore, the extracted\ncoupling rate of gc=2\u0019= 6:38 GHz is comparable to the intrinsic excitation frequency fr=\n(f1+f2)=2 = 17:2 GHz. The normalized coupling rate \u0011=gc=(2\u0019fr) [8, 32] evaluates\nto\u0011= 0:37. Consequently, we observe magnon-magnon hybridization in the ultrastrong\ncoupling regime [1]. Importantly, the measured gcis the intrinsic coupling strength between\nspin-up and spin-down magnons and is independent of geometrical factors, in particular,\nsample volume or \flling factor. This is in stark contrast to the magnon-photon or cavity-\nmediated magnon-magnon coupling typically observed in spin cavitronics [8, 33{37].\nTo demonstrate that the coupling is continuously tunable between the extreme cases\ndiscussed so far, we rotated H0with \fxed magnitude in the (111)-plane at T= 280 K.\nThe background corrected transmission parameter (see Supplemental Material [30]) as a\nfunction of the angle 'is shown in Fig. 3(a) and (b) for \u00160H0= 0:5 T and\u00160H0= 0:8 T,\nrespectively. These magnetic \feld magnitudes correspond to H0slightly below and above\nthe hybridization point at T= 280 K (see Fig. S2 [30]). For both H0values, we observe two\nresonances for each value of ', where the lower resonance frequency depends strongly on '\nwhile the upper one is nearly independent of '. Overall, these results strongly indicate a\n'-dependent level repulsion that allows to continuously adjust the coupling strength.\nTo understand the coupling strength variation with ', we analyze the cubic anisotropy\nlandscape of our GdIG disk by plotting its magnetic free energy density F(c.f. Eq. (S9) [30])\nin Fig. 3(c). The applied \feld directions for the e.a.s. and a.s.b. cases are indicated by the\ntwo grey dots in Fig. 3(c). The sublattice magnetizations as well as the magnon spin axis are\ncollinear with the applied \feld in our considerations. As derived rigorously below, coupling\nbetween the opposite-spin magnons is proportional to the degree of anisotropy in the free\nenergy about the magnon spin axis [24]. Moreover, since they represent small and symmetric\ndeviations of magnetization about the equilibrium con\fguration, the magnons can only sense\nanisotropy variations that are local and averaged over antiparallel directions. Considering\nthe a.s.b. con\fguration \frst, if the magnetization deviates from equilibrium along the orange\n(white) arrows, it experiences an increase (a decrease) in energy. Therefore, the free energy\n6-30 0 30 60 90 1200306090120150180\nA(°)A(°)\n-315-287-258-230-202-173-145\nF(H0=0)(J/m3)90 45 0 -45 -900510152025f(GHz)\n(°)0H0=0.5T\n90 45 0 -45 -90\n(°)-707\nRe(∂DS21/∂)(10-3/°)\n0H0=0.8T\n(c)\ne.a.s.a.s.b.\n[001],\nh.a.xx(a) (b)\n[111],e.a.FIG. 3. Tunable coupling strength and anisotropy landscape. (a),(b) BMR-data obtained with\n\fxed magnetic \feld magnitudes with (a) \u00160H0= 0:5 T (below the hybridization point) and (b)\n\u00160H0= 0:8 T (above the hybridization point) as a function of the H0-orientation 'in the (111)-\ndisk plane at T= 280 K. The blue dashed lines are the results from the numerical simulation.\n(c) Colormap of the free energy density FforH0= 0. The angles 'Aand\u0012Adenote the orientation\nofMA, de\fned analogously to 'and\u0012in Fig. 2(a). The dashed horizontal line at \u0012A= 90\u000e\ncorresponds to the (111)-disk plane. The orange and white arrows at the e.a.s. ( 'A= 90\u000e) and a.s.b.\n('A= 0\u000e) orientations point towards increasing and decreasing free energy density, respectively.\nThe [001]-direction denotes a crystalline hard axis (h.a.) and [ \u0016111] a crystalline easy axis (e.a.).\nchange depends on the direction of deviation and the symmetry about the magnon spin axis\nin this con\fguration is clearly broken by anisotropy. This causes a non-zero mode-coupling\nin the a.s.b. con\fguration. In contrast, for the e.a.s. con\fguration, an averaging over the\ntwo antiparallel directions results in a nearly vanishing and direction-independent change in\nthe free energy, thereby e\u000bectively maintaining axial symmetry. This is prominently seen\n7when considering the direction collinear with the orange and white arrows, which nearly\ncancel each other's e\u000bect on averaging. This con\fguration is thus named e\u000bectively axially\nsymmetric (e.a.s.). The corresponding degree of axial anisotropy, and thus mode-coupling,\nvaries smoothly with 'between these two extreme cases.\nThe two key ingredients in the physics observed herein are (i) nonzero mode-coupling\narising from violation of spin conservation by an axial anisotropy [24], and (ii) a strong\nampli\fcation of the otherwise weak coupling via an exchange-enhancement e\u000bect character-\nistic of (quasi-)antiferromagnetic magnons [26]. We now present a minimalistic, analytically\nsolvable model that brings out both these pillars underlying our experiments, and yields\nresults in good agreement with our data (Fig. 2(d) and (e)). To this end, we employ a\ntwo-sublattice model, which corresponds to the net Fe- and Gd-sublattice in GdIG, within\nthe Landau-Lifshitz framework and macrospin approximation, treating anisotropies as uni-\naxial to enable an analytical solution. In our experiments, both of the distinct anisotropy\ncontributions considered here are provided by the cubic crystalline anisotropy of the mate-\nrial. Parameterizing the intersublattice antiferromagnetic exchange by J(>0) and uniaxial\nanisotropies by K(>0) andKa, the free energy density Fmis expressed in terms of the\nsublattice A and B magnetizations MA;B, assumed spatially uniform, as\nFm=\u0000\u00160H0(MAz+MBz)\u0007K\u0000\nM2\nAz+M2\nBz\u0001\n+Ka\u0000\nM2\nAx+M2\nBx\u0001\n+JMA\u0001MB;(1)\nwhere the \frst term is the Zeeman contribution due to the applied \feld H0^z. We further\nassume an appropriate hierarchy of interactions J\u001dK\u001djKaj, such that Katerms do\nnot in\ruence the equilibrium con\fgurations. The upper and lower signs in Eq. (1) above\nrepresent the cases of an applied \feld along easy and hard axes, respectively. The e.a.s.\n(a.s.b.) direction is magnetically easy (hard) [30]. The axial symmetry is broken by the term\nproportional to Ka, withKa\u00190 for the e.a.s. case and Ka6= 0 to the a.s.b. case. We have\nchoosen coordinate systems for treating the two con\fgurations with the z-direction always\nalong the applied \feld. The equilibrium con\fguration is obtained by minimizing Eq. (1)\nwith respect to the sublattice magnetization directions (see Supplemental Material [30]).\nThe dynamics are captured by the Landau-Lifshitz equations for the two sublattices:\n@MA;B\n@t=\u0000j\rA;Bj\u0014\nMA;B\u0002\u0012\n\u0000@Fm\n@MA;B\u0013\u0015\n; (2)\nwhere\rA;Bare the respective sublattice gyromagnetic ratios, assumed negative. It is conve-\nnient to employ a new primed coordinate system with equilibrium magnetizations collinear\n8with ^z0. The ensuing dynamical equations are linearized about the equilibrium con\fguration\nwhich, on switching to Fourier space (i.e. MAx0=mAx0ei!tand so on), lead to the coupled\nequations describing the eigenmodes expressed succinctly as a 4 \u00024 matrix equation:\n\u0010\n~P0+~Pa\u0011\n~m=0; (3)\nwhere ~m|= [mA+mB+mA\u0000mB\u0000] withmA\u0006\u0011mAx0\u0006imAy0and so on. The matrix ~P0\nis block diagonal in 2 \u00022 sub-matrices and describes the uncoupled spin-up and spin-down\nmodes, distributed over both sublattices. The matrix ~Pacaptures axial-symmetry-breaking\nanisotropy e\u000bects, and provides the spin-nonconserving, o\u000b-diagonal terms that couple the\ntwo modes and underlie the hybridization physics at play. The detailed expressions for the\nmatrices are provided in the Supplemental Material [30].\nFor applied \felds along the easy-axis (e.a.s.), the equilibrium con\fguration is given by\nMA=MA0^zandMB=\u0000MB0^z, withMA0;B0the respective sublattice saturation magneti-\nzations and MA0&MB0. For the case of a su\u000eciently small \feld applied along the hard axis\n(a.s.b.), the equilibrium orientation of MAis orthogonal to the hard axis. With increasing\n\feld strength, MAmoves to align with the applied \feld. In the considered temperature and\n\feld range, MBalways remains essentially anticollinear to MA[38]. The initial decrease\nof the resonance mode with lower frequency (Fig. 2(e)) is associated with this evolution\nof the equilibrium con\fguration. The frequency dip signi\fes alignment of equilibrium MA\nwith thez-axis. Only the Kaanisotropy term breaks axial symmetry about the equilibrium\nmagnetization direction ( z-axis) and leads to o\u000b-diagonal terms in ~Pa, which couples the\ntwo modes. The coupling-mediated frequency splitting \u0001 fres, where uncoupled eigenmode\nfrequencies would cross, is evaluated employing Eq. (3) as:\n2\u0019\u0001fres=!cs\n16JM2\n0\nJ(MA0\u0000MB0)2+Feq; (4)\nwhere!c\u0011j\rjjKajM0is the bare coupling rate, considering \rA\u0019\rB\u0011\randMA0\u0019MB0\u0011\nM0near the compensation point. Feq, given by 16 KM2\n0forH0along an easy axis, is an\nequivalent free energy density comparable to the anisotropy contribution, parametrized by\nK. The bare coupling rate is thus enhanced by a maximum value ofp\nJ=K at the compensa-\ntion point yielding a greatly enlarged coupling. Hereby a small coupling of !c= 2\u0019\u0001160 MHz\noriginating from a weak cubic anisotropy present in GdIG is greatly enhanced as demon-\nstrated by Eq. (4) and the analytical model results displayed in Fig. 2(e), quantitatively\n9describing our experimental observations. The ampli\fcation of coupling from 160 MHz to\nseveral GHz is an exchange-enhancement e\u000bect [26{28, 39]. This (exchange-)enhancement is\nan embodiment of antiferromagnetic quantum \ructuations [26] predicted similarly to amplify\nmagnon-mediated superconductivity [40].\nOur \fndings demonstrate that previously typically neglected details of the magnetocrys-\ntalline anisotropy can lead to giant e\u000bects on spin-dynamics if they have the appropriate sym-\nmetry and are exchange-enhanced. The ultrastrong and size-independent magnon-magnon\ncoupling reported here opens exciting perspectives for studying ultrastrong coupling ef-\nfects in nanoscale devices and exploring quantum-mechanical coupling phenomena beyond\nclassical electrodynamics. The reported e\u000bect also enables the tuning and tailoring of quasi-\nantiferromagnetic dynamics and magnons.\nNote added: During the preparation of the manuscript, we became aware of a related\nstudy showing magnon-magnon coupling in the canted antiferromagnet CrCl 3[41].\nAcknowledgments. { We thank A. Habel, K. Helm-Knapp, and K. Danielewicz for techni-\ncal support. We gratefully acknowledge the \fnancial support of the Deutsche Forschungsge-\nmeinschaft (DFG, German Research Foundation) via Germany's Excellence Strategy EXC-\n2111-390814868 (R.G. and H.H.) and the projects WE5386/4 and WE5386/5 (L.L. and\nM.W.). A.K. acknowledges \fnancial support from the Research Council of Norway through\nits Centers of Excellence funding scheme, project 262633, \\QuSpin\". 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We have observed\nhysteresis with varying magnetic field and the evolution of c oexisting antiferromagnetic and fer-\nrimagnetic state on mesoscopic length scale. These studies show that the magnetic state of the\nsystem at low field depends on the path followed to reach 120 K. The low field magnetic states\nare mesoscopically different for virgin and second field incr easing cycle when 120 K is reached by\nwarming from 5K, but are the same within measurement accurac y when the measuring temperature\nof 120K is reached from 300K by cooling.\nPACS numbers: 75.30.Kz, 72.15.Gd\nI. INTRODUCTION\nFirst ordermagnetic transitions have been of extensive\nscientific interest in recent years. The interest in these\nsystems arise due to their technological importance like\ngiantmagnetoresistance, magnetocaloriceffect, magnetic\nshape memory effect etc. as well as their fundamental\nimportanceto understand variousinterestingphenomena\nlike phase separation, metastability, glass like magnetic\nstate etc. Quench disorderin a system can lead to spread\nof local transition temperature resulting in the broaden-\ning of a first order transition1. This broadening gives\nrise to coexistence of competing phases in the transition\nregion. The metastability of coexisting phases within\n(and below) the supercooling and superheating spinodals\nhas been of wide interest and actively pursued in a wide\nvariety of systems like systems showing metal insulator\ntransitions2,3,4, multiferroics5, intermetallics6etc. The\nunderstanding of magnetic first order transition (due to\neasy control of magnetic field (H) and temperature (T)\n) also has implication to wider class of systems where\nfirst order transition plays a role (like glass transition\nwhere pressure and quenching rate are sometimes diffi-\ncult to control). It has been argued that the glass like\nmetastablestates resultingfromthe slowdynamicsofthe\ntransition are different from the metastable states which\narise due to supercooling and superheating near the first\norder transition7. Both kind of metastable states can\nshow seemingly similar features in some of their physical\nproperties e.g. open hysteresis loop in isothermal R-H\nmeasurement. However, it has been shown recently8that\nsuch open hysteresis loop due to supercooling and super-\nheating will be observed only for T within these spin-\nodals and will be observed only during cooling or only\nduring heating, depending on the sign of the slope of\ntransition band in H-T space. Mesoscopic investigation\nby Scanning Hall Probe Microscopy (SHPM) has shown\ncoexisting antiferromagnetic (AFM) and ferromagnetic(FM) phase around critical field in doped CeFe2and\nGd5Ge49,10. For the T chosen in these studies, it did not\nmatter whether the measurement temperature is reached\nby cooling or by warming. Here we present real space\nmagnetic imaging study by SHPM along with magneti-\nzation and resistivity measurement of Mn1.85Co0.15Sbto\nshow that field induced transition for T (=120 K) lying\nbetweensupercoolingandsuperheatingspinodaldepends\non the path followed to reach the measurement temper-\nature.\nDopedMn2Sbshowsfirstorderantiferro(AFM)tofer-\nrimagnetic (FRI) transition at low temperature11. Below\ntransition temperature ( TN) AFM to FRI transition can\nbe induced with the application of magnetic field. When\nTNis shifted to lower temperature, these systems show\nanomalous magnetic behavior12,13. We have addressed\nsome of these anomalous behavior in our magnetotrans-\nport studies of Co doped Mn2Sb8. In these studies we\nhave shown that anomalous thermomagnetic irreversibil-\nities at low temperature are a result of critically slow\ndynamics of the transition and these are different from\nthe seemingly similar irreversibility that arise due to su-\npercooling and superheating.\nII. EXPERIMENTAL DETAILS\nMn1.85Co0.15Sbsample used in the present study\nis taken from same ingot which has been used for\nearlier resistivity/magnetoresistance studies8. Resistiv-\nity/magnetoresistance measurement were performed us-\ning home made resistivity setup inside Oxford magnet\nsystem. Magnetization measurement were performed us-\ning VSM option of PPMS. Magnetic imaging was carried\nout using Scanning Hall Probe Microscope from Nano-\nMagnetics Instruments, U.K. The microscope incorpo-\nrates a chip sensor, which consists of a 1-micron size\nsquare Hall sensor integrated adjacent to a tunneling tip.2\nThe tunneling tip is used for bringing the Hall sensor in\nclose proximity to sample surface. The sensor chip is\naligned with a small angle ( ≈10) to keep tunneling tip\nclosertosamplesurfacethanHallsensor. Magneticimag-\ning is carried out by scanning the Hall sensor over the\nsample surface while simultaneously measuring the Hall\nvoltage, which is proportional to perpendicular compo-\nnent of the magnetic field at the surface. In the present\nstudy we have carried out magnetic imaging in lift off\nmode. In this mode sample surface is reached by find-\ning tunneling current. After finding the sample surface\ntip is retracted few hundred nanometers, called lift off,\nand scanning is performed at this constant height. For\nlow temperature and high field measurements this insert\nis placed inside 9-Tesla superconducting magnet (Amer-\nican Magnetics) system supported on a two-stage vibra-\ntionisolationstage. Approximately6mmdiameterand 2\nmm thick sampleis polishedto mirrorfinished surfacefor\nSHPM imaging. All the images in the present study are\nof 27µm×27µmscan area and pixel size 128 ×128,which\nwere scanned at 0 .3µmlift off and 5 µm/secscan speed.\nAll the measurements were carried out as a function of\nMagnetic field at 120 K for two protocols; (i) Sample\nis cooled from 300 K to 120K (i.e reached by cooling)\nin zero field and (ii) sample is cooled to 5 K and then\nheated back to 120 K (i.e. reached by warming)in zero\nfield.\nIII. RESULTS AND DISCUSSION\nFigure 1 [a]and [b] show the temperature dependence\nof resistivity ( ρ) in zero magnetic field and magnetiza-\ntion (M) in 0.1 Tesla magnetic field respectively for cool-\ning and then warmingcycle. AFM (higher resistivity and\nlowermagnetization) to FRI (lowerresistivity and higher\nmagnetization) transition is visible as a sharp decrease\nin resistivity (increase in magnetization) with increasing\ntemperatureandshowsahysteresisof 10Kbetweenheat-\ning and cooling cycle. The slightly lower transition tem-\nperatures obtained from magnetization measurement are\nin accordancewith magnetic field dependence of TN. Be-\nside the 10 K hysteresis, transition is broad for both cy-\ncles during cooling as well as warming. This is expected\nfor substitutional alloyswhereinherent chemical disorder\ncan result in distribution of local transition temperature\non the length scale of correlation length1. The spread in\nlocaltransitiontemperatureresultin aband oftransition\ninH-Tspaceandtwophases(hereFRIandAFM)canco-\nexist within this band. Therefore, this broadeningof first\norder transition makes this compound suitable to study\nthe coexistence of phases and their evolution with mag-\nnetic field. The schematic of ( H∗,T∗) and (H∗∗,T∗∗) for\nAFMtoFRItransitionisshownintheinsetoffigure1[a].\nFor the sake of simplicity ( H∗,T∗) and (H∗∗,T∗∗) spin-\nodals are shown well separated in contrast to overlapping\nbands actually observed in the present system. Here,\nisothermal measurement were carried out along path QS/s83\n/s82 /s81 /s80/s84/s32/s42/s32/s42/s84\n/s78\n/s84/s32/s42/s72\n/s84\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48/s49/s46/s50/s49/s46/s54/s50/s46/s48/s91/s97/s93\n/s77/s110 \n/s49/s46/s56/s53/s67/s111\n/s48/s46/s49/s53/s83/s98/s32/s32/s40/s109 /s32/s99/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48/s46/s49/s32/s84/s101/s115 /s108/s97/s91/s98/s93\n/s77/s110 \n/s49/s46/s56/s53/s67/s111\n/s48/s46/s49/s53/s83/s98/s32/s77/s32/s40\n/s66 /s32/s47/s102/s46/s117/s46/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\nFIG. 1:[a]Resistivity in zero field and [b]Magnetization in\n0.1Teslamagneticfieldasafunctionoftemperaturemeasure d\nduring cooling and subsequent warming for Mn1.85Co0.15Sb.\nInsetshows theschematic ofsupercooling ( H∗,T∗) andsuper-\nheating ( H∗∗,T∗∗) spinodal in H-T space. Isothermal mea-\nsurements presented in Figure 2 (Figure 3) were carried out\nalong path QS when point Q is reached by cooling (warming)\nfollowing path PQ (RQ).\nfor two conditions, viz when the point Q is reached (i)\nby following path PQ (cooling from T > T∗∗) and (ii) by\nfollowing path RQ (heating from T < T∗).\nFigure 2 shows some of the representative SHPM im-\nages as a function of magnetic field with increasing and\nthen subsequent decreasing field at 120K when reached\nfrom 300 K (i.e. reached by cooling). All the images,\nshown in figure 2, are plotted on same scale after sub-\ntracting the applied magnetic field. The labels on these\nimages are marked on corresponding ρ−HandM−H\ncurves (plotted in the middle row of figure 2) to corre-\nlate these results. Image (a) of figure 2, taken at 0.5\nTesla shows inhomogeneous magnetic state where both\nFRI (blue) and AFM (red) phases co-exists. The image\ncontrast remains almost same with further increase in\nmagnetic field to 1 Tesla, image (b), which is consistent\nwith almost constant ρandMbetween these field values.\nAt 2 Tesla image (c) shows increased FRI fraction and\nmuch smaller AFM fraction indicating a field induced\nAFM to FRI transition. Further increase in magnetic\nfield to 4 Tesla results in homogeneous FRI state (im-\nage (d)). On reducing magnetic field from 5 Tesla to 2\nTesla image (e) shows almost homogeneous FRI phase in\ncontrastto field increasingcycle where we observedcoex-\nisting FRI and AFM phase image (c). However image (f)\ntakenat1Tesladuringfield decreasingcycle, showsinho-\nmogeneous magnetic state which is similar to image (c)\nobserved during field increasing cycle at 2 Tesla. This ir-\nreversibilityisconsistentwiththefirstordernatureofthe\nfield induced magnetic transition. Both resistivity value\nas well as magnetization value are identical for point ‘c’\nandpoint‘f’asshowninbottomgraphs. Asthemagnetic\nfield reaches 0.5 Tesla the image (g) resembles image (a)\ntaken during field increasing cycle for same field value\ni.e. the magnetic state of the system is same before and\nafter the application of magnetic field at low field. This3\n/s48 /s50 /s52/s48/s46/s55/s48/s46/s57/s49/s46/s49/s49/s46/s51\n/s48 /s50 /s52/s48/s46/s51/s48/s46/s57/s49/s46/s53/s50/s46/s49/s91/s97/s93/s48/s46/s53/s32/s84/s101/s115 /s108/s97\n/s91/s98/s93/s52/s46/s48/s32/s84/s101/s115 /s108/s97 /s50/s46/s48/s32/s84/s101/s115 /s108/s97 /s49/s46/s48/s32/s84/s101/s115 /s108/s97\n/s91/s99/s93\n/s65/s70/s77 /s70/s82/s73/s91/s100/s93\n/s91/s101/s93 /s91/s102/s93 /s91/s103/s93\n/s50/s53 /s53/s48 /s55/s53 /s49/s48/s48\n/s32/s50/s32/s84/s101/s115/s108/s97\n/s91/s101 /s93\n/s91/s99 /s93\n/s32/s32\n/s50/s53 /s53/s48 /s55/s53/s49/s32/s84/s101/s115/s108/s97/s32\n/s91/s102 /s93/s91/s98 /s93\n/s32/s32\n/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s77 /s97/s103/s110/s101/s116/s105/s99/s32/s73/s110/s100/s117/s99/s116/s105/s111/s110/s32/s40/s37/s41/s50/s53 /s53/s48 /s55/s53/s48/s49/s48/s48/s50/s48/s48\n/s48/s46/s53/s32/s84/s101/s115/s108/s97/s67/s111/s117/s110/s116/s115 /s91/s103 /s93/s91/s97 /s93\n/s32/s32\n/s32/s32/s103\n/s102\n/s101/s100/s97\n/s98/s32/s40/s109 /s32/s99/s109 /s41\n/s99/s32\n/s40/s105/s105/s41/s40/s105/s41\n/s32/s32\n/s72/s32/s40/s84/s101/s115/s108/s97/s41\n/s40/s105/s105/s41\n/s40/s105/s41 /s98/s99/s101\n/s103/s100/s102\n/s97\n/s32/s32/s77/s32/s40\n/s66/s32/s47/s32/s102/s46/s117/s46/s41\n/s72/s32/s40/s84/s101/s115/s108/s97/s41\nFIG.2:[a]-[g]SHPMImages of Mn1.85Co0.15Sbasafunction\nof magnetic field at 120 K (reached by cooling from 300 K)\nalong with corresponding resistivity and magnetization cu rve.\nScan area is 27 µm×27µmand image label corresponds to\nrespective point in resistivity and magnetization curves. Bot-\ntom row shows the histograms of magnetic images at labeled\nfield values for increasing and decreasing field cycle. Inhom o-\ngeneous magnetic state and similar magnetic state after fiel d\ncycling at low field is highlighted along with characteristi c\nhysteresis associated with magnetic field induced first orde r\ntransition\nis consistent with the ρ−HandM−Hcurves, where\nzero field resistivity is found to be same before and after\nthe application of magnetic field and virgin curve (curve\ntaken during first field increasingcycle) overlapswith the\nenvelope curve (taken during second field increasing cy-\ncle).\nThe weak contrast in the images arises due to bulk\nsample(thickness2mm), whereastheobservedphasesep-\naration is on the length scale of few µm. To demonstrate\nmagneticinhomogeneitymoreclearly,histogramsofmag-\nnetic field distribution are plotted in the bottom row cor-\nrespondingtomagnetic imagesshownin samefigure. For\nthe sake of comparison, field window for histogram cal-\nculation as well as vertical scale are kept same in all the\nplots. For 0.5 Tesla both the curves (curve ‘a’ and ‘g’)\nare almost identical with slightly higher FRI phase for\ncurve ‘g’. The magnetization at point ‘g’ is only slightly\nhigher than in point ‘a’. At 1 Tesla, curve ‘b’ and curve\n‘f’, indicate entirely different magnetic field distributionon sample surface during field increasing and decreasing\ncycle. Similarto1Tesla, weobserveentirelydifferenthis-\ntograms corresponding to image (c) and image (e) taken\nat 2 tesla.\nWe repeated similar measurement under identical con-\ndition at 120K,when reached by warming from 5K under\nzero field condition. This experiment also shows a field\ninduced AFM to FRI transition with varying field and\nassociated irreversibility. However, our main interest is\nto study the state of the system at zero field before and\nafter field cycling i.e. across virgin and envelope curve.\nTherefore in Figure 3, we show resistivity and magne-\ntization data along with only two sets of images; one\ntaken at 0.1 Tesla and other taken at 1 Tesla. For each\nset, images were taken during first field increasing cy-\ncle i.e. virgin curve , field decreasing cycle and second\nfield increasing cycle. These curves are labeled as (i), (ii)\nand (iii) in ρ−HandM−Hplots along with markers\nat which magnetic images were taken. To compare the\nSHPM images at constant field, magnetic scale is kept\nsame for each set of images separately but varied for dif-\nferent magnetic field. As can be seen in top row (0.1\nTesla); image (a) is distinctly different from other two\nimages (d) and (e) which are identical. Image (a) shows\nalmost homogeneous AFM state whereas other two im-\nages shows coexisting FRI and AFM states. This is con-\nsistent with the ρ−Hcurve where point ‘d’ and ‘e’ have\nalmost same resistivity but much smaller compared to\npoint ‘a’. In case of M−Halso, M is same for point ‘e’\n/s48 /s50 /s52/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s48 /s50 /s52/s48/s46/s55/s49/s46/s51/s50/s46/s48/s48/s46/s49/s32/s84/s101/s115/s108/s97/s91/s97/s93 /s91/s100/s93 /s91/s101/s93/s49/s46/s48/s32/s84/s101/s115/s108/s97/s91/s98/s93/s32\n/s32/s91/s99/s93\n/s32/s32/s91/s102/s93\n/s32/s32/s40/s105/s105/s105/s41\n/s40/s105/s105/s41/s40/s105/s41\n/s102/s101\n/s100\n/s99/s98/s97\n/s32/s32/s32 /s32/s40/s109 /s32/s99/s109/s41/s32\n/s72/s32/s40/s84/s101/s115/s108/s97/s41/s84/s61/s32/s49/s50/s48/s32/s75/s32\n/s114/s101/s97/s99/s104/s101/s100/s32/s98/s121/s32/s119/s97/s114/s109/s105/s110/s103\n/s40/s105/s105/s105/s41/s40/s105/s105/s41\n/s40/s105/s41/s102\n/s101\n/s100/s99\n/s98\n/s97\n/s32/s32/s77/s32/s40/s32\n/s66/s32/s47/s32/s102/s46/s117/s46/s41/s32\n/s72/s32/s40/s84/s101/s115/s108/s97/s41/s84/s61/s32/s49/s50/s48/s32/s75/s32\n/s114/s101/s97/s99/s104/s101/s100/s32/s98/s121/s32/s119/s97/s114/s109/s105/s110/s103\nFIG. 3: [a]-[f]SHPM Images of Mn1.85Co0.15Sbat 0.1 and\n1.0 Tesla taken at 120 K (reached from 5 K) along with cor-\nresponding resistivity and magnetization curve. Scan area is\n27µm×27µmand magnetic scale is same for each row sepa-\nrately.4\nand ‘d’ and smaller for point ‘a’. This is in contrast to\nfigure 2 (measured during cooling) where magnetic state\nof the system at low field is identical before and after the\napplication of magnetic field. Images for 1 Tesla (bot-\ntom row) show that magnetic state of the system are\nmore similar during curve (i) and curve (iii) (image (b)\nand image (f)) compared to that measured during curve\n(ii) (image (c)). Image (c), taken during field reducing\ncycle, has much larger FRI phase fraction compared to\nimage (f) taken during field increasing cycle at same field\nvalue. This is consistent with ρ−Hcurve where point ‘b’\nand point ‘f’ have similar resistivity values compared to\npoint ‘c’. A closer inspection of these images show higher\nFRI fraction in image (f) compared to image (b).\n/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s32/s32/s32\n/s32/s48/s46/s49/s32/s84/s101/s115/s108/s97/s91/s100/s93/s67/s111/s117/s110/s116/s115\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s32 /s32/s32/s32/s69 /s68 /s67 /s66 /s65 /s69 /s68 /s67 /s66 /s65\n/s48/s46/s49/s32/s84/s101/s115/s108/s97/s91/s101/s93\n/s78/s111/s114/s109/s97/s108 /s105 /s122/s101/s100/s32/s77/s97/s103/s110/s101/s116/s105 /s99/s32/s73 /s110/s100/s117/s99/s116/s105 /s111/s110/s32/s40/s37/s41/s49/s46/s48/s32/s84/s101/s115/s108/s97/s32/s91/s99/s93\n/s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s49/s46/s48/s32/s84/s101/s115/s108/s97\n/s32/s32/s91/s102/s93/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s32/s32\n/s32/s91/s97/s93\n/s48/s46/s49/s32/s84/s101/s115/s108/s97/s91/s98/s93\n/s49/s46/s48/s32/s84/s101/s115/s108/s97\nFIG. 4: Histograms of SHPM Images of Mn1.85Co0.15Sbat\n0.1 Tesla and 1.0 Tesla for virgin curve, field decreasing cy-\ncle and second field increasing cycle at 120 K (reached by\nwarming from 5K).These curveshighlight thatthehistogram\ncorresponding to second field increasig cycle are distincly dif-\nferent from first cycle.\nHistograms of magnetic images at low field are shown\nin figure 4 for 0.1 and 1.0 Tesla magnetic field for all the\nthree cycles. Similar to figure 2, field window chosen for\nhistogramcalculationand verticalscalearekept same for\nall the figures. At 0.1 Tesla, histogram corresponding to\nimage(a) hassharppeak and hasdistinctly different field\ndistribution compared to curve (e).In case of histogram\n(a) about 87% of scanned region have magetic induc-\ntion in the range AB where magnetic induction is lessthan 25% of the total scale, indicating almost homoge-\nnous AFM state for image (a). Whereas for histogram\n(d) and (e) more than 90% region has magnetic induc-\ntion in the range BD (25 −75% of total scale). Even at 1\nTesla there is a difference in magnetic field distribution\nforcurve(b) and(f) thoughlessdrasticcomparedto that\nobserved in 0.1 Tesla. Here also, more than 81% region\nfor histogram (b) have magnetic induction in the range\nAB (0-25 % of total scale) comapared to only ≈60%\nregions for histogram (f) in the same range. It shows\nthat there is more FRI phase fraction during second field\nincreasing cycle at same field value compared to that ob-\nserved during virgin curve. These observation show that\nmagnetic state of the system changes from almost ho-\nmogenous AFM state to coexisting AFM and FRI state\nafter field cycling at 120 K (when reached by warming).\nThis explains the origin of open hysteresis loop in ρ−H\nand virgin curve lying outside envelope curve in M−H\nas well as ρ−Hmeasurements.\nIV. CONCLUSIONS\nThe SHPM images of Mn1.85Co0.15Sbat 120 K show\nalmost homogenous AFM state at low field when 120 K\nis reached by warming in contrast to coexisting AFM\nand FRI state when reached by cooling i.e. the mag-\nnetic state of the system on mesoscopic length scale de-\npends on the path followed to reach the measuerement\ntemperature. Almost homogenous low field AFM state\nduring warming is converted to coexisting AFM and FRI\nstate on mesoscopic length scale after isothermal field\ncycling. These studies provide the origin of open hys-\nteresis loop observed in ρ−Hand virgin curve lying\noutside the envelope curve in ρ−HandM−Hmeasure-\nments observed during warming only. Similar studies on\nfrozen glassy magnetic states will provide further insight\non phase seperation and metastability.\nV. ACKNOWLEDGMENTS\nWe acknowledge DST, India for funding of PPMS-\nVSM used in present study and Kranti Kumar for his\nhelp in magnetization measurement.\n1Y. Imry and M. Wortis, Phys. Rev. B 193580 (1979).\n2Y. Tokura, Rep. Prog. Phys. 69, 797 (2006).\n3P.A. Sharma, S. El-Khatib, I. Mihut, J.B. Betts, A.\nMigliori, S.B. Kim, S. Guha and S.-W. Cheong Phys. Rev.\nB78, 134205 (2008).\n4G.Y. Wang, X. H. Chen, T. Wu, X.G. Luo, W. T. Zhang,\nand G. Wu, Phys. Rev. Lett. 100, 146402 (2008).\n5X. S. Xu, M. Angst, T.V. Brinzari, R. P. Hermann, J. L.Musfeldt, A. D. Christianson, D. Mandrus, B. C. Sales, S.\nMcGill, J.-W. Kim, and Z. Islam, Phys. Rev. Lett. 101,\n227602 (2008).\n6S. B. Roy, P. Chaddah, V. K. Pecharsky and K. A. Gschnei-\ndner, Jr., Acta. Mater. 56, 5895 (2008).\n7K. Kumar, A. K. Pramanik, A. Banerjee, P. Chaddah, S.\nB. Roy, S. Park, C. L. Zhang and S. W. Cheong, Phys. Rev.\nB73184435 (2006).5\n8Pallavi Kushwaha, RRawatandPChaddah, J. Phys.: Con-\ndens. Matter 20022204 (2008).\n9S. B. Roy, G. K. Perkins, M. K. Chattopadhyay, A.K.\nNigam, K.J.S. Sokhey, P. Chaddah, A.D. Caplin and L.\nF. Cohen, Phys. Rev. Lett. 92, 147203 (2004).\n10J. D. Moore, G. K. Perkins, Y. Bugoslavsky, L. F. Co-\nhen, M. K. Chattopadhyay, S. B. Roy, P. Chaddah, K. A.\nGschneidner, Jr., and V. K. Pecharsky, Phys. Rev. B 73,\n14426 (2006).11O. Beckman, Lundgren, in: K.H.J. Buschow (Ed.), Hand-\nbook of Magnetic Materials, vol. 6, Elsevier, Amsterdam,\n1991, p. 181 (Chapter 3).\n12M.I. Bartashevich, T.Goto, T. Tomita, N.V. Baranov, S.V.\nZemlyanski, G. Hilscher, H. Michor, Physica B 318, 198\n(2002).\n13Y. Q. Zhang, Z. D. Zhang and J. Aarts, Phys. Rev. B 70,\n132407 (2004); ibid 71, 229902(E) (2005)."    },    {        "title": "1406.1859v1.Interface_dependent_magnetotransport_properties_for_thin_Pt_films_on_ferrimagnetic_Y3Fe5O12.pdf",        "content": "arXiv:1406.1859v1  [cond-mat.mes-hall]  7 Jun 2014Interface-dependent magnetotransport properties for thi n Pt films on ferrimagnetic\nY3Fe5O12\nY. Shiomi1, T. Ohtani1,∗S. Iguchi1, T. Sasaki1, Z. Qiu2, H. Nakayama1,3, K. Uchida1,4, and E. Saitoh1,2,5,6\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan\n3Laboratory for Nanoelectronics and Spintronics,\nResearch Institute of Electrical Communication, Tohoku Un iversity, Sendai 980-8577, Japan\n4PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan\n5CREST, Japan Science and Technology Agency, Tokyo 102-0076 , Japan and\n6Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan\n(Dated: November 5, 2018)\nWe have studied magnetoresistance and Hall effects for 1 .8-nm-thick Pt films grown on a ferri-\nmagnetic insulator Y 3Fe5O12in a wide temperature (0 .46-300 K) and magnetic-field ( −15-15 T)\nregion. In the low-temperature regime where quantum correc tions to conductivity are observed,\nweak antilocalization behavior observed in Pt films is criti cally suppressed when the film is attached\nto Y3Fe5O12. Hall resistance in the Pt film is also affected by Y 3Fe5O12, and it exhibits logarithmic\ntemperature dependence in a broad temperature range. The ma gnetotransport properties in the\nhigh-field range are significantly influenced by the interfac e between Pt and Y 3Fe5O12.\nIn the field of spintronics, a pure spin current, which is\na flow of spin angular momentum without a net charge\ncurrent, has attracted a great deal of attention in view\nof spin-current science and also of practical application\n[1]. For study on spin-current phenomena, Pt |Y3Fe5O12\n(Pt|YIG) bilayers have been used frequently as a typi-\ncal system. YIG is a ferrimagnetic insulator with a large\ncharge gap ( ∼2.7 eV) and a high magnetic-transition\ntemperature ( ∼553K), which enablesspin-currentinjec-\ntion free from spin-polarized currents at room tempera-\nture. Injected pure spin currents are able to be detected\nelectrically in Pt by means of the inverse spin Hall effect\n(ISHE)whichisthe conversionofaninjectedspincurrent\ninto a transverse electric current due to the spin-orbit\ninteraction. Since Pt has strong spin-orbit interaction,\nefficiency of the ISHE is as high as 1-10 percent; hence,\nPt has often been used as a spin-current detector. Us-\ning Pt|YIG systems, many experiments on spin-current\ninjection and detection have been performed, e.g.spin\npumping [2] and the spin Seebeck effect [3].\nRecently, an unconventional magnetoresistance (MR)\neffect was reported for Pt |YIG structures. Although Pt\nis a paramagnetic metal, MR in about 10-nm-thick Pt\nfilms on YIG reflect the magnetization direction of YIG\nand anisotropic MR was observed in a low magnetic-field\nregion(≤0.2T) [4, 5]. ThisanisotropicMR wasfound to\nbe caused mainly by a spin mixing effect at the interface\nbetween Pt and YIG [5]; concerted actions of the direct\nand inverse spin Hall effects generate an additional elec-\ntriccurrentandthusleadtoresistancechangeaffectedby\nthe magnetization direction in YIG. This magnetoresis-\ntance wasnamed the spin-Hallmagnetoresistance(SMR)\n[5] and this mechanism has been supported by following\nreports [6–13].\n∗Y. Shiomi and T. Ohtani contributed equally to this work.In the present paper, we discuss interface-dependent\nmagnetotransport properties in Pt |YIG at low temper-\natures using very thin ( ∼2 nm) Pt films where the in-\nterface effect should be further pronounced owing to the\nreduced Pt volume. By conducting magnetotransport\nmeasurements in a wide temperature (0 .46-300 K) and\nmagnetic field ( −15-15 T) region, we have shown that\nMRandHalleffectsathighmagnetic-fieldsin Pt |YIGex-\nhibit totallydifferentbehaviorfromthosein conventional\nparamagnetic metals. These unconventional magneto-\ntransport properties are prominent at low temperatures\nand at high magnetic-fields, which are clearly irrelevant\nto magnetization change in YIG.\nWe measure magnetotransport properties of Pt thin\nfilms attached to (111) planes of YIG fims or paramag-\nnetic Gd 3Ga5O12(GGG) substrates. Here, Pt |GGG was\nused for control experiments, since GGG has the same\ncrystal structure as YIG and is paramagnetic down to\n0.46 K. Micrometer-thick YIG films were grown on (111)\nGGG substrates by liquid phase epitaxy [14]; the mag-\nnetization of YIG films is saturated for µ0H>∼0.3 T in\na perpendicular magnetic field [Fig. 1(a)]. Before depo-\nsition of Pt, YIG films and GGG substrates were first\ncleaned in organic solvents inside an ultrasound bath,\nfollowing surface treatment with H 2SO4and H 2O2; this\nprocess is important to observe the interface-dependent\nmagnetotransport phenomena in the present work. We\nthen sputtered 1 .8-nm-thick Pt thin films with Hall-bar\ngeometryoncleanedYIGorGGG surfacesinArpressure\nof 7.0 mTorr. The magnetotransport measurements were\nperformed as illustrated in Fig. 1(a). The measurements\nwere carried out in superconducting magnets up to ±15\nT in the temperature range between 0 .46 K and 2 K as\nwell as up to ±9 T in that from 2 K to 300 K.\nWe show, in Fig. 1(b), the temperature ( T) de-\npendence of sheet resistance, Rsheet, for Pt|YIG and\nPt|GGG. The resistance for both the samples shows\nmetallicTdependence with the residual resistance2\n(a) (b)\n(c) (d)\nH\nIxyz\n0 100 200 3000100200300400500\nT (K)Rsheet (Ω)Pt/YIGPt/GGG\n Pt/YIG  Pt/GGG\n-10 0 1005x101x10\nµ0H (T)∆R/R \nPt/YIGPt/GGG\nWAL -2 \n-3 \n-10 0 10-50050\nµ0H (T)RH (mΩ)Pt/YIG\nPt/GGGT=0.46 KT=0.46 K\nT=1 K7+-\n+-7\nYIG, GGG Pt \nx\n1 10325330475480485\nmagnetoresistance Hall effect\n-10 102x10-4 \n0Pt/YIG\n1x10-4 Pt/GGG 5\u001e\u0013\u0011\u0011, \n00 3 923M (10-3emu)H || z\n1\n0T=300 K\nµ0H (T)magnetization\n6\nFIG. 1: (a) A schematic illustration of experimental setup.\nAn electric current is applied along the xaxis. Longitudi-\nnal (Hall) resistance is calculated from the applied electr ic-\ncurrent and voltage produced along the x(y) axis. Magnetic\nfield (H) dependence of magnetization ( M) for a YIG film\n(1.5mm×2mm) at 300Kin H||z. (b)Temperature ( T)depen-\ndence of sheet resistance ( Rsheet) for Pt|YIG and Pt |GGG.\nRsheetin a low- Trange is magnified in the inset to (b); the\ntemperature scale is logarithmic. (c) Magnetic field ( H) de-\npendence of magnetoresistance [∆ R/R={R(H)−R(H=\n0)}/R(H= 0)] in H||zfor Pt|YIG and Pt |GGG at 0 .46 K.\nThe solid line is the fit to weak-anti localization (WAL). The\ninset shows results at 200 K. (d) Magnetic field ( H) depen-\ndence of Hall resistance for Pt |YIG at 0 .46 K and Pt |GGG at\n1 K.\nRsheet∼300-500 Ω. Different magnitudes of Rsheet\nbetween Pt |YIG and Pt |GGG mainly originate from\nslightly different Pt-thicknesses which are inevitable in\nour sputtering system. As shown in the inset to Fig.\n1(b),Rsheetshows a minimum around 20 K and then in-\ncreases with decreasing Tbelow∼20 K. This resistance\nrise is almost proportional to ln T, indicating manifes-\ntation of weak (anti-)localization which is incipient of\nquantum corrections in disordered conductors.\nFigure 1(c) shows magnetic field ( H) dependence of\nMR for Pt |YIG and Pt |GGG atT= 0.46 K. Here, the\nmagnitude of MR is defined as ∆ R/R≡ {R(H)−R(H=\n0)}/R(H= 0). For Pt |GGG, positive MR is observed\nand its magnitude is ∼1% at 15 T. This positive MR\nin Pt|GGG is well explained by weak anti-localization\n(WAL) which appears in disordered conductors with\nstrong spin-orbit interaction [15, 16]. By contrast, Pt\nthin films on YIG show a totally different MR effect from\nPt|GGG, at 0.46 K. The MR is negative and its magni-\ntude is as small as 0 .1%. This clear difference in MR\nbetween Pt |YIG and Pt |GGG is not observed at high\ntemperatures; as shown in the inset to Fig. 1(c), at\n200 K, while SMR reflecting the magnetization process\nof YIG is observed for Pt |YIG in a low- Hregion (<0.5\nT), MR effects in a high- Hregime (>0.5 T) are simi-(a) (b)\n-10 0 10-1x10-5x1005x10\nµ0H (T)∆R/R  2K4K6K10K200K\n0.46K1K- 4\n- 4\n- 3 \n-10 0 10-50050\nµ0H (T)RH (mΩ)10K\n30K\n40K\n60K\n100K\n200K\n300K2K1K0.46Kmagnetoresistance (Pt/YIG) Hall effect (Pt/YIG)\n0.1 1 10 100-100102030\n҃1/T\n҃lnT\nT (K)RH/µ0H (mΩ/T) Curie law\n(local spin)(d)\n0 1 2 3 4 501x102x103x10\n∆σsheet Ω−1)-6 \n-6 \n-6 (\nµ0H (T)WL (c)\nT=0.46K\nFIG. 2: Magnetic field ( H) dependence of (a) magnetore-\nsistance (∆ R/R) inH||zand (b) Hall resistance ( RH) at\nvarious temperatures between 0 .46 K and 300 K. (c) Hde-\npendence of magnetoconductance [∆ σsheet= 1/Rsheet(H)−\n1/Rsheet(H= 0)] for Pt |YIG at 0 .46 K. Here, SMR contri-\nbution observed in a low- Hregion is subtracted. The solid\nline is the fit to weak localization (WL). (d) Temperature ( T)\ndependence of the Hall coefficient in the Trange between 0 .46\nK and 300 K. The Hall coefficient is calculated in the low- H\nregion<1 T. The temperature scale is logarithmic. The dot-\nted lines are curves proportional to ln Tand 1/T, which are\nguides for the eyes.\nlar between Pt |YIG and Pt |GGG. These results clearly\nshow that in a low- Trange where quantum corrections\nare observed, unconventional MR shows up in Pt |YIG at\nhigh magnetic-fields where the magnetization of YIG is\nfully aligned along the Hdirection.\nIn Fig. 2(a), we show MR in Pt |YIG at various tem-\nperatures. At 200 K, positive MR showing quadratic H-\ndependence is observed in a high- Hregion; this is char-\nacteristic of ordinary MR related with Lorentz force [17].\nAsTis decreased, MR hardly changes with Tdown to\n10 K, but, below 10 K, MR in a high- Hregion shows a\nsign change from positive to negative and its magnitude\nabruptlyincreases,whileSMRobservedinalow Hregion\n(<0.5 T) is almost independent of temperature even in\nthisTrange [see also Figs. 1(c) and 4(a)]. We compare\ntheTrange of MR enhancement and that of weak (anti-\n)localization regime determined from T-Rsheetcurve, in\nFigs. 3(a) and (b). As shown in Fig. 3(b), in the weak\n(anti-)localization regime (highlighted in yellow color in\nFig. 3), negative MR at 9 T is enhanced almost in pro-\nportion to ln T, which signals weak localization (WL)\nbehavior [15]. On YIG, WAL in Pt is suppressed and\nWL appears in spite of the strong spin-orbit interaction\nin Pt.\nIn magnetic fields and under strong spin-orbit interac-\ntion, the quantum correction to the sheet conductance,\n∆σsheet(H)≡1/Rsheet(H)−1/Rsheet(H= 0), is given3\n- 4\n)\u0001\u001e\u0001\u001a5 )\u0001\u001e\u0001\u001a5 -5x \u0012\u0011 \n1 10 100-50050\nT (K)RH (mΩ)\nHall effect 0\n∆R/R  \nmagnetoresistance320330340350360370Rsheet (Ω)resistance\n- 3\n-1x \u0012\u0011 (a)\n(b)\n(c)\nFIG. 3: Temperature ( T) dependence of (a) sheet resistance\n(Rsheet), (b) magnetoresistance (∆ R/R) atµ0H= 9 T (||z),\nand (c) Hall resistance ( RH) atµ0H= 9 T. Here, the tem-\nperature scale is logarithmic. Weak (anti-)localization r egime\nis highlighted in yellow color.\nby [15, 18, 19]\n∆σsheet(H) =e2\n2π2¯h/bracketleftBig\nψ/parenleftBig1\n2+B1\nµ0H/parenrightBig\n−3\n2ψ/parenleftBig1\n2+B2\nµ0H/parenrightBig\n+1\n2ψ/parenleftBig1\n2+B3\nµ0H/parenrightBig\n−ln/parenleftBigB1√B3\nB3/2\n2/parenrightBig/bracketrightBig\n,(1)\nwhereB1=Bτ+BSO,B2= (3/4)BSO+Bφ, and\nB3=Bφ. Here,Bτ,BSO, andBφare effective mag-\nnetic fields for elastic, spin-orbit, and inelastic scatter-\nings, respectively. Using eq.(1), we fit MR at 0 .46 K\nfor Pt|GGG and Pt |YIG, as shown with solid lines in\nFigs. 1(c) and 2(c), respectively. The fitted curves al-\nmost reproduce the experimental results in both cases.\nThe fitting parametersare Bτ= 200T,BSO= 20 T, and\nBφ= 0.049T for Pt |GGG, while Bτ= 200T,BSO= 2.2\nT, andBφ= 0.31 T for Pt |YIG. The decrease in BSOin\nPt|YIG is explained by suppression of spin-flip scattering\ncaused by spin-orbit interaction in the presence of mag-\nnetic spin-exchange interaction at the interface [20–22].\nWe note that the change in BSOvalues between Pt |YIG\nand Pt|GGG also manifests itself in Rsheetin the zero\nfield; since the temperature of minimal sheet-resistance\nis proportional to ln(1 /BSO) [15], that temperature is\nhigher in Pt |YIG than Pt |GGG, as shown in the insets toIH\nH\nI IHIHH\nI\nIH(a) (b)\n-5 0 50\nµ0H (T)H // z\nH // y H // x\n-5 0 5-4x10\n-6x10-2x102x10\n0\nµ0H (T)∆R/R \nH // yH // x H // z\n- 4Pt/YIG Pt/GGG\n5x10- 31x10- 2\n- 4- 4\n- 4\nFIG. 4: Magnetic field ( H) dependence of magnetoresistance\n(∆R/R) in three different magnetic-field directions ( H||x,\nH||y, orH||z) for (a) Pt |YIG and (b) Pt |GGG. Measurement\ntemperature is 2 K. In (a), the dotted lines are raw data,\nwhile the symbols indicate the data in which SMR contribu-\ntions observed in a low- Hregion are subtracted.\nFig. 1(b). While BSOis smaller in Pt |YIG,Bφis larger\nin Pt|YIG than Pt |GGG. A possible magnetic scattering\naround the interface [23, 24] may enhance the effective\nBφvalue in Pt |YIG.\nSuch an interface effect also appears in the Hall ef-\nfect. Figure 1(d) shows Hdependence of Hall resistance,\nRH, at 0.46 K for Pt |YIG and at 1 K for Pt |GGG.RH\nfor Pt|GGG shows linear dependence on H; this is the\nnormal Hall effect induced by Lorentz force. In Pt |YIG,\nby contrast, RHshows clearly nonlinear H-dependence\nand its magnitude is much larger than that for Pt |GGG;\nwith increasing Hfrom the zero field, |RH|increasesdra-\nmatically and becomes almost saturated above 5 T. This\nHdependence of RHcorresponds to neither the applied\nmagnetic field nor the magnetization process in YIG. As\nshown in Fig. 1(d), the field value ( ∼5 T) where RH\nbecomes almost saturated is much higher than the sat-\nuration field of YIG magnetization ( ∼0.3 T), which in-\ndicates that the internal magnetic field induced by YIG\nmagnetization is not the origin of the nontrivial Hde-\npendence of RH.\nA plot of Hall resistance ( RH) versusHis shown at\nvarious temperatures between 0 .46 K and 300 K in Fig.\n2(b). WhileMRlargelychangesonlyatlowtemperatures\nbelow 10 K, RHdepends on Teven above 100 K. RHin\nPt|YIG significantly changes with Tand even shows a\nsign change around 60 K. Since the normal Hall effect\nin paramagnetic metals is independent of Tas observed\nin Pt|GGG (not shown), this sign change suggests the\npresence of another contribution to the Hall effect other\nthan the normal Hall effect in Pt |YIG: anomalous Hall\neffect [25] or topological Hall effect [26].\nWe found that the Hall coefficient defined as\nRH/(µ0H) in a low-Hregion below 1 T shows logarith-\nmicTdependence in all the Tregion between 0 .46 K\nand 300 K, as shown in Fig. 2(d). In very recent papers\n[23, 24], the origin of similar nonlinear Hall resistance\nin Pt|YIG [23] and Pd |YIG [24] was attributed to the\nanomalous Hall effect assuming local paramagnetic mo-\nments produced near the interface; the Hdependence\nwas analyzed with Langevin or Brillouin function. The4\nobserved ln Tdependence is, however, different from the\nCurie law (1 /T) expected from the Langevin/Brillouin\nfunction in a low- Hregion, which is the simplest model\nof localized magnetic moments [23, 24]. Also, similarly\nto the low- Hcase, theTdependence of RHat 9 T is\nproportional to ln Tin a high-Tregime, as shown in Fig.\n3(c). With decreasing Tbelow 10 K, however, RHat 9 T\ndeviates from the ln Tbehavior and becomes almost sat-\nurated below 2 K, although the weak(-anti) localization\ndoes not affect the Hall effect at least in the conventional\nframework of weak localization. These results suggest\nthat lnTdependence of RHis observed in a low-field\nlimit,i.e.µBB/kBT≪1; sinceT=µBB/kB≈6 K for\nB= 9 T,RHmeasured at 9 T deviates from the ln T\ndependence in the low- Trange below ∼10 K.\nAt last, anisotropy of MR is shown at 2 K for Pt |YIG\nand Pt|GGG in Figs. 4(a) and (b), respectively, where\nHis applied in three different directions for each sample:\nH||x,H||y, andH||z[see also Fig. 1(a)]. As shown in\nFig. 4(b), in Pt |GGG, MR, i.e.WAL, clearly depends\non theHdirection; |∆R/R|inH||zis larger than that in\nin-planeHcases (H||xandH||y), which is the behavior\nexpected from WAL in nearly two-dimensional electron\nsystems [19, 27, 28]. In contrast, in Pt |YIG, except for\nSMR contribution affected by magnetization direction in\nYIG in a low- Hregion [5], the high- Hbehavior is almost\nisotropic with respect to H, as shown in symbols in Fig.\n4(a). Since anisotropic MR is not observed even at 2 K\nin our Pt |YIG, the possibility of AMR due to proximity-\ninduced ferromagnetism in Pt [4] is ruled out. Isotropic\nWL observed in Pt |YIG indicates that three dimensional\nnature is prominent compared with Pt |GGG owing tothe stronger inelastic scattering (the larger Bφvalue) in\nPt|YIG than Pt |GGG, since the condition for two di-\nmensionality with respect to WL is that film thickness\nis much smaller than the dephasing length,/radicalbig\n¯h/(4eBφ)\n[15].\nIn summary, we have shown unconventional magneto-\ntransport properties which are prominent at low temper-\natures and at high magnetic-fields for 1 .8-nm-thick Pt\nfilms in contact with YIG. Tdependence and Hdepen-\ndence of Hall resistance are clearly affected by the inter-\nface, butnotassociatedwiththoseofYIGmagnetization;\nin fact, Hall resistance shows logarithmic T-dependence\nin a broad T-range and nonlinear Hdependence at low\ntemperatures. Also, magnetoresistance is influenced by\nthe interface at low temperatures where quantum correc-\ntionsareimportant, andWLbehaviorisobserveddespite\nthe strong spin-orbit interaction of Pt. Such unconven-\ntional characteristics were not observed in Pt |GGG, al-\nthough the magnitude of field-induced magnetization for\nGGG is comparable to that for YIG at 2 K.\nWe thank Y. Fujikawa, T. Niizeki, R. Iguchi, and\nK. Takamura for fruitful discussions. This work was\nsupported by CREST-JST “Creation of Nanosystems\nwith Novel Functions through Process Integration”,\nPRESTO-JST “Phase Interfaces for Highly Efficient En-\nergyUtilization”, Grants-in-AidforJSPSFellows, Young\nScientists(B) (26790037and26790038),YoungScientists\n(A) (25707029), and Scientific Research (A) (24244051)\nfrom MEXT, and the Murata Science Foundation. Part\nof high magnetic-field experiments were supported by\nHigh Field Laboratory for Superconducting Materials,\nIMR, Tohoku University.\n[1] S. Maekawa, S.O. 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S ¸a¸ sıo˘ glu†\nInstitut f¨ ur Festk¨ orperforschung, Forschungszentrum J ¨ ulich, D-52425 J¨ ulich, Germany\nand Fatih University, Physics Department, TR-34500, B¨ uy¨ uk¸ cekmece, ˙Istanbul, Turkey\nI. Galanakis‡\nDepartment of Materials Science, School of Natural Science s, University of Patras, GR-26504 Patra, Greece\n(Dated: October 22, 2018)\nIntermetallic Heusler alloys are amongst the most attracti ve half-metallic systems due to the high\nCurie temperatures and the structural similarity to the bin ary semiconductors. In this review we\npresent an overview of the basic electronic and magnetic pro perties of the half-metallic full-Heusler\nalloys like Co 2MnGe. Ab-initio results suggest that the electronic and mag netic properties in these\ncompounds are intrinsically related to the appearance of th e minority-spin gap. The total spin\nmagnetic moment in the unit cell, Mt, scales linearly with the number of the valence electrons, Zt,\nsuch that Mt=Zt−24 for the full-Heusler alloys opening the way to engineer ne w half-metallic\nalloys with the desired magnetic properties. Moreover we pr esent analytical results on the disorder\nin Co2Cr(Mn)Al(Si) alloys, which is susceptible to destroy the pe rfect half-metallicity of the bulk\ncompounds and thus degrade the performance of devices. Fina lly we discuss the appearance of the\nhalf-metallic ferrimagnetism due to the creation of Cr(Mn) antisites in these compounds and the\nCo-doping in Mn 2VAl(Si) alloys which leads to the fully-compensated half-m etallic ferrimagnetism.\nPACS numbers:\nI. INTRODUCTION\nHalf-metallic ferromagnets, initially predicted by de\nGrootandhiscollaboratorsin 19831usingfirst-principles\ncalculations are at the center of scientific research due to\ntheir potential applications. The minority-spin electrons\nin these compounds show a semiconducting electronic\nband-structure while the majority-spin electrons present\nthe usual metallic behavior of ferromagnets. Thus such\nalloys would ideally exhibit a 100% spin polarization at\nthe Fermi level and therefore they should have a fully\nspin-polarized current and should be ideal spin injectors\ninto a semiconductor maximizing the efficiency of spin-\ntronic devices.2\nHeusler alloys are well-know for several decades and\nthefirstHeuslercompoundsstudiedhadthechemicalfor-\nmula X 2YZ, where X is a high valent transition or noble\nmetal atom, Y a low-valent transition metal atom and Z\na sp element, and crystallized in the L21structure which\nconsists of four fcc sublattices.3,4Heusler compounds\npresent a series of diverse magnetic phenomena like itin-\nerant and localized magnetism, antiferromagnetism, heli-\nmagnetism, Pauli paramagnetism or heavy-fermionic be-\nhaviorand thus they offer the possibility to study various\nphenomenain the samefamily ofalloys.3,4Asecondclass\nofHeusleralloyshavethechemicalformulaXYZandthey\ncrystallizein the C1bstructurewhich consistsofthreefcc\nsublattices; they are often called half- or semi-Heusler al-\nloys in literature, while the L21compounds are referred\nto as full Heusler alloys. The interest in these types\nof intermetallic alloys was revived after the prediction,1using first-principles calculations, of half-metallicity in\nNiMnSb, a semi-Heusler compound.\nThe high Curie temperatures3,4exhibited by the half-\nmetallic Heusler alloys are their main advantage with\nrespect to other half-metallic systems (e.g. some ox-\nides like CrO 2and Fe 3O4and some manganites like\nLa0.7Sr0.3MnO3).5While for the other compounds the\nCurie temperature is near the room temperature, for the\nhalf-metal NiMnSb it is 730 K and for the half-metallic\nCo2MnSi it reaches the 985 K.3A second advantage of\nHeusler alloys for realistic applications is their structural\nsimilaritytothezinc-blendestructure, adoptedbybinary\nsemiconductors widely used in industry (such as GaAs\non ZnS). Semi-Heusler alloys have been already incorpo-\nrated in several devices as spin-filters,6tunnel junctions7\nand GMR devices.8Recently, also the half-metallic full-\nHeusler alloys have found applications. The group of\nWesterholt has incorporated Co 2MnGe in the case of\nspin-valves and multilayer structures9and the group of\nReiss managedto create magnetic tunnel junctions based\non Co 2MnSi.10,11A similar study by Sakuraba and col-\nlaborators resulted in the fabrication of magnetic tunnel\njunctions using Co 2MnSi as magnetic electrodes and Al-\nO as the barrier and their results are consistent with\nthe presence of half-metallicity for Co 2MnSi.12Moreover\nDong and collaborators recently managed to inject spin-\npolarized current from Co 2MnGe into a semiconducting\nstructure.13\nIn the present contribution we review our most re-\ncent results on the electronic properties of the half-\nmetallic full-Heusler alloys obtained from first-principles\nelectronicstructurecalculations. InsectionIIwesumma-2\nrize our older results on bulk compounds obtained using\nthe full-potential version of the screened Korringa-Kohn-\nRostoker (KKR) method14(see references 15,16,17 for\nan extended review). In the next sections we overview\nsome of our results on the defects in the half-metallic\nfull Heusler alloys obtained using the full–potential\nnonorthogonal local–orbital minimum–basis band struc-\nturescheme(FPLO)18,19within thelocaldensityapprox-\nimation (LDA)20,21,22and employing the coherent po-\ntential approximation (CPA) to simulate the disorder in\na random way.19In section III we present the physics\nof defects in the ferromagnetic Heusler alloys contain-\ning Co and Mn like Co 2MnSi.23,24The study of defects,\ndoping and disorder is of importance to accurately con-\ntrol the properties of half-metallic full-Heusler alloys.25\nInsectionIVwedemonstratethecreationofhalf-metallic\nferrimagnets based on the creation of Cr and Mn anti-\nsites in Co 2(Cr or Mn)(Al or Si) alloys26,27and we ex-\npand this study to cover the case of Co defects in fer-\nrimagnetic Mn 2VAl and Mn 2VSi alloys leading to full-\ncompensated half-metallic ferrimagnets (also-known as\nhalf-metallic antiferromagnets).28Finally in sectionV we\nsummarize and conclude.\nII. ELECTRONIC AND GAP PROPERTIES –\nSLATER PAULING BEHAVIOR\nThe electronic, magnetic and gap properties of half-\nmetallic full-Heusler compounds have been studied using\nfirst-principles calculations in reference 29. These results\nhave been extensively also reviewed in reference 15, in\nthe introductory chapter of 16 and in a chapter in refer-\nence 17. The reader is directed to them for an extended\ndiscussion. In this section we will only briefly overview\nthese properties.\nThe electronic and magnetic properties of the full-\nHeusler compounds are similar to the half-Heusler\ncompounds30with the additional complication of the\npresence of 2 Co atoms per unit cell as in Co 2MnGe. In\nthe case of the semi-Heusler alloys there are exactly nine\noccupied minority-spin electronic states; one of s char-\nacter and three of p-character provided by the sp atom\nand five bonding d-states originating from the hybridiza-\ntion between the transition metal atoms. In addition to\nthese bands, in the case of full-Heusler alloys there are\nfivestatesexclusivelylocatedat theCo sitesandnearthe\nFermi level and the Fermi level is located among these\nstates so that three out of five are occupied and two of\nthem unoccupied leading to small energy gaps.29Now\nthere are 12 minority-spin occupied states and the com-\npounds with 24 valence electrons like Fe 2VSb are semi-\nconductors. Ifvanadiumissubstituted byahigher-valent\natom, spontaneous spin polarization occurs, and the ex-\nchange splitting shifts the majority states to lower en-\nergies. The extra electrons fill in only majority states\nand the total spin moment per unit cell in µB,Mt, which\nis the number of uncompensated spins is the total num-20 21 22 23 24 25 26 27 28 29 30 31 32\nTotal number of valence electrons (Zt)-3-2-101234567Total spin magnetic moment (Mt)\nMn2VAlFe2VAlFe2CrAlCo2VAl\nFe2MnAlCo2MnSi\nCo2MnGe\nCo2MnAl\nCo2MnGaCo2CrAl\nFe2MnSi\nMt=Zt-24Mn2VGeCo2MnSn\nSlater-Pauling rule:\nFIG. 1: Calculated total spin moment per unit cell in µBas a\nfunction of the total number Ztof valence electrons per unit\ncell for the full Heusler alloys. The dashed line represents the\nSlater-Pauling behavior.\nber of valence electrons, Zt, minus two times the 12,:\nMt=Zt-24. This behavior is the so-called Slater-Pauling\nbehavior and e.g. half-metallic Co 2CrAl (27 valence elec-\ntrons) has a total spin moment of 3 µBand half-metallic\nCo2MnSi (29 valence electrons) a spin moment of 5 µB.\nThis rule provides a direct connection between the half-\nmetallicityandthetotalspinmomentwhichcanbeeasily\ndetermined experimentally. In figure 1 we have plotted\nthe calculated total spin moments for severalfull-Heusler\ncompounds as a function of the total number of valence\nelectrons. The dashed line represents the Slater-Pauling\nrule of half-metallic full Heusler alloys. Since 7 majority\nbands (2 ×Coeu, 5×Mnd) are unoccupied, the largest\npossible moment is 7 µBand would occur if all majority\nd-states were occupied. Of course it would be impossible\nto get a compound with a total spin moment of 7 µBbut\nevenMt= 6µBis difficult to obtain. As it was shown\nby Wurmehl et al.31, the on-site correlations in Co 2FeSi\nplay a critical role for this compound and calculations\nwithin the LDA+U scheme, rather than the LDA, give\na spin moment of 6 µB. We should also note here that\nferromagnetism is stabilized by the inter-sublattice inter-\nactions between the Mn(Cr) and Co atoms and between\nCo atoms belonging to different sublattices as shown by\nfirst-principles calculations.32,33\nBefore closing this section we should discuss also the\nrole of the sp-elements in half-metallic Heusler alloys.\nWhile the sp-elements are not responsible for the ap-\npearance of the minority gap, they are very important\nfor the physical properties of the Heusler alloys and their\nstructural stability. sp atoms provide one s and three\np bands per spin which lay very low in energy and ac-\ncommodate d-electrons of the transition metal atoms.\nThus the effective d-charge, which is accommodated by\ntransition-metal atomic d-hybrids, is reduced stabilizing3\nthe half-metallicity. These s- and p-states of the sp atom\nstrongly hybridize with the transition metal d-states and\nthe charge in these bands is delocalized and locally the\nsp atoms even lose charge in favor of the transition metal\natoms.\nIII. DEFECTS IN FULL-HEUSLERS\nCONTAINING Co AND Mn\nIn this section we discuss results on the defects in the\ncase of Co 2MnZ alloys where Z is Al and Si.23,24To sim-\nulate the doping by electrons we substitute Fe for Mn\nwhile to simulate the doping of the alloys with holes\nwe substitute Cr for Mn and we have considered the\ncases of moderate doping substituting 5% and 10% of\nthe Mn atoms and we present our results in figure 2 for\nthe Co 2MnSi and Co 2MnAl compounds. As discussed\nin the previous section the gap is created between states\nlocated exclusively at the Co sites. The states low in en-\nergy (around -6 eV below the Fermi level) originate from\nthe low-lying p-states of the sp atoms and the ones at\naround -9 eV below the Fermi level are the s-states of\nthe sp-atom. The majority-spin occupied states form a\ncommon Mn-Co band while the occupied minority states\nare mainly located at the Co sites and the minority unoc-\ncupied states at the Mn sites. The extra electron in the\nthe Co 2MnSi alloy occupies majority states leading to an\nincrease of the exchange splitting between the occupied\nmajority and the unoccupied minority states and thus\nto larger gap-width for the Si-based compound. In the\ncase of the Al-based alloy the bonding and antibonding\nminority d-hybrids almost overlap and the gap is substi-\ntuted by a region of very small minority density of states\n(DOS); we will call it a pseudogap. In both cases the\nFermi level fallswithin the gap(Co 2MnSi) orthe pseudo-\ngap (Co 2MnAl) and an almost perfect spin-polarization\nat the Fermi level is preserved.\nDoping the perfect ordered alloys with either Fe or Cr\nsmoothens the valleys and peaks along the energy axis.\nThis is aclearsignofthe chemical disorder; Fe and Crin-\nduce peaks at slightly different places than the Mn atoms\nresulting to this smoothening and as the doping increases\nthis phenomenon becomes more intense. The important\ndetail is what happens around the Fermi level and in\nwhat extent is the gap in the minority band affected by\nthe doping. So now we will concentrate only at the en-\nlarged regions around the Fermi level. The dashed lines\nrepresentthe Cr-dopingwhile the dotted lines arethe Fe-\ndoped alloys. Cr-doping in Co 2MnSi has only a marginal\neffect to the gap. Its width is narrower with respect to\nthe perfect compounds but overall the compounds re-\ntain their half-metallicity. For Co 2MnAl the situation is\nreversedwith respect to the Co 2MnSi compound and Cr-\ndopinghassignificanteffects onthe pseudogap. Its width\nis larger with respect to the perfect compound and be-\ncomesslightly narrowerasthe degreeofdoping increases.\nIn the case of Fe-doping the situation is more com--202Co2MnSi\nCo2[Mn1-xCrx]Si\nCo2[Mn1-xFex]Si\n-0.4 -0.2 0 0.2 0.4-202x=0.05\nx=0.1DOS(states/eV)\nEnergy-EF (eV)\n-202Co2MnAl\nCo2[Mn1-xCrx]Al\nCo2[Mn1-xFex]Al\n-0.4 -0.2 0 0.2 0.4-202x=0.05\nx=0.1DOS(states/eV)\nEnergy-EF (eV)\nFIG. 2: Spin-resolved density of states (DOS) for the case of\nCo2[Mn1−xCrx]Si and Co 2[Mn1−xFex]Si in the upper panel,\nand Co 2[Mn1−xCrx]Al and Co 2[Mn1−xFex]Al in the lower\npanel for two values of the doping concentration x. DOS’s are\ncompared to the one of the undoped Co 2MnSi and Co 2MnAl\nalloys. The zero of the energy axis corresponds to the\nFermi energy. Positive values of the DOS correspond to the\nmajority-spin (spin-up) electrons and negative values to t he\nminority-spin (spin-down) electrons. In the insets we pres ent\nthe DOS for a wider energy range.\nplex. Adding electrons to the system means that, in\norder to retain the perfect half-metallicity, these elec-\ntrons should occupy high-energy lying antibonding ma-\njority states. This is energetically not very favorable in\nthe case of Co 2MnSi and for these moderate degrees of\ndoping a new shoulder appears in the unoccupied states\nwhich is close to the right-edge of the gap; a sign of a\nlarge change in the competition between the exchange4\nsplitting of the Mn majority and minority states and\nof the Coulomb repulsion. In the case of the 20% Fe\ndoping in Co 2MnSi (not shown here) this new peak\ncrosses the Fermi level and the Fermi level is no more\nexactly in the gap but slightly above it. Further sub-\nstitution should lead to the complete destruction of the\nhalf-metallicity.34Recent ab-initio calculations including\nthe on-site Coulomb repulsion (the so-called Hubbard U)\nhave predicted that Co 2FeSi is in reality half-metallic\nreaching a total spin magnetic moment of 6 µBwhich\nis the largest known spin moment for a half-metal.31,35\nFe-doping on the other hand in Co 2MnAl almost does\nnot change the DOS around the Fermi level. The extra-\nelectrons occupy high-energy lying antibonding majority\nstates but, since Co 2MnAl has one valence electron less\nthan Co 2MnSi, half-metallicity remains energetically fa-\nvorable and no important changes occur upon Fe-doping\nand further substitution of Fe for Mn should retain the\nhalf-metallicityevenfortheCo 2FeAlcompoundalthough\nLDA-based ab-initio calculations predict that the limit-\ning case of Co 2FeAl is almost half-metallic.34\nIV. DEFECTS DRIVEN HALF-METALLIC\nFERRIMAGNETISM\nIn the previous section we have examined the case of\ndefects in half-metallic ferromagnets. Half-metallic ferri-\nmagnetism (HMFi) on the other hand is highly desirable\nsince such compounds would yield lower total spin mo-\nments than the correspondingferromagnets. Well-known\nHMFi are the perfect Heusler compounds FeMnSb and\nMn2VAl.36We will present in the first part of this sec-\ntion another route to half-metallic ferrimagnetism based\non antisites created by the migration of Cr(Mn) atoms at\nCo sites in the case of Co 2CrAl, Co 2CrSi, Co 2MnAl and\nCo2MnSi alloys. The ideal case for applications would\nbe a fully-compensated ferrimagnet, also known as half-\nmetallic antiferromagnet (HMA),37since such a com-\npound would not give rise to stray flux and thus would\nlead to smaller energy consumption in devices. In the\nsecobd part of this section we present a way to achieve\nHMA based on Co defects in half-metallic ferrimagnets\nlike Mn 2VAl.\nWe will start our discussion from the Cr-based alloys\nand using Co 2CrAl and Co 2CrSi as parent compounds\nwe create a surplus of Cr atoms which sit at the perfect\nCo sites. In figure 3 we present the total density of states\n(DOS) for the [Co 1−xCrx]2CrAl and [Co 1−xCrx]2CrSi al-\nloys for concentrations x= 0 and 0.1, and in table I we\nhave gathered the spin moments for the two compounds\nunder study. We will start our discussion from the DOS.\nThe perfect compounds show a gap in the minority-spin\nband and the Fermi level falls within this gap and thus\nthe compounds are half-metals. When the sp atom is Si\ninstead of Al the gap is larger due to the extra electron\nwhich occupies majority states of the transition metal\natoms29and increasesthe exchangesplitting between the-2 0 2\nEnergy-EF (eV)-404DOS (states/eV)-404\nx=0\nx=0.1[Co1-xCrx]2CrAl[Co1-xCrx]2CrSi\nFIG. 3: Total density of states (DOS) as a function of the\nconcentration xfor the [Co 1−xCrx]2CrAl (upper panel) and\n[Co1−xCrx]2CrSi (lower panel) compounds.\nTABLE I: Atom-resolved spin magnetic moments for the\n[Co1−xCrx]2CrAl, [Co 1−xCrx]2CrSi, [Co 1−xMnx]2MnAl and\n[Co1−xMnx]2MnSi compounds (moments have been scaled to\none atom). The two last columns are the total spin moment\n(Total) in the unit cell calculated as 2 ×[(1−x)∗mCo+x∗\nmCrorMn(imp)]+mCr(Mn)+mAlorSiand the ideal total spin\nmoment predicted by the Slater-Pauling rule for half-metal s\n(see section II). With IMP we denote the Cr(Mn) atoms sit-\nting at perfect Co sites.\nCompound Co IMP Cr/Mn Al/Si Total Ideal\nCo2CrAl 0.73 – 1.63 -0.09 3.00 3.00\n[Co0.95Cr0.05]2CrAl 0.71 -1.82 1.62 -0.09 2.70 2.70\n[Co0.9Cr0.1]2CrAl 0.69 -1.85 1.61 -0.08 2.40 2.40\nCo2CrSi 0.95 – 2.17 -0.06 4.00 4.00\n[Co0.95Cr0.05]2CrSi 0.93 -1.26 2.12 -0.06 3.70 3.70\n[Co0.9Cr0.1]2CrSi 0.91 -1.26 2.07 -0.05 3.40 3.40\nCo2MnAl 0.68 – 2.82 -0.14 4.04 4.00\n[Co0.95Mn0.05]2MnAl 0.73 -2.59 2.82 -0.13 3.81 3.80\n[Co0.9Mn0.1]2MnAl 0.78 -2.49 2.83 -0.12 3.61 3.60\nCo2MnSi 0.98 – 3.13 -0.09 5.00 5.00\n[Co0.95Mn0.05]2MnSi 0.99 -0.95 3.09 -0.08 4.80 4.80\n[Co0.9Mn0.1]2MnSi 0.99 -0.84 3.06 -0.07 4.60 4.60\nmajority occupied and the minority unoccupied states.\nThis electron increases the Cr spin moment by ∼0.5µB\nand the moment of each Co atom by ∼0.25µBabout.\nThe Cr and Co majority states form a common band\nand the weight at the Fermi level is mainly of Cr char-\nacter. The minority occupied states are mainly of Co\ncharacter. When we substitute Cr for Co, the effect on\nthe atomic DOS of the Co and Cr atoms at the perfect\nsites is marginal. The DOS of the impurity Cr atoms has\na completely different form from the Cr atoms at the per-5\n-2 0 2\nEnergy-EF (eV)-404DOS (states/eV)-404\nx=0\nx=0.1[Co1-xMnx]2MnAl\n[Co1-xMnx]2MnSi\nFIG. 4: Total density of states (DOS) around the gap region\nfor the [Co 1−xMnx]2MnAl and [Co 1−xMnx]2MnSi alloys as a\nfunction of the concentration x: we denote x= 0 with the\nsolid line and x= 0.1 with a dashed thick line.\nfect sites due to the different symmetry of the site where\nthey sit. But although Crimpurity atomsat the antisites\ninduce minority states within the gap, there is still a tiny\ngap and the Fermi level falls within this gap keeping the\nhalf-metallic character of the parent compounds.\nThe discussion above on the conservation of the half-\nmetallicity is confirmed when we compare the calculated\ntotal moments in table I with the values predicted by the\nSlater Pauling rule for the ideal half-metals. Since Cr\nis lighter than Co, substitution of Cr for Co decreases\nthe total number of valence electrons and the total spin\nmoment should also decrease. This is achieved due to\nthe antiferromagnetic coupling between the Cr impurity\natoms and the Co and Cr ones at the ideal sites, which\nwould have an important negative contribution to the\ntotal moment asconfirmed by the results in table I. Thus\nthe Cr-doped alloys are half-metallic ferrimagnets and\ntheir total spin moment is considerable smaller than the\nperfect half-metallic ferromagnetic parent compounds; in\nthe case of [Co 0.8Cr0.2]2CrAl it decreases down to 1.8 µB\nfrom the 3 µBofthe perfect Co 2CrAl alloy. Here we have\nto mention that if alsoCo atoms migrate to Cr sites (case\nofatomicswaps)thehalf-metallityislost, asitwasshown\nby Miura et al.38, due to the energy position of the Co\nstates which have migrated at Cr sites.\nSimilar phenomena to the discussion above occur in\nthe [Co 1−xMnx]2MnZ compounds varying the sp atom,\nZ, which is one of Al or Si. We have taken into account\nfive different values for the concentration x;x= 0, 0.025,0.05, 0.1, 0.2. In figure 4 we have drawn the total density\nof states (DOS) for both families of compounds under\nstudy and for two different values of the concentration\nx: the perfect compounds ( x=0) and for one case with\ndefects,x= 0.1. As mentioned above and in agreement\nwith previous previous electronic structure calculations\non these compounds Co 2MnSi compounds present a real\ngapandCo 2MnAlapseudogap.23,29,39,40When wecreate\na surplus of Mn atoms which migrate at sites occupied\nby Co atoms in the perfect alloys, the gap persists and\nboth compounds retain their half-metallic character as\noccurs also for the Cr-based alloys presented above. Es-\npecially for Co 2MnSi, the creation of Mn antisites does\nnot alter the width of the gap and the half-metallicity\nis extremely robust in these alloys with respect to the\ncreation of Mn antisites. The atomic spin moments show\nbehavior similar to the Cr alloys as can be seen in table I\nand the spin moments of the Mn impurity atoms are an-\ntiferromagnetically coupled to the spin moments of the\nCo and Mn atoms at the perfect sites resulting to the\ndesired half-metallic ferrimagnetism.\nThe ideal case of fully-compensated half-metallic fer-\nrimagnet or half-metallic antiferromagnetic (HMA) can\nbe achieved by doping with Co the Mn 2VAl and Mn 2VSi\nalloys which are well known to be HMFi. The impor-\ntance ofthis routestems from the existence ofMn 2VAl in\nthe Heusler L21phase as shown by several groups.41,42,43\nEach Mn atom has a spin moment of around -1.5 µBand\nV atom a moment of about 0.9 µB.41,42,43\nAll theoretical studies on Mn 2VAl agree on the half-\nmetallic character with a gap at the spin-up band in-\nstead of the spin-down band as for the other half-metallic\nHeusler alloys.29,44,45,46Prior to the presentation of our\nresults we have to note that due to the Slater-Pauling\nrule, these compounds with less than 24 valence elec-\ntrons have negative total spin moments and the gap\nis located at the spin-up band. Moreover the spin-up\nelectrons correspond to the minority-spin electrons and\nthe spin-down electrons to the majority electrons con-\ntrary to the other Heusler alloys.29We have substituted\nCo for Mn in Mn 2V(Al or Si) in a random way and\nin figure 5 we present the total and atom-resolved den-\nsity of states (DOS) in [Mn 1−xCox]2VAl (solid line) and\n[Mn1−xCox]2VSi (dashedline) alloysfor x=0.1. The per-\nfect compounds show a region of low spin-up DOS (we\nwill call it a “pseudogap”) instead of a real gap. Upon\ndoping the pseudogap at the spin-up band persists and\nthe quaternary alloys keep the half-metallic character of\nthe perfect Mn 2VAl and Mn 2VSi compounds. Co atoms\narestronglypolarizedbytheMnatomssincetheyoccupy\nthe same sublattice and they form Co-Mn hybrids which\nafterwards interact with the V and Al or Si states.29The\nspin-up Co states form a common band with the Mn\nones and the spin-up DOS for both atoms has similar\nshape. Mn atoms have less weight in the spin-down band\nsince they accommodate less charge than the heavier Co\natoms.\nIn table II we have gathered the total and atom-6\n-404\n-6 -3 0 3\nEnergy-EF (eV)-404DOS (states/eV)\n-6 -3 0 3[Mn0.9Co0.1]2VAl\n[Mn0.9Co0.1]2VSiCoMn\nVTotal\nFIG. 5: Total and atom-resolved DOS for the\n[Mn0.9Co0.1]2VAl and [Mn 0.9Co0.1]2VSi compounds. Note\nthat the atomic DOS’s have been scaled to one atom.\nresolved spin moments for all the Co-doped compounds\nas a function of the concentration. We have gone up to a\nconcentration which corresponds to 24 valence electrons\nintheunitcell, thusupto x=0.5forthe[Mn 1−xCox]2VAl\nand x=0.25 for the [Mn 1−xCox]2VSi alloys. In the last\ncolumnwehaveincludedthetotalspinmomentpredicted\nby the Slater-Pauling rule for the perfect half-metals. A\ncomparison between the calculated and ideal total spin\nmoments reveals that all the compounds under study are\nhalf-metals with very small deviations due to the exis-\ntence of a pseudogap instead of a real gap. Exactly for\n24 valence electrons the total spin moment vanishes as\nwe will discuss in the next paragraph. Co atoms have\na spin moment parallel to the V one and antiparallel to\nthe Mn moment, and thus the compounds retain their\nferrimagnetic character. As we increase the concentra-\ntion of the Co atoms in the alloys, each Co has more Co\natoms as neighbors, it hybridizes stronger with them and\nits spin moment increases while the spin moment of the\nMn atom decreases (these changes are not too drastic).\nThe sp atoms have a spin moment antiparallel to the Mn\natoms as already discussed in reference 47.\nThe most interesting point in this substitution proce-\ndureisrevealedwhenweincreasetheCoconcentrationto\na value corresponding to 24 valence electrons in the unit\ncell, thus the [Mn 0.5Co0.5]2VAl and [Mn 0.75Co0.25]2VSi\nalloys. The Slater-Pauling rule predicts for these com-\npounds a zero total spin moment in the unit cell and the\nelectrons population is equally divided between the two\nspin-bands. Our first-principles calculations reveal that\nthis is actually the case. The interest arises from the\nfact that although the total moment is zero, these two\ncompounds are made up from strongly magnetic compo-\nnents. Mn atoms have a mean spin moment of ∼-1.4µB\nin [Mn 0.5Co0.5]2VAl and ∼-0.9µBin [Mn 0.75Co0.25]2VSi.\nCo and V have spin moments antiferromagnetically cou-\npled to the Mn ones which for [Mn 0.5Co0.5]2VAl are∼0.6TABLE II: Atom-resolved spin magnetic moments for the\n[Mn1−xCox]2VAl and [Mn 1−xCox]2VSi compounds (moments\nhave been scaled to one atom). The two last columns are\nthe total spin moment (Total) in the unit cell calculated as\n2×[(1−x)∗mMn+x∗mCo]+mV+mAl or Siand the ideal\ntotal spin moment predicted by the Slater-Pauling rule for\nhalf-metals. The lattice constants have been chosen 0.605 n m\nfor Mn 2VAl and 0.6175 for Mn 2VSi for which both systems\nare half-metals (see reference 47) and have been kept consta nt\nupon Co doping.\nCompound Mn Co V Al Total Ideal\nMn2VAl -1.57 – 1.08 0.06 -2.00 -2.0\n[Mn0.9Co0.1]2VAl -1.56 0.340 1.07 0.07 -1.60 -1.6\n[Mn0.7Co0.3]2VAl -1.48 0.46 0.95 0.05 -0.80 -0.8\n[Mn0.5Co0.5]2VAl -1.39 0.59 0.78 0.02 ∼0 0\nCompound Mn Co V Si Total Ideal\nMn2VSi -0.96 – 0.86 0.06 -1.00 -1.0\n[Mn0.9Co0.1]2VSi -0.93 0.82 0.85 0.05 -0.60 -0.6\n[Mn0.75Co0.25]2VSi -0.90 0.94 0.84 0.04 ∼0 0\nand∼0.8µB, respectively, and for [Mn 0.75Co0.25]2VSi\n∼0.9 and∼0.8µB. Thus these two compounds are half-\nmetallic fully-compensated ferrimagnets or as they are\nbest known in literature half-metallic antiferromagnets.\nV. SUMMARY AND CONCLUSIONS\nIn this chapter we have reviewed our results on the de-\nfects in half-metallic full-Heusler alloys. Firstly we have\npresenteda shortoverviewofthe electronicand magnetic\npropertiesofthehalf-metallicfull-Heusleralloysandhave\ndiscussed in detail the Slater-Pauling behavior of these\nalloys (the total spin moment scales linearly with the\ntotal number of valence electrons as a result of the half-\nmetallicity). We have also studied the effect of doping on\nthe magnetic properties of the Co 2MnAl(Si) full-Heusler\nalloys. Doping simulated by the substitution of Cr and\nFe for Mn overall keeps the half-metallicity and has lit-\ntle effect on the half-metallic properties of Co 2MnSi and\nCo2MnAl compounds.\nAfterwards,wehavestudiedtheeffectofdefects-driven\nappearance of half-metallic ferrimagnetism in the case\nof the Co 2Cr(Mn)Al(Si) Heusler alloys. More precisely,\nbased on first-principles calculations we have shown\nthat when we create Cr(Mn) antisites at the Co sites,\nthese impurity Cr(Mn) atoms couple antiferromagnet-\nically with the Co and the Cr(Mn) atoms at the per-\nfect sites while keeping the half-metallic character of the\nparent compounds. The ideal case of half-metallic fully-\ncompensated ferrimagnets (also known as half-metallic\nantiferromagnets) can be achieved by doping of the half-\nmetallic ferrimagnets Mn 2VAl and Mn 2VSi. Co substi-\ntution for Mn keeps the half-metallic character of the\nparent compounds and when the total number of valence\nelectrons reaches the 24, the total spin moment vanishes\nas predicted by the Slater-Pauling rule. Defects are a7\npromising alternative way to create robust half-metallic\nferrimagnets, which are crucial for magnetoelectronic ap-plications.\n∗Electronic address: kozdogan@gyte.edu.tr\n†Electronic address: e.sasioglu@fz-juelich.de\n‡Electronic address: galanakis@upatras.gr\n1R. A. de Groot. F. M. Mueller, P. G. van Engen, and K.\nH. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983).\n2I.ˇZuti´ c, J. Fabian, and A. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004)\n3P. J. Webster and K. R. A. Ziebeck in Alloys and Com-\npounds of d-Elements with Main Group Elements. Part 2. ,\nLandolt-B¨ ornstein, New Series, Group III, vol 19c, ed. by\nH. R. J. Wijn, (Springer, Berlin 1988) pp 75–184.\n4K. R. A. Ziebeck and K. -U. Neumann in Magnetic Proper-\nties of Metals , Landolt-B¨ ornstein, New Series, Group III,\nvol 32/c, ed. by H. R. J. Wijn, (Springer, Berlin 2001) pp\n64–414.\n5R. J. Soulen Jr., J. M. Byers, M. S. Osofsky, B. Nadgorny,\nT. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka,\nJ. 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Matter 18, 2905 (2006)."    },    {        "title": "1607.03617v1.Breakdown_of_a_Magnetization_Plateau_in_Ferrimagnetic_Mixed_Spin__1_2_S__Heisenberg_Chains_Due_to_a_Quantum_Phase_Transition_Towards_the_Luttinger_Spin_Liquid.pdf",        "content": "arXiv:1607.03617v1  [cond-mat.stat-mech]  13 Jul 2016Vol.XXX (201X) CSMAG‘16 No.X\nBreakdown of a Magnetization Plateau in Ferrimagnetic Mixe d Spin-(1/2, S)\nHeisenberg Chains Due to a Quantum Phase Transition Towards the Luttinger Spin\nLiquid\nJ. Streˇ cka1,∗\n1Institute of Physics, Faculty of Science, P. J. ˇSaf´ arik University, Park Angelinum 9, 040 01 Koˇ sice, Slov akia\nMagnetization curves of the ferrimagnetic mixed spin-(1/2 ,S) Heisenberg chains are calculated\nwith the help of density-matrix renormalization group meth od for several quantum spin numbers\nS=1, 3/2, 2 and 5/2. It is shown that the ferrimagnetic mixed sp in-(1/2,S) Heisenberg chains\nexhibit irrespective of the spin value Sexactly one intermediate magnetization plateau, which can\nbe identified with the gapped Lieb-Mattis ferrimagnetic gro und state. The magnetization plateau\ndue to the Lieb-Mattis ferrimagnetism breaks down at a quant um phase transition towards the\nLuttinger spin liquid, which is characterized by a continuo us change of the magnetization with the\nmagnetic field until another quantum critical point is reach ed at the saturation field.\nPACS numbers: 75.10.Pq ; 75.10.Kt ; 75.30.Kz ; 75.40.Cx ; 75. 60.Ej\nIntroduction\nOver the last few years, the ferrimagnetic mixed spin- s\nand spin- SHeisenberg chains with regularly alternating\nspinss= 1/2 andS >1/2 have attracted a great deal\nof attention, since they exhibit a quantum phase transi-\ntionbetweenintriguinggroundstatesthataremanifested\nin respective magnetization curves as quantized magne-\ntization plateaux and Luttinger spin liquids [1–6]. The\nintermediate magnetization plateaux of the mixed spin-\n(1/2,S) Heisenberg chains should obey the quantization\ncondition known as Oshikawa-Yamanaka-Affleck (OYA)\nrulems−m= integer, where ms=S+1/2 andmare\nthe total spin and total magnetization per elementary\nunit [7]. According to OYA rule, one of possible ways to\nincreasethe total number of magnetization plateaux may\nconsist in increasing size of the constitutent spin S. It\nshould be stressed, however, that OYA criterion provides\njust necessary but not sufficient condition for a presence\nof a magnetization plateau, whose actual existence has\nstill to be verified by explicit calculations.\nAny bipartite quantum ferrimagnet (irrespective of\nspin magnitude and spatial dimensionality) should also\nsatisfy the Lieb-Mattis (LM) theorem [8], which assures\nthe following total magnetization m=S−1/2 per unit\ncell within the zero-field ground state of the ferrimag-\nnetic mixed spin-(1/2, S)Heisenbergchains. Hence, OYA\ncriterion in combination with LM theorem would sug-\ngest that the ferrimagnetic mixed spin-(1/2, S) Heisen-\nberg chains may display one and just one quantized\nmagnetization plateau (regardless of the spin size S) at\nthe following fractional value of the total magnetization\nm/ms= (2S−1)/(2S+1) normalized with respect to its\n∗corresponding author; e-mail: jozef.strecka@upjs.sksaturation value. In the present work we will provide a\nsurvey for zero-temperature magnetization curves of the\nferrimagnetic mixed spin-(1/2, S) Heisenberg chains by\nconsidering a few different quantum spin numbers S= 1,\n3/2, 2 and 5 /2, which will prove all aforementioned fea-\ntures on this paradigmatic class of quantum spin chains.\nModel and method\nLet us consider the mixed spin- sand spin- Squantum\nHeisenbergchainwithregularlyalternatingspins s= 1/2\nandS >1/2 given by the Hamiltonian\nˆH=JL/summationdisplay\nj=1ˆSj·(ˆsj+ˆsj+1)−hL/summationdisplay\nj=1(ˆSz\nj+ ˆsz\nj),(1)\nwhereˆsj≡(ˆsx\nj,ˆsy\nj,ˆsz\nj) andˆSj≡(ˆSx\nj,ˆSy\nj,ˆSz\nj) denote\nthe usual spin-1/2 and spin- Soperators, respectively.\nThe first term entering in the Hamiltonian (1) takes\ninto account the antiferromagnetic Heisenberg interac-\ntionJ >0 between the nearest-neighbor spins and the\nsecond term h=gµBHincorporating the equal Land´ e g-\nfactorsgs=gS=gand Bohr magneton µBaccounts for\nthe Zeemann’s energy of individual magnetic moments\nin an external magnetic field. It is noteworthy that the\noverallchain length is 2 Las the elementary unit contains\ntwospins, whereasthe translationalinvarianceis ensured\nby the periodic boundary condition sL+1≡s1.\nOne should turn to some accurate numerical method\nin order to get a reliable survey of magnetization pro-\ncesses of the ferrimagnetic mixed spin-(1/2, S) Heisen-\nberg chains, since the Hamiltonian (1) is not integrable.\nFor this purpose, we have implemented density-matrix\nrenormalization group (DMRG) calculations from ALPS\nproject[9], whichcan bestraightforwardlyusedtoobtain\nthelowest-energyeigenvalue E(Tz\ntot,L,h= 0)oftheferri-Magnetization Plateaux in Ferrimagnetic Mixed-Spin Heise nberg Chains 2\nmagnetic mixed-spin Heisenberg chain within each sector\nwith the total spin Tz\ntot=/summationtextL\nj=1(Sz\nj+sz\nj) in a zero mag-\nnetic field ( h= 0). The lowest-energy eigenstate of the\nferrimagnetic mixed spin-(1/2, S) Heisenberg chains in a\nnon-zero magnetic field can be subsequently calculated\nfromtheformula E(Tz\ntot,L,h) =E(Tz\ntot,L,h= 0)−hTz\ntot,\nbecause the total spin Tz\ntotis conserved quantity due to\na validity of the commutation relation between the re-\nspective operator and the Hamiltonian (1). The finite-\nsize formula for a magnetic-field induced transition be-\ntween the lowest-energy eigenstates with the total spin\nTz\ntotandTz\ntot+ 1 then readily follows from the formula\nh=E(Tz\ntot+ 1,L,h= 0)−E(Tz\ntot,L,h= 0). In this\nway one may obtain the accurate numerical results for\nthe zero-temperature magnetization curves. To avoid ex-\ntrapolation due to finite-size effects we have performed\nDMRG simulations for a sufficiently large system size\nwith up to L= 64 units (128 spins), whereas adequate\nnumerical accuracy was achieved through 16 sweeps at\nthe targeted system size when increasing the number of\nkept states up to 1200 during the final sweeps.\nResults and discussion\nLet us proceed to a discussion of zero-temperature\nmagnetization curves of the ferrimagnetic mixed spin-\n(1/2,S) Heisenberg chains, which are displayed on the\nleft panel of Fig. 1 for a few different quantum spin num-\nbersS=1, 3/2, 2 and 5/2. It is quite evident from Fig. 1\nthat all considered mixed-spin Heisenberg chains indeed\nexhibit exactly one intermediate magnetization plateau\nat the fractional value m/ms= (2S−1)/(2S+1), which\nis consistent with the gapped LM ferrimagnetic ground\nstate. The intermediate plateau due to LM ferrimag-\nnetism breaks down at a quantum phase transition in-\nvoked by the critical magnetic field hc, which closes an\nenergy gap above the ferrimagnetic ground state. It is\nnoteworthy that the height of LM plateau monotoni-\ncally increases with increasing the quantum spin num-\nberSquite similarly as does its width terminating at\nthe critical field hc= 1.76JforS= 1,hc= 2.84Jfor\nS= 3/2,hc= 3.88JforS= 2 and hc= 4.91Jfor\nS= 5/2. Above the critical magnetic field h > h cthe\nferrimagnetic mixed spin-(1/2, S) Heisenberg chains pass\ntowards the Luttinger spin liquid, where the magneti-\nzation rises continuously with the magnetic field until\nanother quantum critical point is reached at the satura-\ntion field hs=J(1 + 2S). The asymptotic behavior of\nthe magnetization in a vicinity of both quantum phase\ntransitions is governed by the relations: m∝√h−hc\nforh→h+\ncandm∝√hs−hforh→h−\ns. Owing to this\nfact, the quantum phase transitions driven by the mag-\nnetic field should be also reflected in anomalous behavior\nof the magnetic susceptibility χclose to quantum critical\npoints:χ∝1/√h−hcforh→h+\ncandχ∝1/√hs−h/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53/s48/s49/s50/s51/s52\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s48/s49/s50/s51\n/s40/s100/s41/s104 /s32/s47/s32 /s74/s32/s40/s97/s41/s32\n/s49/s47/s50 /s51/s47/s50/s32\n/s40/s99/s41/s49/s47/s51 /s112/s108/s97/s116/s101/s97/s117\n/s49/s47/s50 /s51/s47/s50/s32\n/s32/s109/s32/s47/s32/s109\n/s115\n/s104 /s32/s47/s32 /s74/s32\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s52/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s50/s46/s56/s52/s40/s98/s41/s32/s115/s112/s105/s110/s32/s108/s105/s113/s117/s105/s100\n/s32\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s52/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s50/s46/s56/s52\n/s104 /s32/s47/s32 /s74/s32/s49/s47/s50 /s49\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s51/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s49/s46/s55/s54\n/s49/s47/s50 /s112/s108/s97/s116/s101/s97/s117/s49/s47/s50 /s49\n/s32/s32/s109/s32/s47/s32/s109\n/s115\n/s104 /s32/s47/s32 /s74/s32/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s51/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s49/s46/s55/s54\n/s115/s112/s105/s110/s32/s108/s105/s113/s117/s105/s100\n/s32/s32\n/s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48 /s54/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s40/s104/s41/s104 /s32/s47/s32 /s74/s32/s40/s101/s41/s32\n/s49/s47/s50 /s53/s47/s50/s32\n/s40/s103/s41/s51/s47/s53 /s112/s108/s97/s116/s101/s97/s117\n/s49/s47/s50 /s53/s47/s50/s32\n/s32/s109/s32/s47/s32/s109\n/s115\n/s104 /s32/s47/s32 /s74/s32\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s54/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s52/s46/s57/s49/s40/s102/s41/s32/s115/s112/s105/s110/s32/s108/s105/s113/s117/s105/s100\n/s32\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s54/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s52/s46/s57/s49\n/s104 /s32/s47/s32 /s74/s32/s49/s47/s50 /s50\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s53/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s51/s46/s56/s56\n/s50/s47/s51 /s112/s108/s97/s116/s101/s97/s117/s49/s47/s50\n/s32/s32/s109/s32/s47/s32/s109\n/s115\n/s104 /s32/s47/s32 /s74/s32/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s53/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s51/s46/s56/s56\n/s115/s112/s105/s110/s32/s108/s105/s113/s117/s105/s100\n/s32/s32\nFIG. 1: The magnetization (left panel) and susceptibility\n(right panel) of the mixed spin-(1/2, S) Heisenberg chains as\na function of the magnetic field for four different spin values :\n(a)-(b)S= 1; (c)-(d) S= 3/2; (e)-(f) S= 2; (g)-(h) S= 5/2.\nThe displayed results were obtained from DMRG simulations\nof a finite-size chain with L= 64 units (128 spins).\nforh→h−\ns. In accordance with this statement, the\nmagnetic-field dependences of the susceptibility shown\non the right panel of Fig. 1 furnish evidence for both\nfield-inducedquantumphasetransitionstowardstheLut-\ntinger spin liquid through the observed divergence of the3 Magnetization Plateaux in Ferrimagnetic Mixed-Spin Heise nberg Chains\nmagnetic susceptibility.\nConclusions\nThe zero-temperature magnetization curves of the fer-\nrimagnetic mixed spin-(1/2, S) Heisenberg chains were\ncalculated with the help of DMRG method for several\nvalues of the quantum spin number S. It has been\nverified that the magnetization curves involve due to\nthe gapped LM ferrimagnetic ground state one and just\none intermediate plateau at the fractional magnetization\nm/ms= (2S−1)/(2S+ 1), which breaks down at a\nquantum phase transition towardsthe Luttinger spin liq-\nuid driven by the external magnetic field. Subsequently,\nthe magnetization continuously rises with increasing the\nmagnetic field within the Luttinger spin-liquid phase un-\ntil it reaches the full moment at the saturation field\nhs=J(1 + 2S) closely connected with another field-\ninduced quantum phase transition. It has been demon-\nstrated that the magnetization shows a cusp and sus-\nceptibility diverges in a close vicinity of both quantum\ncritical points. Besides, it could be concluded that the\nrising quantum spin number Sincreases in the mag-\nnetization curve of the mixed spin-(1/2, S) Heisenberg\nchainstheheightaswellaswidthoftheferrimagneticLM\nplateau, while the magnetic-field range corresponding to\nthe gapless Luttinger spin-liquid phase is conversely re-\nduced. Last but not least, it is worth noticing that\ntheoretical implications of the present work are of ob-\nvious relevance for series of bimetallic coordination com-\npounds MM’(pba)(H 2O)3·2H2O [10] and MM’(EDTA)\n·6H2O [11] (M,M’ = Cu, Ni, Co, Mn), which represent\nexperimental realization of the ferrimagnetic mixed-spin\nHeisenberg chains. However, the high-field magnetiza-\ntion measurements on these or related series of bimetal-\nlic complexesaredesirableforexperimentaltesting ofthe\npresent theoretical predictions.Acknowledgement\nThisworkwasfinanciallysupportedbyMinistryofEd-\nucation, Science, Research and Sport of the Slovak Re-\npublic provided under the grant No. VEGA 1/0043/16\nand by the grant Slovak Research and Development\nAgency under the contract No. APVV-0097-12.\n[1] S. Yamamoto, T. Sakai, J. Phys.: Condens. Matter 11,\n5175 (1999). DOI: 10.1088/0953-8984/11/26/318.\n[2] T. Sakai, S. Yamamoto, Phys. Rev. B 60, 4053 (1999).\nDOI: 10.1103/PhysRevB.60.4053.\n[3] A. Honecker, F. Mila, M. Troyer, Eur. Phys. J. B 15,\n227 (2000). DOI: 10.1007/s100510051120.\n[4] S. Yamamoto, T. Sakai, Phys. Rev. B 62, 3795 (2000).\nDOI: 10.1103/PhysRevB.62.3795.\n[5] T. Sakai, S. Yamamoto, Phys. Rev. B 65, 214403 (2002).\nDOI: 10.1103/PhysRevB.65.214403.\n[6] A.S.F. Ten´ orio, R.R. Montenegro-Filho, M.D. Coutinho -\nFilho,J. Phys.: Condens. Matter 23, 506003 (2011).\nDOI:10.1088/0953-8984/23/50/506003\n[7] M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett.\n78, 1984 (1997). DOI: 10.1103/PhysRevLett.78.1984.\n[8] E. Lieb, D. Mattis, J. Math. Phys. 3, 749 (1962). DOI:\n10.1063/1.1724276\n[9] B. Bauer, L.D.Carr, H.G.Evertz, A.Feiguin, J.Freire, S .\nFuchs, L. Gamper, J. Gukelberger, E. Gull, S. Guertler,\nA. Hehn, R. Igarashi, S.V. Isakov, D. Koop, P.N.\nMa, P. Mates, H. Matsuo, O. Parcollet, G. Pawlowski,\nJ.D. Picon, L. Pollet, E. Santos, V.W. Scarola, U.\nSchollw¨ ock, C. Silva, B. Surer, S. Todo, S. Trebst, M.\nTroyer, M.L. Wall, P. Werner, S. Wessel, J. Stat. Mech.:\nTheor. Exp. 2011, P05001 (2011). DOI: 10.1088/1742-\n5468/2011/05/P05001.\n[10] O. Kahn, Struct. Bonding (Berlin) 68, 89 (1987). DOI:\n10.1007/3-540-18058-33.\n[11] M.Drillon, E.Coronado, D.Beltran, R.Georges, J. Appl.\nPhys.57, 3353 (1985). DOI: 10.1063/1.335094."    },    {        "title": "1305.3064v3.Electromagnon_in_ferrimagnetic_eps_Fe2O3_nanograin_ceramics.pdf",        "content": "arXiv:1305.3064v3  [cond-mat.mtrl-sci]  27 Aug 2013Electromagnon in ferrimagnetic ε-Fe2O3nanograin ceramics\nChristelle Kadlec,1Filip Kadlec,1,∗Veronica Goian,1Mart´ ı Gich,2Martin Kempa,1\nSt´ ephane Rols,3Maxim Savinov,1Jan Prokleˇ ska,4Milan Orlita,5and Stanislav Kamba1\n1Institute of Physics, Academy of Sciences of the Czech Repub lic,\nNa Slovance 2, 182 21 Prague 8, Czech Republic\n2Institut de Ci` encia de Materials de Barcelona—Consejo Sup erior de Investigaciones Cient´ ıficas,\nCampus UAB, 08193, Bellaterra, Catalunya, Spain\n3Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, Fra nce\n4Faculty of Mathematics and Physics, Department of Condense d Matter Physics,\nCharles University, Ke Karlovu 5, 121 16 Prague 2, Czech Repu blic\n5Grenoble High Magnetic Field Lab, CNRS - 25, avenue des Marty rs, Grenoble Cedex 9, France\nElectromagnons are known from multiferroics as spin waves e xcited by the electric component of\nelectromagnetic radiation. We report the discovery of an ex citation in the far-infrared spectra of\nε-Fe2O3which we attribute to an electromagnon appearing below 110K , where the ferrimagnetic\nstructure becomes incommensurately modulated. Inelastic neutron scattering shows that the elec-\ntromagnon energy corresponds to that of a magnon from the Bri llouin zone boundary. Dielectric\nmeasurements did not reveal any sign of ferroelectricity in ε-Fe2O3down to 10K, despite its acen-\ntric crystal structure. This shows that the activation of an electromagnon requires, in addition to\nthe polar ferrimagnetic structure, a modulation of the magn etic structure. We demonstrate that a\ncombination of inelastic neutron scattering with infrared and / or terahertz spectroscopies allows\ndetecting electromagnons in ceramics, where no crystal-or ientation analysis of THz and infrared\nspectra is possible.\nPACS numbers: 76.50.+g, 77.22.-d, 63.20.kd, 75.85.+t\nI. INTRODUCTION\nIn the last years, there has been an increasing inter-\nest in so-calledmultiferroicmaterials, displayingsimulta-\nneously spontaneous ferroelectric (FE) polarization and\nferro- or antiferromagnetic (AFM) ordering. Multifer-\nroics exhibit a rich variety of fundamental physical phe-\nnomena, and it is generally believed that they have a po-\ntential for novel applications in non-volatile memories1,2,\nmagnonics3and magnetic sensors4. These applications\nwould rely on the coupling of order parameters on vari-\nous time scales, from quasi-static to ultrafast. However,\nthe understanding of the microscopic mechanism of the\nmagnetodielectric coupling is still a fundamental prob-\nlem of solid state physics. The static and dynamic mag-\nnetoelectric (ME) couplings can have different origins.\nOwing to the static ME coupling, the macroscopic FE\npolarization emerges in the cycloidal or transverse coni-\ncal modulated magnetic structures; this polarization can\nchange with magnetic field. In contrast, the dynamic\nME coupling generates an oscillatory polarization and\nleads to a dielectric dispersion in the terahertz (THz) re-\ngion. Indeed, THz studies of multiferroics revealeda new\nkind of electric-field-active spin excitations contributing\nto the dielectric permittivity ε=ε′−iε′′, called electro-\nmagnons (EMs)5. Their characteristic feature is a cou-\npling with polar phonons, which manifests itself in the\nspectra by a transfer of dielectric strength from phonons\nto EMs on cooling6. In contrast to ferromagnetic and\nAFM resonances, which are magnons from the Brillouin\nzone(BZ) centercontributingtothemagneticpermeabil-\nityµ=µ′−iµ′′, the EMs can be activated also outsideof the BZ center7–10. The understanding of this fact is\nnot trivial, because the photons which excite EMs have\nwavevectors much smaller than the EMs. Thus, to date,\nthere are several different theories attempting to explain\nthe observedpropertiesofEMs in variousmaterials7,9–11.\nThe EMs were discovered first in TbMnO 3and\nGdMnO 35which belong to multiferroics denoted11as\ntype II, where the FE order is induced by a special mag-\nnetic ordering. Since then, EMs were confirmed in nu-\nmerous type-II multiferroics6,7,12–18. Other reports of\nEMs in type-I multiferroics (e.g. BiFeO 319–21or hex-\nYMnO 322) appear inconclusive, since no transfer of the\ndielectric strength from polar phonons to EMs was ob-\nserved19,22. Also, recent infrared IR and THz studies did\nnot confirm the EM in hex-YMnO 323.\nHere we report experiments which reveal an excitation\nidentifiedasanEMintheferrimagnetic εphaseofFe 2O3.\nThanks to its chemical simplicity, this phase appearsalso\nas a suitable model system for theoretical studies of elec-\ntromagnonic excitations. While ε-Fe2O3is quite rare\nand less known than the α(hematite) or γ(maghemite)\nphases of ε-Fe2O324, its properties make it attractive\nfor applications, such as electromagnetic-wave absorbers\nand memories25–27. Owing to limited phase stability,\nit can be synthesized only in the form of nanoparticles\ntens of nanometers in size26,28, epitaxial thin films29or\nnanowires a few micrometers long30. Below 480–495K,\nit is ferrimagnetic31,32; at room-temperature, it has a\ncollinear spin structure33and exhibits a coercive field of\nHc≈2T31—the highest known value among metal ox-\nides. The crystal lattice has a temperature-independent\nnon-centrosymmetric orthorhombic structure with the2\nPna21space group34(magnetic space group Pn′a2′\n1).\nIt consists of three crystallographically non-equivalent\nFeO6octahedra, forming chains along the adirection,\nand one type of FeO 4tetrahedra28,35. Compared to\nisostructural GaFeO 3, the low-temperature phase dia-\ngram of ε-Fe2O3is complex—below 150K, a series of\nmagnetic phase transitions occurs. Below Tm= 110K,\nan incommensurate magnetic ordering appears where\nthe magnetic structure modulation has a periodicity of\nabout 10 unit cells35. NearTm, a drop in ε′was ob-\nserved, and magnetocapacitive measurements revealed\na quadratic coupling36. Room-temperature microwave\nmeasurements provided evidence of a strong ferromag-\nnetic resonance (FMR) near 0.74meV (frequency of\n180GHz) which can be tuned by doping with Al, Ga or\nRh25–27. In order to gain insight into the dynamic ME\nproperties of ε-Fe2O3, we obtained THz, IR and inelastic\nneutron scattering (INS) spectra of ε-Fe2O3nano-grain\nceramics upon cooling down to 10K, providing informa-\ntion about polar and magnetic excitations.\nII. SAMPLES AND EXPERIMENTAL\nMETHODS\nThe nanoparticles of ε-Fe2O3were synthesized by sol-\ngel chemistry. SiO 2-Fe2O3composite gels containing 30\nwt.% of Fe 2O3were prepared from iron nitrate non-\nahydrate (Sigma-Aldrich >98%) and tetraethoxysilane\n(TEOS, Sigma-Aldrich 98%) in hydroethanolic medium\nat TEOS:H 2O:EtOH = 1:6:6 molar ratio. Iron nitrate\nwas first dissolved and then TEOS added dropwise to\nthe mixture under stirring. The sol was poured into 5cm\ndiameter petri dishes that were closed with its cover and\ngelation took place for between 4 and 5 weeks. The\ngels were dried overnight in a stove at 70◦C, crushed\nand thermally treated in air atmosphere for 3 hours at\n1100◦C (heating rate 80◦C/h). The resulting material\nwas a composite of ε-Fe2O3nanoparticles of about 25nm\nin diameter dispersed in an amorphous SiO 2matrix as\nchecked by X-ray diffraction (XRD) which did not re-\nveal any trace of other Fe 2O3polymorphs. The silica\nwas removed by stirring the composite powder for 12h\nin a 12M aqueous NaOH solution at 80◦C under reflux.\nXRD patterns recorded after the silica removal revealed\nthat the microstructure and the phase stability of ε-\nFe2O3nanoparticleswerenot affectedby theetching pro-\ncess. The nanoparticles were further processed by spark\nplasma sintering (SPS) in order to prepare a pellet suit-\nable for dielectric, terahertz (THz) and IR measurements\nby pressing the ε-Fe2O3powder in a graphite mould for\n4 minutes at 350◦C under 100MPa. The XRD analy-\nsis of the sintered pellet showed that the SPS process\ndid not induce any grain growth or phase transforma-\ntion. Finally, the SPS pellets were polished to thin disks\nwith a thickness of 1.2mm. Some IR and THz measure-\nments were performed on ε-Fe2O3pellets with a diam-\neter of about 6mm, which were prepared from powderat room temperature using a standard tabletop manual\nhydraulic press (Perkin Elmer). The spectra were quali-\ntativelythe same, onlythe valueofthe high-frequencyIR\nreflectance was affected by the roughness of the sample\nsurface, which could not be polished.\nIR reflectance measurements with the resolution of\n0.25meV were performed using the Fourier transform\ninfrared spectrometer Bruker IFS-113v in near-normal\nreflectance geometry with an incidence angle of 11◦.\nAn Oxford Instruments Optistat optical cryostat with\npolyethylene windows was used for sample cooling down\nto 10K, and a liquid-He-cooled Si bolometer operatingat\n1.6K was applied as a detector. We also measured far-\nIR reflectivity with applied magnetic field up to 13T. To\nthis aim, another Bruker IFS-113v spectrometer and a\ncustom-made superconducting magnetic cryostat allow-\ning the measurements at 2 and 4K were used. Time-\ndomain THz spectroscopy was based on measurements\nof sample transmittance using custom-made spectrome-\ntersbasedonTi:sapphirefemtosecondlasers; onewithan\nOptistat cryostat with mylar windows for measurements\nwithout magnetic field but with a higher frequency res-\nolution, enabling to discern the FMR profile, and one\nwith an Oxford Instruments Spectromag cryostat, en-\nabling measurements with magnetic field of up to 7T.\nHere, the Voigt configuration was used with the external\nstatic magnetic field Bextperpendicular to the magnetic\ncomponent of the THz radiation BTHz. Similar effects\nwere observed also for Bext/bardblBTHz.\nINSexperimentswereperformedbetween10and190K\nusing about 3g of loose ε-Fe2O3nanopowder in the\nIN4 time-of-flight diffractometer at the Institut Laue-\nLangevin in Grenoble, France.\nIII. RESULTS AND DISCUSSION\nA. Broad-band study of the electromagnetic\nresponse.\nFig. 1a shows the far and mid-IR reflectivity spectra\ndisplaying polar optical phonons of ε-Fe2O3between 10\nand300K.Figs. 1b, cshowthe far-IR ε(E) spectracalcu-\nlated from the fits of IR reflectivity together with the ex-\nperimental THz data. To this purpose, we used a model\ninvolving 35 harmonic oscillators; this number is lower\nthan the number of IR active modes provided by the fac-\ntor group analysis (see Appendix A); apparently, a part\nof the modes are too weak to be observed. Upon cooling,\nall phonons above 12meV exhibit the usual behavior—\ntheir intensity increases due to reduced phonon damp-\ning at low temperatures. The TO1 phonon near 11meV\nexhibits an anomalous behavior: on cooling, its inten-\nsity increases only down to 115K. Below this tempera-\nture, it markedly weakens, while a supplementary broad\nreflectivity peak develops below E∼10meV and be-\ncomes more intense upon cooling (see the inset of Fig.\n1a). This transfer of strengths involves also the TO23\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50/s48/s46/s50/s50/s48/s46/s50/s51/s48/s46/s50/s52/s97\n/s32/s51/s48/s48/s32/s75/s32\n/s32/s50/s48/s48/s32/s75\n/s32/s49/s49/s53/s32/s75\n/s32/s32/s82/s101/s102/s108/s101/s99/s116/s105/s118/s105/s116/s121\n/s69/s110/s101/s114/s103/s121/s32/s40/s109 /s101/s86/s41\n/s32/s101/s108/s101/s99/s116/s114/s111/s109 /s97/s103/s110/s111/s110\n/s32/s32/s32/s49/s48/s48/s32/s75\n/s32/s32/s32/s32/s55/s53/s32/s75\n/s32/s32/s32/s32/s53/s48/s32/s75\n/s32/s32/s32/s32/s49/s48/s32/s75/s32/s32\n/s84/s79/s49/s32/s112/s104/s111/s110/s111/s110\n/s54 /s56 /s49 /s48 /s49 /s50 /s48 /s46/s48 /s48 /s46/s52 /s48 /s46/s56 /s49 /s46/s50 \n/s98/s99/s84/s79/s49/s32/s112/s104/s111/s110/s111/s110/s32\n/s32\n/s69 /s110 /s101 /s114 /s103 /s121 /s32/s40 /s109/s101 /s86 /s41 /s32 /s32/s32/s32/s49/s48/s32/s75 \n/s32 /s32/s32/s32/s55/s53/s32/s75 \n/s32 /s32/s49/s49/s53/s32/s75 \n/s32 /s32/s49/s53/s48/s32/s75 \n/s32 /s32/s51/s48/s48/s32/s75 \n/s52 /s54 /s56 /s49 /s48 /s49 /s50 /s55 /s46/s52 /s55 /s46/s54 /s55 /s46/s56 /s56 /s46/s48 /s56 /s46/s50 \n/s39/s32\n/s39/s39/s101/s108/s101/s99/s116/s114/s111/s109/s97/s103/s110/s111/s110\n/s84/s79/s49/s32/s112/s104/s111/s110/s111/s110\n/s32\nFIG. 1. (a) Lines: IR reflectivity spectra showing polar\nphonons. Symbols below 8meV: data calculated from THz\nspectra. The inset shows in detail the low-energy part where ,\nbelow 100K and 10meV, a new reflection band appears due\nto the EM. (b), (c): Fits of the complex permittivity in the\nfar IR region, obtained from the IR reflectivity spectra us-\ning a sum of harmonic oscillators (lines), compared to data\nobtained from THz spectroscopy (symbols).\nphonon (see Fig. 2), evidencing a coupling among these\nthree polar modes. Despite the lattice distortions which\noccur between 150K and 75K, the crystal symmetry of\nε-Fe2O3does not change with temperature35,37. This is\nfurther confirmed by our IR reflectivity spectra, display-\ningatemperature-independentnumberofpolarphonons;\nshould a structural phase transition occur, it would im-\nply a change of the factor group analysis and different\nphonon selection rules. Given the high number of atoms\nin the unit cell, multiple new reflection bands through-\nout the IR range would be observed. Therefore, one can\nexclude the new mode to originate in a structural modi-\nfication.\nAnother option to be considered is the polar phonon\nsplitting due to exchange coupling below AFM phase\ntransitions which was reported in various transition-\nmetalmonoxidesandchromiumspinels38; themodesplit-\nting increased on cooling below the N´ eel temperature.\nHowever, this explanation cannot be valid as we observe\nan opposite temperature dependence—the new mode ap-\npears below Tmat low energies and hardens towards the\nTO1 phonon energy on cooling, i.e. their energy differ-\nence decreases.\nFinally, one cannot a priori exclude the hypothesis of/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s49/s50/s51/s52/s53/s54\n/s69/s77/s61/s49/s48/s32/s109/s101/s86\n/s84/s79/s50/s61/s49/s50/s46/s54/s32/s109/s101/s86\n/s32/s32/s80/s106/s32/s40/s109/s101/s86/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s84/s79/s49/s61/s49/s49/s32/s109/s101/s86\nFIG. 2. Temperature dependence of the plasma frequencies\n(defined as Ω pj=/radicalbig\n∆εjωj) of the 10-meV-mode attributed\nto EM and of the TO1 and TO2 phonons. The dielectric\nstrengths ∆ εjwere evaluated by fitting using a model with\nharmonic oscillators.\n/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s98\n/s32/s32/s32/s32/s32/s32/s32/s53/s75\n/s32/s32/s32/s55/s53/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s49/s49/s53/s32/s75/s32/s73/s109/s32 /s40 /s32 /s41\n/s69/s110/s101/s114/s103/s121 /s32/s40/s109/s101/s86/s41/s32/s49/s50/s53/s32/s75\n/s32/s49/s53/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s50/s53/s48/s32/s75\n/s32/s51/s48/s48/s32/s75/s97/s55/s46/s50/s55/s46/s52/s55/s46/s54/s55/s46/s56/s56/s46/s48\n/s32\n/s32/s32/s82/s101/s32 /s40 /s32 /s41\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s100\n/s32/s32/s39/s39\n/s69/s110/s101/s114/s103/s121 /s32/s40/s109 /s101/s86/s41/s32/s32/s49/s48/s32/s75\n/s32/s32/s55/s53/s32/s75\n/s32/s49/s48/s48/s75\n/s32/s49/s50/s53/s75\n/s32/s49/s53/s48/s75\n/s32/s50/s48/s48/s75\n/s32/s51/s48/s48/s75/s99\n/s48/s46/s57/s48/s48/s46/s57/s53/s49/s46/s48/s48/s49/s46/s48/s53/s49/s46/s49/s48\n/s32/s32/s39\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56\n/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s70/s77/s82/s32/s40/s109/s101/s86/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s101\n/s32/s83/s116/s114/s101/s110/s103/s116/s104/s32/s40/s49/s48/s52\n/s32/s109/s101/s86/s50\n/s41\nFIG. 3. Temperature dependence of the spectra of the (a)\nreal and (b) imaginary parts of the εµproduct, obtained by\nTHz spectroscopy. Spectra of µ′(c),µ′′(d), corresponding\nto the FMR mode, obtained by fitting the THz spectra. (e)\nTemperature dependence of the FMR energy and strength\n∆µω2\nFMRderived from parts c, d.\nactivation of the TO1 phonon branch from the area of\nthe BZ near its edge. This would require a folding of\nthe structural BZ which could be caused by a transfer of\nthe magnetic BZ folding (linked to incommensurability)\nvia magnetostriction. Nevertheless, in the X-ray diffrac-\ntion studies, no appropriate satellite reflections were ob-\nserved. Even supposing these satellite reflections to be\nvery weak, one would expect the off-center phonons to\nactivate also at higher energies, which we did not ob-\nserve. This hypothesis therefore seems unlikely. Based\non further experimental evidence, especially in view of\nan analogous temperature behavior observed by INS, we\nargue below that the reflection band activated below Tm\nis most probably an EM.4\n\u0000\u0001\u0002\u0003\u0004\u0005\u0001\u0006\u0007\n\u0001\u0006\u0005\u0004\u0000\n\u0000\u0001\u0002\u0003\u0004\u0005\u0007\n\b\t\n\u000b\f\r\u000e\u000f\u0010\n\u0011\u0012\u0013\u0006\u0013\u0003\n\u0013\u0006\u0013\u0001\n\u0013/g004f \u0013\u0014\n\u0001\u0014\n\u0003\u0014\n\u0005\u0014\n\u0007\u0014\u0000\u0000\u000e\u0015\u000e\u0000\u0013\u0013\u000e\u0016\u0000\n\u0013\u0001\u0003\u0005\n\u0017\u000e\u000f\u0014\u0012\u0000\u0006\u0001\n\u0013\u0006\u0005/g005b\u0000\u0001\u0002\u000e\u000f\u0010\n\u0011\u0012\n\u0013\u0001\u0003\u0005\u0001\u0006\u0007\u0002\n\u0001\u0006\u0007\u0001\n\u0001\u0006\u0007\u0000\u0000\n\u0001\u0004\u000e\u0016\n\u0004\u0013\u000e\u0016\n\u0007\u0004\u000e\u0016\u0000\n\u0013\u0001\u0003\u0005\n\u0000\u000e\u000f\u0014\u0012\u0013\u0006\u0013\u0002\n\u0013\u0006\u0013\u0001/g004f\u0000\u0013\u0013\u000e\u0016\n\u0000\u0001\u0004\u000e\u0016\n\u0000\nFIG. 4. (a), (b): Spectra of complex refractive index N≡n−iκofε-Fe2O3measured by THz spectroscopy at T= 100K\nas a function of applied magnetic field. Inset: B-dependence of the FMR frequency, determined as the peak in κ(E) spectra.\n(c), (d): Changes of the value of n,κ, determined within ±0.001, for E= 5meV as a function of temperature and increasing\nmagnetic field (except at 75K).\nThe temperature dependent THz spectra (see Fig. 3)\nreveal the sharp FMR which was previously reported at\nroom temperature25,26. To quantify its temperature be-\nhavior, we used the harmonic oscillator model for all\nphonons and one term accounting for the FMR in µ(E),\nwhile assuming a smooth dependence of ε(E) in this in-\nterval. The resulting spectra, matching well the mea-\nsured data, are shown in Fig. 3c,d. From the fit pa-\nrameters, we derived the temperature dependence of the\nmagnon strength and FMR energy (see Fig. 3e). We\nobserve a sharp drop in the resonance energy between\n150K and 75K, very similar to that of the coercive field\nHc(T)39. This can be explained by the fact that the\nFMR energy is proportional to the magnetocrystalline\nanisotropy field Ha. As the sample consists of randomly\noriented particles with a uniaxial magnetic anisotropy,\nHais proportional to the Hcvalue27.\nFurthermore, we measured THz time-domain spectra\nwith external magnetic field ranging from 0 to 7T. Be-\ncause of the high absorption of the EM, lying near 10\nmeV, the sample was opaque above 7meV. Therefore, we\ncould measure only the low-frequency wing of the EM.\nWhen the magnetic field is applied, two types of changes\nin the THz spectra can be observed: an increase of the\nFMR frequency corresponding to the peak of the κ(E)\nspectra, and a change of the slope of both real and imag-\ninary parts of the index of refraction, indicating shifts of\nthe EM frequency with magnetic field. An example of\nthe former behavior at T= 100K is shown in Fig. 4a,\nb; the FMR frequency, upon applying a static magneticfield ofB= 7T, increases from 0.6 to 1.3meV (see inset\nof Fig. 4a, b). The latter phenomenon is illustrated by\nFig. 4c, d which traces the values of the complex refrac-\ntiveindexat E= 5meVasafunctionoftemperatureand\napplied magnetic field. While changes only close to the\nsensitivity level were detected at temperatures of 10 and\n300K(notshowninFig.4), thereisaclear B-dependence\nof the spectra at intermediate temperatures. The highest\nsensitivity was observed at 100K, close to the magnetic\nphase transition. Also, at T= 75K, a marked hysteresis\ninBoccurs, similarly to the temperature hysteresis ob-\nserved by radio-frequency impedance spectroscopy tech-\nniques near this temperature (see Figure 5); this obser-\nvation will be discussed below. At T≪Tm, where the\nmagnetic structure is probably stable, the changes of N\nwith magnetic field are smaller. This explains also why\nwe did not detect any significant changes of the far-IR\nspectra with magnetic field at T= 2K.\nIn the frequency range from f=10Hz to 1MHz, the\ncomplexpermittivity εwasmeasuredbyimpedancespec-\ntroscopy as a function of temperature (see Fig. 6). No\nsign of a FE phase transition was detected. Above\n200K, both ε′(T) andε′′(T) increase due to the leakage\nconductivity and the related Maxwell-Wagner polariza-\ntion. Between 100 and 200K, we observed a step-like de-\ncrease of ε′(T) towards lower temperatures and maxima\nin losses tan δ(T,f) =ε′′(T,f)/ε′(T,f), which is typical\nof a dielectric relaxation. The temperature dependence\nof the relaxation time τ(T) obtained from the peaks\nof tanδ(T,f) follows an Arrhenius behavior, τ(T) =5\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s49/s48/s46/s50/s49/s48/s46/s51/s49/s48/s46/s52/s49/s48/s46/s53/s49/s48/s46/s54/s49/s48/s46/s55/s49/s48/s46/s56/s49/s48/s46/s57/s49/s49/s46/s48\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s97/s39\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s51/s48/s48/s32/s107/s72/s122 \n/s116/s97/s110/s32\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s55/s46/s52/s55/s46/s54/s55/s46/s56/s56/s46/s48\n/s98\n/s49/s32/s84/s72/s122 \n/s104/s101/s97/s116/s105/s110/s103\n/s32/s32/s39\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48/s48/s46/s48/s48/s54/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/s48/s46/s48/s50/s48\n/s49/s32/s107/s72/s122 /s32/s104/s101/s97/s116/s105/s110/s103/s99\n/s32/s32/s40 /s39/s40/s57/s84/s41 /s45 /s39/s40/s48/s84/s41 /s41/s47 /s39/s40 /s48/s84/s41\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\nFIG. 5. (a) Temperature hysteresis of the dielectric permit -\ntivity (black lines, left axis) and losses (red lines, right axis)\nobserved at 300kHz. (b) Temperature dependence of the per-\nmittivity at 1THz measured on heating. The dashed line is\na guide to the eyes. The values at 300kHz are systemati-\ncally higher than at 1THz due to a small dielectric relaxatio n\nbetween these two frequencies; one can see a similar permit-\ntivity peak near 75K in both experiments. (c) Temperature\ndependence of relative changes of the 1kHz-permittivity du e\nto magnetic field with B= 9T (taken on heating).\nτ0eE0/kBTwithkBdenoting the Boltzmann constant,\nτ0= (1.5±0.2)×10−12s andE0= (0.195±0.002)eV.\nThe origin of this relaxation is not clear, however, sim-\nilar effects are known from several perovskite rare-earth\nmanganites, including the multiferroics TbMnO 3and\nDyMnO 340. We attribute the relaxation to thermally\nactivated vibrations of the FE domain walls or magnetic\ndomain walls which can be polar41. The huge room-/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s48 /s49/s48/s48 /s50/s48/s48/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48\n/s45/s54 /s45/s52 /s45/s50 /s50 /s52 /s54\n/s45/s49/s48/s45/s53/s53/s49/s48\n/s51/s48/s48/s32/s107 /s72/s122 /s49/s48/s32/s72/s122 \n/s32/s116/s97/s110/s32\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s32/s63/s49/s32/s77 /s72/s122 /s32/s39\n/s49/s48/s32/s72/s122 /s80 /s32/s40/s110/s67/s47/s99/s109/s50 \n/s41\n/s85/s32/s40/s107 /s86 /s47/s99/s109/s41\nFIG. 6. Temperature dependence of the real permittivity ε′\n(left) and dielectric losses tan δ(right), measured upon heat-\ning by impedance spectroscopy. Inset: dependence of the po-\nlarization on the applied 50Hz ac bias at 120K (black) and\n15K (red).\ntemperature coercive field Hcis the consequence of a\nsingle-domain magnetic structure of the nanograins26.\nBelow 200K, Hcstrongly decreases due to a transition\nto a polydomain structure39which explains why the di-\nelectric relaxation exists only in this temperature range.\nThe inset of Fig. 6 shows the measured dependences\nof the polarization on applied electric field. No open FE\nhysteresisloops nor signs of saturation were observedun-\nder the applied fields. Since the Pna21crystal structure\nofε-Fe2O3corresponds to a pyroelectric space group, we\ncannot exclude that an applied electric field with an in-\ntensity higher than the one we used (beyond 5kV/cm,\nour sample became leaky) would switch the polariza-\ntion and that ε-Fe2O3is in fact FE. Actually, one of\nus recently investigated strained epitaxial ε-Fe2O3thin\nfilms and, under an applied electric field one order of\nmagnitude stronger, observed a room-temperature FE\nswitching.42Since the crystal symmetry of ε-Fe2O3does\nnot change with temperature43, one can not exclude that\ntheε-Fe2O3nanograinsarealsoFEalreadyabovethefer-\nrimagnetic phase transition occurring near 490K; in any\ncase, it is at least pyroelectric. Consequently, ε-Fe2O3\nwould belong to type-I multiferroics.\nNear 75K, a small peak in ε′(T) was observed in\nour impedance spectroscopy measurements (as marked\nby the arrow in Fig. 6). This peak is rather weak on\ncooling, but it becomes more distinct on heating, and\nit exhibits a temperature hysteresis of ≈15K (see also\nFig.5). Thisisreminiscentofadielectricanomalytypical\nfor pseudoproper or improper FE phase transitions, such\nas those in perovskite rare-earth manganites. However,\nthis hypothesis is not confirmed by the polarization mea-\nsurements shown in Fig. 6, and the X-ray and neutron\ndiffraction investigations did not reveal any structural\nchanges near 75K either35,37. In type-II multiferroics, a\nnarrow dielectric peak is seen at Tconly at frequencies\nbelow1MHz andits intensity stronglydecreaseswith ris-\ning frequency40. By contrast, in our impedance spectra,\nthe peak is present at all frequencies up to the THz re-6\ngion (see Fig. 5b), although it is partly covered by the\nstronger dielectric relaxation at low frequencies. There-\nfore, this anomaly must originate from phonons or an\nEM. As the observed dielectric anomaly occurs at a tem-\nperature close to the lowest-temperature magnetic phase\ntransition35, weproposethatit arisesfromthe transferof\nthedielectricstrengthfromtheTO1andTO2phononsto\nthe EM (see Fig. 2). We note that in single-crystal mul-\ntiferroics, often a step-like increase of the permittivity\noccurs below the temperature where the electromagnon\nactivates.13Our observations on nanograin samples are\nsomewhatdifferent—while astep-likeincreaseof ε′below\n≈130K, superimposed with the narrow-range anomaly\nnear 75K, was detected in the THz range (see Fig. 5b),\nonly the anomaly near 75K manifests itself in the kHz\nrange (see Fig. 5a). We suppose that the step in the\nlow-frequency permittivity is screened by the observed\ndielectric relaxation in the microwave range.\nWe also investigated the dependence of the permittiv-\nity at 1kHz on external magnetic field up to 9T. We\nfound that ε′(B) exhibits the highest changes (almost\n2%) near 70 and 130K (see Fig. 5c). Both of these\nanomalies are clearly linked to the changes of magnetic\nstructure35. We suppose that the lower-temperature\nchange corresponds to the EM anomaly observed also\nin THz experiments, while that observed near 130K is\ndue to the relaxation linked to the magnetic and simul-\ntaneously polar domain walls.\nB. Neutron scattering.\nIn order to further explore the hypothesis of an EM,\nwe performed time-of-flight INS experiments which al-\nlow measuring the phonon and magnon density of states\n(DOS) in the meV energy range. As the nanopowder\ndoes not allow us to determine directly the phonon and\nmagnon dispersion branches in the BZ, the data repre-\nsent an orientation-averaged scattering function S(Q,E)\nwhereQis the total momentum transfer and Ethe en-\nergy transferred between the crystal lattice and the neu-\ntrons (see Fig. 7). The data reveal a steep column of in-\ntense scattering, emanating from magnetic Bragg peaks\natQ= 1.4˚A−1, and extending up to E∼10meV. The\nweaker columns at Q >2˚A−1are due to scattering in\nhigher-order BZs. The fact that the area of most intense\nscattering is located at low Qshows unambiguously44\nthat the dominant contribution to the low- Qscattering\ncomes from spin waves.\nA qualitatively similar magnon response was recently\nobserved in INS spectra of polycrystalline BiFeO 345; the\nspin wave character of the excitation was confirmed by\nINS on BiFeO 3crystals, where the magnon dispersion\nbranch was directly measured46. Our scattering from\nthe magnon waves becomes weaker on cooling due to the\ndecreasing Bose-Einstein factor. Around 10meV, a dis-\ntinct scattering peak persists down to low temperatures,\ncorresponding to a maximum of the magnon DOS; this051015\nMomentum transfer Q (Å−1)Energy (meV)10 K\n  \n1 2 345a\n05101520\n051015\nMomentum transfer Q (Å−1)Energy (meV)80 K\n  \n1 2 3 4 5b\n012345\n051015\nMomentum transfer Q (Å−1)Energy (meV)170 K\n  \n1 2 3 4 5c\n012345\n/s48/s46/s53\n/s53/s49/s48\n/s48 /s53 /s49/s48 /s49/s53/s48/s50/s53/s53/s48/s55/s53/s49/s48/s48/s100\n/s32/s32/s68/s79/s83/s32/s40/s97/s114/s98/s46/s117/s46/s41\n/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s32/s32/s49/s48/s32/s75\n/s32/s32/s56/s48/s32/s75\n/s49/s48/s48/s32/s75\n/s49/s55/s48/s32/s75/s69/s110/s101/s114/s103/s121/s32/s40/s109 /s101/s86/s41/s113 /s32/s40/s114/s46/s108/s46/s117/s46/s41\n/s49/s46/s48\nFIG. 7. (a), (b), (c): Bose-Einstein-factor-normalized IN S in-\ntensity as a function of momentum Qand energy Etransfers\nforT= 10, 80and170K.Near Q= 1.4˚A−1, amagnonbranch\nwith a cut-off energy of ≈11meV can be seen. (d): DOS de-\nterminedbyintegratingovertheregionsmarkedbyblacksol id\nlines in (a)–(c). Inset of (d): scheme of the magnon dispersi on\nbranch in reciprocal lattice units, involving the FMR and EM\nnear the BZ center and boundary, respectively.\nis obviously due to a flat end of the branch below the BZ\nboundary. Moreover, the energy at the maximal magnon\nDOS, as well as its temperature evolution, corresponds\nto that of the newly IR-activated mode (see Fig. 7d).\nThe inset of Fig. 7d shows a schematic view of an\nacoustic-like magnon dispersion branch giving rise to\nthe observed excitations, both the one below 10.5meV\n(at the BZ boundary) and the FMR near 0.5meV (in\nthe BZ center). This dispersion behavior is similar to\nthat observed in the ferrimagnetic HoFe 247, which ex-\nhibits a slightly higher Curie temperature of 597K. In\nε-Fe2O3, the optic-like magnon branches lie probably\nabove 12meV, beyond the energy range used in our INS\nexperiments. We suggest that this acoustic-like magnon\nis activated in the IR spectra due to the loss of mag-\nnetic translation symmetry in the incommensurate mag-\nnetic phase below Tm. Such an activation is analogous\nto that of phonons with q/negationslash= 0 in structurally modulated\ncrystals48. We suppose that the large damping of the\nnewly activated excitation can be explained by an ac-\ntivation of the magnon DOS in the IR spectra. Since\nthe observed spin-wave excitation is coupled with the\nlowest-energy TO1 phonon, it must be excited by the\nelectric component of the electromagnetic radiation; at\nthe same time, it has to contribute to dielectric permit-\ntivity. Therefore, the excitation seen near 10meV must\nbe an EM.7\nIV. CONCLUSION\nIn conclusion, in ε-Fe2O3, we have discovered an ex-\ncitation, appearing simultaneously with the modulation\nof the magnetic structure, at energies below the TO1\nphonon. We attribute this excitation to an EM whose\nenergy corresponds to a magnon from the BZ bound-\nary. We did not observe any other excitation at lower\nenergies, in contrast to type-II multiferroics. There,\nthe Dzyaloshinskii-Moriya(D.-M.) interaction breaksthe\ncenter of symmetry, induces ferroelectricity11and the\nEMs are activated thanks to magnetostriction (( Si·Sj)-\ntype interaction)7. Inε-Fe2O3, the crystal structure is\nacentricatalltemperaturesandit permitstoactivatethe\nD.-M. interaction in an originally collinear ferrimagnetic\nstructure49; the D.-M. interaction tilts the spins and fi-\nnally induces an incommensurately modulated magnetic\nstructure below Tm= 110K, where the EM activates due\nto magnetostriction.\nUp to now, EMs were reported mainly in type-II mul-\ntiferroics. Previous reports of EMs in type-I multifer-\nroics were lacking evidence of their coupling with polar\nphonons, e.g. in BiFeO 319–21or hex-YMnO 322. Our re-\nsults indicate that ε-Fe2O3belongs to type-I multifer-\nroics; it is pyroelectric and perhaps FE even above the\nferrimagnetic phase transition43at 490K, but the EM\nis activated only below Tm, corresponding to the onset\nof the incommensurately modulated magnetic structure.\nIn our case, a clear transfer of dielectric strength from\na low-energy phonon to the zone boundary magnon was\nobserved.\nFinally, we would like to stress that EMs were previ-\nously identified only in single crystals using a thorough\npolarization analysis of measured spectra. Here we have\ndetermined an EM from unpolarized IR and THz spectra\nof nanograin ceramics showing its coupling with a TO1\nphonon. Simultaneously, we have shown from INS exper-\niments made on powder that the EM in ε-Fe2O3comes\nfrom the BZ boundary. This combination of experimen-\ntal methods provides a guideline for an unambiguous de-\ntermination of EMs in materials where sufficiently large\nsingle crystals for polarized IR and THz measurements\nare not available.\nACKNOWLEDGMENTS\nThis work was supported by the Czech Science Foun-\ndation (project P204/12/1163). The experiment in ILL\nGrenoble was carried out at the IN4 spectrometer within\nthe project LG11024 financed by the Ministry of Ed-\nucation of the Czech Republic. M.G. acknowledges\nfunding from the Spanish Ministerio de Econom´ ıa yCompetitividad (projects RyC-2009-04335, MAT 2012-\n35324 and CONSOLIDER-Nanoselect-CSD2007-00041)\nand the European Commission (FP7-Marie Curie Ac-\ntions, PCIG09-GA-2011-294168). S.K. thanks Petr\nBr´ azda for his stimulation of our ε-Fe2O3research and\nS. Artyukchin for a helpful discussion.\nAppendix A: Phonons in ε-Fe2O3\nFor the orthorhombic Pna21crystal structure of\nε-Fe2O3with 8 formula units per unit cell35, the factor\ngroup analysis predicts the following phonon counts and\nsymmetries in the BZ center:\nΓ = 30A1(z,x2,y2,z2)+30A2(xy)+\n+30B1(x,xz)+30B2(y,yz). (A1)\nHere,x,yandzmark electric polarizations of the IR\nwave for which the phonons are IR active, while the rest\nof symbols are components of the Raman tensor. After\nsubtraction of the three acoustic phonons, 87 IR-active\nphonons are expected. We have observed 35 of them (see\ntheir parameters in Table I); the remaining ones cannot\nbe identified, either because of low intensities or because\nthey overlap with other ones.\nNo.∆εΩ0[meV]Γ[meV] No.∆εΩ0[meV]Γ[meV]\nEM0.2710.47 4.67 180.0238.40 1.34\n10.0111.05 0.13 190.1840.13 1.40\n20.0112.61 0.87 200.1542.16 1.37\n30.0813.85 0.44 210.1343.37 1.48\n40.2415.25 0.82 220.0246.76 0.99\n50.0616.26 0.49 230.1648.04 1.87\n60.0217.58 1.82 240.0249.39 1.20\n70.0718.64 0.83 250.2552.78 2.24\n80.0319.95 0.76 260.4055.42 4.68\n90.0921.87 0.99 270.0757.45 3.03\n100.0823.42 0.65 280.1160.84 3.32\n110.5627.18 2.33 290.0363.02 1.95\n120.0228.88 0.77 300.1465.66 3.62\n130.0129.59 0.46 310.0770.78 2.29\n140.0930.91 0.96 320.0372.58 1.96\n150.2133.05 1.12 330.0775.29 3.61\n160.0134.94 0.44 340.0578.46 4.71\n170.3736.44 2.60 350.0885.68 5.43\nTABLE I. 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Lett. 100, 167203 (2008)."    },    {        "title": "2110.13644v1.Optical_interface_for_a_hybrid_magnon_photon_resonator.pdf",        "content": "arXiv:2110.13644v1  [physics.app-ph]  23 Oct 2021Optical interface for a hybrid magnon-photon resonator\nBanoj Kumar Nayak*,1Cijy Mathai*,1Dekel Meirom,1Oleg Shtempluck,1and Eyal Buks1\n1Andrew and Erna Viterbi Department of Electrical Engineeri ng, Technion, Haifa 32000 Israel\n(Dated: October 27, 2021)\nWe study optical detection of magnetic resonance of a ferrim agnetic sphere resonator, which is\nstrongly coupled to a microwave loop gap resonator. Optical fibers are employed for coupling the\nsphere resonator with light in the telecom band. We find that m agnetic resonance can be optically\ndetected in the region of anti-crossing between the loop gap and the ferrimagnetic resonances. By\nmeasuring the response time of the optical detection we rule out the possibility that microwave\ninduced heating is responsible for the optical detectabili ty.\nMagnons are widely employed in a variety of devices\n[1–7], includingnarrowbandoscillators[8], filters[9], and\nparametric amplifiers [10]. Magnons can couple with mi-\ncrowave (MW) photons [11, 12], optical photons [13–23],\nphonons [24, 25], and with superconducting qubits [26–\n29]. Hybrid magnon devices may help developing optical\nchannels linking remote quantum computers [30–32].\nHere we study a hybrid system composed of a MW\nloopgapresonator(LGR) stronglycoupledtoaferrimag-\nneticsphereresonator(FSR)madeofyttrium irongarnet\n(YIG) [33, 34]. Optical fibers are employed for transmit-\nting light in the telecom band through the sphere. The\nfrequency of the hybrid FSR-LGR system is controlled\nusing an externally applied magnetic field (generated by\na magnetized Neodymium). We explore magneto-optic\n(MO) coupling and Faraday rotation of optical polariza-\ntion, and demonstrate optical detection of magnetic reso-\nnance (ODMR) of the hybrid FSR-LGR system. ODMR\nof FSR has been demonstratedbefore in [35], by coupling\na tapered optical fiber to whispering gallery modes of an\nFSR. However, the ODMR method that has developed in\n[35]isbasedonheatinginduced byMWdriving, andcon-\nsequently the response time of this method is relatively\nlong (on the order of a second). As is shown below, the\nresponse time of our ODMR method, which is not based\non heating, is significantly shorter (limited by the ring\ndown time of the FSR, which is about 1 µs).\nThe experimental setup, which is schematically shown\nin Fig. 1, is designed to allow exploring the MO coupling\nbetween MW and optical photons, which is mediated by\nFSR magnons. In Fig. 1, optical components and fibers\nare red colored, whereas blue color is used to label MW\ncomponents and coaxial cables.\nA MW cavity made of an LGR allows achieving a rela-\ntively large coupling between magnons and MW photons\n[36–38]. The LGR is fabricated from a hollow concentric\naluminium tube. A sapphire strip of 260 µm thickness is\ninserted into the gap in order to increase its capacitance,\nwhich inturn reducesthe frequency fcofthe LGRfunda-\nmental mode [39]. An FSR made of YIG having radius\nofRs= 125µm is held by two ceramic ferrules inside\n*These authors contributed equally to this work.\nFIG. 1: Experimental setup. Optical components [ErS (Er-\nbium source), TL (tunable laser), PC (polarization con-\ntroller), OSA (optical spectrum analyzer) and PD (photode-\ntector)] and fibers are red colored, and MW components [LA\n(loop antenna), S (splitter), C (circulator), VNA (vector n et-\nwork analyzer), SA (spectrum analyzer), SG (signal genera-\ntor), and LIA (lockin amplifier)] and coaxial cables are blue\ncolored. The LA is weakly coupled to the FSR-LGR system.\nOptical fibers are installed on both sides of the FSR for trans -\nmission of light through the sphere. Components outlines by\na thick black line (TL, PD and LIA) are used only for the\nmeasurements presented in Fig. 3.\nthe LGR. The applied static magnetic field His con-\ntrolled by adjusting the relative position of the magne-\ntized Neodymium using a motorized stage. The LGR-\nFSR coupled system is encapsulated inside a metallic\nrectangularshield made ofaluminum (represented by the\nblack colored rectangle in Fig. 1). The cavity is weakly\ncoupled to a loop antenna (LA). More information about\nthe FSR-LGR hybrid system, including its fabrication\nand magnetic energy density distribution, can be found\nin Ref. [38].\nAvectornetworkanalyzer(VNA) is employedformea-\nsuring the MW reflection coefficient |S11|2. The plot\nshown in Fig. 2(b) exhibits |S11|2in dB units as a\nfunction of the externally applied magnetic field Hand\nVNA frequency f. The measurement is performed in2\nFIG. 2: Continuous wave measurements. In both (a) and (c) the ErS is employed as a source, and the optical transmission\nmeasurement is performed using the OSA. (a) Measured optica l spectrum data as a function of SG frequency fwith SG power\nof 20 dBm relative to measured optical spectrum correspondi ng to smallest SG frequency in the plot, with a fixed magnetic\nfield ofH= 140.13mT, showing Fabry–P´ erot oscillation near the FSR-LGR re sonance. (b) VNA reflection |S11|2in dB units\nas a function of frequency fand magnetic field Hwith input power of −30 dBm. (c) Optical intensity (in arbitrary units)\nmeasured at a specific wavelength of λ= 1524.292nm as a function of SG frequency fand magnetic field H, with SG power of\n20 dBm.\nthe regionofanti-crossingbetween the LGRfundamental\nmode at frequency fc= 3.9235GHz and the Kittel mode\nFSR frequency fs, which is given by fs=γgH/(2π),\nwhereγg/2π= 27.98 GHz T−1is the gyromagnetic ratio\n[40, 41].\nThe frequencies f±of the hybrid FSR-LGR eigen\nmodes are given by [42]\nf±=fc+fs\n2±/radicalBigg/parenleftbiggfc−fs\n2/parenrightbigg2\n+g2.(1)\nwheregis the FSR-LGR coupling coefficient [43, 44].\nThe overlaid black dashed lines in Fig. 2(b) are calcu-\nlated using Eq. (1). A fitting procedure yields the value\ng/(2π) = 16MHz. Note that in general, gis proportional\nto the FSR volume.\nIn the telecom band (wavelength λ≃1.5µm) YIG\nhas an optical absorption coefficient αof about α=\n(0.5m)−1[45], a Verdet constant AVof about AV=\n5×10−5mT−1µm−1[46, 47], and a polarization beating\nlengthlPin magnetic saturation of about lP= 7.0mm\n[46, 48, 49]. YIG spherescan be employed formaking op-\ntical circulators and isolator in the telecom band [50, 51].\nAn optical cavity can be constructed to enhance Faraday\nrotation [52, 53].\nIn our setup telecom light is transmitted through the\nFSR using single mode optical fibers. The FSR serves\nas a thick optical lens having focal length of FFSR=\n(1/2)n0Rs/(n0−1), where n0= 2.19 is YIG refractive\nindex in the telecom band [51, 54], A graded index fiber\n(GIF) is attached to the tip of one the the fibers that are\ninstalled near the FSR (see Fig. 1). The length of the\nGIF is 0.1pg, wherepg= 1mmis the GIF pitch. Focusingis achieved by displacing both fibers along the optical\naxis and maximizing the fiber to fiber transmission TF,\nwhich for the current device under study is TF= 0.59.\nSpontaneous emission from an Erbium doped fiber is\nusedastheopticalsourceforthemeasurementspresented\nin Fig. 2. The Erbium source (ErS) intensity peaks near\nwavelength of 1530nm. A polarization controller (PC)\nis employed to manipulate the light transmitted through\nthe FSR. An optical spectrum analyzer (OSA) having\nresolution of 0 .004nm is used to probe the transmitted\nlight. All fibers are single mode having 125 µm clad di-\nameter and 9 µm core diameter.\nThe OSAis employedforprobingthe transmitted light\nin the range 1520nm to 1540nm, as a function of MW\ndriving frequency fapplied to the LA, with a fixed mag-\nnetic field of 140 .13mT [see Fig. 2(a)]. In this mea-\nsurement, a signal generator (SG) operating at 20 dBm\nserves as a source [see Fig. 1, and note that a circula-\ntor (C) and a MW spectrum analyzer (SA) are used to\nprobethe backreflectedMWsignal]. The measuredopti-\ncal transmission shown in Fig. 2(a) reveals Fabry–P´ erot\noscillation near the FSR resonance fs. The wavelength\nperiod of the Fabry–P´ erot oscillation is observed to be\n1.77nm. The oscillation is attributed to an optical cavity\nformed between both fibers coupled to the FSR due to\nFresnel reflection at the fibers’ tips. The measured spac-\ning of 1.77nm allows estimating the distance between the\nfibers to be 700 µm.\nTo study the dependence on both the MW frequency\nfas well as magnetic field H, the optical intensity is\nrecorded at wavelength 1524 .292nm [see Fig. 2(c)], at\nwhich the transmission is maximized [see Fig. 2(a)]. The\nSG frequency is varied from 3 .8GHz to 4 .05GHz, and the3\nFIG. 3: LIA measurements. (a) VNA reflection |S11|2in dB units as a function of frequency fand magnetic field H. The\nVNA input power is −10 dBm. (b) LIA measured voltage amplitude (in arbitrary uni ts) as a function of SG frequency fand\nmagnetic field H, with SG power of −10 dBm. In both (a) and (b) the TL wavelength is 1530 .87nm and power is −2.6 dBm.\n(c) The dependence of LIA measured voltage amplitude VLIA(in arbitrary units) on LIA modulation frequency fAM. For this\nmeasurement, the TL wavelength is set to 1538 .556nm and TL power is set to 6 dBm.\npower is set to 20 dBm. The measured optical intensity\npeaks near the Larmorresonance, i.e. when f≃fs. Note\nthat the splitting between f+andf−cannot be resolved\nin Fig. 2(c) due to anisotropy-induced Kerr nonlinearity\n[38].\nNext we explore the response time of the above-\ndiscussed ODMR method. This is done in order to deter-\nmine the role played by MW induced heating, which has\na relatively long time scale [35]. To that end, we perform\nexperiments using a lockin amplifier (LIA). Components\noutlines by a thick black line in the setup sketchshown in\nFig. 1 (TL, PD and LIA) are used only for the LIA mea-\nsurements presented in Fig. 3. A tunable laser (TL) is\nused instead of the high bandwidth ErS. The OSA is re-\nplaced with a photodetector (PD) to measure the optical\nintensity. The LIA reference signal is used to amplitude\nmodulate (AM) the SG signal at a modulation frequency\nfAM, and the PD signal output is fed into the LIA in-\nput port. For LIA measurement shown in Fig. 3(c),\nthe tunable laser wavelength is set to 1538 .556nm, which\ncorresponds to the second highest optical intensity in the\nspectrum shown in Fig. 2(a).\nBoth the VNA measurement shown in Fig. 3(a) andthe LIA measurement shown in Fig. 3(b) are performed\nwith MW power of −10 dBm and TL optical power of\n−2.6 dBm. Figure 3(c) shows a plot of LIA voltage am-\nplitudeVLIA(in arbitraryunits) as a function of modula-\ntion frequency fAM. The measured dependency on fAM\nindicates that the ODMR response time is on the order\nof a microsecond. This observation suggests that the re-\nsponse time is limited by FSR damping, and it rules out\nthe possibility that heating plays a dominant role in the\nunderlying mechanism allowing the ODMR.\nIn summary, ODMR of FSR is demonstrated in the\ntelecomband, and thepossibility that MWinduced heat-\ningistheunderlyingmechanismisruledout. 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Poon1,a)  \n \n1 Physics D epartment, University  of Virginia , Charlottesville , Virginia,22903 , USA \n2 Material Science Engineering , University of Virginia, Charlottesville, Virginia,22903, \nUSA \n \nFerrimagnetic alloy thin films that exhibit perpendicular (out -of-plane) magnetic anisotropy (PMA) with \nlow saturation magnetization, such as GdCo  and Mn 4N, were predicted to be favorable for hosting small \nNéel skyrmions for room temperature applications. Due to the exponential decay of interfacial  \nDzyaloshinskii -Moriya interaction ( DMI ) and the limited range of spin -orbit -torques, which can be used \nto drive skyrmion  motion , the thickness of the ferrimagnetic layer has to be small, preferably under  20 \nnm. W hile there are examples of sub -20 nm, rare earth -transition metal  (RE-TM), ferrimagnetic thin \nfilms fabricated by sputter  deposition , to date  rare-earth -free sub-20 nm Mn 4N films with PMA have \nonly been reported to be achieved by molecular beam epitaxy, which is not suitable for massive \nproduction. Here we report the successful thermal growth of sub -20 nm Mn 4N films with PMA at 400 -\n450 °C substrate temperatures on MgO substrates by reactive sputtering. The Mn 4N films were achieved \nby reducing the surface roughness of MgO substrate through a high -temperature vacuum annealing \nprocess .  The optimal films showed low saturation magnetization (M s = 43 emu/cc), low magnetic \nanisotropy energy (0.7 Merg/cc), and a remanent magnetization to saturation magnetization ratio \n(Mr/Ms) near 1 at room temperature . Preliminary ab -initio density functional theory (DFT) calculations \nhave confirmed the ferrimagnetic ground state of Mn 4N grown on MgO. The magnetic properties, along \nwith the high thermal stability of Mn 4N thin films in comparison with RE -TM thin films , provide the \nplatform for future studies of practical s kyrmion -based spintronic materials.  \n \n \n \n \n \n \na) Authors to whom correspondence should be addressed :  wz8he@virginia.edu  and \nsjp9x@virginia.edu  \n  2 \n As industry rapidly transition s to using big data in their operations and decision making, there is an \nurgent need to develop technologies that accommodate the increasing requirements of high -density data \nstorage 1. One promising candidate that has received increasing attention is using  magnetic skyrmions as \ninformation carriers and manipulating them with current by spin -orbit torque (SOT) for logic or memory \noperations 2,3 (e.g. racetrack memories). Magnetic skyrmions are swirling spin configurations of \nneighbor atoms in a magnetic mate rial with topologic protection. The size of the skyrmion could be as \nsmall as a few nanometers 4 and manipulation of skyrmions could be energy efficient  5,6 in high -density \ndata storage applications 7,8. To date, most RT skyrmions have been discovered in f erromagnetic (FM) \nmultilayer stacks 7-10. The use of ferromagnets as skyrmions have certain shortcomings. For instance, \nferromagnets suffer from large stray fields  and large saturation magnetization (M s), which makes the \nregion of parameter space for small  skyrmion less accessible , especially at room temperature (RT) 11.  \nUnlike ferromagnets, the ferrimagnetic counterparts have small stray fields and the low  Ms, which \nmakes them suitable for hosting small RT skyrmions 11. This makes ferrimagnet a promising candidate \nmaterial for high -density data storage applications. One prototypical example is the amorphous GdCo \nferrimagnetic thin film. Small (10 –30 nm) room temperature skyrmions have been reported in Pt/GdCo  \n(6 nm) /TaO x 12. While the amorphous GdCo ferrimagnet shows promising characteristics relative to the \nferromagnets, it suffers from poor thermal stability. It has been shown that in amorphous GdCo, PMA is \nlost after annealing at 300 -400 °C 13, the temperature range applied in complementary metal –oxide –\nsemiconductor ( CMOS ) fabrication  14. Since Néel skyrmions can only exist in a heterostructure with \nPMA, the poor thermal stability of RE -TM films will have a deleterious effect in both the fabrication \nand performance of amorphous RE -TM in devices that leverage skyrmions for storage technology. One \nof the potential solutions to this problem is to explore crystalline  ferrimagnets  with PMA  synthesized by \ndeposition at high -temperature near 400 °C, thus ensuring compatibility  with conventional CMOS  \nprocessing .  3 \n In addition to the intrinsic effects that we have discussed thus far, film thickness is an important \nextrinsic effect that impacts the overall device performance. The importance of film thickness applies to \nboth ferro - and ferrimagnets. It is now well -established that the interfacial  Dzyaloshinskii -Moriya \ninteraction (DMI ), which stabilizes the magnetic skyrmions in multilayers and heterostructures, decays \nexponentially with increasing film thickness 15-17. Further, the SOT scales inversely with the thickness 18. \nThis necessitates the growth of thin film magnets with sub -20 nm thickness, for the realization of small \nskyrmions in practical applications 12,19,20. \nAnti-perovskite Mn 4N has been known as a crystalline, rare-earth -free ferrimagnetic material. In \nMn 4N, the Mn -atoms have a face -centered cubic structure with one N -atom at the body center. A \nschematic of the unit cell is shown in  Figure  1. The Mn -atoms at the corner and the face center have \ninequivalent magne tic moments and are ferrimagnetically coupled 21. Although the easy axis of bulk \nMn 4N is along the [111] direction, PMA is repeatedly and reproducibly observed in crystalline Mn 4N \nfilms 22-28. As a result, ferrimagnetic Mn 4N thin films have also attracted increasing interest in \nspintronics applications. Compared to the amorphous RE -TM ferrimagnetic alloys, the Mn 4N system has \nbetter thermal stability for two main reasons. First, in most thin film studies, it has been shown that the \nanti-perovskite crystal s tructure is formed at 400 -450 °C. No loss of PMA is reported in thin films after \nannealing, which is an encouraging outcome. Second, there is no known evidence for any structural \nphase transformation on cooling to room temperature. Thus, we interpret that the anti -perovskite crystal \nstructure is tolerant of high -temperature device fabrication processes  (unlike the GdCo amorphous \nalloys). As noted earlier, the emergence of PMA is an important magnetic property for its use in \nspintronic applications. One of t he plausible reasons for the PMA in Mn 4N thin films could be attributed \nto the deviation of the out -of-plane lattice constant, c, to the in -plane lattice constant, a, (c/a)  ratio from \n1, 22-28 due to the in -plane epitaxial strain. A recent study also showe d that the anisotropy energy is 4 \n correlated to the c/a ratio 27. We note that further studies are warranted to understand the interplay of \nmagnetic and strain effects on  the PMA and this is beyond the scope of this paper.  \nSeveral groups have grown crystalline Mn 4N epitaxial films (30 -100 nm) on MgO, SrTiO 3, or \nLaAlO 3 substrates by magnetron sputtering 22,23, molecular beam epitaxy (MBE) 24-27, and pulsed laser \ndeposition (PLD) 28, and they reported similar c/a ratios of ~0.99. The reported uniaxial magnetic \nanisotropy constant (K u) and M s of those Mn 4N films were about 0.5 –1 Merg/cc and 50 –100 emu/cc, \nrespectively 22-28, comparable with the data observed for amorphous GdCo thin films (K u ~ 0.25 Merg/cc \nand M s ~ 50 emu/cc) 12.  \nTo date, however, only a few groups have grown sub -20 nm Mn 4N thin film s with good magnetic \nhysteresis (M(H)) loops, which has remanent magnetization M r to M s ratio larger than 0.5 (M r/Ms > 0.5), \nby MBE 26,27. For those reported Mn 4N film by sputtering , the thickness es were 30 n m 22 to 100 nm 23, \nsome of them even up to hundr eds nanometers 29,30.  Similar results on sputter deposited sub -20 nm \nMn 4N thin film have not been reported. C ompared to MBE, sputter deposition is a more widely adopted \nmethod in CMOS  technology . In this work, we have epitaxially grown sub-20 nm Mn 4N thin film s on \nMgO (001) substrate with PMA by reactive sputtering . The effect of MgO substrate morphology on the \nquality of Mn 4N film is studied. We then compare the magnetic properties of the Mn 4N films on MgO \nsubstrate from experiment and first principles -based density functional theory (DFT) calculations. The \nmain contribution of this paper lies in the demonstration of the growth of high magnetic quality sub -20 \nnm crystalline Mn 4N films on MgO subs trate using reactive sputtering that is more promising for scale -\nup production.  \n 5 \n  \nFigure  1. Mn 4N crystal structure  \n \nMn 4N thin films with nominal thickness es of ~15 nm and 10 nm were deposited  on MgO(001) \n5x5x0.5 mm substrate s by reactive rf -sputtering at 400 °C  and 450 °C  substrate temperature s and a base \npressure of 7x10-8 Torr. The flow rates of Ar and N 2 gases were controlled by a mass flow meter and we \nmaintained a flow rate Ar:N 2 ratio of 93:7. A 3  nm Pt capping layer was  deposited on Mn 4N layer at \nroom temperature to prevent oxidization. Before loading the MgO  (001) substrate into the vacuum \nchamber, it was wet -cleaned with 2% diluted Hellmanex III alkaline detergent, acetone, and \nisopropanol, sealed in a vacuum tube with  pressure 30 mTorr . The MgO substrate was  baked at high \ntemperatures 1,000 °C or 1,100 °C for 4 hours . Then, the substrates were annealed inside the chamber at \n500 0C for one hour to remove the surface contaminations. The Mn target was pre -sputtered with A r gas \nfor 20 minutes to remove the surface oxide. Surface roughness of the MgO substrate was measured \nusing atomic force microscopy (AFM).  The deposition rate was measured by X -ray reflect ometry  \n(XRR). The film compositions were determined with X -ray phot oelectron spectroscopy (XPS). The film \nthickness was verified by XRR  with an XRR  simulation/calculation program (Rigaku, GXRR) 31.  \n         The epitaxial growth of Mn 4N crystal layer was demonstrated using X -ray diffraction (XRD) with \nCu-Kα radiation. The  magnetic properties of the samples were measured at room temperature with \nvibrating sample magnetometry (VSM). The diamagnetic component of the substrate was deduced from \nthe slope of raw M -H curves at large H region and subtracted from the raw data.  K u was calculated  from \neffective anisotropy K  with Eq. (1) and (2)  \n6 \n 𝐾𝑢=𝐾+2𝜋𝑀𝑠2                                                                               (1) \n𝐾=∫(𝑀𝑒𝑎𝑠𝑦(𝐻)−𝑀ℎ𝑎𝑟𝑑(H))𝑑𝐻∞\n0                                                          (2) \nWhere M easy and M hard are the magnetization in out -of-plane  applied field  and in -plane applied field, \nrespectively.     \nAb-initio  electronic structure calculations were carried out in the Density Functional Theory (DFT) \nframework using the plane -wave pseudopotential Quantum ESPRESSO code 32, 33. Core and valence \nelectrons were treated using the ultrasoft pseudopotential method 34,35. The exchange -correlation \nfunctionals were described using the Perdew -Burke -Ernzerhof parameterization of the genera lized \ngradient approximation modified for solids (PBEsol) 36. The plane -wave cutoff energy was set to 60 Ry \nand a Γ -centered Monkhorst -Pack k-point mesh of 12x12x12 was used to sample the Brillouin Zone 37. \nLattice parameters were fixed to experimentally m easured value ( 𝑐/𝑎 = 0.987) and Ms was calculated \nusing the formula, 𝑀𝑠=|𝜇𝑡𝑜𝑡𝑎𝑙|\n𝑉, where |𝜇𝑡𝑜𝑡𝑎𝑙| is the absolute value of the total magnetization from DFT \ncalculations given in Bohr magnetons and 𝑉 is the unit cell volume 38. Self-consistent spin -polarized \ncalculations were performed for collinear spin structures with no spin -orbit coupling term in the \nHamiltonian. The Mn -atoms (Mn  I) located in the cell corners with coordinates (0,  0, 0) were \nferromagnetically coupled to Mn -atoms (Mn  IIa) located at (0.5, 0.5, 0). Both Mn I - and Mn  IIa-atoms \nwere anti -ferromagnetically coupled to the Mn -atoms (Mn  IIb) in the face centers with coordinates (0,  \n0.5, 0.5) and (0.5,  0, 0.5). Since the total atomic magnetic moments at all three Mn -sites do not cancel \neach other out (Mn  I = 3.47 μ B, Mn  IIa = 0.75 μ B and Mn  IIb = -2.36 μ B,), the ground state is a \nferrimagnet.  \nTable I lists the five samples (S1 -S5) that were examined in this study . We begin by investigating \ndifferent substrate annealing  temperatures to understand the impact of the MgO surface on epitaxial \ngrowth  of Mn 4N. The S1 -S3 were deposited at 400 °C, and S4 and S5 were deposited at a higher 7 \n temperatu re of 450 °C.  Film thickness was determined by XRR measurement. In the case of S2 -S5, the \nMgO substrates were pre -baked before loading into the sample deposition chamber.  \nTABLE I. List of Mn 4N thin films on MgO substrates  \nSample No.   MgO a nnealing \nTemperature  MgO a nnealing \nTime  Film t hickness  Deposition temperature  \nS1 No annealing  (RT) -- 16.6 ± 0.6 nm  400 °C \nS2 1000 °C 4 hours  16.8 ± 0.5nm  400 °C \nS3 1100 °C 4 hours  16.8 ± 0.5 nm  400 °C \nS4 1100 °C 4 hours  16.8 ± 0.5 nm  450 °C \nS5 1100  °C 4 hours  11.5  ± 0.6 nm  450 °C \n \nThe film thicknesses measured by the XRR technique indicated that the thicknesses of all five thin \nfilm samples  were less than 20 nm (see Table I). Analysis of XRR results is shown in Supplementary \nFigure  1. S1 -S4 had a thickness of 16.8 ± 0.5nm, whereas S5 had a thickness of 11.5 ± 0.6 nm. Figure  \n2(a) shows the normalized M -H curves with out -of-plane applied magnetic field for 1 6.8 nm Mn 4N film \non different heat -treated MgO substrates (S1 -S3). The M -H curves  show a strong dependence on the \nannealing temperature. All out -of-plane hysteresis loops w ere open, indicating PMA. The substrate \nannealing temperature of S3 ( 1100 °C) is higher than that of  S2 (1000 °C), whereas S1 is unannealed. \nThe squareness of the M(H) hysteresis loop improves from S1 to S2 to S3.  Figure  2(b) shows the \ndependence of the substrate annealing temperature on M r/Ms. The M r/Ms ratio increased from 0.47 to \n0.72 as the annealing temperature increased  from RT to 1100 °C . The increment of M r/Ms is attributed \nto the improved epitaxial growth of the thin film. High -temperature annealing not only decomposes the \nMg(OH) 2 and MgCO 3 contaminants , it also reconstructs the surface of the MgO substrate 39,40. During \nthis process, the substrate forms an atomically smooth surface 39,40, which is beneficial for epitaxial \ngrowth  41. The difference between S 1 and S3 is mainly attributed to the different substrate annealing 8 \n temperatures. Thin films annealed at 1100 °C produced a smoother surface than 1000 °C.  The surface \nmorphology as characterized by the atomic force microscopy revealed an average root -mean -square \n(rms) roughness of 0.592  nm, 0.308  nm, 0.278  nm for the as -received, 1000 °C annealed  and 1100 °C \nannealed thin films, respectively (See Supplement ary Figure 2).  \nTo explore further improvement, we increased the deposition temperature to 450 °C, while retaining \nthe substrate annealing temperature and time at 1100 °C and 4 hours . It has been shown that the  \nsquareness of the Mn 4N film can be improved by  increasing deposition temperature  29 .We found that \nthe Mr/Ms ratio  of these samples , S4 and S5 , increased  further to near 1 , as shown in Figure  2(c). The \ndifference between S4 and S5 is in the film thickness: the S4 and S5 thin films were 16.8 ± 0.5 nm nm \nand 11.5 ± 0.6 nm thick, respectively (see Table I). S5, t he thinnest film,  kept a high Mr/Ms ratio ~0.8, \nwhich is an encouraging outcome.  \nFigure  2(d) shows the out -of-plane and in -plane M -H curves for the S4 film. The M s and K u for the \nS4 film is 43  ± 1 emu/cc and 0. 70 Merg/cc, respectively . All five samples had similar M s (40-60 \nemu/cc).  Although the M s of the se films are smaller than the reported Ms (~100 emu/cc ) in thick er films \n(30-100 nm) 22-25,28 , they are  comparable with that of the MBE -grown sub-20 nm Mn 4N film on MgO, \n50-80 emu/cc  26,27. The low er Ms of sub -20 nm Mn 4N films may  be due to surface oxidation  25.  \n 9 \n   \n \nFigure  2 (a) Normalized out -of-plane M(H) loops of 16.8 nm Mn 4N film s deposited at 400  °C on MgO \nsubstrates  annealed at different temperatures ; S1 (red) was unannealed, S2 (green) was annealed at 1000 \n°C, and  S3 (blue) was annealed at 1100 °C, (b) M r/Ms ratio of 16.8 nm Mn 4N film (S1-S3) deposited at \n400 °C on MgO substrates, which have been annealed at different temperatures,  (c) Normalized out -of-\nplane M(H) loops of 16.8 nm (S4, cyan ) and 11.5 nm (S5 , brown ) thick Mn 4N film s deposited at 450  °C. \non MgO substrates, which have been annealed at 1100 °C , (d) out-of-plane and in -plane M(H) loops of \nthe S 4 Mn 4N film.  \n \nFigure  3 shows the out -of-plane 2θ -θ XRD profile (a) and φ scan  (b) of S3. Besides the MgO \nsubstrate peaks, only Mn 4N (002) peak is observed in the 2θ -θ XRD profile, which indicates the Mn 4N \n(00l) orientation is parallel to the MgO  (00l). In the XRD 360o φ scan, both Mn 4N and MgO shows four \npeaks with 90o interval, which corresponding to (202),  (022),  (-202), and (0 -22). The overlap of Mn 4N \npeaks and MgO peaks in φ -scan confirm their epitaxial relat ionship to be MgO  (001) [100]// Mn 4N \n(001) [100]. The out -of-plane lattice constant c and in -plane lattice constant a deduced from the (002) \nand (202) diffraction peaks are 0.386 nm and 0.391 nm , respectively. Therefore, the c/a ratio was \ncalculated to be 0.987, which is close to the value reported earlier  22-28.  \n10 \n  \nFigure  3 . (a) 2θ- θ profile of S3 and MgO substrate  annealed at 1100 °C. (b)φ scan of Mn 4N (101) and \nMgO  (101) peaks.  \n \nBased on the experimental lattice constant, DFT predicts an Ms value of 153 emu/cc, which is larger \nthan our experiment results.  However, DFT is a zero -temperature electronic structure calculation. Li et \nal have shown that 𝑀𝑠 of the bulk Mn 4N would decrease as a function of increasing measurement \ntemperature 40.  DFT also does not consider the possible effects of the mixing layer at the interface as \nwell as defects . Future investigation will address the temperature dependence of saturation \nmagnetization and magnetic anisotropy energy.  \nWe have grown sub-20 nm ferrimagnetic Mn 4N epitaxial thin film s with high M r/Ms ratios at 400 -\n450 °C substrate temperatures on MgO substrates  by reactive sputtering . The quality of the epitaxial \ngrowth was optimized by ex-situ pre-annealing  of MgO substrates  at temperatures  as high as 1100 °C. \nThe annealing was found to reduce the surface roughness of the MgO substrate s at the atomic level, \nwhich was found to improve the quality of the out -of-plane magnetization hysteresis loops. The present \nresults established Mn 4N thin film as a thermally stable ferrimagnet for further investigation as \npromising skyrmion -based  spintronic material.  \n \n \n11 \n Suppl ementary material  \nSee su pplementary material for the XRR results  of the film  and the AFM results of the MgO \nsubstrate s surface.  \nAcknowledgment  \nThis work was supported by the DARPA Topological Excitations in Electronics (TEE) program \n(grant  D18AP00009).   The content of the information does not necessarily reflect the position or the \npolicy of the Government, and no official endorsement should be inferred. Approved for public release; \ndistribution is unlimited.  \nData Availability Statements  \nThe data that support the findings o f this study are available from the corresponding author upon \nreasonable request.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  12 \n Reference:  \n \n1 A. Fert, F . N. V. 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J. \nPhys. 10(1) , 181-188 (2012) . \n \n \n \nSupplementary Materia l for Rare -earth free ferrimagnetic Mn 4N sub-20 nm  thin \nfilms as potential high - temperature spintronic material  \n \n \n \nSupplementary Figure  1.  XRR curve of the S3(a) and S5(b) . the blue line is the raw data , and  the red line is the fitting line.  \n \nNotes on Supplementary Figure 1.  \n \n15 \n X-ray reflectivity (XRR) was performed on the MgO /Mn 4N/Pt samples to confirm that the Mn 4N layer \nis thinner than  20 nm. The blue line is the raw data collected with a Cu -Kα source (λ = 1.5406 Å) and \nthe red line is a fitting line produced by the software Rigaku -GXRR31. Supplementary Figure 1(a) shows \nthe XRR curve and fitting line  of S3.  The fit gave a Mn 4N film thickness of 16.8  ± 0.5nm (nominal \nthickness was 15 nm) and a Pt capping layer thickness of 3.4   ± 0.6 nm nm. Supplementary Figure 1(b) \nshows the XRR curve and fitting line  of S5.  The fit gave a Mn 4N film thickness of 11.5   ± 0.6 nm \n(nominal thickne ss was 10 nm) and a Pt capping layer thickness of 4.2   ± 0.5 nm .  The discrepancy \nbetween the XRR fitting model and the measured data at small angle (< 0.8) is because these angles are \nsmaller than the critical angle (approx. 0.8 degree) for total reflecti on. \n \nSupplementary Figure 2.  AFM surface morphology of pure MgO (100) substrate surface (a)  as received,  and after annealing \nfor 4 hours at (b) 1000  °C, (c) 1100  °C. The lower  panels (d-f) show the corresponding line scans of the AFM topographies.  \n \nNotes  on Supplementary Figure 2.  \n \nAFM was performed on the MgO substrates, which have been annealed at different temperatures, to \ncheck the surface morphology of the MgO (001) substrates.  Supplementary Figure 2 (a), (b), (c) shows \nthe AFM images of as -received  MgO substrate, annealed at 1000 °C, and annealed at 1100 °C, \nrespectively. Supplementary Figure 2 (d), (e), (f) shows the corresponding line scans of the AFM \ntopographies.  It is clear that as -received MgO has the roughest surface. However, from the image s, it is \nhard to tell the difference between the 1000 °C annealed MgO substrate surface (Supplementary Figure \n2(b), (e)) and the 1100 °C annealed MgO substrate surface (Supplementary Figure 2(c), (f)). Gwyddion \n43 was employed to analyze the roughness, whi ch gave the 1000 °C annealed MgO substrate surface rms \nroughness of 0.308 nm and the 1100 °C annealed MgO substrate surface rms roughness of 0.278 nm. \nFor comparison, the surface rms roughness of the as -received substrate was 0.592 nm.  \n \n \n \n \n \n   \n \n"    },    {        "title": "2312.07678v2.Dzyaloshinskii_Moriya_interaction_inducing_weak_ferromagnetism_in_centrosymmetric_altermagnets_and_weak_ferrimagnetism_in_noncentrosymmetric_altermagnets.pdf",        "content": "Dzyaloshinskii-Moriya interaction inducing weak ferromagnetism in centrosymmetric\naltermagnets and weak ferrimagnetism in noncentrosymmetric altermagnets\nCarmine AutieriID,1,∗Raghottam M SattigeriID,1,†Giuseppe CuonoID,1,‡and Amar Fakhredine2\n1International Research Centre Magtop, Institute of Physics,\nPolish Academy of Sciences, Aleja Lotnik´ ow 32/46, 02668 Warsaw, Poland\n2Institute of Physics, Polish Academy of Sciences, Aleja Lotnik´ ow 32/46, 02668 Warsaw, Poland\nThe Dzyaloshinskii-Moriya interaction (DMI) has explained successfully the weak ferromagnetism\ninsome centrosymmetric antiferromagnets. However, in the last years, it was generally claimed that\nthe DMI is not effective in centrosymmetric systems. We reconciled these views by separating the\nconventional antiferromagnets and altermagnets. Altermagnets represent collinear antiferromagnetic\ncompounds with spin-up and spin-down sublattices connected only by mirror and roto-translational\nsymmetries. Consequently, the system shows even-parity wave spin order in the k-space lifting the\nKramer’s degeneracy in the non-relativistic band structure. We emphasize that the DMI can create\nweak ferromagnetism in centrosymmetric altermagnets while it is not effective in centrosymmetric\nconventional antiferromagnets. Additionally, DMI can create weak ferromagnetism or weak ferri-\nmagnetism in noncentrosymmetric altermagnets. Once the spin-orbit coupling is included in an\naltermagnetic system without time-reversal symmetry, the components of spin moments of the two\nsublattices along the N´ eel vector are antiparallel but the other two spin components orthogonal\nto the N´ eel vector can be either parallel or antiparallel for centrosymetric systems. For noncen-\ntrosymmetric systems, we can have different bands showing parallel or antiparallel spin components\nresulting in weak ferrimagnetism. We can divide the altermagnetic compounds into classes based on\nthe weak ferromagnetism or weak ferrimagnetism properties. In the case of weak ferromagnetism,\nthe Hall vector is orthogonal to the N´ eel vector while in the case of weak ferrimagnetism the Hall\nvector has a component parallel to the N´ eel vector too. Therefore, an electric field breaking the in-\nversion symmetry could generate a magnetization parallel to the N´ eel vector. We find a sign change\nof the magnetization, and possibly of the anomalous Hall effect, as a function of the band filling and\nN´ eel vector. We describe the dependence of the weak ferromagnetism from the charge doping and\nelectric field. The weak ferromagnetism and weak ferrimagnetism induced by DMI is a property\nexclusively of the altermagnets, not present in either ferromagnets or conventional antiferromagnets\nso we propose that altermagnets should be classified based on this property.\nI. INTRODUCTION\nIt was discovered that non-relativistic spin-splitting\nin electronic bands can be a feature in antiferromag-\nnetic materials having two equivalent sublattices with\nmagnetic moments related by crystal-symmetry1,2. This\nmagnetic phase has been named altermagnetism (AM)3–7\nor antiferromagnetism with non-interconvertible spin-\nstructure motif pair8. The existence of AM necessi-\ntates that the electronic charge of spin-up (down) atoms\nshould be mapped into spin-down (up) atoms not ex-\nclusively via translations, inversions, or their combina-\ntions but through the application of roto-translations or\nmirrors6,7. As a result, AM can be observed in specific\nspace groups, and more precisely, in the magnetic space\ngroups of type-I and type-III9. Numerous transition\nmetal oxides9and rare-earth compounds have been docu-\nmented to present altermagnetism10. Altermagnets have\na wide range of potential applications, they are promis-\ning to enable highly efficient spin-current generation11,\nas well as contributing to the development of giant and\ntunneling magnetoresistance12. It was shown that spin-\nindependent conductance in altermagnets can be effi-\nciently exploited in spintronics13–15. Furthermore, alter-\nmagnets can play a significant role in spincaloritronics16\nand can be practical use in Josephson junctions17.The Dzyaloshinskii-Moriya interaction (DMI) is a\nrelativistic magnetic interaction that has successfully\nexplained the weak ferromagnetism described for the\nfirst time in centrosymmetric antiferromagnets18–20with\nspace group 167. This weak ferromagnetism has a rel-\nativistic origin and is the only kind of weak ferromag-\nnetism discussed in this paper. Some compounds of\nthe centrosymmetric space group 62 exhibit weak ferro-\nmagnetism and lead to a significant ferromagnetic be-\nhavior, which was later explained by the mechanism\nof the Dzyaloshinskii Moriya exchange based on torque\nmeasurements21,22. The spin canting has been attributed\nto the DMI in most cases, although other effects share\nthe same requirements to be imposed on the crystal\nsymmetry23. The weak ferromagnetism can also be\npresent in complex systems like solids with molecules\nas cations24and biphasic systems25. In other litera-\nture, the weak ferromagnetism from DMI in bulk systems\nwas associated with noncentrosymmetric systems26. Fur-\nthermore, in ferroelectric BiFeO 3, a subtle DMI which\nconstitutes about 1% of the antiferromagnetic exchange\nstrength causes a tilting of the two spin sublattices, con-\nsequently giving rise to ”weak ferromagnetism”20,27. The\nDMI as explained by definition, appears as an effect\nof antisymmetric interactions, which necessitates a lack\nof inversion symmetry between two spins. Neverthe-\nless, a structure can possess an overall centrosymmet-\nTypeset by REVT EXarXiv:2312.07678v2  [cond-mat.mtrl-sci]  26 Dec 20232\nEFEF\nEFEF\nEFK1K2\nK1K2Sz\nSz0\n0Г\nГMz = 0 ; σxy = 0\nMz ≠ 0 ; σxy ≠ 0a) b)\nc)\nd) e)f)\nFIG. 1. Origin of weak ferromagnetism in altermagnets. (a)\nSchematic band structure of an altermagnet with SOC, N´ eel\nvector in the xy plane and Fermi level in the gap. The red and\nblue colors represent the negative and positive S zcomponent\nof the total system. (b) Local S zcomponent for the magnetic\natom as a function of the energy for the energy spectrum (a),\nin the centrosymmetric case S zfor the two atoms are equal.\n(c) Magnetic configuration without M zand AHE ρxy. (d)\nSame as (a) with metallicity. (e) S zcomponent as a function\nof the energy for the energy spectrum (d). (f) Magnetic con-\nfiguration with non-zero M zcomponents and non-zero AHE\nρxy.\nric crystal symmetry while also accommodating a DMI\ninteraction28. DMI plays an important role in forming\nskyrmions with a fixed chirality and vorticity in the case\nof noncentrosymmetric magnets since it allows the tilting\nof adjacent spins under special rules related to crystal\nsymmetry. In contrast, in centrosymmetric structures,\nother mechanisms are believed to be responsible for com-\nplex magnetism29.\nThe non-relativistic spin-splitting is proportional to\nthe hopping parameters which tend to increase with the\natomic number. Large spin-splittings have been found in\ncompounds with heavy elements as MnTe, RuO 2, CrSb,\netc ... which are also the most representative and most\nstudied compounds among the altermagnetic systems.\nBeyond the large spin-splitting, these heavy atoms own\na huge spin-orbit coupling (SOC). Therefore, the study\nof the spin-orbital effects is extremely relevant, especially\nfor the most representative altermagnets. Beyond that,\nthe anomalous Hall effect (AHE) strongly depends on\nthe size of the SOC. In the case of a metallic system, the\nDMI produces weak ferromagnetism in AM compounds\nthat display spontaneous AHE3,30which can be enhanced\nby Weyl points31. The orientation of the Hall vector\nstrongly depends on the N´ eel vector, which is defined as\nthe difference between the magnetization vectors of the\ntwo distinct magnetic sublattices15,32,33. While the direc-\ntion of the Hall vector has been explored using symmetry\nconsiderations, there is a notable absence of information\nin the literature regarding how to establish this orienta-tion based on fundamental electronic properties through\ndensity functional theory calculations.\nIn this work, our objective is to have a general under-\nstanding of the appearance of weak ferromagnetism in\ncentrosymmetric systems, while we found weak ferromag-\nnetism and weak ferrimagnetism in noncentrosymmetric\nsystems. We discuss why weak ferromagnetism and weak\nferrimagnetism are relativistic effects that can appear\nonly in altermagnets. Once the time-reversal symmetry\nis broken and the Kramers degeneracy is lifted, the spin\ncomponents of the two sublattices orthogonal to the N´ eel\nvector are related by symmetry in centrosymmetric sys-\ntems and therefore must be parallel or antiparallel. The\nscenario changes in noncentrosymmetric systems where\ndifferent sets of bands can generate at the same time\nparallel or antiparallel spin components in which we es-\ntablish that the direction of the Hall vector is orthogonal\nto the N´ eel vector. A schematic representation of our\nmain results is presented in Fig. 1. We propose that the\nband structure reported in Fig. 1(a) can be present in\nall altermagnets for a suitable N´ eel vector, as a conse-\nquence of this band structure, the local density of states\n(DOS) for the component of the spin orthogonal to the\nN´ eel vector have a relativistic spin-splitting as shown in\n1(b). If the Fermi level is in the gap, we still observe a\nnet zero magnetic configuration M zand zero anomalous\nHall effect in altermagnets owing to a complete antiparal-\nlel spin configuration in two sublattices. However, when\nbands cross the Fermi level (as in Fig. 1(d)) which is\nalso evident from the increase in the contribution from\nthe S zcomponent (as in Fig. 1(e)), the SOC creates\nan effective non-zero magnetic configuration giving rise\nto ferromagnetic spins aligned perpendicular to the N´ eel\nvector (as in Fig. 1(f)) leading to a non-zero anoma-\nlous Hall effect in altermagnetic compounds. Integrating\nin energy the spin component orthogonal to the N´ eel in\na realistic multiband system we will obtain the magne-\ntization as a function of the energy, this magnetization\ncan be zero at the same energies by compensation of the\nspins, however, the compensation of the spins does not\nmean that we have a compensation of the Berry curva-\nture. Therefore, a null magnetization is not associated\nwith a null anomalous Hall effect34. The weak ferromag-\nnetism is difficult to detect especially on the nanoscale\nlevel, so our work provides a guide to theoretically pre-\ndict the direction of the weak ferromagnetism to support\nthe experimental investigation. Using density functional\ntheory calculations, the results of the self-consistent cal-\nculation will report parallel or antiparallel components\ndepending on which of them minimizes the total energy.\nWe calculate the spin components of the two sublattices\nas a function of the N´ eel vector for the centrosymmetric\nd-wave RuO 2, the centrosymmetric g-wave CrSb and the\nnoncentrosymmetric g-wave MnSe. The paper is orga-\nnized as follows: the second Section describes the com-\nputational framework while the third Section is dedicated\nto the results for the three chosen compounds. Finally,\nin the fourth Section, the authors draw their conclusions.3\n-0.3 0 0.3\n S1  Γ  S2 Energy (eV) (a)Sx\nn || x\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n S1  Γ  S2 Energy (eV) (b)Sy\nn || x\nMY≠0\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n S1  Γ  S2 Energy (eV) (c)Sz\nn || z\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. 2. Band structure of d-wave RuO 2along the S 1-Γ-S 2low-symmetry k-path. (a) The positive and negative S xcomponents\nare plotted in red and blue, respectively, while the N´ eel vector is along the x-axis. (b) The positive and negative S ycomponents\nare plotted in red and blue, respectively, while the N´ eel vector is along the x-axis. (c) The positive and negative S zcomponents\nare plotted in red and blue, respectively, while the N´ eel vector is along the z-axis. S x, Syand S zrepresent the spin components\nfor the total system. The Fermi level is set to zero. The coordinates of the k-points are S 1=(0.5,0.5,0) and S 2=(-0.5,0.5,0) in\ndirect coordinates.\n-0.3 0 0.3\n R1  Γ  R2 Energy (eV) (a)Sy\nn || x\nMY≠0\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n R1  Γ  R2 Energy (eV) (b)Sy\nn || z\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. 3. Band structure of d-wave RuO 2along the R 1-Γ-R 2\nlow-symmetry k-path with the N´ eel vector along (a) the x-\nand (b) z-axis. The positive and negative S yand S xcompo-\nnents are plotted in red and blue, respectively. S x, Syand\nSzrepresent the spin components for the total system. The\nFermi level is set to zero. The coordinates of the k-points are\nR1=(0.5,0,0.5) and R 2=(-0.5,0,0.5) in direct coordinates.II. COMPUTATIONAL DETAILS\nThe electronic properties were computed within the\nframework of density functional theory calculations based\non plane wave basis set and projector augmented wave\nmethod using VASP35–37package. A plane-wave energy\ncut-off of 400 eV was chosen and the generalized gra-\ndient approximation of Perdrew, Burke, and Ernzerhof\nhas been used38. The Hubbard U effects for the 4d-\norbitals of Ru4+, the 3d-orbitals of Cr and 3d-orbitals\nof Mn2+have been adopted39. We have used the value\nof U = 2 eV40,41keeping J H= 0.15U. The value of U\nstrengthens the gap in MnSe and establishes a robust\nmagnetic moment in RuO 2, however, all qualitative re-\nsults are independent of the value of U. We have per-\nformed the calculations using 20 ×20×28 and 24 ×24×18\nk-points centered in Γ for RuO 2and CrSb, respectively.\nA denser k-grid of 28 ×28×18 was used for wurtzite MnSe,\nwhich is necessary to have an accurate description of the\nDOS, especially in the presence of relatively small spin-\norbit coupling. The lattice constants are a=4.4825 ˚A\nand c=3.1113 ˚A for the RuO 2, a=4.103 ˚A and c=5.463\n˚A42for the CrSb and a=4.12 ˚A with c=6.72 ˚A for the\nMnSe43. The experimental N´ eel vectors are along the\nz-axis for RuO 2and CrSb44. All calculations were per-\nformed in the presence of SOC. The information on the\nindividual spinor components is available for the site’s\nprojected density of states in the VASP code. The site-\nprojected orbital- and energy-resolved spin density S x, Sy\nand S zare available. We sum over all orbitals to obtain\nthe total S x, Syand S zcomponents. The smearing value\nfor the DOS was 50 meV. Regarding the magnetic space\ngroup (MSG), the RuO 2has MSGs 58.398 and 136.499\nfor the N´ eel vector along the x-axis and z-axis, respec-\ntively. CrSb has MSGs 63.457, 63.462 and 194.268 for\nthe N´ eel vector along the x-axis, y-axis, and z-axis, re-\nspectively. Wurtzite MnSe has the magnetic space group\n194.268 when the N´ eel vector is along the z-axis. The4\n-6-5-4-3-2-1 0 1 2 3 4 5 6\n-0.3  0  0.3DOS (10-1states/eV)\nEnergy (eV)(a)\nn || xSx Ru1\nSx Ru2\nTotal DOS\n-5-4-3-2-1 0 1 2 3 4 5\n-0.3  0  0.3DOS (10-2states/eV)\nEnergy (eV)(b)\nn || xSy Ru1\nSy Ru2\nTotal DOS\nEnergy (eV)-6-5-4-3-2-1 0 1 2 3 4 5 6\n-0.3  0  0.3DOS (10-1states/eV)(c)\nn || zSz Ru1\nSz Ru2\nTotal DOS\nFIG. 4. DOS for the spin-components as a function of the energy for the d-wave RuO 2. (a,b) DOS of the S xand S ycomponents\nwith the N´ eel vector along the x-axis, respectively. (c) DOS of the S zcomponent with N´ eel vector along the z-axis. The other\ncomponents are antiparallel. The Fermi level is set to zero. The total DOS is plotted in green and renormalized to fit within\nthe plot.\nmagnetic point group without spin-orbit coupling32,45of\nthe wurtzite MnSe is 6′m′m, different from that of the\nhexagonal CrSb in the NiAs structure which is 6′/m′m′m.\nHowever, both groups belong to the same magnetic Laue\nclass (6′/m′m′m) proving that the spin splitting of elec-\ntron bands without SOC is qualitatively the same in both\nsystems. This could be established using the formalism\nof spin groups6,7.\nIII. RESULTS\nIn a centrosymmetric conventional antiferromagnet, we\nhave Kramers degeneracy therefore the DOS of the spin-\nup sublattice and the spin-down sublattice are always\nequal for all components. The two sublattices have al-\nways antiparallel spin components and no weak ferromag-\nnetism is possible. In a noncentrosymmetric conventional\nantiferromagnet with SOC, there is an additional Rashba\nspin-splitting where the spin components belonging to\ntwo sublattices are antiparallel.46Therefore, in noncen-\ntrosymmetric conventional antiferromagnets, there is no\nweak ferromagnetism as well.\nWe move to the study of the electronic properties re-\nsponsible for weak ferromagnetism in centrosymmetric\nand noncentrosymmetric altermagnets. We study the\nelectronic properties of d-wave centrosymmetric alter-\nmagnet RuO 2in the first subsection. In the second and\nthird subsections, we compare the g-wave altermagnets\nCrSb and wurzite MnSe, where the former is centrosym-\nmetric and the latter is noncentrosymmetric. We will see\nhow SOC acts differently on centrosymmetric and non-\ncentrosymmetric space groups. All selected compounds\nhave two atoms per unit cell which we will call M 1and\nM2(M= Ru, Cr, Mn) to represent the properties of the\ntwo sublattices with opposite spins.A. Weak ferromagnetism in centrosymmetric\ntetragonal altermagnet RuO 2\nIn this subsection, we describe the metallic centrosym-\nmetric d-wave RuO 2. Several authors described the al-\ntermagnetic properties of RuO 23,47, however, we will fo-\ncus on the band structure and the DOS for the spin-\ncomponents which were never investigated before. The\nappearance of the weak ferromagnetism derives from the\nbreaking of the time-reversal symmetry, therefore, there\nis the chance for the spins to be not antiparallel when\nSOC is included. The self-consistent DFT calculations\nprovide the energetic ground state for the parallel or an-\ntiparallel configuration of the spin components orthogo-\nnal to the N´ eel vector. Since RuO 2is metallic, we expect\nthat this weak ferromagnetism could be observed in the\nDFT calculations. Indeed, the magnetic moment orthog-\nonal to the N´ eel vector is 0.022 µBper Ru atom along\nthe y-axis when the N´ eel vector is along the x-axis. In\nFigs. 2 and 3, we report the band structures of RuO 2pro-\njected on different spin components for the two inequiva-\nlent directions of the N´ eel vector (x-axis and z-axis) along\nthe S 1-Γ-S 2and R 1-Γ-R 2low-symmetry k-paths, respec-\ntively. When the N´ eel vector is along the x-axis, the S x\nspin component of the total system for every energy has\na value along S 1-Γ and an equal but opposite value along\nΓ-S2as reported in Fig. 2(a). As a result, the magne-\ntization component M xof the entire system that is the\nintegral over the energy up to the Fermi level of S xis\nalways zero. On the contrary, the spin components S yof\nthe total system show a band structure where the value\nof Syalong S 1-Γ is equal to the value along Γ-S 2. For\nevery value of the energy, the value of the magnetization\nMyas integral of S ywill be in general non-zero resulting\nin a weak magnetization in the y-direction as we can see\nfrom Figs. 2(b) and 3(a). The origin of the canted spin\nmoment derives exclusively from the spin-orbit coupling\nbut this picture is equivalent to the DMI picture. In\nthe case of RuO 2with spins along the z-axis, we have a\npure antiparallel order without weak ferromagnetism, as\nshown in Figs. 2(c) and 3(b). The other results were not5\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(a)Sx\nn || x\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(b)Sy\nn || x\n-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(d)Sx\nn || y\n-0.6-0.4-0.2 0 0.2 0.4 0.6\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(e)Sy\nn || y\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(g)Sx\nn || z\nMX≠0\n-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(h)Sy\nn || z\nMY≠0\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(c)Sz\nn || x\nMZ≠0\n-0.6-0.4-0.2 0 0.2 0.4 0.6\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(f)Sz\nn || y\nZ≠\n-0.6-0.4-0.2 0 0.2 0.4 0.6\n0 M\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(i)Sz\nn || z\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. 5. Band structure of centrosymmetric CrSb with SOC along the L 1-Γ-L 2low-symmetry k-path. With the N´ eel vector\nalong the x-axis, we report the projection of the spin-component (a) S x, (b) S yand (c) S z. With the N´ eel vector along the\ny-axis, we report the projection of the spin-component (d) S x, (e) S yand (f) S z. With the N´ eel vector along the z-axis, we\nreport the projection of the spin-component (g) S x, (h) S yand (i) S z. Sx, Syand S zrepresent the spin components for the\ntotal system. The Fermi level is set to zero. The coordinates of the k-points are L 1=(0.5,0,0.5) and L 2=(-0.5,0,0.5) in direct\ncoordinates.\nshown because they are either equivalent to the shown\nresults or with null spin components. The band degen-\neracy is broken at Γ for the N´ eel vector along the x-axis,\nbut not for the N´ eel vector along to z-axis. Therefore,\nwe have lifting of the degeneracy at Γ when the weak\nferromagnetism rises.\nTo further extend our consideration to the entire Bril-\nlouin zone, we also show the density of states for the\nnon-zero and inequivalent spin components in Fig. 4 for\nthe N´ eel vector along the x- and z-axis. When the N´ eel\nvector is along the x-axis, the local DOSs of the S xcom-\nponents for the two sublattices are equal and opposite as\ndescribed by the local DOS related to atoms Ru 1and Ru 2\nin Fig. 4(a), therefore, the S xcomponents are equal andopposite for the two sublattices. From the DOS of the S y\ncomponent reported in 4(b), it is further clear that there\nis a magnetic moment along the y-axis on the Ru 1and\nRu2atoms that are equal and in the same direction re-\nsulting in a weak ferromagnetism. The DOSs of the spin\ncomponent S yfor the atoms Ru 1and Ru 2are perfectly\nequal, therefore, the size of the local magnetic moments\nof Ru 1and R 2are equal as expected since the two atoms\nare connected by roto-translation symmetries. We can\nsee that we have a sign change in M yas a function of\nthe energy, this could be accompanied by a sign change\nof the AHE. Comparing the DOS of the spin-component\nSxwith the total DOS, we can see a strong reduction\nof Sxaround -0.4 eV together with the maximum of S y,6\n-0.4-0.2 0 0.2 0.4\n-0.3  0  0.3DOS (states/eV)\nEnergy (eV)(a)\nn || xSx Cr1\nSx Cr2\nTotal DOS\n-0.3-0.2-0.1 0 0.1 0.2 0.3\n-0.3  0  0.3DOS (states/eV)\nEnergy (eV)(b)\nn || zSz Cr1\nSz Cr2\nTotal DOS\n-2-1.5-1-0.5 0\n-0.3  0  0.3DOS (10-3states/eV)\nEnergy (eV)(c)\nn || x Sz Cr1\nSz Cr2\nTotal DOS-0.2 0 0.2 0.4 0.6 0.8 1 1.2\n-0.3  0  0.3DOS (10-3states/eV)\nEnergy (eV)(d)\nn || ySz Cr1\nSz Cr2\nTotal DOS\n 0 1\n-0.3  0  0.3DOS (10-3states/eV)\nEnergy (eV)(e)\nn || zSx Cr1\nSx Cr2\nTotal DOS\n 0 0.2 0.4\n-0.3  0  0.3DOS (10-3states/eV)\nEnergy (eV)(f)\nn || zSy Cr1\nSy Cr2\nTotal DOS\nFIG. 6. DOS for the spin-components S x, Syand S zas a\nfunction of the energy for the centrosymmetric CrSb. (a,b)\nDOS for the components S xand S zwith the N´ eel vector along\nthe x-axis and along the z-axis, respectively. (c,d) DOS for\nthe spin component S zwith the N´ eel vector along the x-axis\nand along the y-axis, respectively. Both of them show weak\nferromagnetism. (e,f) DOS for the spin component S xand\nSywith the N´ eel vector along the z-axis. Both of them show\nweak ferromagnetism. The Fermi level is set to zero. The\ntotal DOS is renormalized to fit within the plot.\ntherefore, there is a strong rotation of the S xcomponent\ninto the S ycomponent at -0.4 eV. In the case of the N´ eel\nvector along the z-axis, the DOS of all spin-components\nrelative to Ru 1and Ru 2are equal and opposite, there-\nfore, the Ru spins are perfectly compensating each other\nas it is shown in Fig. 4(c), consequently, there is no weak\nferromagnetism. From the DOS we can also understand\nthat the system presents ferromagnetic or antiferromag-\nnetic features depending on the orientation of the N´ eel\nvector, and this property in the k-space is the same in\nthe whole Brillouin zone and for all bands.\nB. Weak ferromagnetism in centrosymmetric\nhexagonal altermagnet CrSb\nWe repeat the investigation for the g-wave centrosym-\nmetric altermagnet CrSb. In Figs. 5, we report the band\nstructures of CrSb projected on different spin compo-\nnents (S x, Syand S z) for the three different directions of\nthe N´ eel vector along the L 1-Γ-L 2focusing on the region\nnear Γ point. The ground state is altermagnetic phase\nwith the N´ eel vector along the z-axis, we have an energydifference of 0.2 meV per formula unit with respect to\nthe in-plane N´ eel vector. When the N´ eel vector is along\nthe x-axis as in Fig. 5(a,b,c), we have weak ferromag-\nnetism only for S zresulting in a non-zero value of M z.\nWe have qualitatively similar results when the N´ eel vec-\ntor is along the y-axis as in Fig. 5(d,e,f). We have weak\nferromagnetism only for S zresulting in a non-zero value\nof M z. When we have the N´ eel vector along the z-axis,\nwe have weak ferromagnetism for the S xand S ycompo-\nnents resulting in a non-zero M xand M yas shown in\n5(g,h,i). Overall, we find a weak ferromagnetism out-of-\nplane when the N´ eel vector is in-plane and a weak ferro-\nmagnetism in-plane when the N´ eel vector is out-of-plane.\nAdditionally, we find a strong change in the relativistic\nband structure as a function of the N´ eel vector, especially\nbetween in-plane and out-of-plane directions of the N´ eel\nvector. Qualitatively similar results are observed for the\nhigh-symmetry k-path K 1-Γ-K 2(not reported) with the\ndifference that S zis always zero when the N´ eel vector is\nalong the z-axis since the non-relativistic spin splitting is\nabsent. We check the electronic properties in the DOS\nas well where we have information along the entire Bril-\nlouin zone and not for a limited number of k-points. The\nDOS for S xand S zwhen the N´ eel vector is along the x-\nand z-axis are reported in 6(a) and (b), respectively. We\ncompare the DOS of the spin components with the total\nDOS illustrated in green. At energy 0.2 eV above the\nFermi level, the spin-component resolved DOS with spin\nalong the N´ eel vectors has less weight with respect to the\ntotal DOS. This could mean that 0.2 eV above the Fermi\nlevel there is DOS from non-magnetic atoms or there is a\nrotation of the spins with the weight of the DOS moving\ntowards the spin components orthogonal to the N´ eel vec-\ntor. When the N´ eel vector is along the x- and y-axis, we\nhave non-zero magnetization along the z-axis. The values\nof DOS for S zare reported in 6(c) and (d) for the N´ eel\nvector along the x- and y-axis. The case with N´ eel vec-\ntor along the x-axis gives the larger weak ferromagnetism\nfor CrSb with a value of -0.002 µBper formula unit. We\ncan observe a value of the total DOS above 0.2 eV larger\nthan the DOS of the spin-component, therefore, the spins\nin this region are rotated from the main component by\nspin-orbit. We can see that the DOS of S zis positive or\nnegative depending on whether the N´ eel is along the y-\nor x-axis. This means that by rotating the N´ eel vector\nin the xy plane, we will have a sign change of M z. This\nsign change of M zcould also produce a sign change of the\nanomalous Hall effect. Recent studies on the sign change\nof the AHE in ferromagnets have explained this effect\nwhich is relatively rare in experimental conditions48,49.\nWhen the N´ eel vector is along the z-axis, we observe\nweak ferromagnetism for both spin components for S x\nand S yreported in 6(e) and (f).7\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (g)Sx\nn || z\nMX≠0\n-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (d)Sx\nn || y\n-0.3-0.2-0.1 0 0.1 0.2 0.3-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (a)Sx\nn || x\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (b)Sy\nn || x\n-0.3-0.2-0.1 0 0.1 0.2 0.3\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (e)Sy\nn || y\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (c)Sz\nn || x\nMZ≠0\n-0.3-0.2-0.1 0 0.1 0.2 0.3\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (f)Sz\nn || y\nMZ≠0\n-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (h)Sy\nn || z\nMY≠0\n-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (i)Sz\nn || z\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. 7. Band structure of noncentrosymmetric MnSe with SOC along the L 1-Γ-L 2low-symmetry k-path. With the N´ eel vector\nalong the x-axis, we report the projection of the spin-component (a) S x, (b) S yand (c) S z. With the N´ eel vector along the\ny-axis, we report the projection of the spin-component (d) S x, (e) S yand (f) S z. With the N´ eel vector along the z-axis, we\nreport the projection of the spin-component (g) S x, (h) S yand (i) S z. Sx, Syand S zrepresent the spin components for the\ntotal system. S x, Syand S xrepresent the spin components for the total system. The Fermi level is set to zero. The coordinates\nof the k-points are L 1=(0.5,0.0.5) and L 2=(-0.5,0,0.5) in direct coordinates.\nC. Weak FM and Weak ferrimagnetism in\nnoncentrosymmetric hexagonal altermagnet MnSe\nTo understand the effect of the breaking of the in-\nversion symmetry, we analyze MnSe which is a hexag-\nonal g-wave altermagnet like CrSb. Indeed, MnSe with a\nwurtzite crystal structure represents the case of an alter-\nmagnet without inversion symmetry with a small number\nof atoms in the unit cell. The structural and electronic\nproperties without SOC for wurtzite MnSe are reported\nin the literature43. The non-relativistic spin-splitting in\nwurzite MnSe can reach a maximum of 200-300 meV in\nthe Mn-bands, the SOC of Mn is of the order of 30 meV\nwhile the SOC of Se is 140 meV. Therefore, the SOC\nwould be a perturbation at least in the region where the\nspin-splitting is maximum. For MnSe, the top of the va-\nlence band at Γ and A are mainly p x/pywith a sizeablecontribution from Mn orbitals. When the N´ eel vector is\nmoved from in-plane to out-of-plane, the top of the va-\nlence band rises. We have a reduction of the band gap of\n80 meV (from 1.72 to 1.64 eV) when we move from AFM\nwith the N´ eel vector in the plane to AFM with the N´ eel\nvector out-of-plane. The same sensitivity of the band gap\nfrom the N´ eel vector was found for the centrosymmetric\ng-wave altermagnet MnTe.50\nWe calculate the band structure for the three different\nspin components along the low symmetry k-path L 1-Γ-\nL2where we have the non-relativistic spin-splitting. The\ntop of the valence band at the Γ point without SOC is\nfour-fold degenerate. Starting from the low-symmetry k-\npath L 1-Γ-L 2reported in Fig. 7, we find a relativistic\nspin-splitting in all cases. When the N´ eel vector is in\nthe plane, the eigenvalues at the Γ point split into two\neigenvalues twofold degenerate with zero magnetization8\nEFEF\nL1L2 0 ГMy ≠ 0 ; σxz ≠ 0\nMy = 0 ; σxz = 0a)\nb)\nc)B1B1\nEFEF\nL1L2 Sy0 ГMn1\nMn2\nB2\nB2Mn1\nMn2\nEFEF\nL1L2 Sy0 ГB1\nB2 Mn1\nMn2B2\nB1\nMy ≠ 0 ; σxz ≠ 0Sy\nFIG. 8. Schematic band structure of noncentrosymmetric al-\ntermagnets with N´ eel vector along the z-axis. The red and\nblue colors represent the negative and positive S ycomponents\nof the total system in the band structure. DOS of the spin\ncomponent S yand magnetic configuration for three selected\ncases. a) Weak ferromagnetism in some bands (B1) of non-\ncentrosymmetric altermagnets. b) Weak antiferromagnetism\nin some bands (B2) of noncentrosymmetric altermagnets. c)\nWeak ferrimagnetism as the superposition of weak FM and\nweak AFM from bands B1 and B2.\nin the plane (see Figs. 7(a,b,d,e)) and non-zero magneti-\nzation along the z-axis (see Figs. 7(c,f)). When the N´ eel\nvector is along the z-axis, there is a strong splitting with\nthe two highest eigenvalues single-degenerate and the one\neigenvalue double-degenerate at the Γ point. The single-\ndegenerate highest eigenvalues produce non-zero M xand\nMyas shown in Fig. 7(g) and 7(h), respectively. How-\never, the double degenerate bands show antiparallel spin\ncomponents. Without SOC, there is a fourfold degener-\nacy in the top of the valence band43. When the system\nis noncentrosymmetric as in the wurzite structure, we\ncan have a combination of parallel and antiparallel spin\norientation despite both sets of bands coming from the\nsame fourfold degenerate point without SOC. As a con-\nsequence in the DOS, we will see that we do not have\na completely parallel or antiparallel DOS weight. Fi-\nnally, 7(i) shows the sign and weight of S zon the two\nsublattices that are equal and opposite when the N´ eel\nvector is along the z-axis. Qualitatively similar results\nare observed for the K 1-Γ-K 2path (not reported) with\nthe difference that S zis always zero when the N´ eel vector\nis along the z-axis since the non-relativistic spin splitting\nis absent. We find weak ferromagnetism for g-wave com-\npounds along the same direction with or without inver-\nsion symmetry. The main difference is the coexistence of\n-0.1 0 0.1\n-0.3 -0.2 -0.1  0DOS (states/eV)\nEnergy (eV)(a)\nn || xSx Mn1\nSx Mn2\nTotal DOS\n-0.1 0 0.1\n-0.3 -0.2 -0.1  0DOS (states/eV)\nEnergy (eV)(b)\nn || zSz Mn1\nSz Mn2\nTotal DOS\n-0.1 0 0.1\n-0.3 -0.2 -0.1  0DOS (10-4states/eV)\nEnergy (eV)(c)\nn || xSz Mn1\nSz Mn2\nTotal DOS\n-0.2 0 0.2 0.4\n-0.3 -0.2 -0.1  0DOS (10-4states/eV)\nEnergy (eV)(d)\nn || ySz Mn1\nSz Mn2\nTotal DOS\n-0.1 0\n-0.3 -0.2 -0.1  0DOS (10-4states/eV)\nEnergy (eV)(e)\nn || zSx Mn1\nSx Mn2\nTotal DOS Total DOS\n-0.04-0.02 0 0.02 0.04\n-0.3 -0.2 -0.1  0DOS (10-4states/eV)\nEnergy (eV)(f)\nn || zSy Mn1\nSy Mn2FIG. 9. DOS for the spin-components S x, Syand S zas a\nfunction of the energy for the wurtzite MnSe. (a,b) DOS for\nthe spin components S xand S zwith the N´ eel vector along\nthe x-axis and along the z-axis, respectively. (c,d) DOS for\nthe spin component S zwith the N´ eel vector along the x-axis\nand along the y-axis, respectively. (e,f) DOS for the spin\ncomponent S xand S ywith the N´ eel vector along the z-axis,\nrespectively. The solid and dashed lines represent the local\nspin components S x, Syand S zfor the two sublattices repre-\nsented by Mn1 and Mn2. The Fermi level is set to zero. The\ntotal DOS is renormalized to fit within the plot.\nboth parallel and antiparallel magnetic configurations for\nthe different bands of the in-plane spin-components when\nthe N´ eel is along the z-axis. Basically, we have the super-\nposition of weak ferromagnetism [see Fig. 8](a)] and weak\nantiferromagnetism [see Fig. 8](b)]. The superposition\nof both kinds of magnetism as reported schematically in\nFig. 8(c) ends up forming weak ferrimagnetism (weak\nFiM). In this case, there is no exact compensation of the\nmagnetization along the N´ eel vector and the Hall vector\nhas a component along the N´ eel vector too.\nTo further check the results from the band structure,\nwe report in Fig. 9 the resolved DOS for the spin-\nsublattice and spin components S x, Syand S zfor dif-\nferent directions of the N´ eel vector, namely along the x-,\ny- and z-axis. The DOSs for the spin components along\nthe N´ eel vector are reported in Fig. 9(a,b). The two\nsublattices, represented by the Mn 1and Mn 2local DOS,\nhave equal and opposite DOS of the spin components\nsince the spins are antiparallel. Summing the DOSs of\nthe components along the N´ eel vector for the two sub-\nlattices, we would obtain zero. Now, we focus on the\ncomponents orthogonal to the N´ eel vector. The size of9\nthese DOSs is 3-4 orders of magnitude smaller than the\nsize of the DOSs for the spin-components along the N´ eel\nvector. When the N´ eel vector is along the x-axis, we ob-\nserve from Fig. 9(a,c) that the in-plane S xcomponents of\nthe two sublattices are antiparallel, while the spin com-\nponents S zare parallel. Since the filling was completed\nand the MnSe is an insulator, the weak ferromagnetism\nwas not obtained in the computational details but in the\ncase of p-doping, the weak ferromagnetism would be ob-\nservable. Assuming a p-doping, the magnetization M z\nobtained integrating the spin density up to the Fermi\nlevel will be non-zero and the system will host a weak\nferromagnetism along the z-direction. The same happens\nfor the N´ eel vector along the y-axis, the in-plane compo-\nnents are antiparallel while we have weak ferromagnetism\nalong S zas shown in Fig. 9(d). When the N´ eel vector\nis along the z-axis, we have both M xand M ycompo-\nnents different from zero, as shown in Fig. 9(e,f). When\nthe N´ eel vector is in the xy plane, the weak ferromag-\nnetism is along the z-axis. When the N´ eel vector is along\nthe z-axis, the weak ferromagnetism is in the xy plane.\nThis weak ferromagnetism generates AHE with the Hall\nvector along the direction of the weak ferromagnetism.\nWe observe that the Hall vector is always orthogonal to\nthe N´ eel vector. Even if some band structures resemble\nthe Rahsba-like dispersions, the spin textures of the al-\ntermagnets are different and more complicated than the\nRashba effect9. Of course, a Rashba spin texture rises on\ntop of the altermagnetic spin texture when the system\nlacks inversion symmetry like in the wurtzite structure.\nFor the wurtzite with N´ eel vector along the z-axis, we\nhave a coexistence of bands with parallel and antipar-\nellal components regarding S xand S yas shown in Fig.\n7(g) and (h), respectively.\nIV. DISCUSSION AND CONCLUSIONS\nIn this paper, we propose that all compounds host-\ning weak ferromagnetism induced by the relativistic\nDzyaloshinskii-Moriya interaction18,19belong to the fam-\nily of the altermagnets. While Dzyaloshinskii and\nMoriya focused on the interplay between magnetic in-\nteractions and symmetries, the recent studies on alter-\nmagnetism provided a deeper knowledge of the crystal\nsymmetries, extended the investigation on the magnetic\nproperties and discovered new effects on the electronic\nproperties1,4–7,33. The connection between these two\nfields emphasized in this paper paves the way to simplify\nthe physical scenario and merge the two fields giving the\nchance to connect the literature developed from two dif-\nferent perspectives which until now had a minor overlap.\nIn Table I, we report several altermagnetic compounds\nwith different space groups where the non-relativistic\nspin-splitting and the weak ferromagnetism were theoret-\nically predicted or experimentally detected. All the com-\npounds described in the seminal papers of Dzyaloshin-\nskii and Moryia are altermagnets present in this table19.Compound NO SOC Weak FM\n(No. space group) splitting from DMI\n(Sr,Ca) 3Ru2O7(36) - Exp.51\nCa3Mn2O7(36) Th.9Exp.52\nMn(N(CN) 2)2(58) Th.9Exp.53\nCa2RuO 4(61) Th.54-\nYVO 3(62) Th.54Exp.55\nLaMnO 3(62) Th.56Exp.57\nCaMnO 3(62) Th.9Exp.58\nYCrO 3(62) Th.9Exp.59\nCaCrO 3(62) Th.60Th.60\nBa2CoGe 2O7(113) Th.9Exp.61\nRuO 2(136) Th.6Th.*, Exp.47\nLa2NiO 4(138) Th.9Exp.62\nLiFeP 2O7(140) Th.9Exp.63\nKMnF 3(140) Th.9Exp.64\nMnTiO 3(161) Th.9Exp.65\nα-Fe2O3(167) Th.9Th.19, Exp.66\nMnCO 3(167) Th.9Th.19, Exp.67\nCoCO 3(167) Th.9Th.19, Exp.68\nCrF 3(167) Th.9Th.19, Exp.69\nCoF 3(167) Th.9Exp.70\nFeF 3(167) Th.9Exp.70\nFeBO 3(167) Th.9Exp.71\nNi1/3NbS 2(182) Th.72-\nNi1/3TaS 2(182) Th.72-\nMnSe (186) Th.43Th.*\nMnTe (194) Th.6, Exp.73Th.74, Exp.74\nCrSb (194) Th.6, Exp.42Th.*, Exp.75\n* Present work.\nTABLE I. List of selected altermagnetic compounds where\nthe non-relativistic spin-splitting appears with weak ferro-\nmagnetism induced by DMI. The words ”Th.” and ”Exp.”\nare reported for the theoretical and experimental verification\nof the non-relativistic spin splitting or weak ferromagnetism.\nThe space groups 36, 182 and 186 lack inversion symmetry.\nDue to the lifting of the Kramers degeneracy, all alter-\nmagnetic systems could potentially show weak ferromag-\nnetism, however, for particular N´ eel vector orientations\nthe DMI is zero as in the case of the RuO 2with N´ eel vec-\ntor along the z-axis. Once separating the centrosymmet-\nric and noncentrosymmetric compounds, we propose to\ndivide the centrosymmetric altermagnets into two classes\nas shown in Fig. 10. The g-wave CrSb falls into the first\ncategory which exhibits weak ferromagnetism for every\nN´ eel vector. In the second category, are some altermag-\nnets that do not display weak ferromagnetism for a given\nN´ eel vector, such as the d-wave CaCrO 360and RuO 2\nwhere we have shown that the N´ eel vector is oriented\nalong the z-axis. In the DMI picture, the symmetries\ndo not allow the presence of the DMI vector for partic-\nular N´ eel vector orientation. The compounds belonging\nto this second class are probably a limited number and\nseem to have higher symmetry with respect to the first10\nclass. As a consequence of this proposed classification,\nthe family of altermagnets is wider than the family of\nweak ferromagnets since there are altermagnets that do\nnot exhibit weak ferromagnetism. For the noncentrosym-\nmetric systems, we have found that weak FM or weak\nFiM are present depending on the N´ eel vector orienta-\ntion. To extend this classification, further studies are\nneeded especially for the noncentrosymmetric systems.\nIn this prospect, interesting compounds are the metal-\nlic d-wave Ca 3Ru2O7and the metallic g-wave Ni 1/3MS2\n(M=Nb, Ta). These results will be presented elsewhere.\nBeyond the crystal symmetries and the N´ eel vector\norientation, the only other requirement for weak FM or\nweak FiM is to have a non-compensated magnetization\nas happens in metallic systems or doped semiconduc-\ntors. As shown in Fig. 1, when the system is insulat-\ning no weak ferromagnetism is observable since there is\na spin compensation, therefore, a small doping is neces-\nsary as in the case of MnTe74and other semiconductive\naltermagnets. From the DOS of the spin-components,\nwe can understand that doping can change the Fermi\nlevel, strongly modify the weak ferromagnetism, and can\neven reverse the sign of the DMI. This explains the de-\npendence of the weak ferromagnetism from electron- or\nhole-doping concentration in several cases76,77. What\nwas defined as ”weak altermagnetism” is the band struc-\nture effect that gives rise to the DMI in centrosymmet-\nric altermagnets.73The electric field increases or acti-\nvates the non-relativistic spin-splitting depending on the\nfield orientation56,78, therefore, the electric field can in-\ncrease and manipulate the DMI in altermagnets79. We\ncan make another observation regarding centrosymmet-\nric altermagnetic systems that can host an active DMI.\nIn the case of magnetic systems with frustrated mag-\nnetic interactions, the presence of an active DMI in al-\ntermagnetic crystal structure can further push the sys-\ntem towards helical magnetic structures, spin spiral and\ncomplex magnetism as observed for instance in transi-\ntion metal monopicnitides80,81. The component of the\nweak ferromagnetism is proportional to the spin-orbit,\ntherefore, to look for a large canting angle we need to\nsearch among systems with a strong spin-orbit and a\nsmall magnetic moment. In the weak ferromagnets, an\nextremely relevant role is given by the magnetocrystalline\nanisotropy (MCA) that decides a key ingredient as the\ndirection of the N´ eel vector. Additionally, the size of the\nMCA determines how easy it is for the spin to rotate,\ntherefore a low MCA is fundamental to favor the weak\nferromagnetism. Even if there are altermagnets where\nthe weak ferromagnetism is zero in the bulk as the com-\npounds belonging to the second class, the weak ferromag-\nnetism can be obtained by breaking crystal symmetries\non the surfaces. Further studies are necessary for the\nevolution of the DMI and weak ferromagnetism on the\nsurfaces of the altermagnets.56,82\nIn conclusion, we propose that weak ferromagnetism\ninduced by DMI can appear in centrosymmetric and non-\ncentrosymmetric altermagnetic compounds but not in\nAltermagnetic system\nCentrosymmetric Non -centrosymmetric\nMnSe\nCCaCrO3CrSbRRuO2Weak \nferromagnetism \npresent for all \ndirectionsWeak \nferromagnetism  \npresent for suitable \ndirections of the \nNéel vector Weak ferromagnetism \nor weak \nferrimagnetism \ndepending on the N éel \nvectorFIG. 10. Altermagnets classified into centrosymmetric and\nnoncentrosymmetric. We further classify the centrosymmet-\nric systems into two classes; the first contains compounds in\nwhich the weak FM is always present, and the second contains\ncompounds in which the weak FM is present for suitable direc-\ntions of the N´ eel vector. For the noncentrosymmetric MnSe,\nwe have weak ferromagnetism with the N´ eel vector in the\nxy plane and weak ferrimagnetism with the N´ eel vector out-\nof-plane. Likely, this classification would be improved and\nextended in coming years.\ncentrosymmetric and noncentrosymmetric conventional\nantiferromagnets. Weak FiM can appear in noncen-\ntrosymmetric altermagnets. This weak ferromagnetism\noriginates exclusively from the spin-orbit coupling in al-\ntermagnets and this picture is equivalent to the DMI pic-\nture. The weak ferromagnetism and the AHE are always\northogonal to the N´ eel vector for centrosymmetric sys-\ntems. The weak ferromagnetism depends on the filling,\ncan be tuned by an electric field and can change its sign\nby rotating the N´ eel vector in hexagonal systems. From\nour results, the weak ferromagnetism is absent for insula-\ntors and in lightly doped semiconductive systems would\nbe proportional to the carriers. We provide a recipe to\nestablish the presence of weak ferromagnetism and the\ncomponents of the Hall vector from a simple relativistic\ndensity of state and band structure calculations without\ncalculating the DMI vector. There is a correlation be-\ntween the rise of weak FM and weak FiM with the lifting\nof the electronic degeneracy at the Γ point. We have the\ncentrosymmetric altermagnetic and the noncentrosym-\nmetric altermagnets. We further divided the centrosym-\nmetric altermagnets into two classes, the first class always\nhosts weak ferromagnetism for every direction of the Hall\nvector while the other class hosts weak ferromagnetism\nfor suitable directions of the N´ eel vector. For noncen-\ntrosymmetric systems, we have weak ferromagnetism in\nthe x-y plane while parallel and antiparallel spin compo-\nnents for different bands can coexist at the same energy\nresulting in weak ferrimagnetism and a component of the\nHall vector along the N´ eel vector. Therefore, an exter-11\nnal electric field breaking the inversion symmetry could\ngenerate a weak ferromagnetic component and an AHE\nalong the N´ eel vector. We have predicted weak ferro-\nmagnetism in CrSb and weak ferrimagnetism in wurztite\nMnSe. The size of weak ferromagnetism, weak ferrimag-\nnetism and AHE strongly depends on the spin-orbit cou-\npling, therefore, further research directions should focus\non heavy elements. In summary, the family of the alter-\nmagnets is mainly composed of weak ferromagnets with\na minor number of compounds that are weak ferrimag-\nnets (MnSe with N´ eel vector along the z-axis) and zero\nmagnetization compounds (RuO 2with N´ eel vector along\nthe z-axis). It was shown that altermagnetism displays\nproperties that either belong to ferromagnets or conven-\ntional antiferromagnets1, the weak FM and weak FiM\ninduced by DMI are properties belonging exclusively to\naltermagnets not present neither in ferromagnets or in\nconventional antiferromagnets. If we want to classify al-\ntermagnets based on their macroscopic physical proper-\nties, we can define altermagnets as magnetic systems with\nspin-up and spin-down sublattices connected by symme-\ntry where the DMI allows the rise of weak ferromagnetism\nfor a suitable N´ eel vector orientation.ACKNOWLEDGMENTS\nC. A., R. M. S. and G. C. contributed equally to\nthis work. We acknowledge M. Cuoco, K. V´ yborn´ y,\nM. Gryglas-Borysiewicz and M. Grzybowski for useful\ndiscussions. The work is supported by the Founda-\ntion for Polish Science through the International Re-\nsearch Agendas program co-financed by the European\nUnion within the Smart Growth Operational Programme\n(Grant No. MAB/2017/1). A.F. was supported by\nthe Polish National Science Centre under project no.\n2020/37/B/ST5/02299. We acknowledge the access to\nthe computing facilities of the Interdisciplinary Center\nof Modeling at the University of Warsaw, Grant g91-\n1418, g91-1419 and g91-1426 for the availability of high-\nperformance computing resources and support. We ac-\nknowledge the CINECA award under the ISCRA initia-\ntive IsC99 ”SILENTS”, IsC105 ”SILENTSG” and IsB26\n”SHINY” grants for the availability of high-performance\ncomputing resources and support. 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Giri∗\nDepartment of Solid State Physics and Center for Advanced Ma terials,\nIndian Association for the Cultivation of Science, Jadavpu r, Kolkata 700 032, INDIA\nWe report the new results of exchange bias effect in Nd 1−xSrxCoO3forx= 0.20 and 0.40, where\ntheexchange bias phenomenon is involvedwith the ferrimagn etic (FI)state in a spontaneously phase\nseparated system. The zero-field cooled magnetization exhi bits the FI ( TFI) and ferromagnetic ( TC)\ntransitions at ∼23 and∼70 K, respectively for x= 0.20. The negative horizontal and positive\nvertical shifts of the magnetic hysteresis loops are observ ed when the system is cooled through TFI\nin presence of a positive static magnetic field. Training effe ct is observed for x= 0.20, which could\nbe interpreted by a spin configurational relaxation model. T he unidirectional shifts of the hysteresis\nloops as a function of temperature exhibit the absence of exc hange bias above TFIforx= 0.20. The\nanalysis of the cooling field dependence of exchange bias fiel d and magnetization indicates that the\nferromagnetic (FM) clusters consist of single magnetic dom ain with average size around ∼20 and∼\n40˚A forx= 0.20 and 0.40, respectively. The sizes of the FM clusters ar e close to the percolation\nthreshold for x= 0.20, which grow and coalesce to form the bigger size for x= 0.40 resulting in a\nweak exchange bias effect.\nPACS numbers: 75.70,Cn, 75.60.-d, 75.50.Gg\nINTRODUCTION\nThe exchange bias (EB) coupling in a heterogeneous\nsystem gives rise to the unidirectional anisotropy at the\ninterface when the samples are cooled down to a temper-\nature below a critical temperature in an external mag-\nnetic field. [1] It manifests itself by a shift of the hys-\nteresis loop and enhancement of coercivity, which at-\ntract considerable attention for the potential applica-\ntions in magnetic memories, spin-electronics, and devel-\nopment of permanent magnet. Numerous reports are\nfound on EB phenomenon at artificial interfaces, which\nhave been mainly focused to develop advanced materi-\nals for the applications and understanding of the com-\nplex EB phenomenon. [2, 3, 4, 5] Nevertheless, the\nsignature of EB phenomenon is rarely observed in com-\npounds with a unique crystal structure and having spon-\ntaneous interface. Long back the first example of EB ef-\nfect without artificial interface was reported for Cu(Mn)\nand Ag(Mn) alloys, which have been recognized as typ-\nical spin-glass (SG) and cluster-glass (CG) systems de-\npending on the dilution limit. [6] Recently, the signature\nof EB effect has been reported at the spontaneous in-\nterfaces for few mixed-valent manganites and cobaltites\nwith perovskite structure. [7, 8, 9, 15, 23, 24] The first\nreport was found in a charge ordered (CO) compound\nPr1/3Ca2/3MnO3, where ferromagnetic (FM) droplets\nwere spontaneously embedded in an antiferromagnetic\n(AFM) background creating the FM/AFM interface. [7]\nThe EB phenomenon has also been reported for another\n∗Electronic address: sspsg2@iacs.res.inCO manganite Y 0.2Ca0.8MnO3, where a strong cooling\nfield dependence of EB is observed due to a considerable\nchange of phase fraction between FM/AFM layers. [8]\nRecently, we observed the signature of EB phenomenon\nin the cluster-glass (CG) compounds LaMn 0.7Fe0.3O3\nand La 0.87Mn0.7Fe0.3O3, [9, 10] where short range FM\nclusters are embedded in a SG-like matrix creating spon-\ntaneous FM/SG interface. [9, 11, 12, 13] The average\nsizeoftheFMclustersisfoundtodecreasesystematically\nwith decreasingparticlesize, whichhasastronginfluence\non the EB phenomenon. The EB field increases signifi-\ncantlywith decreasingparticle size, which wasattributed\nto the increase of interface area by decreasing particle\nsize. [14] The exchange bias-like phenomenon was also\nreported for another CG compound, La 0.80Ba0.20CoO3,\nwhich wassuggested due to the freezing effect ofthe local\nanisotropy. [15]\nThe mixed-valent cobaltites with perovskite structure\nexperience the delicate interplay among charge, spin\nstate, transport, magnetic, and structuraldegreesoffree-\ndom exhibiting the complex phase separation scenario.\nAn interesting phase diagram has been proposed for the\nhole doped compound Nd 1−xSrxCoO3depending on the\ndegree of hole doping. [16] For low doping range (0\n< x < 0.18) the SG or CG state has been proposed\nwith semiconducting temperature dependence of resis-\ntivity. With further increase in hole doping the short\nrange FM clusters begin to coalesce above a percolation\nthreshold ( x >0.18) to attain the magnetic long range\nordering and start to show metallic conductivity in the\nordered state. Metallic conductivity in both the param-\nagnetic and ordered states is observed for x≥0.28. The\ncoexistence of ferrimagnetic (FI) and FM ordering is re-\nported for 0.20 ≤x≤0.60. Neutron powder diffraction2\nstudies on Nd 0.67Sr0.33CoO3confirm that the FM ( Tc)\nand FI (TFI) ordering temperatures are ∼200 and ∼40\nK, respectively, where ferrimagnetism was interpreted in\nterms of an induced antiparallel ordering of the Nd spins\nin close proximity of Co sublattice. [17] Several reports\non electrical, magnetic, and thermodynamic studies of\nNd1−xSrxCoO3also suggest the coexistence of FI and\nFM ordering for x= 0.33 and 0.50. [18, 19, 20]59Co\nNMR studies on Nd 1−xSrxCoO3(0≤x≤0.50) confirm\ndifferent spin states of Co3+and Co4+ions. [21] The\nparent compound NdCoO 3shows the low spin (LS) state\nofCo3+ion in the paramagneticstate. As a result ofhole\ndoping (0.10 ≤x≤0.20) an intermediate spin (IS) state\nof Co3+and Co4+appears in addition to the LS state of\nCo3+and the LS state of Co3+no more exists with the\nfurther increase in hole doping (0.30 ≤x≤0.50). The\nreportedresultsindicatethatthemagneticandelectronic\nphase separation scenario of Nd 1−xSrxCoO3is very sim-\nilar to that in La 1−xSrxCoO3, where the dissimilarity is\nthat here Nd ion carries moment unlike La ion. [22]\nRecently, the signature of EB phenomenon ascribed\nto the intrinsic inhomogeneous phase separation was re-\nportedintheCGcompoundsLa 1−xSrxCoO3(0.12≤x≤\n0.30), where EB was suggested due to the cluster-glass\nstate consisting of FM and SG phases. [23, 24] In order\nto observeEB phenomenon the system must involvewith\ntwo exchange coupled phases, the reversible and rigid\nphases, where magnetization of the first one can be re-\nversedandthesecondonecannotbe. TheEBisobserved\nfor La 1−xSrxCoO3at the reversible FM and rigid SG in-\nterface. ThephaseseparatedcompoundsNd 1−xSrxCoO3\nalso exhibit the necessary ingredient for the EB effect,\nwhere spontaneous phase separation between FM and\nFI states exists giving rise to the FM/FI interface for\nx≥0.18. Here, the EB phenomenon is investigated in\nNd1−xSrxCoO3withx= 0.20 and 0.40, where the first\none is close to the percolation threshold with semicon-\nducting behavior and the second one exhibits the metal-\nlic conductivity. We observe the EB effect for both the\nsamples, where the effect is strong for x= 0.20 and weak\nforx= 0.40. In order to explain different EB effects for\nboth the compounds the nanoscale phase separation sce-\nnario has been proposed in Nd 1−xSrxCoO3forx≥0.20.\nSo far EB phenomenon has been reported for manganites\nand cobaltites with perovskite structures at spontaneous\nFM/AFM and FM/SG interface. Here, we present a new\nexample of EB effect in the spontaneously phase sepa-\nrated compounds Nd 1−xSrxCoO3forx≥0.20, where\nEB effect is involved with ferrimagnetic states.\nEXPERIMENTAL\nThe polycrystalline samples of Nd 1−xSrxCoO3with\nx= 0.20 and 0.40 were prepared by the chemical cit-\nrate route, which is described in our earlier report. [11]/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48\n/s120 /s32/s61/s32/s48/s46/s52/s48/s120 /s32/s61/s32/s48/s46/s50/s48\n/s50 /s32/s40/s100/s101/s103/s46/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s32/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\nFIG. 1: X-ray powder diffraction patterns at room tempera-\nture for Nd 1−xSrxCoO3withx= 0.20 and 0.40.\nThe stoichiometric proportion of Nd 2O3, SrCO 3, and Co\npowders were dissolved in an aqueous solution of nitric\nacid, mixed thoroughly, and then citric acid was added\nto achieve a homogeneous mixture of the metal citrates.\nThe metal citrates were dried and decomposed at 873 K\nfor 6 h. Finally, the powdered samples were pressed into\npellets and heated at 1273 K for 6 h followed by slow\ncooling at a rate of 0.7 K/min. The single orthorhombic\n(Pbnm) structure at room temperature is confirmed for\nboth the cases, where x-ray powder diffraction (Seifert\nXRD 3000P) was recorded using CuK αradiation. We do\nnot observe any impurity phase in the x-ray diffraction\npatterns, which are shownin Fig. 1. The values oflattice\nparameters are 5.35 ( a), 5.36 (b), and 7.56 ( c)˚A forx=\n0.20, while for x= 0.40 the parameters are 5.40 ( a), 5.35\n(b), and 7.61 ( c)˚A. The lattice parameters are consis-\ntent with the earlier report. [21] The average size of the\nparticles is found around ∼150.0 nm for both the cases,\nwhich was observed by a Transmission Electron Micro-\nscope (TEM), model ZEOL JEM-2010. The dc magneti-\nzation was measured using a commercial superconduct-\ning quantum interference device (SQUID) magnetometer\n(MPMS, XL). The sample was cooled down to the lowest\ntemperature in zero magnetic field and the magnetiza-\ntion was measured in the warming cycle by applying an\nexternal magnetic field for zero-field cooled (ZFC) mag-\nnetization. On the other hand, the sample was cooled in\nnon-zero field for the measurement of field-cooled (FC)\nmagnetization and the measurement was performed in\nthe warming cycle with field kept switched on.\nEXPERIMENTAL RESULTS\nTemperature dependence of field-cooled effect of mag-\nnetization measured at 100 Oe is shown in Fig. 2 for\nNd0.80Sr0.20CoO3. The inset of the figure exhibits ZFC\nmagnetization highlighting the ferrimagnetic ( TFI) and3\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s50/s51/s52/s53\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50\n/s32/s32\n/s32/s84/s32/s40/s75/s41/s32/s77/s40/s101/s109/s117/s47/s103/s41/s84\n/s67\n/s32/s77 /s40/s101/s109/s117/s47/s103/s41/s32/s77 /s40/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41/s32/s72/s32/s61/s32/s49/s48/s48/s32/s79/s101/s32\n/s32/s32/s72/s32/s61/s32/s53/s48/s32/s107/s79/s101\n/s84\n/s67\n/s32/s32\n/s84\n/s70/s73\nFIG. 2: Temperature dependence of magnetization measured\nat 100 Oe under zero field cooled (ZFC) and field cooled (FC)\nconditions for x= 0.20. The inset shows the ZFC magnetiza-\ntion highlighting ferrimagnetic ( TFI) and ferromagnetic ( Tc)\ntransitions for the measurements at 100 Oe and 50 kOe.\nferromagnetic ( Tc) transitions at ∼23 and∼70 K, re-\nspectively, where TFIis indicated by a shoulder around\n∼23 K for the low-field measurement at 100 Oe. A\nsharp increase in the ZFC magnetization is observed be-\nlowTFIwhen the measurement was performed at a high\nfield with 50 kOe. Tcis defined around ∼70 K, where\na sharp increase in FC magnetization is observed for the\nmeasurement at 100 Oe. The values of TFIandTcin\nthe present observation are reproduced exactly as com-\npared to the values reported by Stauffer et al. [16] The\nreported experimental results including the neutron and\n59Co NMR results clearly indicate the coexistence of FM\nand FI phases at low temperature in Nd 1−xSrxCoO3for\nx≥0.20. [16, 17, 18, 19, 20, 21] Thus, the magnetization\nloops at 5 K should have two components corresponding\nto the FM and FI phases. Recently, Niebieskikwiat and\nSalamon reported the EB effect in Pr 1/3Ca2/3MnO3con-\nsistingofFMandAFMphasesatlowtemperature, where\nthe AFM background exhibiting linear M−Hresponse\nwas subtracted from the overall magnetization loop at 5\nK to get the evident features of the horizontal and verti-\ncal shifts. [7] In the present investigation it is difficult to\nsubtract the FI background. Nevertheless, the horizon-\ntal and vertical shifts of the hysteresis loop are clearly\nobserved exhibiting the typical feature of EB effect when\nthe sample was cooled in a non-zero field. For non-zero\npositive cooling field a negative shift in the field axis and\na positive shift in the magnetization axis are observed\nat 5 K with respect to the hysteresis loop for Hcool=\n0, which is shown in the top panel of Fig. 3. We de-/s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48/s48/s49/s50/s51/s52\n/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s45/s51/s48 /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48 /s51/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48/s53/s49/s48/s49/s53/s50/s48/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s54/s45/s51/s48/s51/s54\n/s72 \n/s69/s32/s40/s107/s79 /s101/s41\n/s72\n/s109/s97/s120/s32/s40/s107 /s79/s101/s41\n/s77\n/s69/s32/s40/s107/s79 /s101/s41/s32/s32\n/s84/s32/s61/s32/s53/s32/s75\n/s32/s72\n/s109/s97/s120/s32/s61/s32/s51/s48/s32/s107 /s79/s101\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s51/s53/s32/s107 /s79/s101\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s52/s48/s107 /s79/s101\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s52/s53/s32/s107 /s79/s101\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s53/s48/s32/s107 /s79/s101\n/s32/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s72\n/s99/s111/s111/s108/s32/s61/s32/s53/s48/s32/s107/s79/s101/s32/s72\n/s99/s111/s111/s108/s32/s61/s32/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s50/s48/s32/s107/s79/s101\n/s32/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41/s84/s32/s61/s32/s53/s32/s75\nFIG. 3: (Color online) Inthetop panel central part of thehys -\nteresis loops at 5 K measured after cooling the sample in zero\nfield and field at 20 kOe, where loop measured after field cool-\ning are indicated by the broken curve. In the bottom panel\ncentral part of the minor loops at 5 K for the measurements in\nbetween ±30,±35,±40,±45, and±50 kOe after cooling\nthe sample with 50 kOe. The inset exhibits the variation of\nexchange bias field ( HE) and magnetization ( ME) withHmax\nforx= 0.20.\nfine the coercivity, HC=|(Hleft−Hright)/2| ≈13 kOe\nand EB field, - HE= (Hleft+Hright)/2≈- 500 Oe for\nthe measurement of magnetic hysteresis in between ±50\nkOe. Here, HrightandHleftare the positive and nega-\ntive values of field at magnetization, M= 0, respectively.\nThe remanence asymmetry is defined as EB magnetiza-\ntion (ME), which is estimated from the vertical shifts at\nthe saturation. [25] The value of ME/MSis∼0.53×\n10−2, whereMSis the saturation of magnetization. The\nvalues of the asymmetry parameters associated with the\nFM and FI states are comparable to those involved with\nthe FM and SG statesfor the cobaltiteLa 0.88Sr0.12CoO3,\nwhere the maximum effect of EB was reported in the se-\nries. [23, 24] The values of HEandME/MSwere 500 Oe4\nTable 1: The maximum values of exchange bias field ( HE)\nand relative vertical shift ( ME/MS) at different types of\ninterfaces for phase separated cobaltites measured in\nbetween different ±Hmax.\nSystem HmaxHEME/MSInterface\nkOe kOe (10−2)\nLa0.80Ba0.20CoO3[15] 3 0.54 - FM/SG\nLa0.88Sr0.12CoO3[26] 20 2.95 10.40 FM/SG\nLa0.88Sr0.12CoO3[24] 50 0.50 1.10 FM/SG\nNd0.80Sr0.20CoO3∗30 3.23 4.15 FM/SG\n50 0.50 0.53 FM/FI\nNd0.60Sr0.40CoO3∗50 0.08 0.55 FM/FI\n∗Current investigation\nand 1.10 ×10−2, respectively when hysteresis was mea-\nsured in between ±50 kOe for La 0.88Sr0.12CoO3. [24]\nThe reported values of the shifts for the phase separated\ncobaltites are compared in Table 1 with the present ob-\nservationsfor x=0.20and0.40fordifferent Hmax, where\nHmaxis the maximum field used for the measurement of\nhysteresis loops.\nThe characteristics of EB phenomenon strongly de-\npend on the delicate interplay among Zeeman energy\n(EZ) associated with the FM clusters, anisotropy energy\n(EA) associated with the FI clusters, and exchange en-\nergy (Eex) at the FM/FI interface. The first term can\nbe experimentally controlled by tuning HcoolandHmax.\nThemagnetichysteresisloopsweremeasuredfor x=0.20\nat 5 K in between ±Hmax=±30,±35,±40,±45,\nand±50 kOe when the sample was cooled down to 5 K\nfrom 250 K in presence of 50 kOe field. The central parts\nof the hysteresis loops are shown in the bottom panel of\nFig. 3. The values of HEandMEas a function of Hmax\nare shown in the inset of the figure. The values of HE\nandMEare decreased with increasing Hmax. The plots\nfurther indicate that the values show a tendency of stabi-\nlization for Hmax≥45 kOe. We note that the hysteresis\nloops measured in between ±45 and±50 kOe almost\nmerge together. The Hmaxdependence of HEandME\nhas been reported by Tang et al. for La 0.82Sr0.18CoO3,\nwhereHEandMEvanish even at small value of Hmax\n= 30 kOe. [23] The results are analogous to the minor\nloops effect, which is explicitly described by Geshev. [27]\nHowever, the signature of EB effect in La 0.82Sr0.18CoO3\nwas confirmed by the training effect. [23] The Hmaxde-\npendence of HEandMEhas also been investigated by\nSalazar-Alvarez et al. for MnO/Mn 3O4core/shell struc-\nture, where the plots show the non-zero asymptotic value\nof the horizontal shift while the vertical shifts virtually\nvanishat Hmax>70kOe. [28]In the presentobservation\nthe plots show that HEandMEstabilize around Hmax\n= 50 kOe, where the values of HEandMEare consid-\nerably large. The results indicate that the plots are not\nthe typical minor loops effect described by Geshev [27],\nrather it exhibits significant EB effect./s45/s49/s51/s46/s54 /s45/s49/s51/s46/s50 /s49/s50/s46/s48 /s49/s50/s46/s52/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s48/s46/s49/s55/s48/s46/s49/s56/s48/s46/s49/s57/s48/s46/s50/s48/s48/s46/s50/s49/s48/s46/s50/s50\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s52/s48/s48/s52/s50/s48/s52/s52/s48/s52/s54/s48/s52/s56/s48/s53/s48/s48\n/s32/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41\n/s77\n/s69/s32/s40/s101/s109/s117/s47/s103/s41/s32\n/s32/s72\n/s69/s32/s40/s79/s101/s41\nFIG. 4: (Color online) The decrease of exchange bias field\n(HE) and magnetization ( ME) with the consecutive number\n(λ) of cycling of the hysteresis loop for x= 0.20 exhibiting\ntraining effect. The central part of the 1st, 2nd, 8th, and\n16th loops are shown in the inset, where arrow indicates the\nincreasing direction of λ.\nTraining effect is one of the important features of the\nexchange bias system, which describes the decrease of\nHC,HE, andMEwhen the system is successively field\ncycled at a particular temperature after the field cool-\ning. [4, 7, 23] The training effect is investigated for x\n= 0.20, where the sample was cooled down to 5 K from\n250 K in presence of 20 kOe field and then the hysteresis\nloops were measured successively in between ±50 kOe\nup to 16 times. We note that the training effect is ev-\nident in the system. The central parts of the 1st, 2nd,\n8th, and 16th loops are shown in the inset of Fig. 4. The\nsystematic decrease of HEandMEwith the consecutive\ncycling number ( λ) are shown in Fig. 4 by the open sym-\nbols. The values of HEandMEare decreasedup to ∼10\n% and∼11 %, respectively for λ= 2, which are compa-\nrable to the reported results for the spontaneously phase\nseparated systems. For Pr 1/3Ca2/3MnO3[7] bearing the\nFM/AFM interface the value of MEwas decreased up to\n∼20 % for the second cycle, whereas the values of ME\nandHEwere decreased up to ∼10 % and ∼19 %, re-\nspectively for La 0.82Sr0.18CoO3having FM/SG interface.\n[23]\nThe decrease of HE(ME) is fitted satisfactorily with\nthe following empirical relation\nHE(λ)−H∞\nE∝1√\nλ, (1)\nwhereH∞\nEis the value for λ→ ∞. The solid lines in Fig.\n4 exhibit the best fit of HEandMEvariation with λfor\nλ≥2. The values of the fitted parameters are H∞\nE≈\n377 Oe and M∞\nE≈0.157 emu/g. The above empirical\nrelation does not fit the sharp decrease between first and5\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s52/s56/s49/s50/s48/s50/s52/s54/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s40/s99/s41\n/s32/s32/s72\n/s67/s32/s40/s107/s79/s101/s41\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s32/s32/s77\n/s69/s47/s77\n/s83/s32/s40/s49/s48/s45/s51\n/s41\n/s32/s32/s72\n/s69/s32/s40/s79/s101/s41/s40/s97/s41\nFIG. 5: Temperature dependence of (a) exchange bias field\n(HE), (b) relative vertical shift ( ME/MS), and (c) coercivity\n(HC) withHcool= 20 kOe for x= 0.20.\nsecond loops, which is in accordance with the reported\nresults. [4, 7, 23] Recently, Binek proposed a recursive\nformula in the framework of spin configurational relax-\nation to understand the training effect for a FM/AFM\nheterostructure, which describes the ( λ+1)th loop shift\nwith respect to the λth one as [29]\nHE(λ+1)−HE(λ) =−γ[HE(λ)−H∞′\nE]3,(2)\nwhereγis a sample dependent constant. Using γ= 2.28\n×10−5Oe−2andH∞′\nE= 369.4 Oe the whole set of data\nis reproduced for HE, while the values of γandM∞′\nEare\n79.5 (emu/g)−2and 0.153 emu/g, respectively for ME.\nThe calculated values are shown by the filled symbols in\nFig. 4, which matches satisfactorily with the experimen-\ntal data (open symbols). Thus, the spin configurational\nrelaxation model can describe our experimental results\nsatisfactorily, where successive reversal of the FM spins\ntriggers the configurational relaxation of the interfacial\nFI spins toward equilibrium giving rise to the training\neffect.\nThe magnetic hysteresis loops were measured in\nbetween ±50 kOe at different temperatures for\nNd0.80Sr0.20CoO3with the sample been cooled down to\nthe desired temperatures from 250 K with Hcool= 20\nkOe. The values of HC,HE, andME/MSas a func-\ntion of temperature are shown in Fig. 5. HCdecreases\nslowlyandvanishesatthe onsetofthe Curietemperature\naround∼70 K. On the other hand, HEandME/MSdecrease sharply with increasing temperature and dis-\nappear for T≥20 K, where a shoulder in the ZFC\nmagnetization is observed indicating the signature of fer-\nrimagnetic ordering. The temperature dependence of\nthe asymmetry parameters are typical for the exchange\nbiased systems viz., charge ordered Pr 1/3Ca2/3MnO3,\n[7] cluster-glass LaMn 0.7Fe0.3O3, [9] and cluster-glass\ncobaltites La 1−xSrxCoO3, [23, 24] where EB effect van-\nishes above AFM ( TN) or spin freezing ( Tf) transition\ntemperature. In the cases of EB systems the rigid AFM\norSG spins apply a couplingforceon the FM spins at the\ninterfaceand alayerofpinned orfrozenFM spinsarecre-\natedontheoutersurfaceoftheFMclusterswhenthesys-\ntem is cooled in non-zero field. The pinned or frozen FM\nspins give rise to the unidirectional shift of the hysteresis\nloops and reveal the EB effect. For Nd 1−xSrxCoO3with\nx≥0.20 the coexistence of FM and FI phases has been\nsuggested by Stauffer et al., [16] which is also indicated\nhere by the ZFC magnetization for x= 0.20 (inset of\nFig. 2). Therefore, the EB effect below TFIis suggested\ndue to the pinning effect. The rigid FI spins operate the\npinning force on the reversible FM spins at the FM/FI\ninterface and the pinned FM spins lead to the EB effect.\nThe signature of EB effects involving with the ferrimag-\nnetic phase are reported at the different combinations\nof artificial interfaces viz., FM/FI, FI/AFM, and FI/FI\ninterface for different bilayer heterogeneous structures.\n[4, 28, 30, 31, 32, 33] Here, we observe the new results of\nEB effect at the intrinsic FM/FI interface, where the EB\neffect is ascribed to the spontaneous separation between\nFM and FI phases.\nThe cooling field dependence of EB effect was investi-\ngated for x= 0.20, where the sample was cooled down to\n5 K from 250 K with different cooling fields and the hys-\nteresis loops were measured in between ±50 kOe. The\nHcooldependence of HEis shown in the top panel of Fig.\n6. The value of HEincreases sharply with Hcoolup to 10\nkOe and then shows a slight decreasing trend above 20\nkOe. As seen in the inset of the top panel of the figure\ntheHcooldependence of HCfollows similar behavior like\ntheHcooldependence of HE, where the sharp increase of\nHCis associated with the increase of HE. The plots of\nHEandHCwithHcoolexhibit that it follows nearly lin-\near dependence above 20 kOe, which is shown in Fig. 7.\nRecently, Qian et al. have shown the linear Hcooldepen-\ndence of HEabove 10 kOe, where strong correlation be-\ntweenHEandHCwas reported. [8] The authors further\nsuggested that the decrease of HEwith increasing Hcool\nisassociatedwith the increaseofthe sizeoftheFM layers\nforY0.2Ca0.8MnO3. Consideringthe magnetizationat50\nkOe close to the saturation value at 5 K, the increase of\nM50indicates the increase of the average size of the FM\nclusters, where M50is determined from the averagevalue\nof the positive and negative values of the magnetization\nat 50 kOe. The linear plot of HEagainstM−1\n50is shown\ninthe insetofFig. 7, whichindicatesthat HEisinversely6\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s48/s49/s50/s51/s52/s53/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s50/s51/s52/s53/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s49/s50/s46/s54/s49/s50/s46/s55/s49/s50/s46/s56/s49/s50/s46/s57/s49/s51/s46/s48\n/s32/s32/s77\n/s69/s47/s77\n/s83/s40/s49/s48/s45/s51\n/s41\n/s72\n/s69/s40/s79/s101/s41\n/s32/s32/s72\n/s69/s32/s40/s79/s101/s41\n/s32/s32/s77\n/s69/s47/s77\n/s83/s32/s40/s49/s48/s45/s51\n/s32/s41\n/s72\n/s99/s111/s111/s108/s32/s40/s107/s79/s101/s41/s72\n/s67/s32/s40/s107/s79/s101/s41\n/s72\n/s99/s111/s111/s108/s32/s40/s107/s79/s101/s41\n/s32/s32\n/s32/s32\nFIG. 6: (Color online) Cooling field ( Hcool) dependence of\n(top panel) exchange bias field ( HE) and (bottom panel)\nME/MSat 5 K for x= 0.20, where the hysteresis loops\nwere measured between ±50 kOe. MEandMSare the\nexchange bias magnetization and saturation magnetization .\nTheHcooldependence of coercivity ( HC) and the plot of\nME/MSagainstHEare shown in the inset of top and bottom\npanels, respectively. The broken curve in the bottom panel i s\nthe fitted curve using simplified exchange interaction model .\nThe solid straight line in the inset of the bottom panel shows\nthe linear fit.\nproportional to the size of the FM clusters. Note that a\nvery small decrease of HEaround∼0.6 % is observed\nat 5 K for the increase of Hcoolfrom 20 kOe to 50 kOe.\nIn the high Hcoolrange the feature of HEis reminiscent\nto that observed for FM/AFM bilayers, [34] AFM-core\nand FI-shell structure, [28] and unlike to that observed\nfor FM/SG systems. [9, 14, 23, 24, 35] For the exchange\nbiased systems involved with a SG state the values of HE\nare decreased considerably with increasing Hcoolabove a\ncertain value of Hcool, where large Hcooltypically affects\nthe frozen spins in the SG phase and frozen FM spins by\npolarizing the glassy magnetic spins toward the direction\nofHcool. In fact, HEdecreases up to ∼40 % for the\nincrease of Hcoolfrom 20 to 50 kOe in La 0.88Sr0.12CoO3,\n[24]∼47 % for the increase of Hcoolfrom 6 to 12 kOe in\nLaMn0.7Fe0.3O3, [9] and ∼25 % for the increase of Hcool\nfrom 2 to 4 kOe in La 0.87Mn0.7Fe0.3O3, [10] where the\nsystems are involved with the SG states exhibiting EB\neffect. The frozen FM spins and pinned FM spins may/s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48/s49/s50/s46/s57/s53/s49/s50/s46/s57/s54/s49/s50/s46/s57/s55/s49/s50/s46/s57/s56/s49/s50/s46/s57/s57/s49/s51/s46/s48/s48\n/s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48/s52/s57/s52/s52/s57/s53/s52/s57/s54/s52/s57/s55/s52/s57/s56\n/s52/s49/s46/s54/s50 /s52/s49/s46/s54/s52 /s52/s49/s46/s54/s54 /s52/s49/s46/s54/s56/s52/s57/s52/s52/s57/s53/s52/s57/s54/s52/s57/s55\n/s72\n/s67/s40/s107/s79/s101/s41/s32\n/s72\n/s99/s111/s111/s108/s32/s40/s107/s79/s101/s41/s72\n/s69/s32/s40/s79/s101/s41\n/s32/s32/s72\n/s69/s40/s79/s101/s41\n/s77\n/s53/s48/s45/s49\n/s32/s40/s49/s48/s45/s51\n/s32/s101/s109/s117/s47/s103/s41/s45/s49\nFIG. 7: (Color online) Plots of exchange bias field ( HE) and\ncoercivity ( HC) against cooling field ( Hcool) at 5 K for x=\n0.20. Inset shows the plot of HEas a function of inverse of\nthe average magnetization at 50 kOe ( M−1\n50). Solid straight\nlines indicate the linear fits.\nhave different features of cooling field dependence, where\nthe large values of Hcoolcan not influence the pinned\nFM spins significantly, which is indicated by the minor\nchange of HEfor AFM or FI state. The large reduc-\ntion ofHEaround∼37 % is exceptionally observed in\nY0.2Ca0.8MnO3for the increase of Hcoolfrom 10 kOe to\n60 kOe, which is associated with the FM/AFM interface.\nIn the abovecasethe largedecreaseof HEwasattributed\nto the increase of the phase fraction of the FM layer at\nthe cost of AFM layer, which was pointed out by Qian\net al. [8] Herein, the minor reduction of HEaround∼\n0.6 % may indicate that small increase of the size of FM\nclusters is ascribed to the reduction of pinned FM layers\nat the FM/FI interface, which does not involve with the\nconversion of phase fraction.\nRecently, a simplified exchange interaction model has\nbeen proposed by Niebieskikwiat and Salamon consider-\ning the FM clusters to be embedded in the AFM host for\ncharge ordered compound Pr 1/3Ca2/3MnO3. [7] The FM\nclusters were assumed to have a single domain structure,\nwhich exhibit magnetization reversal like FM nanopar-\nticles consisting of single magnetic domain. The model\ngives the simplified relation between HEandME/MS\nasHE∝ME/MSforµHE<kBT, where MSandµ\nare the saturation magnetization and average moment of\nthe FM clusters, respectively. We observe nearly linear\nbehavior between HEandME/MSin the inset of the\nbottom panel of Fig. 6, which has also been verified for\nPr1/3Ca2/3MnO3, [7] CaMnO 3−δ, [36] LaMn 0.7Fe0.3O3,\n[9, 14] and La 0.87Mn0.7Fe0.3O3[10] exhibiting the EB ef-\nfect. Here, the value of µHE/kBT≈0.87, where the\nvalue of µis obtained from the fit of ME/MSagainst7\nHcoolplot, which is described below. Note that the value\nofµHE/kBTwas∼0.9forPr 1/3Ca2/3MnO3. [7] Thelin-\near dependence of the ME/MSagainstHEplot indicates\nthat FM clusters may have a single domain structure for\nNd0.80Sr0.20CoO3. The model further defines the Hcool\ndependence of ME/MSas\nHE∝ME/MS∝ji/bracketleftbiggjiµ0\n(gµB)2L(µHcool\nkBTf)+Hcool/bracketrightbigg\n,(3)\nwhich was verified successfully for spontaneously phase\nseparated Pr 1/3Ca2/3MnO3, [7] LaMn 0.7Fe0.3O3, [9, 14]\nand La 0.87Mn0.7Fe0.3O3. [10] The first term in the ex-\npression dominates for small Hcool, whereas the second\nterm dominatesforlarge Hcool, which varieslinearlywith\nHcool. In the above relation Jiandµare adjustable pa-\nrameters, where Jiis the interface exchange constant.\nTfin the above expression was defined as the freezing\ntemperature below which ME/MSwas found to increase\nsteeply for Pr 1/3Ca2/3MnO3. Here, we assume the value\nofTf∼15 K below which ME/MSincreases sharply.\nThe broken curve in the bottom panel of Fig. 6 exhibits\nasatisfactoryfit oftheexperimentaldatausingthe above\nexpression. The number density of FM clusters ( n) is es-\ntimated from the saturation magnetization, MS≈nµ,\nwhere the value of MSat 5 K is obtained from the ex-\ntrapolation of the magnetization to 1/ H→0. The value\nofnis≈15.5×10−5˚A−3which further gives the rough\nestimate of the size of the FM clusters around ∼20˚A.\nThe value of the size of FM cluster is consistent with\nthose around ∼10˚A for Pr 1/3Ca2/3MnO3[7] and∼10 -\n30˚A for LaMn 0.7Fe0.3O3depending on the particle size.\n[14]\nWe also investigate the EB effect for x= 0.40, where\nthemaximumvaluesof HEandME/MSareshowninTa-\nble 1. The value of ME/MSis nearly the same, whereas\nHEis much smaller for x= 0.40 than that of the value\nforx= 0.20. The Hcooldependence of EB effect was also\nmeasured for x= 0.40. The plots of HEandME/MSas\na function of Hcoolat 5 K are shown in Fig. 8. HEis\nfound to increase sharply up to ∼20 kOe and shows a\ndecreasingtrend with the furtherincreaseof Hcool, where\nME/MSfollows almost similar behavior. The linear de-\npendence of ME/MSagainstHEis observed for x= 0.40\nanalogous to x= 0.20, which indicates that the FM clus-\nters ofx= 0.40 also have a single domain structure. We\nobservethe linear Hcooldependence of HEandHCabove\n20kOe, whichisshownintheinsetofFig. 8. Here, avery\nsmall decrease of HE(∼0.5 %) is also observed for x=\n0.40likex=0.20. ThemodelproposedbyNiebieskikwiat\nand Salamon [7] was used to fit the Hcooldependence of\nME/MS, where the satisfactory fit is shown in Fig. 8 by\nthe broken curve. The number density of FM clusters is\nthus obtained from the fit as n≈32.5×10−5˚A−3, which\ngives the rough estimate of the size of the FM clusters\naround∼40˚A. The analysis indicates that the average/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/s55/s48/s56/s48\n/s48/s49/s50/s51/s52/s53\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s56/s46/s56/s51/s56/s46/s56/s52/s56/s46/s56/s53/s56/s46/s56/s54/s56/s46/s56/s55/s56/s46/s56/s56\n/s55/s50/s55/s52/s55/s54/s55/s56/s56/s48\n/s77\n/s69/s47/s77\n/s83/s32/s40/s49/s48/s45/s51\n/s41/s32/s72\n/s69/s32/s40/s79/s101/s41\n/s72\n/s99/s111/s111/s108/s32/s40/s107/s79/s101/s41\n/s32/s32/s72\n/s67/s32/s40/s107/s79/s101/s41\n/s72\n/s99/s111/s111/s108/s32/s40/s107/s79/s101/s41/s32\n/s32/s72\n/s69/s32/s40/s107/s79/s101/s41/s32\nFIG. 8: (Color online) Plots of exchange bias field ( HE) and\nrelative vertical shift ( ME/MS) against cooling field ( Hcool)\nat 5 K for x= 0.40. The broken curve is the fit of the Hcool\ndependence of ME/MSplot. Inset shows the plot of HEand\nHCas a function of Hcool. Solid straight lines indicate the\nlinear fits.\n/s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s49/s50/s51\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s53/s54/s55/s56/s57/s32/s120 /s32/s61/s32/s48/s46/s50\n/s32/s120 /s32/s61/s32/s48/s46/s52\n/s32/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s84 /s32/s61/s32/s53/s32/s75\n/s72\n/s99/s111/s111/s108/s61/s32/s50/s48/s32/s107/s79/s101\n/s32/s32\n/s32/s84/s32/s40/s75/s41/s32 /s32/s40/s49/s48/s45/s50\n/s32 /s45/s99/s109/s41/s120 /s32/s61/s32/s48/s46/s50\n/s120 /s61/s32/s48/s46/s52\n/s32/s84/s32/s40/s75/s41/s32 /s32/s40/s49/s48/s45/s52\n/s32 /s45/s99/s109/s41\n/s32/s32\n/s32/s32\nFIG. 9: Magnetic hysteresis loop at 5 K after field cooling\nwith 20 kOe for x= 0.20 (continuous curve) and x= 0.40\n(broken curve) for Nd 1−xSrxCoO3. The upper and lower in-\nsets exhibits the temperature dependence of resistivity ( ρ) for\nx= 0.20 and 0.40, respectively.\nsize of the FM clusters is increased considerably for x=\n0.40 than x= 0.20.\nTemperature variation of resistivity ( ρ) for the sam-\nples with x= 0.20 and 0.40 are shown in the upper and\nlower insets of Fig. 9, respectively. The semiconducting\ntemperature dependence of ρis observed for x= 0.20,\nwhile for x= 0.40 nearly linear temperature dependence\nofρis observed in the paramagnetic state and a sharp8\ndecrease of ρis observed below TCat 200 K. Tempera-\nture dependence of ρfor both the samples are similar to\nthe reported results [16] except for the magnitude of ρ.\nThe slightly larger values of the resistivity in the present\ninvestigations might be attributed to the particle size ef-\nfect, where the particle size in the present observation\n(∼150 nm) is less than that of the results reported by\nStauffer et al. [16] For x= 0.20 the system is close to\nthe percolation limit, where the hole rich FM clusters are\nof small size, which grow and coalesce with increasing x.\nTherefore,theaveragesizeoftheFMclustersfor x=0.40\nis larger than that of the sample with x= 0.20, which\nis reflected in the temperature dependence of resistivity\nand the analysis of the Hcooldependence of ME/MS.\nThe size of the FM clusters is also crucial for the EB\neffect. The hysteresis loops measured at 5 K under iden-\ntical condition with Hcool= 20 kOe are shown in Fig. 9\nforx= 0.20and 0.40. The values of HEandME/MSare\n82 Oe and 0.55 ×10−2forx= 0.40, while for x= 0.20\nthe values are 500 Oe and 0.53 ×10−2, respectively (see\nTable 1). In addition, the magnetization at 50 kOe and\nHCare 28.55 emu/g and 8.7 kOe for x= 0.40, and 23.92\nemu/g and 13.0 kOe for x= 0.20, respectively. The val-\nues of magnetization at 50 kOe indicate that the size and\ndensityoftheFMclustersareincreasedfor x=0.40com-\npared to x= 0.20, which is reflected in HE, whereHEis\nmuch smaller for x= 0.40. Similar features of EB effect\nwas observed for the particle size dependence of EB phe-\nnomenon in the cluster-glasscompound LaMn 0.7Fe0.3O3,\nwhere FM clusters are embedded in the SG matrix. [14]\nThe sizes of the FM clusters were found to increase with\nincreasing particle size, which resulted in a decrease in\nthe EB effect. Negligible EB effect was observed for the\nparticles with average size ∼300 nm. The increase of\nthe size of the FM clusters decreases the effective in-\nterface area, which might be the origin of weakening of\nthe effective exchange coupling at the FM/FI interface in\nthe present investigations. The explanation is consistent\nwith the model proposed by Meiklejohn, which predicts\nthe relation, HE≈Jex/(MFM×tFM), where Jexis the\nexchangeconstantacrosstheFM/AFMinterfaceperunit\narea. [37] MFMandtFMare the magnetization and the\nthickness of the FM layer, respectively. The increase of\nMFMandtFMin the denominator of the above expres-\nsion decreases the EB field. In the present investigation\nthe average size of the FM clusters, analogous to tFM,\nis increased for x= 0.40 compared to x= 0.20, which\nresults in the decrease of EB field.\nSUMMARY\nAs none before, we observe the signature of exchange\nbias effect in Nd 1−xSrxCoO3forx= 0.20 and 0.40 at\nthe spontaneous ferromagnetic/ferrimagnetic interface.\nWhen the sample was cooled in a static magnetic field,the systematic shifts of the magnetic hysteresis loops are\nobserved as a function of temperature and cooling field.\nTheexchangebiaseffectvanishesabovetheferrimagnetic\ntransition ( TFI) temperature, which indicates that the\nferrimagneticspinsapplyapinningforceonthereversible\nferromagnetic spins at the interface below TFIand the\npinned ferromagnetic spins give rise to the exchange bias\nphenomenon. The exchange bias is further confirmed by\nthe training effect, which could be explained satisfacto-\nrily by the spin configurational relaxation model. The\nexchange bias field is found to increase sharply ( ≤10\nkOe) with increasing cooling field and then it shows a\nvery small decreasing trend ( ≥20 kOe) for high cooling\nfield. The coercivity almost follows a similar trend of ex-\nchange bias field, where a linear dependence of exchange\nbias field and coercivity against cooling field is observed\nabove 20 kOe for both the compounds. The cooling field\ndependenceofexchangebiaseffectisanalysedbythesim-\nplified exchange interaction model, which gives a rough\nestimate of the average size of the ferromagnetic clusters\naround∼20 and∼40˚A, where the FM clusters con-\nsisting of single magnetic domain are suggested for x=\n0.20 and 0.40, respectively. The sizes of the ferromag-\nnetic clusters are close to the percolation threshold for x\n= 0.20, which grow and coalesce with increasing x. 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J.ˇSaf´ arik University, Park Angelinum 9, 040 01 Koˇ sice, Slov ak Republic\nbDepartment of Physics, Faculty of Science, Adnan Menderes U niversity,\n09010 Aydın, Turkey\nAbstract\nMixed-spin Ising model on a decorated Bethe lattice is rigorously solv ed by\ncombining the decoration-iteration transformation with the metho d of exact re-\ncursion relations. Exact results for critical lines, compensation te mperatures,\ntotal and sublattice magnetizations are obtained from a precise ma pping rela-\ntionship with the corresponding spin-1/2 Ising model on a simple (und ecorated)\nBethe lattice. The effect of next-nearest-neighbour interaction and single-ion\nanisotropy on magnetic properties of the ferrimagnetic model is inv estigated in\nparticular. It is shown that the total magnetization may exhibit mult icompen-\nsation phenomenon and the critical temperature vs. the single-ion anisotropy\ndependence basically changes with the coordination number of the u nderlying\nBethe lattice. The possibility of observing reentrant phase transit ions is related\nto a high enough coordination number of the underlying Bethe lattice .\nKeywords: Ising model, Bethe lattice, reentrant phase transitions,\ncompensation points\nPACS:05.50.+q, 75.10.Hk, 75.10.Jm\n1. Introduction\nFerrimagnetism has been extensively studied in the past both exper imen-\ntally as well as theoretically. The magnetic structure of insulating fe rrimagnets\nisbuilt up fromseveral(at leasttwo)sublattices, whichareoccupie dbydifferent\ncarriers of unequal magnetic moments interacting with each other through the\nantiferromagnetic interaction. Mixed-spin Ising models probably re present the\nsimplest theoreticalmodels, which maybringan insight intothe mostim portant\nEmail addresses: jozef.strecka@upjs.sk (Jozef Streˇ cka), ekizc@yahoo.com (Cesur\nEkiz)\nPreprint submitted to Physica A June 25, 2018features accompanying ferrimagnetic ordering such as for instan ce a compensa-\ntion phenomenon. A compensation point denotes the temperature Tcomp, at\nwhich the resultant magnetization vanishes due to a complete cance llation of\nsublattice magnetizations below the critical temperature Tc, i.e.Tcomp< Tc\n[1]. An existence of compensation temperatures has obvious techn ological sig-\nnificance in the field of thermomagnetic recording, because only a sm all driving\nfield is needed at a compensation point in order to achieve a magnetic p ole\nreversal [2, 3]. Besides, the mixed-spin Ising models belong to the mo st inter-\nesting extensions of the standard spin-1/2 Ising model, which may d isplay more\ndiverse critical behaviour compared with their single-spin counterp arts.\nDecorated mixed-spin Ising models, which were originally introduced b y\nSyoziand co-workers(see Ref. [4] and referencescited there in), havebeen inten-\nsively studied many years ago as theoretical models of ferrimagnet ism. Thermal\nvariations of the magnetization in the decorated Ising ferrimagnet s have been\ninvestigated in detail and these results shed light on some characte ristic fea-\ntures of real ferrimagnetic materials [5, 6]. The effect of single-ion a nisotropy\non magnetic properties of the mixed spin-1/2 and spin- S(S >1/2) Ising ferri-\nmagnet on decorated planar lattices has been examined by Kaneyos hi with the\nhelp of differential operator technique and the effective-field theo ry [7, 8]. By\nmaking use of these theoretical tools, Kaneyoshi has predicted m any interesting\nfeatures of the decorated Ising ferrimagnets such as an appear ance of two com-\npensationpointsand/ormultiplereentrantphasetransitions. Itis worthwhileto\nremark that Kaneyoshi’s predictions were subsequently confirmed by Jaˇ sˇ cur [9]\nand Dakhama [10] when solving the ferrimagnetic version of the deco rated Ising\nmodel rigorously. The significance of this model also lies in the fact th at tem-\nperature dependences of the total magnetization show numerou s characteristic\nfeatures not predicted in the standard N´ eel theory of ferrimag netism [1].\nRecently, the magnetic properties of various decorated mixed-sp in Ising\nmodels have been explored in detail by variety of mathematical tech niques\nwith some exact [9, 10, 11, 12, 13, 14, 15, 16] as well as approximat e results\n[17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. The present work deals with magnetic\nproperties of the mixed-spin Ising ferrimagnet on the decorated B ethe lattice\naccounting for the next-nearest-neighbor interaction and the u niaxial single-ion\nanisotropy. The exact solution will be obtained by combining the deco ration-\niteration transformation with the another rigorous method based on recursion\nrelations. First, the decoration-iteration transformation will be a pplied to es-\ntablish the precise mapping correspondencebetween the mixed-sp in Ising model\non the decorated Bethe lattice and its equivalent spin-1/2 Ising mod el on the\nsimple (undecorated) Bethe lattice. It is well known that this latter model\ncan be subsequently treated by making use of recursion relations [2 7], which\nwill help us to complete our exact calculation for the original mixed-sp in Ising\nmodel on the decorated Bethe lattice. Within the framework of the se two ex-\nact methods, we will focus on the effect of the single-ion anisotropy and the\nnext-nearest-neighbour interaction on the phase diagrams, crit ical behaviour\nand compensation phenomenon of the model under investigation.\nThe outline of the present paper is as follows. In Section 2, the deta iled\n2description of the model system will be presented and the basic ste ps of both\nexactmethodswillbebrieflyclarified. Themostinterestingresultsa represented\nanddiscussedindetailin Section3. Inparticular,ourattentionisfo cusedonthe\nfinite-temperature phase diagrams, reentrantphase transition s and temperature\nvariations of both sublattice as well as total magnetizations. Finally , some\nconclusions are mentioned in the last Section IV.\n2. Model and method\nLetusbeginbyconsideringthemixedspin-1/2andspin- SIsingmodelonthe\ndecorated Bethe lattice with a quite general coordination number qas schemat-\nically illustrated in Fig. 1 on the particular example of the Bethe lattice w ith\nthe coordination number q= 3. The magnetic structure of the investigated\nmodel system constitutes the Ising spins σ= 1/2 placed on lattice sites of a\ndeep interior of infinite Cayley tree (Bethe lattice) and the Ising spin s of a quite\ngeneral magnitude Splaced on each bond of the original Bethe lattice. The\ntotal Hamiltonian of the mixed-spin Ising model on the decorated Be the lattice\nthen reads\nH=−Jnn/summationdisplay\ni,jSiσj−J′nnn/summationdisplay\nk,jσkσj−DNq/2/summationdisplay\ni=1S2\ni, (1)\nwhereσj=∓1/2 andSi=−S,−S+ 1,...,S, the relevant subscript specifies\nthe lattice position and the exchange constants JandJ′represent the nearest-\nneighbour interaction between the spin-1/2 and spin- Satoms and, respectively,\nthe next-nearest-neighbour interaction between the spin-1/2 a toms. Further,\nthe parameter Dstands for the uniaxial single-ion anisotropy acting on the\nspin-Satoms only. As one can see from Fig. 1, the magnetic structure of t he\ninvestigated model system constitutes two interpenetrating sub lattices. The\nformersublattice A is formed by the sites of original(undecorated ) Bethe lattice\nthat is occupied by the atoms with the fixed spin σ= 1/2, while the latter\nsublattice B is occupied by the decorating atoms with an arbitrary sp in value\nS. The calculation on the Bethe lattice is done recursively [27].\nNow, wewillturnourattentiontomainpointsofthetransformation method,\nwhich provides a rigorous mapping equivalence between the investiga ted spin\nmodel and its corresponding spin-1/2 Ising model on undecorated Bethe lattice.\nFirst, it is quite useful to rewrite the total Hamiltonian (1) as a sum o f the bond\nHamiltonians, i.e. H=/summationtextNq/2\nk=1Hk, whereNdenotes the total number of the\nspin-1/2 atoms from the sublattice A, qis the coordination number of the un-\ndecorated Bethe lattice and the summation runs over all bonds of t he decorated\nBethe lattice. Thus, each bond Hamiltonian Hkinvolves all the interaction\nterms including the decorating Ising spin Skand the next-nearest-neighbour\ninteraction between its two nearest-neighbouring Ising spins σk1andσk2(for\nbetter illustration see spin cluster enclosed by ellipse in Fig. 1)\nHk=−JSk(σk1+σk2)−DS2\nk−J′σk1σk2. (2)\n3Sk\nDIT/c115k1/c115k2\n/c115k1/c115k2\nJJ\nJ´DR\nFigure 1: The mixed spin-1/2 and spin- SIsing model on the decorated Bethe lattice (figure on\nthe left)and its relation to the corresponding spin-1/2 Isi ng model on the simple (undecorated)\nBethe lattice with the coordination number q= 3 (figure on the right). The blue (red) circles\ndenote lattice positions of the spin-1/2 (spin-1) Ising spi ns and the ellipse demarcates all the\ninteraction terms belonging to the kth bond Hamiltonian.\nA crucial step in our procedure represents the evaluation of the p artition func-\ntion. For further convenience, the partition function of the mode l under consid-\neration can be written in the following useful form\nZ=/summationdisplay\n{σi}Nq/2/productdisplay\nk=1/summationdisplay\nSkexp(−βHk), (3)\nwhereβ= 1/(kBT),kBis the Boltzmann’s constant and Tis the absolute\ntemperature. The symbol/summationtext\nSkdenotes a summation over 2 S+ 1 spin states\nof thekth decorating Ising spin from the sublattice B, while the symbol/summationtext\n{σi}\nmarks a summation over all available spin configurations of the comple te set\nof the spin-1/2 Ising atoms from the sublattice A. It is noteworthy that the\nsummation over spin states of the decorating spin- Satoms can be performed\nindependently of each other because of the lack of any direct inter action among\nthem. After performing the summation over the spin states of the decorating\nIsingspin Sk, onegetsthe effectiveBoltzmann’s weightthat implies apossibility\nof applying the generalized decoration-iteration transformation [ 28, 29, 30]\n/summationdisplay\nSkexp(−βHk) = exp( βJ′σk1σk2)S/summationdisplay\nn=−Sexp(βDn2)cosh[βJn(σk1+σk2)]\n=Aexp(βRσk1σk2). (4)\nThe physical meaning of the mapping transformation (4) lies in remov ing all\nthe interaction parameters associated with the decorating spin Skand replac-\ning them by a new unique effective interaction Rbetween its two nearest-\nneighbouring Ising spins σk1andσk2. It is worthwhile to mention that the\nboth mapping parameters AandRare ’self-consistently’ given directly by the\ntransformation formula (4), which must hold for all four possible sp in combina-\ntions of two Ising spins σk1andσk2that provide just two independent equations\n4from the mapping transformation (4). In the consequence of tha t, two yet un-\nknown mapping parameters AandRcan be unambiguously determined by the\nfollowing formulas\nA= (V1V2)1\n2, βR =βJ′+2ln/parenleftbiggV1\nV2/parenrightbigg\n, (5)\nwhich contain two newly defined functions\nV1=S/summationdisplay\nn=−Sexp(βDn2)cosh(βJn), V 2=S/summationdisplay\nn=−Sexp(βDn2). (6)\nAfter straightforward substitution of the mapping transformat ion (4) into\nthe formula (3), one easily obtains an exact relation between the pa rtition func-\ntion of the mixed-spin Ising model on the decorated Bethe lattice an d respec-\ntively, the partition function of the spin-1/2 Ising model on the simp le (un-\ndecorated) Bethe lattice with the temperature-dependent effec tive interaction\nR\nZ(β,J,J′,D,S,q) =ANq/2ZBethe(βR). (7)\nThe mapping relation (7) between both partition functions represe nts a central\nresult of our calculation, because this relationship can in turn be emp loyed for\na rigorous calculation of some important quantities such as sublattic e magneti-\nzations and free energy. It is worthwhile to recall that this mapping correspon-\ndenceisvalidforarbitraryvaluesofthedecoratingspin Sandalsoindependently\nof the lattice coordination number q.\nExact mapping theorems developed by Barry et al. [31, 32, 33, 34] in com-\nbination with Eq. (7) yield a straightforward relation between the sp ontaneous\nmagnetization of the sublattice A of the mixed-spin Ising model on th e dec-\norated Bethe lattice and respectively, the spontaneous magnetiz ation of the\ncorresponding spin-1/2 Ising model on the undecorated Bethe lat tice\nmA≡1\n2(/angbracketleftσk1/angbracketright+/angbracketleftσk2/angbracketright) =1\n2(/angbracketleftσk1/angbracketright+/angbracketleftσk2/angbracketright)Bethe≡mBethe(βR).(8)\nHere, the symbols /angbracketleft···/angbracketrightand/angbracketleft···/angbracketrightBethemean the standard canonical ensemble\naverage performed in the mixed-spin Ising model on the decorated Bethe lattice\ndefined through the Hamiltonian (1) and respectively, its correspo nding spin-\n1/2 Ising model on the undecorated Bethe lattice. According to Eq . (8), the\nsublattice magnetization mAdirectly equals to the spontaneous magnetization\nmBetheof the corresponding spin-1/2 Ising model on the simple Bethe lattic e\nprovided that Eqs. (5)-(6) lead to the effective ferromagnetic int eraction R >0.\nInthe caseofthe effectiveantiferromagneticinteraction R <0, onealternatively\ngains a quite similar exact result for the staggered magnetization ms\nAof the\nsublattice A of the mixed-spin Ising model on the decorated Bethe la ttice\nms\nA≡1\n2(/angbracketleftσk1/angbracketright−/angbracketleftσk2/angbracketright) =1\n2(/angbracketleftσk1/angbracketright−/angbracketleftσk2/angbracketright)Bethe≡mBethe(β|R|),(9)\n5which can also be expressed only in terms of the spontaneous magne tization\nmBetheof the corresponding spin-1/2 Ising model on the undecorated Be the\nlattice. The other spontaneous sublattice magnetization of the mix ed-spin Ising\nmodel on the decorated Bethe lattice can be descended from the g eneralized\nCallen-Suzuki identity [35, 36, 37], which enables to relate it to the pr eviously\nderived spontaneous sublattice magnetization mA\nmB≡ /angbracketleftSk/angbracketright= 2mAS/summationdisplay\nn=−Snexp(βDn2)sinh(βJn)\nS/summationdisplay\nn=−Sexp(βDn2)cosh(βJn). (10)\nTo complete our calculation of spontaneous sublattice magnetizatio ns, one\nstill needs the exact closed-form formula for the spontaneous ma gnetization\nmBetheof the corresponding spin-1/2 Ising model on the simple (undecora ted)\nBethe lattice. This quantity can be rigorously found within the frame work of\nexact recursion relations [27]. If the undecorated Bethe lattice (s ee Fig. 1) is\n’cut’ at a central site with the spin σk1, it will consequently split into qidentical\nbranches. The partition function of the spin-1/2 Ising model on th e simple\nBethe lattice will take the form\nZBethe=/summationdisplay\nσk1[gn(σk1)]q=/bracketleftbigg\ngn/parenleftbigg1\n2/parenrightbigg/bracketrightbiggq\n+/bracketleftbigg\ngn/parenleftbigg\n−1\n2/parenrightbigg/bracketrightbiggq\n, (11)\nwheregn(σk1) is the partition function of the one individual branch and nis\nthe total number of generations in the recursively built Bethe lattic e. Cutting\neach branch at the subsequent site σk2allows one to find the recursive relation\nbetween the partition function of the branch gn(σk1) withngenerations of spins\nand respectively, the partition function of the sub-branch gn−1(σk2) withn−1\ngenerations of spins\ngn(σk1) =/summationdisplay\nσk2exp(βRσk1σk2)[gn−1(σk2)]q−1. (12)\nThe canonical ensemble average of the central spin σk1in the spin-1/2 Ising\nmodel on the Bethe lattice can readily be calculated from the relation\nmBethe=/angbracketleftσk1/angbracketrightBethe=1\nZBethe/summationdisplay\nσk1σk1[gn(σk1)]q=1\n2/parenleftbigg1−xq\nn\n1+xq\nn/parenrightbigg\n,(13)\nwhich contains the newly defined parameter xn=gn(−1\n2)\ngn(1\n2). Even though the\nparameter xndoes not have a direct physical meaning, it directly determines\nthe overall thermodynamics of the spin-1/2 Ising model on the Bet he lattice in\nthe limit n→ ∞. With the help of Eq. (12), one may easily derive for xnthat\n6enters the magnetization formula (13) the following recursion relat ion\nxn=exp/parenleftBig\n−βR\n4/parenrightBig\n+exp/parenleftBig\nβR\n4/parenrightBig\nxq−1\nn−1\nexp/parenleftBig\nβR\n4/parenrightBig\n+exp/parenleftBig\n−βR\n4/parenrightBig\nxq−1\nn−1, (14)\nwhich can be regarded as the equation of state. The equation (14) forms an\niteration sequence and converges to stable fixed points in the ther modynamic\nlimit for non-staggered phases. In our case this situation appears if the effective\nnearest-neighbour coupling of the corresponding spin-1/2 Ising m odel on the\nBethe lattice is ferromagnetic (i.e. R >0). On the other hand, the recursion\nrelation (14) leads to two-cycle double points in the case of stagger ed antifer-\nromagnetic phase, which emerges in the corresponding spin-1/2 Is ing model on\nthe Bethe lattice provided that the effective interaction is antiferr omagnetic(i.e.\nR <0) [39, 40, 41, 42]. However, onemay takeadvantageofthe bipart ite nature\nof the Bethe lattice to overcome this difficulty under the simultaneou s change\nof the nearest-neighbour interaction R→ −Rand the spins from each second\ngeneration σi→(−1)i+1σi. Hence, it follows that the staggered magnetization\n(9) may alternatively be calculated from the relevant exact result ( 13) for the\nferromagnetic spin-1/2 Ising model on the Bethe lattice with the eff ective in-\nteraction |R|. In this way, our exact calculation of all spontaneous sublattice\nmagnetizations of the mixed-spin Ising model on the decorated Bet he lattice\nis completed, because it is now sufficient to substitute Eqs. (13)-(1 4) into the\nformerly derived expressions (8)-(10) for the sublattice magnet izationsmA,ms\nA\nandmB. In doing so, one should bear in mind that the effective temperature -\ndependent coupling βRgiven by Eqs. (5)-(6) must enter into the recurrence\nrelation (14) in order to gain the correct exact result for the spon taneous mag-\nnetization of the correspondingspin-1/2 Ising model on the simple B ethe lattice\ngiven by Eq. (13).\nFinally, let us examine in detail the critical and compensation behaviou r of\nthe model under investigation. It can be readily understood from t he mapping\nrelation (7) between both partition functions that the mixed-spin I sing model\non the decorated Bethe lattice may exhibit a critical point only if the c orre-\nsponding spin-1/2 Ising model on the undecorated Bethe lattice is a t a critical\npoint as well. This result is taken to mean that the critical temperatu re of the\nmixed-spin Ising model on the decorated Bethe lattice can easily be o btained by\ncomparing the effective nearest-neighbour coupling of the corres ponding spin-\n1/2 Ising model on the simple (undecorated) Bethe lattice with its cr itical value\nβc|R|= 2ln(q\nq−2). The critical condition for the the mixed-spin Ising model on\nthe decorated Bethe lattice then reads\nexp/parenleftbiggβcJ′\n2/parenrightbiggS/summationdisplay\nn=−Sexp(βcDn2)cosh(βcJn)\nS/summationdisplay\nn=−Sexp(βcDn2)=/parenleftbiggq\nq−2/parenrightbigg±1\n, (15)\n7whereβc= 1/(kBTc),Tcdenotes the critical temperature and the plus (minus)\nsign applies when the effective coupling βRgiven by Eqs. (5)-(6) is being\npositive (negative). The sign ambiguity to emerge in the critical cond ition (15)\ncomes from the fact that the critical temperature of the spin-1/ 2 Ising model\non the undecorated Bethe lattice remains the same independently o f whether\nthe effective interaction is being ferromagnetic ( R >0) or antiferromagnetic\n(R <0). It is worthy to notice, moreover, that the nearest-neighbou r spin-1/2\nIsing atoms from the sublattice A should consequently exhibit ferro magnetic\n(antiferromagnetic) spin alignment in the phases delimited by critical phase\nboundaries (15) with plus (minus) sign.\nLast but not least, the compensation phenomenon may also come int o play\nif the total magnetization of the mixed-spin Ising model disappears at tempera-\nture(s) below the critical point. To explore this effect, let us there fore define the\ntotal magnetization per one spin mT= (2mA+qmB)/(1+qS), which is normal-\nized with respect to its saturation value ms= (1+qS)/(2+q). In this respect,\none easily finds the necessary (but not sufficient) condition for an a ppearance\nof the compensation point(s)\n1+qS/summationdisplay\nn=−Snexp(βcompDn2)sinh(βcompJn)\nS/summationdisplay\nn=−Sexp(βcompDn2)cosh(βcompJn)= 0, (16)\nwhereas βcomp= 1/(kBTcomp) andTcomplabels the compensation temperature.\nIt is quite obviousfrom Eq. (16) that the compensationpoint may po ssiblyarise\nonly in the ferrimagnetic model with the antiferromagnetic ( J <0) nearest-\nneighbour interaction between the spin-1/2 and spin- Satoms. In addition,\nit also turns out from Eq. (16) that the compensation temperatur e does not\napparently depend on a strength of the next-nearest-neighbou r interaction J′\nbetween the spin-1/2 Ising atoms, which merely determines a magnit ude of the\ncritical temperature according to Eq. (15).\n3. Results and Discussion\nLet us proceed to a discussion of the most interesting results. Eve n though\nall the results derived in the foregoing section are valid for arbitrar yvalue of the\ncoordinationnumber qofthe underlying Bethe lattice, regardlessofwhether the\ninteractions JandJ′are ferromagnetic or antiferromagnetic and also irrespec-\ntive of the value of the decorating spins, our further analysis will be henceforth\nrestricted only to the particular case of the ferrimagnetic mixed sp in-1/2 and\nspin-1 Ising model on the decorated Bethe lattice with a quite gener al coordi-\nnation number q, the antiferromagnetic ( J <0) nearest-neighbour interaction\nand either ferromagnetic ( J′>0) or antiferromagnetic ( J′<0) next-nearest-\nneighbour interaction. It is noteworthy that the investigated mod el displays all\n8general features of ferrimagnetism found in the mixed spin-(1/2, S) Ising models\nwith the integer-valueddecoratingspins S, while the respective behaviourof the\nmixed spin-(1/2, S) Ising models with the half-odd-integer decorating spins Sis\nmuch less pronounced as convincingly evidenced in Refs. [7, 8, 9, 10 ] and will\nbe therefore postponed from the present study. Note further more that exact\nresults presented hereafter may also be confronted with severa l known rigor-\nous results, which have been previously obtained for the mixed spin- 1/2 and\nspin-1 Ising ferrimagnets on the decorated planar lattices [9, 10], o r Bethe lat-\ntices [38, 43, 44]. To reduce the number of free parameters, all th e interaction\nconstants will be hereafter normalized with respect to a strength of the nearest-\nneighbourinteraction |J|, whichmeans that the ratio J′/|J|,D/|J|andkBT/|J|\nwill accordingly measure a relative magnitude of the next-nearest- neighbour in-\nteraction, single-ion anisotropy and temperature, respectively.\nLet us begin with the analysis of the ground state. The ground-sta te phase\ndiagram of the mixed spin-1/2 and spin-1 Ising ferrimagnet on the de corated\nBethe lattice of arbitrary coordination number qis displayed in Fig. 2. In\nagreement with our expectations, the typical ferrimagnetic phas e (FI) with\nthe antiparallel alignment [ σi,Si] = [1/2,−1] between the nearest-neighbour\nspin-1/2 and spin-1 atoms is being the ground state provided that a relative\nstrength of the easy-plane (negative) single-ion anisotropy D/|J|and/or the\nantiferromagnetic next-nearest-neighbour interaction J′/|J|is not too strong.\nOn assumption that the next-nearest-neighbour interaction is be ing ferromag-\nnetic (J′/|J|>0), the decrease in the single-ion anisotropy causes a first-order\nphase transition from FI towards the remarkable ferromagnetic p hase (FII)\nwith the respective spin alignment [ σi,Si] = [1/2,0]. Note that the sponta-\nneous long-range order inherent to FII is merely kept by the ferro magnetic\nnext-nearest-neighbour interaction between the spin-1/2 atom s, whereas all the\nspin-1atomsarenon-magneticasaresultofsufficientlystrongeas y-planesingle-\nion anisotropy ( D/|J|<−1.0) that energetically favours the non-magnetic spin\nstateSi= 0 before two magnetic ones Si=±1. If one considers the anti-\nferromagnetic next-nearest-neighbour interaction from the int ermediate range\nJ′/|J| ∈(−2.0,0.0), the antiferromagnetic phase (AFI) characterized through a\nmorecomplexspinarrangement[ σi,Si;σi+1,Si+1] = [1/2,0;−1/2,0]arisesfrom\nFI upon lowering the single-ion anisotropy. All the decorating spin-1 atoms re-\nside in AFI the non-magnetic spin state Si= 0 quite similarly as in FII, but\nthe spin-1/2 atoms from the nearest-neighbour shells of the deco rated Bethe\nlattice align antiparallel with respect to each other on account of th e antifer-\nromagnetic nature of the next-nearest-neighbour interaction b eing responsible\nfor the symmetry breaking (period doubling). It should be noted he re that\nthe emergence of both FII and AFI is restricted to the parameter space with\nthe easy-plane character of the single-ion anisotropy D/|J|<0. Considering\nthe easy-axis single-ion anisotropy D/|J|>0 and reinforcing the antiferromag-\nnetic next-nearest-neighbour interaction, the mixed-spin syste m undergoes at\nJ′/|J|=−2.0 a first-order phase transition from FI towards the partially anti-\nferromagnetically ordered and partially disordered phase (AFII) c haracterized\nthrough the spin arrangement [ σi,Si;σi+1,Si+1] = [1/2,±1;−1/2,±1]. The pe-\n9/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49\n/s95\n/s32/s32/s32/s45/s49/s47/s50/s44/s32/s43/s49/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s47/s50/s44/s32/s45/s49/s93/s65/s70/s73/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s70/s73\n/s68\n/s47/s124\n/s74\n/s124/s61/s32/s45/s49/s32/s45/s32\n/s74/s39\n/s47/s124/s50\n/s74\n/s124/s91/s49/s47/s50/s44/s32/s43/s49/s59/s32/s32\n/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s32\n/s68/s32 /s47/s32 /s124/s74/s124\n/s91/s49/s47/s50/s44/s32/s48/s59/s32/s45/s49/s47/s50/s44/s32/s48/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s47/s50/s44/s32/s48/s93/s65/s70/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s70/s73/s73/s95\nFigure 2: Ground-state phase diagram of the mixed spin-1/2 a nd spin-1 Ising model on the\ndecorated Bethe lattice in the J′/|J| −D/|J|plane. Square brackets [ σ,S] demonstrate the\nrelevant spin alignment in each sector of the ground-state p hase diagram. For the more\ndetailed description of individual phases see the text.\nculiarcoexistenceofapartialorderandapartialdisorderinAFIIc omesfromthe\nantiferromagnetic alignment of the next-nearest-neighbouring s pin-1/2 atoms,\nwhich is simultaneously responsible for a unusual spin frustration of the deco-\nrating spin-1 atoms. Hence, it follows that the spin-1 atoms reside in AFII with\nthe same probability one of its two magnetic spin states Si=±1 due to the\neasy-axis character of the single-ion anisotropy D/|J|>0. It should be pointed\nout that the established ground-state phase diagram holds for th e mixed-spin\nIsing models on the decorated planar lattices as well. It surprisingly a ppears\nthat the characterizationofthe phase FII has been notably over lookedin several\nprevious approximative [7, 8] as well as exact [9, 10] studies on the m ixed-spin\nIsing model on decorated planar lattices. In addition, it seems that the partic-\nular case with the antiferromagnetic next-nearest-neighbour int eraction has not\nbeen dealt with hitherto and hence, theoretical predictions of the se two phases\nrepresent the completely new finding not reported in the literature yet.\nNow, let us focus our attention on a discussion of finite-temperatu re phase\ndiagrams, which are plotted in Figs. 3-6 in the form of critical temper ature\nversus single-ion anisotropy dependences for several values of t he coordina-\ntion number qand different strengths of the next-nearest-neighbour interac -\ntionJ′/|J|. It is noteworthy that all displayed phase boundaries are the lines\nof second-order phase transitions and the ordered (disordered ) phases can be\nlocated below (above) depicted critical boundaries. Fig. 3(a) show s the crit-\nical temperature as a function of the single-ion anisotropy in an abs ence of\nthe next-nearest-neighbour interaction (i.e. J′/|J|= 0.0). In this particu-\nlar case, the spontaneously long-range ordered FI represents t he ground state\nforD/|J|>−1.0, while the disordered paramagnetic phase (PP) becomes the\nground state for D/|J|<−1.0. As one can see, the critical line approaches the\nzero-temperature phase boundary between FI and PP with a nega tive slope for\n10/s45/s49 /s48 /s49 /s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s80/s80\n/s113 /s61/s51/s113 /s61/s52/s113 /s61/s53/s113 /s61/s54\n/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s61/s32 /s48/s46/s48/s32\n/s107\n/s66/s32/s84\n/s99/s32/s47/s32 /s124/s74/s124\n/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s68 /s32/s47/s32 /s124/s74/s124/s70/s73\n/s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s45/s49/s46/s48\n/s65/s70/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s70/s73/s45/s48/s46/s53/s45/s48/s46/s50\n/s45/s48/s46/s49/s113 /s61/s52/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s61/s32 /s48/s46/s48/s32\n/s107\n/s66/s32/s84\n/s99/s32/s47/s32 /s124/s74/s124\n/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s68 /s32/s47/s32 /s124/s74/s124\nFigure 3: Critical temperature of the mixed spin-1/2 and spi n-1 Ising model on the decorated\nBethe lattice as a function of the single-ion anisotropy D/|J|for: (a) J′/|J|= 0.0 andq=3,\n4, 5 and 6; (b) q= 4 and J′/|J| ≤0.\nthe decorated Bethe lattices with the coordination number q <4, with a posi-\ntive slope for the decorated Bethe lattices with the coordination nu mberq >4\nand with an infinite gradient for the particular value of the coordinat ionnumber\nq= 4. These observationswould suggest that reentrantphase tra nsitionscan be\nobservedin a close vicinity of the ground-stateboundary between FI and PP for\nthe decorated Bethe lattices with a sufficiently high coordination num berq >4,\nwhereas the reentrance becomes the more pronounced, the high er the coordi-\nnation number qof the underlying Bethe lattice is. The displayed exact results\nmay be contrasted with the approximative results reported by Kan eyoshi [7, 8],\nwhich imply either more [7] or less [8] robust reentrant region for the decorated\nsquare lattice. On the other hand, the present results are in acco rdance with\nthe previously reported exact results for mixed-spin Ising model o n the deco-\nrated planar lattices, among which the decorated square [9, 10], ho neycomb and\nkagom´ e lattice [15] do not show reentrance in contrast to the dec orated trian-\ngular lattice with a pronounced reentrant region [15]. Note further more that\nthe investigated model cannot exhibit the compensation phenomen on when the\nnext-nearest-neighbour interaction is absent.\nNext, we will pay our attention to the effect of the antiferromagne tic next-\nnearest-neighbour interaction on the critical behaviour of the mix ed-spin Ising\nmodelonthedecoratedBethelatticewiththecoordinationnumber q= 4. Itcan\nbe clearly seen from Fig. 3(b) that the critical temperature of FI m onotonon-\nically decreases with decreasing the single-ion anisotropy until it rea ches the\nzero-temperature phase boundary between FI and AFI (see the right wing of\neach displayed critical lines in Fig. 3(b)). Afterwards, the new critic al line\ndevelops for a more negative values of the single-ion anisotropy, wh ereas the op-\nposite trend is observed; the critical temperature monotonically in creases with\ndecreasing the single-ion anisotropy until it tends towards its maxim um value in\nthe limit D/|J| → −∞. The left wing of each critical line obviously corresponds\nto AFI phase with the zero spontaneous magnetization but the non -zero stag-\ngered magnetization of the spin-1/2 Ising atoms (see the subsequ ent discussion\n11/s45/s51 /s45/s50 /s45/s49 /s48 /s49/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s70/s73/s73/s80/s80\n/s70/s73/s50/s46/s48\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s50 /s48/s46/s48/s48/s46/s49\n/s113 /s61/s52/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s61/s32 /s51/s46/s48/s32\n/s107\n/s66/s32/s84\n/s99/s32/s47/s32 /s124/s74/s124\n/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s68 /s32/s47/s32 /s124/s74/s124/s45/s49/s46/s53/s48 /s45/s49/s46/s50/s53 /s45/s49/s46/s48/s48 /s45/s48/s46/s55/s53 /s45/s48/s46/s53/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s70/s73/s73/s80/s80\n/s70/s73/s48/s46/s48/s48/s46/s49/s113 /s61/s52\n/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s61/s32 /s48/s46/s50/s32\n/s107\n/s66/s32/s84\n/s99/s32/s47/s32 /s124/s74/s124\n/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s68 /s32/s47/s32 /s124/s74/s124\nFigure 4: Critical temperature of the mixed spin-1/2 and spi n-1 Ising model on the decorated\nBethe lattice with the coordination number q= 4 as a function of the single-ion anisotropy\nD/|J|for several values of the ferromagnetic next-nearest-neig hbour interaction J′/|J| ≥0.\nBroken curve shows the line of compensations points and Fig. 4(b) is a detailed plot from the\nregion near the ground-state phase boundary between FI and F II.\nfor more details). It should be noticed that the antiferromagnetic next-nearest-\nneighbour interaction can neither induce the reentrant phase tra nsitions, nor\nthe compensation phenomenon for the decorated Bethe lattices w ith the coor-\ndination number q <5.\nFor completeness, the finite-temperature phase diagram of the m ixed spin-\n1/2 and spin-1 Ising model on the decorated Bethe lattice with the c oordination\nnumberq= 4 is plotted in Fig. 4 for different strengths of the ferromagnetic\nnext-nearest-neighbour interaction. The critical temperature generally exhibits\na smooth monotonous decline with decreasing the single-ion anisotro py and at\nfirst sight, there is no clear evidence of the phase transition betwe en FI and FII\nphases in the displayed critical lines. It will be later demonstrated, h owever,\nthat the sublattice magnetization of the spin-1 atoms approaches in the zero-\ntemperature limit its saturation value only if D/|J|>−1.0, while it starts from\nzero in the reverse case. With this in mind, one may conclude that the critical\nline ofFI continuouslymergesatsome finite temperature with the cr iticalline of\nFII at the specific value of the single-ion anisotropy D/|J|=−1.0, whereas the\nformer FI phase is located below the right part ( D/|J|>−1.0) and the latter\nFII phase below the left part( D/|J|<−1.0)ofthe overallcriticalline. Lastbut\nnot least, it is worth mentioning that a presence of the weak (stron g) ferromag-\nnetic next-nearest-neighbour interaction may lead to an existenc e of one (two)\ncompensation points in the temperature dependence of the total magnetiza-\ntion as it can be clearly seen from the respective behaviour of the br oken curve,\nwhich displaysthe numericalsolution ofthe necessarycondition (16 ) forappear-\nance of the compensation points. It should be remembered that th e necessary\ncondition for appearance of compensation points does not depend according to\nEq. (16) on a relative strength of the next-nearest-neighbour in teraction J′/|J|.\nFrom this point of view, the line of compensation temperatures star ts at the\nground-state boundary between FI and FII phases at D/|J|=−1.0 and it\nends up at a respective crossing point between the necessary con dition (16) and\n12/s45/s50 /s45/s49 /s48 /s49/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s45/s49/s46/s48\n/s65/s70/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s70/s73/s45/s48/s46/s53/s45/s48/s46/s50\n/s45/s48/s46/s49/s113 /s61/s54/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s61/s32 /s48/s46/s48/s32\n/s107\n/s66/s32/s84\n/s99/s32/s47/s32 /s124/s74/s124\n/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s68 /s32/s47/s32 /s124/s74/s124/s45/s49/s46/s50/s53 /s45/s49/s46/s48/s48 /s45/s48/s46/s55/s53 /s45/s48/s46/s53/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s65/s70/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s70/s73/s45/s48/s46/s53/s45/s48/s46/s50/s45/s48/s46/s49\n/s113 /s61/s54/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s61/s32 /s48/s46/s48/s32\n/s107\n/s66/s32/s84\n/s99/s32/s47/s32 /s124/s74/s124\n/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s68 /s32/s47/s32 /s124/s74/s124\nFigure 5: Critical temperature of the mixed spin-1/2 and spi n-1 Ising model on the decorated\nBethe lattice with the coordination number q= 6 as a function of the single-ion anisotropy\nD/|J|for several values of the antiferromagnetic next-nearest- neighbour interaction J′/|J| ≤\n0. Broken curve shows the line of compensations points and Fi g. 5(b) is a detailed plot from\nthe observed reentrant region near the ground-state phase b oundary between FI and AFI.\nthe relevant critical line depending basically on a relative strength of the next-\nnearest-neighbour interaction (above this line the spin system is dis ordered and\nthere is no spontaneous ordering). On the other hand, it also turn s out that\nthe ferromagnetic next-nearest-neighbour interaction cannot induce reentrant\nphase transitions for the decorated Bethe lattices with the coord ination number\nq <5.\nTo provide a deeper insight into how the coordination number affects the\noverall critical behaviour, the critical temperature of the mixed s pin-1/2 and\nspin-1 Ising model on the decorated Bethe lattice with the higher co ordination\nnumberq= 6 is plotted against the single-ion anisotropy in Figs. 5-6. First, let\nus briefly comment on the most significant changes of the relevant c ritical lines\nwhen considering the influence of the antiferromagnetic next-nea rest-neighbour\ninteraction (Fig. 5). In the absence of the next-nearest-neighb our interaction,\nthe investigated spin system exhibits a relatively wide reentrant reg ion for the\nsingle-ion anisotropy close to but slightly below the ground-state bo undary be-\ntween FI and PP at D/|J|=−1.0. In this special case, the spin system is\ndisordered in the ground state, then it orders at a lower critical te mperature\nto display the spin order inherent to FI, which finally disappears at th e upper\ncritical temperature (PP-FI-PP reentrance). On the other han d, there are quite\nfundamental differences in the respective reentrant behaviour o f the spin system\nwith the antiferromagnetic next-nearest-neighbour interaction . First of all, the\nspin system exhibits a more striking reentrance with three consecu tive critical\npoints when starting from the spontaneously long-range ordered ground state,\nwhich either corresponds to AFI or FI depending on an interplay bet ween the\nsingle-ion anisotropy and the next-nearest-neighbour interactio n. The spon-\ntaneous ordering of AFI (or FI) disappears at the first critical te mperature,\nthen there appears PP in a relatively narrow region in between the fir st and\nsecond critical temperature, at which the spontaneous order of FI occurs (or\n13/s45/s51 /s45/s50 /s45/s49 /s48 /s49/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s70/s73/s73/s80/s80\n/s70/s73/s48/s46/s53/s49/s46/s48\n/s48/s46/s50\n/s48/s46/s48/s48/s46/s48/s53\n/s48/s46/s49\n/s113 /s61/s54/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s61/s32 /s50/s46/s48/s32\n/s107\n/s66/s32/s84\n/s99/s32/s47/s32 /s124/s74/s124\n/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s68 /s32/s47/s32 /s124/s74/s124/s45/s49/s46/s53/s48 /s45/s49/s46/s50/s53 /s45/s49/s46/s48/s48 /s45/s48/s46/s55/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s70/s73/s73/s80/s80\n/s70/s73\n/s48/s46/s48/s53/s48/s46/s48\n/s113 /s61/s54/s74/s39 /s32/s47/s32 /s124/s74/s124/s32/s61/s32 /s48/s46/s49/s32\n/s107\n/s66/s32/s84\n/s99/s32/s47/s32 /s124/s74/s124\n/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s68 /s32/s47/s32 /s124/s74/s124\nFigure 6: Critical temperature of the mixed spin-1/2 and spi n-1 Ising model on the decorated\nBethe lattice with the coordination number q= 4 as a function of the single-ion anisotropy\nD/|J|for several values of the ferromagnetic next-nearest-neig hbour interaction J′/|J| ≥0.\nBlack broken curve shows the line of compensations points, w hereas Fig. 4(b) depicts a detail\nfrom the region, where reentrant phase transitions can be ob served.\nre-appears) and finally vanishes at the third critical temperature . The observed\nreentrantphasetransitions canbe thus characterizedby two fo llowingsequences\nof second-order phase transformations AFI-PP-FI-PP and FI- PP-FI-PP. It can\nbe also clearly seen from Fig. 5(b) that the reentrant region moves towards a\nmorepositive values ofthe single-ionanisotropy(together with the ground-state\nboundary between FI and AFI) and it gradually vanishes upon stren gthening\nthe antiferromagnetic next-nearest-neighbour interaction.\nFinally, the last couple of phase diagrams from Fig. 6 illustrates the eff ect of\nthe ferromagnetic next-nearest-neighbour interaction on the c ritical behaviour\nof the mixed spin-1/2 and spin-1 Ising model on the decorated Beth e lattice\nwith the coordination number q= 6. As one can see, the character of reen-\ntrant phase transitions substantially changes by taking the ferro magnetic next-\nnearest-neighbour interaction into consideration even although o ne still detects\nthe reentrance with three consecutive critical points. However, the observed\nreentrant phase transition is basically different in that both low-tem perature as\nwell as high-temperature spontaneously long-range ordered pha se is always FII.\nAccording to this, the spin system undergoes the following sequenc e of second-\norder phase transitions FII-PP-FII-PP when starting from the lo west and end-\ning up at the highest temperature. To summarize, the only common f eature of\nall three kinds of reentrant phase transitions FII-PP-FII-PP, F I-PP-FI-PP and\nAFI-PP-FI-PP is that the reentrance is quite sensitive and easily dim inishes by\nchanging a relative strength of the next-nearest-neighbour inte raction.\nTo provide an independent check of the aforedescribed critical be haviour,\nlet us proceed to a discussion of thermal dependences of the orde r parame-\nters as depicted in Figs. 7-9. First, let us confirm the phase transit ion from\nFI towards FII by considering the constant value of the ferromag netic next-\nnearest-neighbour interaction J′/|J|= 3.0 and changing a relative strength of\nthe single-ion anisotropy. Fig. 7(a)–(b) illustrate some typical tem perature\ndependences of the total and sublattice magnetizations when FI is being the\n14/s48 /s49 /s50 /s51/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s109\n/s84 /s32/s109\n/s66\n/s109\n/s65/s32/s109\n/s65/s32/s44/s32 /s109\n/s66/s32/s44/s32 /s109\n/s84/s32\n/s113 /s61/s52/s74/s39 /s32/s47/s32 /s124/s74/s124/s32 /s61 /s32/s51/s46/s48/s32\n/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74/s124/s68 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s49/s46/s48\n/s48 /s49 /s50 /s51/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s50/s46/s50 /s50/s46/s51 /s50/s46/s52/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52\n/s124/s109\n/s84/s124\n/s32/s109\n/s66\n/s109\n/s65/s32/s109\n/s65/s32/s44/s32 /s109\n/s66/s32/s44/s32 /s109\n/s84/s32\n/s113 /s61/s52/s74/s39 /s32/s47/s32 /s124/s74/s124/s32 /s61 /s32/s32/s51/s46/s48/s32\n/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74/s124/s68 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s48/s46/s57\n/s32/s32/s32\n/s32\n/s48 /s49 /s50 /s51/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/s50/s46/s49 /s50/s46/s50 /s50/s46/s51 /s50/s46/s52/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52\n/s124/s109\n/s84/s124\n/s32/s109\n/s66\n/s109\n/s65/s32/s109\n/s65/s32/s44/s32 /s109\n/s66/s32/s44/s32 /s109\n/s84/s32\n/s113 /s61/s52/s74/s39 /s32/s47/s32 /s124/s74/s124/s32 /s61 /s32/s32/s51/s46/s48/s32\n/s40/s99/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74/s124/s68 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s49/s46/s48\n/s32/s32/s32\n/s32\n/s48 /s49 /s50 /s51/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s124/s109\n/s84/s124\n/s32\n/s109\n/s66/s109\n/s65/s32/s109\n/s65/s32/s44/s32 /s109\n/s66/s32/s44/s32 /s109\n/s84/s32/s113 /s61/s52\n/s74/s39 /s32/s47/s32 /s124/s74/s124/s32 /s61 /s32/s51/s46/s48/s32\n/s40/s100/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74/s124/s68 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s49/s46/s53\nFigure 7: Thermal variations of the total and sublattice mag netizations for the mixed spin-\n1/2 and spin-1 Ising model on the decorated Bethe lattice wit h the coordination number\nq= 4, the fixed value of the ferromagnetic next-nearest-neigh bour interaction J′/|J|= 3.0\nand four different values of the single-ion anisotropy: (a) D/|J|= 1.0; (b)D/|J|=−0.9; (c)\nD/|J|=−1.0; (b)D/|J|=−1.5.\nground state. The resultant magnetization exhibits for D/|J|= 1.0 and−0.9\nthe standard R-type dependence and the slightly deformed N-typ e dependence\nwith one compensation point, respectively, whose unusual shape c omes from\na steeper temperature-induced decrease in the sublattice magne tizationmBof\nthe decorating spin-1 atoms. Furthermore, Fig. 7(c) shows temp erature varia-\ntions of the total and sublattice magnetizations exactly at the pha se boundary\nbetween FI and FII. In agreement with this statement, the sublat tice magne-\ntizationmBtends towards the intermediate value 0 .5 when approaching zero\ntemperature, whichindicatesacoexistenceofFIandFII inthegro undstate. Fi-\nnally, thermal dependences of the total and sublattice magnetiza tions displayed\nin Fig. 7(d) provide an exact evidence of FII. As a matter of fact, t he sublat-\ntice magnetization mBvanishes in the zero-temperature limit and its anoma-\nlous temperature-induced increase can be attributed to spin excit ations from\nthe non-magnetic spin state Si= 0 preferably towards the excited spin state\nSi=−1. In the consequence of that, the resultant magnetization might display\nthe striking W-type dependence with two successive compensation points not\n15/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s109\n/s84 /s32/s109\n/s66\n/s109\n/s65/s32/s109\n/s65/s32/s44/s32 /s109\n/s66/s32/s44/s32 /s109\n/s84/s32\n/s113 /s61/s52/s74/s39 /s32/s47/s32 /s124/s74/s124/s32 /s61 /s32/s45/s48/s46/s50/s32\n/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74/s124/s68 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s48/s46/s56\n/s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56 /s48/s46/s49/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s115\n/s32\n/s32/s115/s32\n/s109\n/s65/s32/s109\n/s65/s32\n/s32\n/s113 /s61/s52/s74/s39 /s32/s47/s32 /s124/s74/s124/s32 /s61 /s32/s45/s48/s46/s50/s32\n/s40/s98/s41/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74/s124/s68 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s49/s46/s48\nFigure 8: Thermal variations of the total, sublattice and st aggered magnetizations for the\nmixed spin-1/2 and spin-1 Ising model on the decorated Bethe lattice with the coordination\nnumber q= 4, the fixed value of the antiferromagnetic next-nearest-n eighbour interaction\nJ′/|J|=−0.2 and two different values of the single-ion anisotropy: (a) D/|J|=−0.8; (b)\nD/|J|=−1.0.\nincluded in the standard N´ eel theory of ferrimagnetism (for the c haracterization\nof temperature dependences of the total magnetization see Ref s. [45, 46]). The\nformer compensation point appears owing to a higher number of the decorating\nspins (the overall contribution of the sublattice magnetization mBto the to-\ntal magnetization consequently exceeds the one of the sublattice magnetization\nmAprovided that mBbecomes sufficiently high), while the latter compensation\npoint emerges as a result of more subtle character of the decorat ing spins with\nrespect to thermal fluctuations (the sublattice magnetization mBtends steeper\nto zero than the sublattice magnetization mAin a vicinity of the critical point).\nNext, let us turn to thermal variations of the total and sublattice magnetiza-\ntionsofthemixedspin-1/2andspin-1Isingmodelonthedecorated Bethelattice\nwith the coordination number q= 4 and the antiferromagnetic next-nearest-\nneighbour interaction J′/|J|=−0.2. It has been demonstrated that the an-\ntiferromagnetic next-nearest-neighbour interaction competes with the nearest-\nneighbour interaction and hence, the weaker easy-plane single-ion anisotropy is\nthus generally needed in order to destroy the spontaneous order of FI. However,\none still detects temperature dependences of the total and sub lattice magneti-\nzations quite typical for FI whenever the single-ion anisotropy is se lected above\nits critical value Db/|J|=−1−J′/(2|J|) (see Fig. 8(a)). Contrary to this, both\nspontaneous sublattice magnetizations equal zero for any single- ion anisotropy\nbelow this boundary value D < D band the system surprisingly exhibits the\nunusual antiferromagnetic long-range order AFI with the non-ze ro staggered\nmagnetization of the spin-1/2 atoms as the relevant order parame ter (see Fig.\n8(b)). Hence, it actually turns out that the spin-1/2 Ising atoms f rom nearest-\nneighbour shells of the decorated Bethe lattice align antiparallel with respect to\neach other due to the antiferromagnetic next-nearest-neighbo ur interaction and\nowing to this fact, the decorating spin-1 atoms become frustrate d and all tend\ntowards their non-magnetic spin state Si= 0.\n16/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s109\n/s84/s32/s109\n/s66\n/s109\n/s65/s32/s109\n/s65/s32/s44/s32 /s109\n/s66/s32/s44/s32 /s109\n/s84/s32\n/s113 /s61/s54/s74/s39 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s48/s46/s50/s32\n/s40/s97/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74/s124/s68 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s48/s46/s56/s57\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s115\n/s109\n/s65\n/s109\n/s84/s32/s109\n/s66\n/s109\n/s65/s32/s109\n/s65/s32/s44/s32 /s109\n/s66/s32/s44/s32 /s109\n/s84/s32/s44/s32 /s109\n/s65/s113 /s61/s54\n/s74/s39 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s48/s46/s50/s32\n/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74/s124/s68 /s32/s47/s32 /s124/s74/s124/s32 /s61/s32/s45/s48/s46/s57/s49/s115\nFigure 9: Thermal variations of the total, sublattice and st aggered magnetizations for the\nmixed spin-1/2 and spin-1 Ising model on the decorated Bethe lattice with the coordination\nnumber q= 6, the fixed value of the antiferromagnetic next-nearest-n eighbour interaction\nJ′/|J|=−0.2 and two different values of the single-ion anisotropy: (a) D/|J|=−0.89; (b)\nD/|J|=−0.91.\nFor completeness, let us also confirm an existence of the reentran t phase\ntransitions by analyzing thermal dependences of the spontaneou s magnetization\nofthe mixed spin-1/2and spin-1Isingmodel onthe decoratedBeth e lattice with\nthe higher coordination number q= 6. For this purpose, we depict in Fig. 9\ntemperaturevariationsofthetotalandsublatticemagnetization sforonespecific\nvalue of the antiferromagneticnext-nearest-neighbourinterac tionJ′/|J|=−0.2\nand two different values of the single-ion anisotropy. In the former case with\nD/|J|=−0.89, one observes three successive reentrant phase transitions just\nin between two phases FI and PP in the following order FI-PP-FI-PP. On the\nother hand, another type of reentrance can be detected in the la tter case with\nD/|J|=−0.89, which reveals three consecutive phase transitions between AF I,\nFI and PP in the following order AFI-PP-FI-PP. It should be stresse d that\nthe latter reentrance would be misinterpreted as the reentrant p hase transition\nwith only two consecutive critical points if the staggered magnetiza tion would\nnotbetakenintoaccountasthepossibleorderparameteralongwit htheuniform\nmagnetization.\n4. Conclusion\nIn this work, we have exactly calculated magnetic properties of the mixed\nspin-1/2 and spin- SIsing model on the decorated Bethe lattice with a quite\ngeneral coordination number qwhen combining the decoration-iteration trans-\nformation with the exact recursion relations. The model Hamiltonian includes\nthe nearest-neighbour interaction between the spin-1/2 and spin -Satoms, the\nnext-nearest-neighbourinteraction between the spin-1/2 atom s and the uniaxial\nsingle-ionanisotropy. By making use ofthe decoration-iterationtr ansformation,\nwe have established a precise mapping equivalence between the mixed spin-1/2\nand spin- SIsing model on the decorated Bethe lattice and its corresponding\nspin-1/2 Ising model on the simple (undecorated) Bethe lattice sub sequently\n17treated with the aid of exact recursion relations. Furthermore, t he rigorous\nmapping theorems and exact spin identities have been employed in ord er to get\nthe precise results for both spontaneous sublattice magnetizatio ns as well as\nstaggered magnetization.\nOur main attention was focused on magnetic properties of the mixed spin-\n1/2 and spin-1 Ising ferrimagnet, which exhibits remarkably diverse critical\nbehaviour and compensation phenomena. The four different phase s have been\nfound in the ground-state phase diagram, two of them have the un iform mag-\nnetization and another two staggered magnetization as the order parameter.\nIn addition, it has been demonstrated that the investigated mixed- spin sys-\ntem may exhibit either reentrant phase transitions with two consec utive critical\npoints (for zero next-nearest-neighbour interaction) or three different types of\nreentrant phase transitions with three successive critical points (for non-zero\nnext-nearest-neighbour interaction). It is worthy to recall tha t the reentrance\nmay appear just for a sufficiently high coordination number q >4 of the under-\nlying Bethe lattice and this observation seems to be of a quite genera l validity as\nit has been already proved in the Ising-Heisenberg models on the diam ond-like\ndecorated Bethe lattices [47, 48]. Finally, it has been also illustrated t hat the\nnext-nearest-neighbour interaction may cause a presence of on e or two compen-\nsation points in the temperature dependence of the total magnet ization.\nLet us conclude our study by mentioning that the present formalism can\nbe rather straightforwardly extended for the mixed spin-1/2 and spin-SIsing\nmodelonthedecoratedBethelatticesinapresenceoftheextern almagneticfield\nand this exactly soluble model might display a quite intriguing magnetiza tion\nprocess including several non-trivial magnetization plateaus. In t his direction\nwill continue our further efforts.\nReferences\n[1] L. N´ eel, Ann. 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McGuire,3\nPramod Kumar,4Dino Sulejmanovic,2Jian He,5and Shiou-Jyh Hwu2\n1Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\n2Department of Chemistry, Clemson University, Clemson, South Carolina 29634-0973, USA\n3Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n4Indian Institute of Information Technology, Allahabad U.P. 211012, India\n5Department of Physics and Astronomy, Clemson University, Clemson, South Carolina 29634-0978,USA\n(Dated: March 2, 2022)\nResults of magnetic \feld and temperature dependent neutron di\u000braction and magnetization mea-\nsurements on oxy-arsenate Rb 2Fe2O(AsO 4)2are reported. The crystal structure of this compound\ncontains pseudo-one-dimensional [Fe 2O6]1sawtooth-like chains, formed by corner sharing isosce-\nles triangles of Fe3+ions occupying two nonequivalent crystallographic sites. The chains extend\nin\fnitely along the crystallographic b-axis and are structurally con\fned from one another via dia-\nmagnetic (AsO 4)3\u0000units along the a-axis, and Rb+cations along the c-axis direction. Neutron\ndi\u000braction measurements indicate the onset of a long range antiferromagnetic order below approxi-\nmately 25 K. The magnetic structure consists of ferrimagnetic chains which are antiferromagnetically\ncoupled with each other. Within each chain, one of the two Fe sites carries a moment which lies\nalong the b-axis, while the second site bears a canted moment in the opposite direction. Exter-\nnally applied magnetic \feld induces a transition to a ferrimagnetic state, in which the coupling\nbetween the sawtooth chains becomes ferromagnetic. Magnetization measurements performed on\noptically-aligned single crystals reveal evidence for an uncompensated magnetization at low mag-\nnetic \felds that could emerge from to a phase-segregated state with ferrimagnetic inclusions or from\nantiferromagnetic domain walls. The observed magnetic states and the competition between them\nis expected to arise from strongly frustrated interactions within the sawtooth chains and relatively\nweak coupling between them.\nPACS numbers: 75.50.Gg, 75.60.Jk, 75.25.-j, 71.27.+a, 61.05.F-\nI. INTRODUCTION\nLow-dimensional magnetic materials have drawn con-\ntinued attention in condensed matter physics, owing to\ntheir distinct electronic and magnetic properties. In par-\nticular, the oxyanion systems based on transition metal\n(M) oxides sublattices that are magnetically isolated\nby closed shell nonmagnetic oxyanions (SiO4\u0000\n4, PO3\u0000\n4,\nAsO3\u0000\n4) show great potential for exploring and charac-\nterizing new emergent phenomena.1This class of com-\npounds has been shown to exhibit a variety of magnetic\nground states ranging from ferromagnetism, to ferrimag-\nnetism and antiferromagnetism.2On the other hand, due\nto their impressive structural diversity in terms of archi-\ntecture these materials have shown potential applications\nin the \felds of energy storage, chemical sensors, catalysis\nand biomedical research.3{6\nRecent e\u000borts have been directed towards the synthe-\nsis of new members of the family of oxy-arsenates with\ngeneral formula A2M2O(AsO 4)2, where A= K, Rb, and\nMis a transition metal. These compounds exhibit in-\ntriguing one-dimensional triangular-shaped chains made\nof edge-sharing MO 6octahedra (see Fig. 1). Antifer-\nromagnetic sawtooth chain model (or \u0001-chain), as that\nrealized in these systems, is known to be highly frus-\ntrated and has drawn considerable attention from a the-\noretical standpoint.7{9For example, the ground state of\ntheS=1/2 sawtooth chain is two-fold degenerate witheither left pair or right pair of spin at each triangle form-\ning spin-singlet states. The lowest excitation in a chain\nwith periodic boundary conditions is given by a kink-\nantikink pair which has a dispersionless gap \u0001E '0.234J\nwhereJis the coupling between pairs of spins.7,8Exper-\nimental realization of magnetic lattices corresponding to\nthe sawtooth chain is only found in a very small num-\nber of compounds, including the delafossite YCuO 2:5,10\neuchroite Cu 2(AsO 4)(OH)\u00013H2O11and ZnL2S4olivines,\nwithL=Er, Tm, Yb.12\nWe report here on the magnetic behaviour of oxy-\narsenite Rb 2Fe2O(AsO 4)2, containing sawtooth chains of\nmagnetic Fe3+ions in high-spin state S=5/2. Although\nthis compound was \frst reported \fve years ago,13a thor-\nough investigation of its magnetic properties has not been\nconducted so far. The development of well-controlled\nmethods for the growth of high quality single crystals\nhas allowed us for detailed orientation-dependent mag-\nnetic measurements. Magnetization measurements have\nbeen complemented with x-ray and neutron di\u000braction\nmeasurements. Temperature and magnetic \feld depen-\ndent neutron di\u000braction measurements demonstrate con-\nclusively the development of a non-collinear antiferro-\nmagnetic ground state below approximately 25 K, which\nunder relatively low magnetic \felds, transforms to a ferri-\nmagnetic state. These transitions are also observed in dc\nmagentization measurements, which reveal strong mag-\nnetic anisotropy and complex \feld and temperature de-\npendent behaviors associated with the multiple magneticarXiv:1401.1539v1  [cond-mat.str-el]  7 Jan 20142\nFIG. 1: (Color online) (top) Polyhedral view of the extended\nstructure of Rb 2Fe2O(AsO 4)2showing the structural con-\n\fnement of the chains. (bottom) Partial structure of the\n[Fe4O14]1sawtooth chain de\fned by unequal base-base ( Jbb)\nand base-vertex ( Jbv) exchange interactions.\nstates.\nII. EXPERIMENTAL DETAILS\nSingle crystals of Rb 2Fe2O(AsO 4)2were grown using\n\rux methods in the RbCl/RbI (50:50 wt%) molten-salt\nmedia. The reactants KO 2(Alfa Aesar, 96.5%), Fe 2O3\n(Alfa Aesar, 99.945%) As 2O5(Alfa Aesar, 99.9%) were\nmixed and ground together in a nitrogen-purged dry box.\nThe reaction mixture was sealed in an evacuated fused-\nsilica ampoule and then heated to 650\u000eC for one day,\nfollowed by another heating to 800\u000eC at 1\u000eC/min for\ntwo days, slowly cooled to 300\u000eC at 0.05\u000eC/min, and\nthen furnace-cooled to room temperature. Brown colum-\nnar crystals (up to about 2.6 mm x 0.6 mm x 0.4 mm)\nwere recovered by washing the product with de-ionized\nwater using vacuum \fltration method. Single crystals\nwere physically examined and selected under an optical\nmicroscope equipped with a polarizing light attachment.\nThe crystals were then mounted on a glass \fber and sin-\ngle crystal x-ray di\u000braction data was collected using a\nRigaku Mercury CCD detector on an AFC-8S di\u000brac-\ntometer equipped with graphite monochromated Mo K \u000b\nradiation (\u0015= 0.7107 \u0017A). The structure was solved by di-rect methods using the SHELX-9714program and re\fned\non F2by least-squares, full-matrix technique.\nPolycrystalline samples were prepared by the solid\nstate reaction of stoichiometric amounts of Rb 2CO3\n(Alfa Aesar, 99.8%), Fe 2O3(Alfa Aesar, 99.945%) and\n(NH 4)H2AsO 4(Alfa Aesar, 99.9%). The mixture was\nloaded into an alumina crucible, heated to 800\u000eC at a\nrate of 2\u000eC/min, and held at that temperature for 2 days\nbefore being furnace cooled to room temperature. All\npolycrystalline samples were characterized using Rigaku\nUltima IV multipurpose X-ray di\u000braction system.\nTemperature and \feld-dependent magnetic measure-\nments were carried out with a Quantum Design Magnetic\nProperty Measurement System (MPMS). The measure-\nments were taken on collections optically co-aligned sin-\ngle crystals at temperatures from 2 K to 300 K in the\napplied \felds up to 50 kOe. Additional magnetization\nmeasurements were performed on polycrystalline sample\nusing a physical property measurement system (PPMS)\nat \felds up to 120 kOe.\nNeutron di\u000braction measurements were carried out us-\ning the HB2A high-resolution powder di\u000bractometer at\nthe High Flux Isotope Reactor (HFIR). Di\u000braction pat-\nterns were collected using the 2.408 \u0017A wavelength pro-\nduced by the (113) re\rection of a vertically focusing ger-\nmanium wafer-stack monochromator. The beam collima-\ntion was 120{400{60. More details about the HB2-A in-\nstrument and data collection strategies can be found in\nRef. 15. For the measurements, polycrystalline samples\nwith a total mass of approximately 5 grams were com-\npacted in pellets to prevent grain reorientation by the\napplied \feld. The pellets were loaded into a cylindrical\nvanadium can, and placed in a vertical-\feld cryomagnet.\nData were collected at various temperatures from 40 K\nand down to 4 K, and magnetic \felds up to 40 kOe.\nRietveld re\fnements were carried out with the FullProf\nSuite program.16\nIII. RESULTS AND DISCUSSION\nA. Crystal structure\nThe single crystal X-ray and neutron powder di\u000brac-\ntion data were well \ft by using an orthorhombic struc-\nture of space group Pnma , similar to that proposed\nby Chang et al. .13The re\fned structural parameters of\nRb2Fe2O(AsO 4)2, such as atomic coordinates and dis-\nplacement parameters are listed in Table I, while se-\nlected bond lengths and angles are given in Table II. Ri-\netveld analysis indicate that the powder samples contain\na Fe 2O3impurity phase in an amount of less than 3%\nweight.\nThe structure of Rb 2Fe2O(AsO 4)2consists of iron-\narsenate layers parallel to the abplane, separated by\nRb+cations. In each layer, there are two crystallo-\ngraphically distinct Fe3+sites (Fe1 and Fe2) which are\nbridged together to form triangle-based chains, as shown3\nTABLE I: Re\fned structural parameters of Rb 2Fe2O(AsO 4)2\nfrom single-crystal x-ray data collected at room temperature.\nAtom (Wyck.) x y z Ueq\nRb1 (4 c) 0.8888(1) 1/4 0.8628(1) 0.023(1)\nRb2 (4 c) 0.0379(1) 1/4 0.2679(1) 0.028(1)\nFe1 (4 c) 0.2844(1) 1/4 0.4347(1) 0.009(1)\nFe2 (4 b) 0 0 1/2 0.009(1)\nAs1 (4 c) 0.8204(1) 1/4 0.6426(2) 0.010(1)\nAs2 (4 c) 0.6758(1) 1/4 0.4440(2) 0.008(1)\nO1 (4 c) 1.0580(1) 1/4 0.4375(3) 0.009(1)\nO2 (4 c) 0.9610(1) 1/4 0.5760(3) 0.010(1)\nO3 (8 d) 0.7666(1) 0.0108(1) 0.4785(2) 0.015(1)\nO4 (4 c) 0.4923(1) 1/4 0.4744(4) 0.014(1)\nO5 (8 d) 0.7043(1) 0.0127(1) 0.6338(2) 0.018(1)\nO6 (4 c) 0.6843(1) 1/4 0.3556(3) 0.018(1)\nO7 (4 c) 0.9130(2) 1/4 0.7197(3) 0.025(1)\nPnma ,a= 8.5331(2) \u0017A,b= 5.7892(2) \u0017A,c= 18.611(4) \u0017A\nR1= 0.0347, wR 2= 0.0936, GOF = 1.038\nTABLE II: Selected bond distances and angles in\nRb2Fe2O(AsO 4)2, as determined from x-ray analysis.\nFe1 - O1 (x 1) 1.932(6) \u0017A Fe2 - O1 (x 2) 1.922(4) \u0017A\nFe1 - O3 (x 2) 2.254(4) \u0017A Fe2 - O2 (x 2) 2.051(4) \u0017A\nFe1 - O4 (x 1) 1.923(6) \u0017A Fe2 - O3 (x 2) 2.032(4) \u0017A\nFe1 - O5 (x 2) 1.986(4) \u0017A\nd(Fe1 - Fe2) = 3.076(1) \u0017A,Jbvvia O1, O3\nFe1 - O3 - Fe2 = 91.5(1)\u000eFe1 - O1 - Fe2 =105.9(1)\u000e\nd(Fe2 - Fe2) = 2.895(1) \u0017A,Jbbvia O1, O2\nFe2 - O1 - Fe2 = 97.7(1)\u000eFe2 - O2 - Fe2 = 89.7(1)\u000e\nin Fig. 1. These [Fe 4O14]1chains extend along the crys-\ntallographic b-axis, and are structurally con\fned from\none another via diamagnetic As5+cations in AsO 4tetra-\nhedral units along the crystallographic a-axis (Fig. 1).\nDue to the non-magnetic nature of As5+cation, the mag-\nnetic interactions between adjacent [Fe 4O14]1\u0001-chains\nvia Fe-O-As-O-Fe connection are expected to be subdom-\ninant compared to the interactions within the chains.\nA comparison of the re\fned bonds and angles in\nRb2Fe2O(AsO 4)2indicates di\u000berent interactions occur-\nring between spins on the base ( Jbbcoupling) and be-\ntween the base-vertex ( Jbv) of the triangles. The Fe1-Fe2\nand the Fe2-Fe2 distances are 3.076(1) and 2.895(1) re-\nspectively. The bond angles Fe2{O1{Fe2 and Fe2{O2{\nFe2 de\fning the super-exchange paths along the base\nof triangles ( Jbb) are 97.7(3)\u000eand 89.7(7)\u000e, respectively,\nwhile the base-vertex angles ( Jbv) Fe1-O1-Fe2 and Fe1-\nO3-Fe2 are 105.9(1)\u000eand 91.5(1)\u000e.\nFIG. 2: Evolution of the magnetic susceptibility (M/H) as a\nfunction of temperature, for H = 100 Oe applied parallel and\nperpendicular to the b-axis. The inserted picture displays the\nsingle crystals assembly used in the measurements. The inset\nshows the inverse susceptibility versus temperature above the\ntransition temperature, and a linear (Curie-Weiss) \ft.\nB. Macroscopic magnetic measurements\nMagnetization measurements were performed on single\ncrystals with the b-axis (chain direction) aligned parallel\nand perpendicular to the magnetic \feld. Due to the small\nsize of the crystals, several specimens were optically co-\nmounted on a Kapton tape as displayed in Fig. 2. Two\nsamples were prepared with total masses of 4.0 and 8.4\nmg. The measured magnetization was con\frmed to be\nconsistent between the two samples. The temperature\ndependence of magnetic susceptibility ( \u001f=M=H ) mea-\nsured with an applied \feld of 100 Oe, is shown in Fig. 2.\nThe sharp rise in magnetization data reveals the onset\nof long range magnetic order below approximately 25 K.\nThere is strong anisotropy between the two crystal ori-\nentations near the transition temperature. The inverse\nsusceptibilty (H/M) for temperature well above this tem-\nperature is shown in the inset, along with a \ft using the\nCurie Weiss model, M/H = C/(T - \u0002 CW). The data are\nnot strictly linear over this wide temperature range. A\nbetter \ft can be obtained by including a temperature in-\ndependent term; however, such a \ft returns unreasonable\nvalues for the \ftting parameters. The observed devia-\ntion from Curie Weiss behavior may indicate some short\nranged or low dimensional magnetic ordering above the\nlong-ranged ordering transition at 25 K. Using the \ft\nshown in Fig. 2, an e\u000bective moment of 6.7(1) \u0016Band a\nWeiss temperature of -510 K are determined. Restrict-\ning the \ftted range to the more linear region between\n250 and 350 K gives an e\u000bective moment of 6.3 \u0016Band\na Weiss temperature of -426 K. These e\u000bective moments\nare somewhat larger than the spin only value for high4\nFIG. 3: (Color online) Magnetization curves for co-align\nsingle crystals of Rb 2Fe2O(AsO 4)2measured on warming at\nthe indicated \felds after \feld-cooling from 50 K. Results are\nshown for H parallel and perpendicular to the b-axis along\nwhich the sawtooth chain of Fe are oriented.\nFIG. 4: Zero-\feld-cooled magnetization curves measured in\n10 Oe magnetic \feld (H kb) upon warming from 2 to 50 K,\nafter having reduced the \feld to zero at 50 K from (a) 10 kOe,\nwhich leaves a small negative residual \feld, and (b) -10 kOe\nwhich leaves a small positive residual \feld.\nspind5, 5.9\u0016B. The large and negative Weiss tempera-\ntures indicates strong antiferromagnetic interactions.\nAdditional magnetization measurements were per-\nformed near the ordering temperature. Data were col-\nlected upon warming from 2 to 50 K in zero-\feld-cooled\n(ZFC) and \feld-cooled (FC) modes. Figure 3 displays\nthe FC results for the four selected \felds of 10 Oe, 100\nOe, 1 kOe and 10 kOe. The temperature dependence\nchanges signi\fcantly as the applied \feld is varied, and\nthe data show strong anisotropy between the two mea-\nsurement directions at all \felds. For H along the direc-tion of the sawtooth chains (the b-axis), a sharp cusp,\nsuggesting a canted antiferromagnetic order, is seen at\nthe magnetic ordering transition for H \u00141 kOe, while\nat 10 kOe the susceptibility increases and then saturates\nupon cooling, similar to the behavior expected at a ferro-\nmagnetic transition. For H aligned perpendicular to the\nb-axis, a smaller cusp is seen at all of the \felds shown.\nThe qualitative di\u000berence between the behavior in the\ntwo directions at H = 10 kOe is due to a \feld induced\nmagnetic transition which occurs at a di\u000berent \feld in\nthe di\u000berent orientations and is discussed in more detail\nbelow.\nNegative magnetic susceptibilities were observed in\nsome ZFC measurements at low \felds. An example is\nshown in Figure 4(a), where the magnetization is neg-\native at low temperatures, and switches sign abruptly\nupon warming as the phase transition is approached.\nThis was observed at \felds up to 100 Oe and in both\norientations of the \feld. Temperature induced magne-\ntization reversal phenomena is well known to occur in\nferrimagnetic systems with strong magnetic anisotropy\ndisplaying a partial cancellation of antiferromagnetically\ncoupled magnetic sublattices with di\u000berent magnitudes\nof magnetic moments and/or di\u000berent temperature de-\npendence of magnetization.17,18This scenario is certainly\nplausible in our system where Fe3+occupies two non-\nequivalent sites and are exposed to di\u000berent molecular\n\felds. However, a negative ZFC magnetization curve has\nalso been attributed in some cases to the existence of a\nsmall trapped \feld in the superconducting magnet.19,20\nSuch an experimental artefact is likely to occur in some\nferromagnetic materials with strong anisotropy and a\nsigni\fcant coercive \feld that can drastically a\u000bect the\nZFC process. To investigate this possibility we per-\nformed experiments in which we reducing the \feld to\nzero from both 10 kOe and -10 kOe using the MPMS\n\"no-overshoot\" mode at 50 K, then cooling in \"zero \feld\"\nto 2 K, applied 10 Oe, and then measured upon warm-\ning. As discussed in Ref.19, the sign of the trapped\n\feld is opposite in sign when reducing the \feld to zero\nfrom a positive or negative \feld, and hence, a starting\n\feld of 10 kOe is expected to generate a small nega-\ntive trapped \feld, while -10 kOe will create a small pos-\nitive trapped \feld. Our measurements on a Pd reference\nsample using similar conditions con\frm this and indicate\nthe magnitude of the \feld is several Oe. As apparent\nin Fig. 4, the ZFC magnetization curves are indeed fol-\nlowing the direction of the trapped \feld in the super-\nconducting magnet. This prompts us to conclude that\nthe observed negative magnetization under ZFC condi-\ntion is not a ferrimagnetic e\u000bect, but rather a feature\ninduced by the presence of pinned uncompensated spins\noriginated from inhomogeneities,20or from the antifer-\nromagnetic domain walls,21,22or slight canting in the\nantiferromagnetic structure, although no canting could\nbe resolved in the neutron di\u000braction results presented\nbelow. We note that several ZFC measurements using\na \feld of 10 Oe were also conducted after resetting or5\nFIG. 5: (Color online) Magnetic \feld dependence of magne-\ntization for oriented single crystals of Rb 2Fe2O(AsO 4)2with\nHkbat four di\u000berent temperatures. Shown in the insert is\nthe isothermal magnetization curve at 2 K for H ?b.\nFIG. 6: Isothermal magnetization data measured on a com-\npacted powder sample at T = 4 K. The upper inset shows\na blow up of the region near the metamagnetic transition (\n\u00194 kOe). The lower insert displays the magnetization data\nmeasured upon reducing the \feld from 120 kOe at 5 K.\nquenching the magnet at 50 K. Some of these measure-\nments resulted in behavior like that shown in Fig. 4(a),\nand others like Fig. 4(b). This indicates that the residual\n\feld, which is expected to be very small ( <1 Oe) after\nresetting the magnet had unpredictable polarity and that\nthe low temperature state that is adopted by the mate-\nrial upon cooling through the transition temperature is\nextremely sensitive to even very small magnetic \felds.\nThe isothermal magnetization curves at 2 K, 10 K and50 K in magnetic \felds up to 15 kOe applied along baxis\nare shown in Fig. 5. The presence of hysteresis in the\nmagnetization data demonstrates the presence of a fer-\nroic component in the high-\feld magnetic phase. Apart\nfrom a hysteresis separating the ascending and descend-\ning branches, there is a readily noticeable stepwise mag-\nnetic behavior. The metamagnetic transition occurs near\n3 kOe at T = 2 K and shifts to 6 kOe at T = 10 K.\nWhen the \feld is applied perpendicular to the baxis the\nmetamagnetic phase transition occurs at a much higher\n\feld of 30 kOe, at T = 2 K (as shown in the inset of\nFig. 5). As expected, the magnetization measured on\ncompacted powder sample shows a more gradual \feld\ntransition, arising at approximately 4 kOe at a tempera-\nture of 4 K (see Fig. 6). It is also worth noting that the\nmagnetization curves does not reach the full saturation\neven for the highest applied external \felds of 120 kOe.\nThe largest absolute value of the magnetization, of about\n0.3\u0016B/Fe, is smaller by over an order of magnitude than\nthe predicted value corresponding to a parallel alignment\nof the Fe3+magnetic moments ( gS= 5\u0016B).\nC. Neutron di\u000braction\n1. Measurements in zero magnetic \feld\nNeutron di\u000braction patterns collected in the paramag-\nnetic state at 40 K are well described by the structural\nmodel described in the previous section. Upon cooling\nbelow 25 K, magnetic Bragg peaks (e.g. (100), (102))\nstart to appear at low angles (see Fig. 7) indicating a\ntransition to an antiferromagnetic long-rage order. The\nevolution with temperature of (100) magnetic peak inten-\nsity is shown in Fig. 8(upper panel). Near the transition\ntemperature the order parameter follows a power law be-\nhaviour.\nRepresentational analysis was conducted to determine\nof the symmetry allowed magnetic structures that can\nresult from a second-order magnetic phase transition\ngiven the crystal structure and the propagation vec-\ntork=(0, 0, 0) of the magnetic ordering. These cal-\nculations were carried out using the program SARA h-\nRepresentational Analysis.23There are eight possible\nirreducible representations (IRs) associated with the\nPnma space group and k=(0, 0, 0). The decompo-\nsition of the magnetic representation for the Fe1 site\n(:715; :25; :565) is \u0000 Mag = 1\u00001\n1+ 2\u00001\n2+ 2\u00001\n3+ 1\u00001\n4+\n1\u00001\n5+ 2\u00001\n6+ 2\u00001\n7+ 1\u00001\n8, while for the Fe2 site (0 ;0; :5) is\n\u0000Mag= 3\u00001\n1+0\u00001\n2+3\u00001\n3+0\u00001\n4+3\u00001\n5+0\u00001\n6+3\u00001\n7+0\u00001\n8. The\nlabeling of the IRs follows the scheme used by Kovalev.24\nTable III presents only two of representations (\u00001, \u00005)\nthat are relevant for our discussion here.\nThe 4 K di\u000braction data was \ft by a magnetic struc-\nture model based on the representation \u0000 1, de\fned in\nTable III. \u0000 1representation is equivalent to the Shub-\nnikov magnetic space group Pnma . Within this model,\nthe Fe1 moments are constrained to lie along the b-axis6\nFIG. 7: (Color online) Upper panel: Rietveld plot of neuron\npowder di\u000braction data collected at 4 K in zero magnetic \feld.\nThe inset displays a comparison of the low- Qdata, near the\n(100) magnetic peak, measured at 4 K and 30 K. Lower panel:\nRietveld plot of neuron data measured at 4 K in magnetic \feld\nof 40 kOe.\ndirection and are ferromagnetically coupled to each other\ninside the chain. In contrast, the Fe2 moments have\ncomponents in all three coordinate directions, and form\nan undulating pattern with the relative angles between\nneighboring spins of approximately 60\u000e. The spin canting\noccurs in such a way that they remain mostly con\fned\nto the plane of the sawtooth chain. The projection along\nthe chain axis of the Fe2 moment is oriented in oppo-\nsite direction to the Fe1 moment, thus producing a non-\nzero net moment. The polarity of the chains alternates\nalong thec-axis direction, giving an overall antiferromag-\nnetic arrangement. A stereographic view of the magnetic\nstructure is illustrated in Fig. 9(a).\nThe re\fned magnitudes of the ordered moments for\nFe1 and Fe2 sites are 3.64(6) and 3.19(8) \u0016B, respec-\ntively. The projections of Fe2 magnetic moment on the\nthree axes of coordinates are: \u0016a= 1.19(8)\u0016B,\u0016b=\nFIG. 8: (Color online) Upper panel: Temperature depen-\ndence of the (100) and (101) peaks intensities measured in\nzero magnetic \feld and 40 kOe, respectively. The lines repre-\nsent a guide to the eye indicating power law behavior. Lower\npanel: Variation with magnetic \feld of the (100) and (101)\nmagnetic peaks at 4 K. The dashed line is a guide to the\neye. The vanishing of (100) re\rection at the expense of (101)\nmagnetic is caused by an induced transition from antiferro-\nmagnetic to ferrimagnetic state.\n2.88(4)\u0016Band\u0016c= 0.68 (9)\u0016B. The re\fned values are\nsigni\fcantly lower than those expected for the Fe3+ions\nwith the e\u000bective spin S=5/2. Considering the high value\nof the frustration parameter f=j\u0002CWj=TNof approx-\nimately 18, one can infer that the reduced static mag-\nnetic moment is due to the presence of spin frustration.\nThe frustration is reasonably expected in this system due\nto competing superexchange interactions within the tri-\nangular chains. Furthermore, the non-collinear spin ar-\nrangement is reminiscent of the predicted non-collinear\nspin con\fgurations in triangular-lattice Heisenberg anti-\nferromagnet where the frustration of the three nearest-\nneighbor spins on a triangular plaquette is resolved by a\n120\u000erotation of neighboring spins.25\nPerhaps the most important aspect that needs to be\nemphasized here is the fully compensated character of\nthe magnetic structure, which makes the magnetiza-\ntion reversal process observed in the ZFC magnetiza-\ntion data di\u000ecult to comprehend. Ferrimagnetic-like\nresponse has previously been observed only in a hand-\nful of antiferromagnetic compounds where a weak fer-\nromagnetic moment is due to the tilting of sublattice\nmagnetizations.26{29One of the mechanisms that has\nbeen proposed to explain the magnetization reversal in7\nTABLE III: Basis vectors (BV) of the IRs \u0000 1and \u0000 5, of the\nspace group Pnma with k= (0, 0, 0). The equivalent Fe1\natoms of the nonprimitive basis are de\fned according to 1:\n(:715; :25; :565), 2: ( :215; :25; :934), 3: ( :284; :75; :434), 4:\n(:784; :75; :065). The Fe2 atoms are de\fned as follows: 1:\n(0;0; :5), 2: ( :5; :5;0), 3: (0 ; :5; :5), 4: ( :5;0;0).\nIR BV Atom Fe1 Fe2\nmkamkbmkcmkamkbmkc\n\u00001 1 1 0 1 0 0 1 0\n2 0 -1 0 0 -1 0\n3 0 1 0 0 1 0\n4 0 -1 0 0 -1 0\n 2 1 1 0 0\n2 1 0 0\n3 -1 0 0\n4 -1 0 0\n 3 1 0 0 1\n2 0 0 -1\n3 0 0 -1\n4 0 0 1\n\u00005 4 1 0 1 0 0 1 0\n2 0 1 0 0 1 0\n3 0 1 0 0 1 0\n4 0 1 0 0 1 0\n 5 1 1 0 0\n2 -1 0 0\n3 -1 0 0\n4 1 0 0\n 6 1 0 0 1\n2 0 0 1\n3 0 0 -1\n4 0 0 -1\nantiferromagnets relies on the competition between sin-\ngle ion anisotropy and the DzyaloshinskyMoriya (D-M)\ninteraction. For our system, the antisymmetric DM ex-\nchange coupling can exist between the magnetically or-\ndered Fe2 ions located at 4 bsite ofPnma , and the mag-\nnetic spin corresponding to that site has indeed been de-\ntermined to be canted. However, because the chains are\nantiferromagnetically coupled, any ferromagnetic compo-\nnent due to canting is compensated by the nearest neigh-\nbor chain. These results come to con\frm our conjecture\nin the previous section that the negative magnetization\nis not an inherent ferrimagnetic feature, but likely an ef-\nfect of an uncompensated magnetization originating from\nantiferromagnetic domain walls or from other magnetic\ninhomogeneities present in the sample.\n(a)\n(b)\nFIG. 9: (Color online) (a) Schematic view of the magnetic\nstructure of Rb 2Fe2O(AsO 4)2at 4 K and zero magnetic \feld.\nThe Fe1 magnetic moments are collinear with the b-axis, while\nthe Fe2 moments are forming undulating patterns, running in\nopposite direction with respect to the Fe1 moments. The fer-\nrimagnetic chains are coupled antiferromagnetically to neigh-\nbouring chains in the cdirection (b) Magnetic structure at 4\nK and applied magnetic \fled of 40 kOe. In contrast to the\nzero-\feld magnetic order, the polarity of the ferrimagnetic\nchains remains the same throughout the lattice yielding an\noverall ferrimagnetic ordering.\n2. Measurements in external magnetic \felds\nTo understand the e\u000bect of magnetic \feld on the mag-\nnetic structure of Rb 2Fe2O(AsO 4)2, polycrystalline sam-\nple in the form of pressed pellets has been measured un-\nder di\u000berent magnetic \felds up to 40 kOe. The di\u000braction\ndata measured at 4 K reveals a progressive decrease in\nthe amplitude of (100) magnetic peak with increasing the\napplied magnetic \feld, and at the same time, an increase\nin intensity of the (101) peak. The change in peaks in-8\ntensities only manifests at low momentum transfer ( Q),\nin accordance with the Fe3+magnetic form factor. The\nfact that the high- Qre\rections are not a\u000bected by the\nmagnetic \feld gives assurance that the sample grains do\nnot re-align in the direction of the \feld. As displayed\nin Fig. 8(lower panel), the (100) peak intensity starts\nto decrease just above 3 kOe, and it nearly vanished at\n40 kOe. The (101) peak follows a very similar trend to\nthat seen for the isothermal magnetization curve mea-\nsured for the powder sample in the magnetically ordered\nstate at 4 K, in Fig. 6. The intensity of the (101) peak\nhas also been monitored as a function of temperature in a\nconstant magnetic \feld of 40 kOe. The order parameter\nversus temperature shown in Fig. 8(upper panel) follows\na power-law dependence, similar as that observed for the\n(100) peak in zero magnetic \feld.\nRietveld re\fnement has been performed using the\ndi\u000braction data collected at 40 kOe and 4 K, and the\n\ft result is presented in Fig. 7. The magnetic scattering\nhas been well reproduced by a ferrimagnetic model based\non the irreducible representation \u0000 5given in Table III.\nThis representation is equivalent to the Shubnikov mag-\nnetic space group Pn0ma0. In this new magnetic state\nall chains exhibit the same polarity, with all symmetry\nequivalent Fe ions having the b-axis component of mag-\nnetic moment aligned in the same direction. On the other\nhand, Fe1 and Fe2 sub-lattices maintain opposite mag-\nnetizations while not compensating each other. Another\ne\u000bect of the magnetic \feld is the change in the canting di-\nrection of Fe2 spins with respect to the local plane of the\nsawtooth-chain. The spins rotate around the b-axis from\nan in-plane canting to an out-of-plane canting. The rel-\native angle between Fe2 and Fe1 moments remains close\nto 120\u000e. The high-\feld magnetic structure is shown in\nFig. 9(b).\nThe magnitude of the ordered magnetic moments at\n40 kOe is determined to be very close to that observed\nfor the antiferromagnetic state in zero magnetic \feld.\nThe re\fned Fe1 moment is 3.5(1) \u0016B, while the ampli-\ntude of the Fe2 moment is 3.1(1) \u0016Bwith the three axes\ncomponents \u0016a=0.5(1)\u0016B,\u0016b= 2.76(6)\u0016Band\u0016c=\n1.3(1)\u0016B. These values appear to rule out the possibility\nthat the external magnetic \feld may polarize additional\ndisordered Fe spins into the novel ordered state. On the\nother hand, the net magnetization resulted from such am-\nplitudes is about 0.37 \u0016Bper Fe3+, which is reasonably\nconsistent to what has been determined from magnetiza-\ntion measurements (see Figs. 5 and 6). Interestingly, the\nmagnetization data shown in Fig. 5 suggest that at su\u000e-\nciently low temperatures, the \feld induced ferrimagnetic\nstate can be maintained even after reducing the \feld back\nto zero.\nIt appears that the metamagnetic transition corre-\nsponds to the transition from the antiferromagnetic to\nferrimagnetic state where all magnetic couplings between\nadjacent sawtooth chain become exclusively ferromag-\nnetic while the intra-chain exchange interactions remain\nnearly unaltered. The \feld induced ferrimagnetic order-ing provides an interesting insight into the tendency of\nthe system to produce a net ferromagnetic component,\nand opens up the question whether local ferrimagnetic\ncluster may start to form within the antiferromagnetic\nhost even at very low magnetic \felds. The contribu-\ntion to the scattering of such weak uncompensated mag-\nnetic order could be well below the detection limit of our\nneutron scattering measurements. Magnetic clustering\ncould plausibly explain the observed negative magnetiza-\ntion but it implies the existence of inhomogeneities in the\nsample. Note that in our case the inhomogeneities would\nlikely have their origin in oxygen vacancies, which could\nnot be either con\frmed or ruled out by our structural\nre\fnement. As mentioned before, the uncompensated\nspins might also result from antiferromagnetic domain\nwalls, as demonstrated in the case of the HoMnO 3multi-\nferroic system by means of polarized small angle neutron\nscattering experiments.21\nIV. SUMMARY\nThe magnetic properties of the new oxy-arsenate\nRb2Fe2O(AsO 4)2have been studied by means of magne-\ntization and neutron powder di\u000braction measurements.\nThe system structure consists of triangle-based magnetic\nchains made of S= 5/2 Fe3+ions occupying two crys-\ntallographically distinct tetrahedral sites (Fe1 and Fe2).\nMagnetization measurements indicate a complex mag-\nnetic behavior with a temperature induced magnetization\nreversal at low magnetic \felds, and a step-like metamag-\nnetic transition. Neutron di\u000braction experiments per-\nformed in zero magnetic \feld reveal an antiferromag-\nnetic long-range magnetic order below the N\u0013 eel tem-\nperature of about 25 K. The magnetic structure con-\nsists of ferrimagnetic sawtooth chains which are mutually\ncompensating each-other along the c-direction. Within\neach chain, the Fe1 moments are collinear lying along\nthebdirection, while the Fe2 moments are reversely\ncanted by approximately 30\u000eforming a zigzag pattern.\nA reduced static magnetic moments of Fe has been at-\ntributed to a magnetic frustration caused by the trian-\ngle geometry. Neutron data collected under externally\napplied magnetic \feld reveal a transition from an an-\ntiferromagnetic to a ferrimagnetic state. In the ferri-\nmagnetic state the individual chains remain ferrimag-\nnetic with similar spin topology but their coupling be-\ncomes exclusively ferromagnetic. The ferrimagnetic or-\ndering at at magnetic \felds greater than 3 kOe cannot\nbe invoked to explain the negative ZFC magnetization\nobserved in the macroscopic study. Instead, the present\ndata suggests that the negative magnetization could be\nan e\u000bect of a phase-segregated state with ferrimagnetic\ninclusions, or of an uncompensated magnetization arising\nwithin antiferromagnetic domain walls. On the basis of\nour results, the sawtooth-like lattice model displayed by\nthe Rb 2Fe2O(AsO 4)2system deserves further experimen-\ntal and theoretical investigations to fully understand its9\nmagnetic behaviour. While the magnetic phase segrega-\ntion antiferromagnetic domain walls are at present only\nhypothetical scenarios, future polarized neutron scatter-\ning experiments able to provide increased sensitivity to\nweek ferromagnetism may shed light on the origin of the\nnegative magnetization phenomenon.\nAcknowledgments\nWork at the Oak Ridge National Laboratory, was spon-\nsored by the Scienti\fc User Facilities Division (neutrondi\u000braction) and Materials Sciences and Engineering Divi-\nsion (magnetization measurements), O\u000ece of Basic En-\nergy Sciences, US Department of Energy (DOE). The au-\nthors acknowledge the \fnancial support by grants from\nthe National Science Foundation: DMR-0706426, CHE-\n9808165 and CHE-9808044. L.D.S. and D.S. travel to\nORNL was supported by DOE through the EPSCoR\nGrant, DE-FG02-08ER46528.\n\u0003Electronic address: garleao@ornl.gov\n1S-J. Hwu, Chem. Mater. 10, 2846 (1998).\n2W. L. Qeen, S.-J. Hwu, L. Wang, Angew. Chem. Int. 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Sundaresan, Solid State Commun. 150, 1162\n(2010).\n20A. A. Belik, Inorg. Chem. 52, 8529 (2013).\n21B. G. Ueland, J. W. Lynn, M. Laver, Y. J. Choi, and S.-W.\nCheong, Phys. Rev. Lett. 104, 147204 (2010).\n22M. Bode, E. Y. Vedmedenko, K. von Bergmann, A. Ku-\nbetzka, P. Ferriani, S. Heinze and R. Wiesendanger, Nat.\nMat. 5, 477 (2006).\n23A. S. Wills, Physica B 276, 680 (2000), Program available\nfrom www.ccp14.ac.uk.\n24O. V. Kovalev, Representations of the Crystallographic\nSpace Groups Edition 2 (Gordon and Breach Science Pub-\nlishers, Switzerland, 1993).\n25L. Capriotti, A. E. Trumper, S. Sorella, Phys. Rev. Lett.\n82, 3899 (1999).\n26H. Kageyama, D. I. Khomskii, R. Z. Levitin, M. M. Mark-\nina, T. Okuyama, T. Uchimoto, A. N. Vasilev, J. Magn.\nMagn. Mater. 262, 445 (2003).\n27R. P Singh and C. V. Tomy, J. Phys.: Condens. Matter\n20, 235209 (2008).\n28Y. Ren, T. T. M. Palstra, D. I. Khomskii, A. A. Nugroho,\nA. A. Menovsky, G. A. Sawatzky, Phys. Rev. B 62, 6577\n(2000).\n29Y. Kimishima, Y. Chiyanagi, K. Shimizu and T. Mizuno\nJ. Magn. Magn. Mater. 210, 244 (2000)."    },    {        "title": "2205.14346v2.Magnetic_collapse_in_Fe__3_Se__4__under_high_pressure.pdf",        "content": "Academic Editors: Liwei Geng and\nYongke Yan\nPublisher’s Note: MDPI stays neutral\nwith regard to jurisdictional claims in\npublished maps and institutional affil-\niations.Article\nMagnetic Collapse in Fe 3Se4under High Pressure\nLyudmila V . Begunovich1,2\n, Maxim M. Korshunov1,2,*\nand Sergey G. Ovchinnikov1,2\n1Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Akademgorodok 50/38,\n660036 Krasnoyarsk, Russia; lyuda.illuzia@gmail.com (L.V .B.); sgo@iph.krasn.ru (S.G.O.)\n2Siberian Federal University, Svobodny Prospect 79, 660041 Krasnoyarsk, Russia\n*Correspondence: mkor@iph.krasn.ru\nAbstract: Electronic structure and magnetic properties of Fe 3Se4are calculated using the density func-\ntional approach. Due to the metallic properties, magnetic moments of the iron atoms in two nonequivalent\npositions in the unit cell are different from ionic values for Fe3+and Fe2+and are equal to M1=2.071 mB\nand M2=\u00002.042 mB, making the system ferrimagnetic. The total magnetic moment for the unit cell is\n2.135 mB. Under isotropic compression, the total magnetic moment decreases non-monotonically and\ncorrelates with the non-monotonic dependence of the density of states at the Fermi level N(EF). For 7%\ncompression, the magnetic order changes from the ferrimagnetic to the ferromagnetic. At 14% compres-\nsion, the magnetic order disappears and the total magnetic moment becomes zero, leaving the system in a\nparamagnetic state. This compression corresponds to the pressure of 114 GPa. The magnetic ordering\nchanges faster upon application of an isotropic external pressure due to the sizeable anisotropy of the\nchemical bondings in Fe 3Se4. The ferrimagnetic and paramagnetic states occur under pressures of 5.0 and\n8.0 GPa, respectively. The system remains in the metallic state for all values of compression.\nKeywords: band structure; magnetic moment; DFT; pressure; ferrimagnet; ferromagnet; iron selenide\nPACS: 62.50.-p; 71.15.Mb; 75.50.Gg; 75.50.Bb\n1. Introduction\nMagnetic collapse associated with the disappearance of magnetic moments in 3dions is\nobserved in many insulating transition metal oxides (Mn–O, Fe–O, Co–O and Ni–O systems).\nA review of experimental data for iron oxides [ 1] revealed that the spin crossover between the\nhigh-spin and low-spin states of the cation in most cases takes place with increasing pressure\nand critical pressure is close to 50–70 GPa. The crystals with Fe2+ions have a low-spin state\nwith a zero spin ( S=0) that leads to the appearance of a nonmagnetic phase. In the case\nof crystals with Fe3+ions, the low-spin state exhibit S=1/2, so the magnetic state can be\npreserved, albeit at a lower critical temperature, such as in FeBO 3[2]. For magnetite Fe 3O4\ncontaining both Fe2+and Fe3+ions, a new nonmagnetic phase was experimentally found [ 3,4]\nat pressures above 25 GPa and room temperature.\nThe iron selenides FeSe x(1\u0014x\u00141.33) form phases with iron vacancies that crystallize\ninto structures derived from the hexagonal NiAs-type structure. Among them, two compounds,\nFe7Se8and Fe 3Se4, have superstructures with ordered Fe vacancies [ 5]. Fe 7Se8has a hexagonal\nstructure whereas Fe 3Se4has a monoclinic structure isomorphic to Cr 3X4(X = S, Se, Te). In\nan Fe 3Se4unit cell, the vacancies of Fe appear in every second iron layer. The presence of\nordered Fe vacancies facilitates the appearance of the ferrimagnetic state in Fe 3Se4[6–9]. The\nexperimental values of the total magnetic moment are in the range from 0.69mBto1.17mBper\nformula unit (f.u.) [ 6–8]. The magnetic moments on Fe ions estimated within the framework of\nthe ionic model [ 7] are too large ( 3.25mBfor site 1 and 1.94mBfor site 2) and are not validated by\nexperimental data [ 7,9,10]. Neutron diffraction on Fe 3Se4gives smaller effective spin values for\ntwo Fe positions, S1=1.08mBandS2=0.71mB[10]. In addition, a study by Mössbauer revealedarXiv:2205.14346v2  [cond-mat.mtrl-sci]  4 Jul 20222 of 14\nthe very low average internal magnetic fields for Fe sites [ 9], which do not correspond to regular\nhigh-spin Fe3+or Fe2+states. All of the abovementioned studies indicate the delocalization of\n3delectrons of iron ions and the inapplicability of the model of localized magnetic moments to\nthe system.\nIn the series Fe 3O4–Fe 3S4–Fe 3Se4, on the one hand, the presence of two non-equivalent\npositions of cations and ferrimagnetic properties are preserved, on the other hand, an increase\nin covalence enhances the metallic properties. Despite the structural similarity between greigite\nFe3S4and magnetite Fe 3O4, their magnetic properties [ 11] and magneto-optical spectra [ 12]\ndiffer significantly. The differences from magnetite become even greater for Fe 3Se4, which has\ninteresting magnetic and electrical properties [ 8–10]. Fe 3Se4is a metallic ferrimagnetic material.\nIts electronic structure was calculated within the density functional theory (DFT) [13]. Tewari\net al. [ 6] declare that the Fe 3Se4material possess half-metallic properties. The spin-down band\ngap ( Eg) and half-metallic energy gap ( EHM) calculated using the HSE06 hybrid functional [ 14]\nwere found to be 1.8 eV and 0.17 eV , respectively. However, the analysis of experimental data\nreveled the extremely small EHM(1.3 meV and 34 meV , depending on the Se stoichiometry),\nthus the low energy is required to occupy minority spin states by the majority spin carriers\nduring spin flip processes. The discrepancy between the theoretical and experimental EHM\nvalues can be associated with an increase in the localization of electronic states when a fraction\nof the Hartree–Fock exchange energy is included. This results in the inaccurate description of\nelectronic structure near the Fermi level. It is known that the HSE06 functional can overestimate\nband gap [ 15,16]. Moreover, Gao et al. [ 17] suggest that hybrid functionals, such as HSE and\nPBE0 are not suitable for studying metal systems, and the LDA or GGA give better results in\ndescribing metallic properties.\nHere we study the properties of Fe 3Se4under high pressure within DFT. We calculated the\nchanges of the magnetic moment on each sublattice with the increasing isotropic compression.\nThe values of the magnetic moments decrease non-monotonically and eventually vanish for\nboth inequivalent positions. Since Fe 3Se4possess metallic properties, such a magnetic collapse\ncannot be represented as the energy-level crossing of the high-spin and the low-spin cation\nstates. In this case, the magnetic collapse is caused by the alignment of the numbers of spin-up\nand spin-down electrons on each cation such that one can call this the itinerant analogue of\nthe spin crossover. The spin crossover occurs faster under compression by isotropic pressure\ncompared to when compression by isotropic strain.\n2. Computational Details\nCalculations of atomic and electronic structure and magnetic properties were performed\nin the framework of density functional theory using the Vienna ab initio simulation package\n(VASP) [ 18]. Exchange-correlation effects were described by the Perdew–Burke–Ernzerh (PBE)\nof generalized gradient approximation (GGA) [ 19]. The ion-electron interactions were rep-\nresented by the projector-augmented wave method (PAW) [ 20], and the plane-wave cutoff\nenergy of 600 eV was applied. The criteria for the total energy minimization and interatomic\nforces were set to 10\u00004eV and 10\u00002eV/Å, respectively. The energy convergence criteria were\ndecreased to 10\u00005eV to obtain a more accurate electronic structure. The first Brillouin zone\n(1BZ) was sampled by 24\u000214\u00028grid using the Monkhorst–Pack scheme [ 21]. The isotropic\ncompressive strain was modelled as the change of the structural parameters. The value of strain\nis defined as x= (l0\u0000l)/l0, where l0andlare the equilibrium and strained lattice constant,\nrespectively. The strain was uniform on three lattice directions, i.e., all lattice constants a,b,\nandcwere changed at the same time by the same value of x. The strained lattice constants did\nnot change during the geometry optimization. Compressive strain ranging from 0 to 14% is\nconsidered. The lattice shape remains unchanged during the compression. Compression by an\nisotropic pressure was studied by adding the external pressure to the diagonals of stress tensors.3 of 14\nThe value of the external pressure was changed from zero up to 8 GPa. Lattice constants, cell\nshape, cell volume, and ionic positions were optimized for these structures. The Visualization\nfor Electronic and Structural Analysis (VESTA) [ 22] software was used for representation of\natomic structures. The “vaspkit” software [23] was used for post-processing.\n3. Results and Discussion\nThe crystal structure of the bulk Fe 3Se4is shown in Figure 1a. The unit cell relates to the\nI2/mspace group with the structural parameters a=6.071 Å,b=3.377 Å,c=11.174 Å, and\nb=92.818\u000e, which are well in agreement with previously reported experimental data [ 6,9,10],\nsee Table 1 for comparison. The unit cell contains six Fe atoms and eight Se atoms, with\niron atoms occupying two non-equivalent positions, namely, octahedral (Fe 2) and distorted-\noctahedral (Fe 1) sites. The Se\u0000Fe2\u0000Se angles for distorted-octahedral sites are equal to 104.0\u000e,\n89.8\u000e, 89.9\u000eand 75.5\u000e. The bond lengths between Fe 2and surrounding Se ions are 2.468 Å and\n2.413 Å (bonds 1 and 2 in Figure 1a, respectively) and Fe 1–Se bond lengths are 2.390 Å,\n2.374 Å, 2.545 Å, and 2.581 Å (bond 3, 4, 5, and 6 in Figure 1a, respectively), see Table 2.\nThe distances between Se atoms closest to the iron vacancy are 2.996 Å and 5.277 Å. The\nmagnetic structure was found to be ferrimagnetic. The coupling between spins in Fe 3Se4\nis accomplished by the mechanism of “through-bond spin polarization” that is a specific\nrealization of the superexchange cation–anion–cation interaction, i.e., an iron atom with spin-\nup (spin-down) density induces spin-down (spin-up) density on the p-orbital of the adjacent\nSe atom directly bonded to it. Thus an iron atom is bonded to a selenium atom through the\np-orbital with the direction of the electron spin opposite to its spin density. The distribution\nof the magnetisation density is shown in Figure 1b. The calculated magnetic moments on\niron in sites 1 and 2 are 2.071 mBand\u00002.042 mB, respectively. The magnetic moments on Se\natoms are very small and equal to \u00000.018 mBand 0.015 mB. The total magnetic moment of\nFe3Se4is2.135 mB/f.u., i.e., in the range of the previously DFT-calculated values [ 6,13,24] but\nlarger than the known experimental values [ 6–8], see Table 1. The discrepancy in magnetic\nmoment between experimental and DFT-predicted values relates to the delocalization of 3d\nelectrons of Fe ions. Our calculations showed that the electronic structure of Fe 3Se4is metallic,\nsee Figure 2a,b, as in [ 13,25], and half-metallic state was not found. The density of states\n(DOS) at the Fermi level are predominantly formed by iron states. The bulk modulus of\nFe3Se4is determined by performing six finite distortions of the lattice with \u00060.5,\u00061.0,\u00061.5%\nmagnitude. The calculated P\u0000Vdata are fitted to the Birch–Murnaghan equation of state\nP=3/2B0h\nu\u00007/3\u0000u\u00005/3i\n\u0001h\n1\u00003/4(4\u0000B0)(u\u00002/3\u00001)i\n, where B0,u=V/V0,V0,V, and B0\nare the bulk module, the dimensionless volume, the reference volume (the initial volume of\nFe3Se4), the deformed volume, and the derivative of the bulk modulus with respect to pressure,\nrespectively. Bulk modulus as a function of volume is shown at Figure 3. The obtained value of\nB0is 68 GPa, which is very similar to that of Fe 3S4(62.8 GPa ) [26] and is in the range of values\nfor isomorphic Cr 3Se4(57.7 GPa) and Cr 3S4(72.9 GPa) [ 27] structures. Its pressure derivative\nwas found to be six.4 of 14\nFigure 1. Atomic structure ( a) and the spin-resolved magnetization density ( b) of Fe 3Se4. The unit cell is\nmarked with black lines. Brown and yellow-green colors correspond to Fe and Se atoms, respectively.\nIn panel ( b), yellow (blue) areas indicate the spin-up (spin-down) density. Isosurface value is 0.002\na\u00003\n0,where a0is the Bohr radius.\nTable 1. Structural and magnetic parameters of monoclinic phase of Fe 3Se4.\nStructural Parameters Magnetic Ref.\nLattice Constant (Å) Angle (\u000e) Moment\na b c b (mB/f.u.)\n6.202 3.532 11.331 91.825\u000e- [9]\n6.113 3.486 11.139 91.66\u000e- [10]\n6.16 3.53 11.10 92.0\u000e- [5]\n- - - - 1.17 [6]\n- - - - 0.9 [7]\n- - - - 0.69 [8]\n6.071 3.377 11.174 92.818\u000e2.128 This work\nTable 2. Bond lengths (Å) and distance (Å) between Se atoms closest to the iron vacancy of the Fe 3Se4\nmonoclinic phase. The atomic numbering scheme is shown in Figure 1a.\nBond or distance (Å)\nStrain (%) 1 2 3 4 5 6 7 8\n0 2.468 2.413 2.390 2.374 2.545 2.581 2.996 5.277\n2 2.419 2.364 2.349 2.355 2.503 2.516 2.895 5.151\n3 2.396 2.342 2.325 2.344 2.480 2.481 2.849 5.091\n3.5 2.384 2.333 2.312 2.338 2.467 2.465 2.828 5.066\n4 2.381 2.335 2.291 2.324 2.440 2.446 2.825 5.068\n4.5 2.370 2.319 2.283 2.323 2.426 2.428 2.796 5.044\n5 2.346 2.301 2.281 2.316 2.424 2.429 2.756 4.994\n5.5 2.338 2.280 2.268 2.330 2.416 2.393 2.729 4.967\n6 2.325 2.276 2.264 2.305 2.397 2.400 2.700 4.943\n6.5 2.314 2.262 2.253 2.303 2.387 2.382 2.671 4.916\n7 2.306 2.245 2.239 2.311 2.379 2.352 2.645 4.888\n8 2.282 2.220 2.220 2.298 2.358 2.327 2.588 4.829\n9 2.258 2.196 2.202 2.283 2.337 2.301 2.535 4.767\n10 2.235 2.172 2.183 2.266 2.315 2.274 2.483 4.705\n11 2.212 2.147 2.164 2.250 2.294 2.248 2.432 4.644\n12 2.189 2.123 2.145 2.231 2.272 2.222 2.385 4.582\n13 2.166 2.100 2.127 2.211 2.250 2.195 2.337 4.519\n14 2.143 2.078 2.109 2.189 2.277 2.169 2.292 4.4555 of 14\nFigure 2. Density of states ( a,b) of Fe 3Se4. Total DOS of Fe 3Se4is shown by black curves. In panel ( a),\npartial DOS of Fe and Se atoms are shown by green and red curves, respectively. In panel ( b), partial\nDOS of iron atoms on site 1 and on site 2 are shown by green and red curves, respectively. Positive and\nnegative values of DOS corresponds to spin-up and spin-down channels, respectively. The Fermi level\ncorresponds to zero.\nFigure 3. Pressure as a function of Fe 3Se4volume. Blue dots indicate values for structures with finite\nstrains\u00060.5,\u00061.0,\u00061.5%. The curve through this values fitted to the Birch–Murnaghan equation is\nmarked by red.\nNext we study the effect of strain on the magnetic and electronic properties of Fe 3Se4.\nFirst approach is to model the isotropic compressive strain via the change of the structural\nparameters. The lattice of Fe 3Se4remains monoclinic and the angles between vectors remain\nunchanged during compression. Isotropic compression along the lattice constant up to 4%\nreduces the total magnetic moment of the system to 0.157 mB/f.u., see Figure 4a and Table 3,\nwhich is caused by a decrease in bond lengths, see Table 2. The reduction in bond lengths\nmakes the covalent character greater than the ionic character, which causes a decrease in the\nspin polarization of atoms. Further compression to 5% increases the magnetic moment by\n0.904 mB/f.u. In this compression range, the bond lengths with iron atoms located in the vacant\nlayer (Fe 2\u0000Se) decrease faster (by near 1.5%) than Fe 1\u0000Se bond lengths (by 0.44, 0.34, 0.66,\n0.70%). This leads to the fact that the magnetic moment on the Fe 2atoms drastically decreases\nby 65.4% upon compression from 4 to 5%. At the same time, the magnetic moment on Fe 1\natoms increases from 0.602 to 0.764 mB. A significant decrease in the magnetization on Fe 2,\nopposite in direction to the magnetization of Fe 1, leads to an increase in the total magnetic\nmoment in the system. With further compression, the change in the Fe 2\u0000Se bond lengths\nslows down and the change in Fe 1\u0000Se bond lengths increases, causing the magnetic moment\nto decrease to 0.124 mB/f.u. (6.5% compression).6 of 14\nTable 3. Magnetic moments (in mB) of iron atoms on two sites, Fe 1and Fe 2, and total magnetic moments\nper Fe 3Se4formula unit of Fe 3Se4under compression. The Wigner–Seitz radius for Fe ions is 1.302 Å\nStrain (%) Fe 1Moment Fe 2Moment Total Moment\n0 2.071 \u00002.042 2.135\n2 1.778 \u00001.631 1.969\n3 1.556 \u00001.417 1.759\n3.5 1.387 \u00001.318 1.460\n4 0.602 \u00001.334 0.157\n4.5 0.282 \u00001.214 0.692\n5 0.764 \u00000.462 1.061\n5.5 0.493 \u00000.297 0.673\n6 0.266 \u00000.209 0.368\n6.5 0.053 \u00000.041 0.075\n7 0.463 0.666 1.626\n8 0.352 0.540 1.286\n9 0.282 0.425 1.038\n10 0.225 0.328 0.823\n11 0.162 0.230 0.585\n12 0.068 0.094 0.243\n13 0.004 0.005 0.009\n14 0.000 0.000 0.0007 of 14\nFigure 4. The dependence of the total magnetic moment ( a) and DOS at the Fermi level ( b) on isotropic\ncompression.\nCompression to 7% leads to a sharp jump in the magnetic moment up to 1.626 mBand the\nmagnetic order changes from ferrimagnetic to ferromagnetic. The magnetic moments on Fe 1\nand Fe 2ions are 2.071 mBand\u00002.042 mB, respectively. The magnetic moments on Se atoms are\npractically absent and equal to 0.006 mB. The bond lengths of Fe 3Se4under 7% strain is unevenly\ndecreased by 2.26–8.87%. The distances between selenium atoms located near the iron vacancy\nare reduced by 7.4 and 11.7% compared to the initial distance, and equal to 4.888 Å and 2.645 Å,\nTable 2. The magnetic moment decreases linearly with the further compression from 7% to 13%\nand vanishes under 14% of compression. The pressure corresponding to this strain is equal to\n114 GPa and the volume of the cell is 145.55 Å3. In this structure Fe–Se bond lengths are 7.79–\n15.96% shorter compared to the bond lengths in the original structure. The distances between\nselenium atoms near the vacancy decrease by 15.6 and 23.5% and are equal to 4.455 Å and\n2.292 Å. Octahedrons formed by Fe 2-Se bonds are disodered, so that Se \u0000Fe2\u0000Se angles are\nequal to 85\u000eand 95\u000e. Selenium atoms located at opposite corners of the octahedron are still in\nthe same plane (Se \u0000Fe2\u0000Se angles are 180\u000e). The Se\u0000Fe2\u0000Se angles are equal to 108.3\u000e, 89.5\u000e,\n86.6\u000e, and 74.7\u000e. Values of the iron magnetic moments on sites 1 and 2 are shown in Table 3.\nThe electronic structure of Fe 3Se4remains metallic throughout the studied compression\nrange. The DOS at the Fermi level N(EF)as a function of strain is presented at Figure 4b.\nSlight compression up to 3.5% leads to an increase in the density of states at the Fermi level.\nFurther, the N(EF)non-monotonically depends on the strain and correlates with the non-\nmonotonic dependance of the total magnetic moment. An increase and decrease in the total8 of 14\nmagnetic moment is accompanied by an increase and decrease in the N(EF). The decrease in\nthe magnetic moment during compression from 8 to 14% is accompanied by an increase in\ntheN(EF). Figure 5 shows DOS of Fe 3Se4at critical points. The redistribution of DOSs are\nobserved, compared with the DOS of the original structure (Figure 2b). Compression up to 7%\nresults in N(EF)decreasing, and the vacant states shifting by 0.6eV to higher energies. The\ndensity of states with spin-up and spin-down is equalized on each Fe cation in the structure at a\ndeformation of 14% (Figure 5b), which leads to the disappearance of the magnetic order. It is the\nitinerant analogue of the spin crossover in the metallic system. The metallic system of itinerant\nelectrons lacks for the long-range magnetic order thus ending in the Pauli paramagnetic state.\nIn this case, the DOS at the Fermi level is increased. The occupied states decrease significantly\nat\u00000.25 eV and increase at lower energies. The vacant states decrease at 0.19 eV and increase\nat higher energies.\nFigure 5. Total (TDOS) and partial (PDOS) densities of states for Fe 3Se4under 7% ( a) and 14% ( b)\ncompression. Positive and negative values corresponds to spin-up and spin-down channels, respectively.\nThe Fermi level corresponds to zero.\nThe energy difference DEper formula unit between compressed and original Fe 3Se4was\ncalculated as DE=Ecompressed\u0000Eoriginal . Here Ecompressed andEoriginal are the total energy of the\nFe3Se4cell under isotropic compression and the total energy of the non-compressed Fe 3Se4cell,\nrespectively. The dependence of DEon compressive strain is shown in Figure 6. The increase\nin the energy is described by the cubic polynomial f=0.0032 x3\u00000.0048 x2+0.0759 x\u00000.0241 .\nThe structure at critical points of 7 and 14% is higher in energy by 1.38 and 8.86 eV/f.u.,\nrespectively.9 of 14\nFigure 6. Energetic stability of the compressed Fe 3Se4structure relative to the original structure, DEin eV\nper Fe 3Se4formula unit.\nSecond approach to studying the effect of compression of Fe 3Se4cells is to apply the\nisotropic pressure by adding the external pressure to the diagonals of the stress tensors. The\nresults of calculations show that the cell shape of Fe 3Se4does not change over the entire interval\nof the external pressure. Lattice constants vary nonuniformly because the cell is anisotropic,\nsee Table 4. The lattice constant cdecreases faster than aandbdue to the presence of vacant\nlayers. As a result the Fe 2–Se bonds and the magnetic moments on Fe 2atoms decrease rapidly,\nsee Tables 5 and 6. This difference is not so significant when the cell is compressed by small\npressure. The cconstant changes by more than 7% at an external pressure of more than 4.8 GPa.\nAt the same time, the bconstant is slightly increased compared to the original structure. In this\ncase, the magnetic moments on iron in both sublattices are co-directed and oppositely directed\nto the magnetic moments on the Se ions, which are equal to \u00000.025 mBand\u00000.003 mB. This\nferrimagnetic ordering is accomplished by the through-bond spin polarization. Iron atoms\nwith a spin-up density induce a spin-down density on the adjacent Se atoms. This leads to\nan increase in the magnetic moment by 0.608 mB. An increase in pressure up to 5.0 GPa leads\nto the disappearance of magnetic moments on Se atoms and the magnetic order of Fe 3Se4\nbecomes ferromagnetic. Then, the total magnetic moment decreases with increasing external\npressure that is associated with a further decrease in bond lengths, see Table 5. At a pressure of\n5.0 GPa, the value of the magnetic moment sharply decreased from 2.617 mBto0.146 mBsince the\nFe1–Se bond lengths starts to decrease faster than previously. The magnetic order disappears\nand the magnetic moment becomes zero at a pressure of 8.0 GPa. The Fe–Se bond lengths are\n2.02–7.59% shorter compared to those in the original structure. The distances between selenium\natoms near the vacancy decrease by 8.9 and 6.0%. Se–Fe 1–Se angles are equal to 97.7\u000e,91.2\u000e,\n93.5\u000e, and 76.8\u000e. Se–Fe 2–Se angles are 86.7\u000eand 93.3\u000e. Selenium atoms located at opposite\ncorners of this octahedron remain in the same plane. The volume of the cell is 201.61 Å3. Thus,\nthe spin crossover in the monoclinic phase of Fe 3Se4occurs faster under compression by the\nisotropic pressure compared to by the isotropic strain. The dependence of the total magnetic\nmoment under compression by an external pressure is shown in Figure 7a.\nTable 4. Lattice constants (Å) of Fe 3Se4under isotropic external pressure (GPa).\nExternal Pressure (GPa)Lattice Constant2.0 3.0 4.5 4.8 5.0 7.5 8.0\na(Å) 6.016 5.993 5.962 5.883 5.892 5.857 5.850\nb(Å) 3.341 3.327 3.311 3.423 3.477 3.460 3.455\nc(Å) 11.065 11.014 10.929 10.382 10.048 10.002 9.98710 of 14\nTable 5. Bond lengths (Å) and distance (Å) between Se atoms closest to the iron vacancy of the Fe 3Se4\nunder isotropic external pressure. The atomic numbering scheme is shown in Figure 1a.\nPressure (GPa) Bond or Distance (Å)\n2.0 2.440 2.391 2.371 2.364 2.529 2.543 2.936 5.232\n3.0 2.429 2.382 2.363 2.360 2.521 2.528 2.912 5.214\n4.5 2.414 2.369 2.350 2.353 2.511 2.505 2.878 5.184\n4.8 2.380 2.339 2.310 2.337 2.499 2.432 2.795 5.071\n5.0 2.377 2.345 2.307 2.333 2.457 2.403 2.772 4.993\n7.5 2.366 2.332 2.296 2.450 2.388 2.736 2.968 4.968\n8.0 2.362 2.329 2.293 2.326 2.447 2.385 2.730 4.959\nTable 6. Magnetic moments (in mB) of iron atoms on two sites, Fe 1and Fe 2, and total magnetic moments\nper Fe 3Se4formula unit of Fe 3Se4under external pressure. The Wigner–Seitz radius for Fe ions is 1.302 Å.\nPressure (GPa) Fe 1Moment Fe 2Moment Total Moment\n2.0 1.942 \u00001.829 2.097\n3.0 1.884 \u00001.738 2.068\n4.5 1.786 \u00001.596 2.009\n4.8 1.065 0.544 2.617\n5.0 0.077 0.005 0.146\n7.5 0.013 0.001 0.017\n8.0 0.000 0.000 0.00011 of 14\nFigure 7. The dependence of the total magnetic moment ( a) and DOS at the Fermi level ( b) on\nexternal pressure.\nDOS at the Fermi level N(EF)correlates with the non-monotonic dependance of the total\nmagnetic moment, see Figure 7b. A decrease in the total magnetic moment is accompanied by\nan increase in the N(EF). A change in the magnetic order from ferrimagnetic to ferromagnetic\nleads to a decrease in N(EF)that was also observed in the structure under compression by the\nisotropic strain. The electronic structure of Fe 3Se4remains metallic at all values of external\npressure. DOS with spin-up and spin-down become equal on each Fe cation under 8.0 GPa\n(Figure 8), resulting in a paramagnetic state. DOS in the vicinity of the Fermi level are increased\ncompared to the DOS of the original structure shown in Figure 2b. DOS redistribution in this\ncase differs from that for the paramagnetic Fe 3Se4structure at 14% compressive strain, since\nthe structural parameters change differently.\nThe energy difference DEbetween compressed and uncompressed Fe 3Se4changes faster\nwith the increasing external pressure than with the increasing strain, compare Figures 9 and 6 .\nUnder the external pressure, the increase in energy is described by the linear dependence12 of 14\nf=1.3141 x+0.1717 . Energy of Fe 3Se4at critical points of 5.0 GPa and 8.0 GPa is higher by\n7.10 and 10.55 eV/f.u., respectively. Thus, the ferromagnetic structure obtained under the\nisotropic strain compression is more stable than the one under the isotropic external pressure\nby\u00005.72 eV/f.u. The energy difference of the paramagnetic states in two regimes is not so\nsignificant and equals to \u00001.69 eV/f.u.\nFigure 8. Total and partial DOS for Fe 3Se4under isotropic external pressure of 8.0 GPa. Positive and\nnegative values correspond to spin-up and spin-down channels, respectively. The Fermi level corresponds\nto zero.\nFigure 9. Energetic stability of the Fe 3Se4compressed by the external pressure relative to the uncom-\npressed structure, DEin eV per Fe 3Se4formula unit.13 of 14\n4. Conclusions\nDFT calculations within GGA for Fe 3Se4show that the ground state is metallic and the\nsystem is not in the half-metal state. That agrees with the conclusions of [ 13,25]. The value\nof the bulk modulus was found to be 68 GPa. By studying the compression effect on Fe 3Se4\nwithin DFT, we found the itinerant analogue of the spin crossover in the metallic system. In\nparticular, we calculated the magnetic moment in each of two iron sublattices of Fe 3Se4in two\nregimes: (a) assuming an increasing isotropic compression by the compressive strain and (b)\nthe isotropic external pressure. For crystals with such an anisotropic chemical bonding as in\nFe3Se4, the regime (b) is more relevant to the experiments where the isotropic external pressure\nis applied. If the deformation of the crystal were isotropic as in the regime (a), the values of the\nmoments would vanish for both inequivalent positions at the 14% of strain that corresponds\nto the pressure of 114 GPa. On the other hand, in the regime (b) under the isotropic external\npressure, the magnetic collapse is expected to occur at a much smaller value of pressure, namely,\nat 8 GPa. Under the compression, the system evolves from the uncompressed ferrimagnetic\nstate first to the ferromagnetic state and then to the paramagnetic state; the process is sketched\nin Figure 10.\nFigure 10. Illustration of the Fe 3Se4transformation under the applied isotropic strain xor the isotropic\nexternal pressure Pexbetween the ferrimagnetic and the paramagnetic states through the ferromagnetic\nstate. Brown and yellow-green colors correspond to Fe and Se atoms, respectively. The arrows indicate\nthe direction of magnetic moments on iron atoms.\nAuthor Contributions: Conceptualization, S.G.O.; calculations, L.V .B.; formal analysis, M.M.K.; writing,\nL.V .B., S.G.O., and M.M.K.; funding acquisition, S.G.O. All authors have read and agreed to the published\nversion of the manuscript.\nFunding: L.V .B. and S.G.O. acknowledge the support of the Russian Science Foundation (Project 18-12-\n00022P).\nInstitutional Review Board Statement: Not applicable.\nInformed Consent Statement: Not applicable.\nData Availability Statement: Not applicable.\nAcknowledgments: We acknowledge the useful discussions with M.A. Vysotin. L.V .B. would like to\nthank the Information Technology Center, Novosibirsk State University, for providing access to their\nsupercomputer facilities.\nConflicts of Interest: The authors declare no conflicts of interest.14 of 14\nReferences\n1. Lyubutin, I.S.; Gavriliuk, A.G. High and Ultra-High Pressure Research on Phase Transformations in 3d-Metal Oxides: Current\nProgress. Uspekhi Fiz. Nauk 2009 ,179, 1047. doi:10.3367/UFNe.0179.200910b.1047.\n2. Gavriliuk, A.G.; Trojan, I.A.; Lyubutin, I.S.; Ovchinnikov, S.G.; Sarkissian, V .A. 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Lett. 2009 ,\n88, 67007. doi:10.1209/0295-5075/88/67007"    },    {        "title": "1907.04540v1.Temperature_dependence_of_magnetic_resonance_in_ferrimagnetic_GdFeCo_alloys.pdf",        "content": "1 \n Temperature dependence of magnetic resonance in ferrimagnetic \nGdFeCo alloys  \nTakaya  Okuno1, Se Kwon Kim2,3†, Takahiro  Moriyama1, Duck-Ho Kim1, Hayato  \nMizuno1,4, Tetsuya  Ikebuchi1, Yuushou  Hirata1, Hiroki  Yoshikawa5, Arata Tsukamoto5, \nKab-Jin Kim6, Yoichi  Shiota1, Kyung -Jin Lee7,8, Teruo Ono1,9† \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 611 -0011, Japan  \n2Department of Physics and Astronomy, University of California  Los Angeles, California \n90095, USA  \n3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, \nUSA \n4Institute for Solid State Physics, University of Tokyo, Kashiwa  277-8581 , Japan  \n5College of Science and Technology, Nihon University, Funabashi, Chiba 274 -8501, Japan  \n6Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea  \n7Department of Materials Science & Engineering, Korea University, Seoul 02841, \nRepublic of Korea  \n8KU-KIST Graduate School of Converging Science and Technology, Korea University, \nSeoul 02841, Republic of Korea  \n9Center for Spintronics Research Network (CSRN), Graduate School of Engineering \nScience, Osaka University, Osaka 560 -8531, Japan  \n†E-mail: kimsek@missouri.edu , ono@scl.kyoto -u.ac.jp  \n \n  2 \n We provide a macroscopic theory and experimental results for magnetic resonances of \nantiferromagnetically -coupled ferrimagnets. Our theory, which interpolates the dynami cs of \nantiferromag nets and ferromagnets smoothly, can describe ferrimagnetic resonance s across \nthe angular momentum compensation point . We also present experimental results for spin-\ntorque induced ferrimag netic resonance at several temperatures. The spectral analysis based \non our theory reveals that the Gilbert damping parameter, which has been considered to be \nstrongly temperature dependent, is insensitive to temperature. We env ision that our work \nwill facilitate further investigation of ferrimagnetic dynamics by providing a theoretical \nframework  suitable for a broad range of temperatures .  \n \n  3 \n Antiferromagnets have been gaining much attention in spintronics because of their \npotential utility for high -speed ultra -dense spintronic devices.1-4) Due to the antiparallel \nalignment of adjacent spins , their dynamics  is different  from that of ferromagnets.5) One \nemerging material platform for studying antiferromagnetic dynamics is \nantiferromagnetically -coupled ferrimagnets ,6-11) for which we can use conventional \ntechniques for ferromagnets owing to small but finite magnetizations . Indeed, r ecent \nexperiments in such ferrimagnets have found that both field -driven and current -driven \ndomain -wall dynamics are fastest at the angular momentum compensation point 𝑇A where \nthe magnetic dynamics are antiferromagnetic .12-15) However, the magnetic resonance \nphenomenon of ferrimagnets (FiMR) has not been fully clarified so far because of \ninsufficient experimental investigations. In the literature, Stanciu et al . have studied the \nlaser-induced precession and its decay to equilibrium in ferrimagnets and concluded that the \neffective Gilbert damping parameter 𝛼 , which governs the dissipation rate of angular \nmomentum,  is strongly temperature dependent and increases signif icantly  at 𝑇A.6) However, \nsome recent studies have provided a new perspective on 𝛼  of ferrimagnets by full y \nconsidering the antiferromagnetic dynamics in ferrimagnets . Kamra et al. have theoretically \nrevealed that the temperature dependence of FiMR occurs because of the temperature \ndependence of magnetic dynamics , not because of temperature dependence of  𝛼.16) Kim et \nal. have also reported the temperature -insensitive 𝛼 of ferrimagnets  through the  DW motion \nexperiment s.17) In this paper, we provide an additional evidence of temperature -insensitive \n𝛼 of ferrimagnets by performing the FiMR experiment analyzed by our macroscopic FiMR \ntheory.  \nFirst, we derive the equations for FiMR in a ferrimagnet consisting of two \nantiferromagnetically -coupled sublattices. Throughout the manuscript, we will focus on the \nregime where the ferrimagnet is away from the magnetization compensation temperature 𝑇M  \nso that th e magnetization is finite and well defined. Our experiments are also performed well \nwithin the considered regime  as detailed below.  To this end, we  expand the L andau -Lifshitz -\nGilbert -like (LLG -like) equation  for ferrimagnet films12,18-20) at the uniform ground state \nalong the positive in-plane z directio n to linear order in the small  fluctuations  |𝑛𝑥|,|𝑛𝑦|≪\n1, where the unit vector 𝒏 represents the N éel order parameter . The resultant equations are \ngiven by  \n𝑠net𝑛̇𝑥−𝛼𝑠total𝑛̇𝑦−𝜌𝑛̈𝑦=𝑀𝐻ext𝑛𝑦,     (1) 4 \n  \n𝑠net𝑛̇𝑦+𝛼𝑠total 𝑛̇𝑥+𝜌𝑛̈𝑥=−𝑀(𝐻ext+𝐻ani)𝑛𝑥,     (2) \n \nwhere 𝑠net=𝑠1−𝑠2 is the net spin density  of two sublattices  𝑠1>0 and 𝑠2>0, 𝛼 is the \nGilbert damping parameter , 𝑠total =𝑠1+𝑠2 is the sum of the magnitudes of the two spin \ndensities , 𝜌>0 is the  moment of inertia  for the dynamics  (which is inversely proportional \nto the microscopic exchange field between the two sublattices  and describes the \nantiferromagnetic dynamics of the magnet ),2) 𝐻ext is the external  field along the z direction,  \n𝐻ani  is the effective  anisotropy field along the x direction perpendicular to the film \n(including the effect of the demagnetizing field),  and 𝑀 is the magnetization . Here, we are \nneglecting the terms that are quadratic or higher order in 𝐻ext and the time derivative of the \norder parameter. The damping term is added by considering the Rayleigh dissipation \nfunction  𝑅=𝛼𝑠total ∫𝑑𝑉 𝒏̇2/2, which is the half of the ener gy dissipation rate through the \nmagnetic dynamics.20) Note that the Rayleigh function is defined in terms of 𝑠total, not in \nterms of 𝑠net, so that it is well defined even in the vicinity of 𝑇A where 𝑠net vanishes.17)  \nTo the zeroth -order in the damping paramete r 𝛼 , the resonance  frequencies for \nmonochromatic solutions to the above equations are given by  \n𝑓±2\n=𝑠net2+𝜌𝑀(2𝐻ext+𝐻ani)±√𝑠net4+2𝜌𝑀(𝑠net)2(2𝐻ext+𝐻ani)+𝜌2𝑀2𝐻ani2\n8𝜋2𝜌2,     (3) \n \nwhere 𝑓+ and 𝑓− are the frequencies for  higher and lower resonance  frequencies for the given \nfield. Far away from 𝑇A, where the  net spin density |𝑠net| is suf ficiently large, Eq. (1) and \nthe corresponding dynamics are dominated by the first -order time derivative term and thus \nwe can neglect the second -order term by setting 𝜌=0 . In that ferromagnetic limit, the \nexpression for the lower frequency is reduced to that for the ferromagnet resonance \nfrequency :21) \n         𝑓FiM= 𝑀\n2𝜋|𝑠net|√𝐻ext(𝐻ext+𝐻ani).     (4) \nNote that 𝑀/|𝑠net|  is the effective gyromagnetic ratio  𝛾eff  of the ferrimagnets. As the \ntemperature approaches 𝑇A , the net spin density | 𝑠net|  decreases and thus  the resonance \nfrequency  is expected to increase. However, this formula cannot be used in the vicinity of 5 \n 𝑇A, where 𝑠net vanishes and thus the second -order term cannot be neglected. Exactly at 𝑇A, \nthe net spin density vanishes 𝑠net=0, which reduces the obtained resonance  frequencies \n[Eq. (3)] to  \n   𝑓+= 1\n2𝜋√𝑀(𝐻ext+𝐻ani)\n𝜌,    𝑓−= 1\n2𝜋√𝑀𝐻ext\n𝜌.  (5) \nInclusion of the second -order time derivative term ∝𝜌 in the LLG -like equations [Eq. ( 1) \nand Eq. ( 2)] is necessary to obtain finite resonance  frequencies  at 𝑇A; otherwise, the LLG -\nlike equations lack in the reactive dynamic term ∝𝑠net at 𝑇A and becom e unable to describe \nthe ferrimagnetic dynamics properly therein.   \n Since our experimental results, which are presented below, are performed away from \n𝑇A, let us derive the resonance linewidth for ferrimagnets in the ferromagnetic regime. When \nwe include the Gilbert damping term, the resultant linewidth of  ferrimagnets ∆𝐻  (half -\nwidth -half-maximum) is given by  \n∆𝐻≈2𝜋𝛼\n𝛾eff 𝑠total\n|𝑠net| 𝑓FiM.     (6) \n \nTherefore 𝛼 in ferrimagnet is given by  \n𝛼FiM≈(𝛾eff\n2𝜋)|𝑠net|\n𝑠total (∆𝐻\n𝑓FiM).     (7) \nNote that b oth 𝑠total  and 𝑠net  appear in the lin ewidth expression because 1) the energy \ndissipation rate is proportional to 𝑠total since two lattices contribute additively and 2) the \nresonance frequency is inversely  proportional to 𝑠net. On the other hand, in conventional \nexpressions for ferromagnetic  resonance, the two spin -density parameters are assumed to be \nidentical , 𝑠total =𝑠net, and the corresponding expre ssion  𝛼FM≈(𝛾eff\n2𝜋)(∆𝐻\n𝑓FiM) was used to \nanalyze the magnetic resonance of ferrimagnets  in the previous reports.6,7) Below, these two \nexpressions for the Gilbert damping parameters, 𝛼FiM and 𝛼FM, will be compared based on  \nour experimental results.  \nWe experimentally  investigate d the FiMR in the GdFeCo compounds  by using  the \nhomodyne technique22-24) as shown in Fig 1 . For this study, we used a 5-nm SiN/10 -nm \nGd25.0Fe65.6Co9.4/5-nm Pt/100 -nm SiN /Si substrate  film. The film was patterned into a 10 -\nµm-wide  and 10 -µm-long strip pattern structure using optical lithography and Ar ion milling. \nA coplanar waveguide made of 100 -nm Au /5-nm Ti  were de posited at the ends of the strip.  6 \n The measurements were performed by sweeping an external magnetic field  𝐻ext at a fixed \nrf current  𝐼rf  (frequency 𝑓=4−18  GHz). 𝐻ext  was applied  in-plane  45°  away from the \nlong axis of the strip.  \nFigure 2a shows the FiMR  spectra at several temperatures 𝑇 between  220 K and  295 \nK. Although a single peak was clearly observed at 295 K, a second peak was also observed \nat 𝐻ext≈50 mT when 𝑇 is lower than 240 K. Note that the spontaneous magnetization lies \nin the sample plane at 𝑇=295  K while  it becomes perpendicular to the plane when 𝑇≤\n240  K . Thus, the two resonan ce peaks when 𝑇≤240  K  originat e from the magnetic \nresonance of perpendicular  (𝐻ext≈50 mT ) and in -plane (higher field) magnetization s, \nrespectively . Here we focus on the resonan ce peak originating from in -plane magnetization, \nso we cut off the low -field regime to exclude the resonance peak  from perpendicular \nmagnetization  and fit those spectra in Fig. 2a by the combination of symmetric and anti -\nsymmetric Lorentzian  function s, from which the resonan ce parameters are obtained .22,23)  \nFigure s 2b and 2c  show the resonance frequency  𝑓res as a function of  the resonance  \nfield 𝐻res and the spectral linewidth  ∆𝐻 (half -width -half-maximum) as a function of 𝑓res, \nrespectively . Firstly, we analyze th ese data using the conventional expressions of \nferromagnetic resonance,21,25) \n𝑓res=𝑔eff𝜇B\nℎ√𝐻res(𝐻res+𝐻ani),     (8) \n \n∆𝐻=𝛼FM\n(𝑔eff𝜇Bℎ⁄)𝑓res+∆𝐻0.     (9) \n \nHere, 𝑔eff  is the  effective  Landé  g-factor, 𝜇B  is the Bohr magneton,  ℎ  is the Planck ’s \nconst ant, 𝐻ani is the effe ctive  anisotropy field including  the demagnetization field, 𝛼FM is \nthe effective Gilbert damping parameter  defined as in Ref. 6, and ∆𝐻0  is a frequency -\nindependent linewidth known as the inhomogeneous broadening, which originates from \nmagnetic non -uniformity.25) Equation (8) can be matched with Eq. (4) once we identify \n𝑔eff𝜇B/ℏ as the effective gyrom agnetic ratio  𝑀/|𝑠net| (ℏ=ℎ2𝜋⁄ is the reduced Planck ’s \nconstant ) and 𝐻res  as 𝐻ext . The 𝐻res  vs 𝑓res  shown in Fig. 2b are well fitted b y Eq. (8), \nindicated by the solid lines, and  𝑔eff and 𝐻ani are obtained as the fitting parameters. Figures \n3a and 3b show 𝑔eff and 𝐻ani as a function of 𝑇, respectively.  It is found that 𝑔eff remarkably \nincreases as 𝑇 decreases. Since the 𝑇A of the device is estimated  to be 160 K (see below the 7 \n estimation method) , the result shows that 𝑔eff increases as 𝑇 approaches 𝑇A. Note that the \ndrastic decrease in 𝐻ani with decreasing 𝑇 (Fig. 3b) is attributed to the change in magnetic \nanisotropy from in -plane (295 K) to perpendicular (220 K) direction as mentioned above. \nThe 𝑓res vs ∆𝐻 show n in Fig. 2c are well fitted by Eq. (9), indicated by the solid lines, and  \n𝛼FM and ∆𝐻0 are obtained as the fitting parameters. Figures 3c and 3d show 𝛼FM and ∆𝐻0 \nas a function of 𝑇, respectively.  It is found that 𝛼FM increases significantly as 𝑇 decreases, \ni.e. as 𝑇 approaches 𝑇A. The 𝑇 dependences of 𝑔eff and 𝛼FM are in good agreement with the \nprevious paper s.6,7,2 7) According to the previous paper s,6,7) the 𝑇  dependences of 𝑔eff  and \n𝛼FM are understood in terms of that of the net angular momentum 𝑠net; both 𝑔eff𝜇Bℏ⁄=\n𝑀net 𝑠net⁄  and 𝛼FM [from Eq. (9)] are ill -defined  at 𝑇A where 𝑠net vanishes , which makes \nthe theory based on ferromagnets invalid therein . However, as shown in the discussion of \nour theory  for FiMR , by defining the Gilbert damping parameter in the Rayleigh dissipation \nfunction 𝑅=𝛼𝑠total ∫𝑑𝑉 𝒏̇2/2 in such a way that the damping parameter is always well -\ndefined, the resonance frequency  and the li newidth of FiMR can be described properly \nacross 𝑇A . In order to test whether our theory can explain the experimental results, we \nanalyze those data in Figs. 2b and 2c based on our theory.  \n As mentioned in the theory part, the ferrimagnetic resonance frequency  in the \nferromagnetic limit is reduced to the conventional ferromagnetic case, while the spectral \nlinewidth is modified by including the additional term 𝑠net 𝑠total⁄  . Therefore, the Gilb ert \ndamping parameter [Eq. (7)] in our theory [Eq. (1) and Eq. (2)] for the dynamics of \nferrimagnets can be obtained by the following expression:  \n𝛼FiM=𝛼FM|𝑠net\n𝑠total|.     (10) \nTo obtain 𝛼FiM from Eq. (7) and Fig. 2c, 𝑠net 𝑠total⁄  needs to be acquired.  Although the net \nspin density 𝑠net is easy to obtain from the effective gyromagnetic ratio, the total spin density \n𝑠total  is not straightforward to obtain.  To solve this problem , we perform the following \nanalysis . The effective net gyromagnetic ratio satisfies the following equation;6,7) \n𝑔eff𝜇B\nℏ=𝑀net\n𝑠net=𝑀FeCo −𝑀Gd\n𝑀FeCo\n(𝑔FeCo 𝜇Bℏ⁄)−𝑀Gd\n(𝑔Gd𝜇Bℏ⁄).     (11) \n \nHere, 𝑀FeCo (𝑀Gd) is the magnetizations of transition metal (rare -earth metal), and 𝑔FeCo \n(𝑔Gd) is the Landé  g-factor  of transition metal (rare -earth metal) sublattice. 𝑔eff is shown in 8 \n Fig. 3a, 𝑔FeCo  and 𝑔Gd  are obtained from literature  (𝑔FeCo ~2.2  and 𝑔Gd~2.0 ).28-30) Two \nquantities can be measured directly : 𝑀net is independently  measured  by SQUID  as shown \nin Fig. 4a  and 𝑠net  can be obtained from the effective gyromagnetic ratio  when the \nferrimagnet is well within the ferromagnetic regime.  With the measured values of 𝑀net  = \n𝑀FeCo −𝑀Gd and 𝑠net= 𝑀FeCo\n(𝑔FeCo 𝜇Bℏ⁄)−𝑀Gd\n(𝑔Gd𝜇Bℏ⁄), we can obtain the magnetizations of two \nsublattices,  𝑀FeCo  and 𝑀Gd , and also the spin densities of two sublattices,  𝑠FeCo  and 𝑠Gd .  \nFrom these results, we can obtain the total spin density 𝑠total =𝑠FeCo +𝑠Gd. Figures 4a and \n4b show 𝑀FeCo and 𝑀Gd, and 𝑠net=𝑠FeCo −𝑠Gd and 𝑠total =𝑠FeCo +𝑠Gd as a function of \n𝑇, respectively . Note that the 𝑇M (110 K ) determined by SQUID ( Fig. 4a ) and the 𝑇A (160 \nK) roughly estimated from the 𝑇  dependence of  𝑠net  (Fig. 4b) are clearly different,12,31) \nwhich supports the validity of this analysis. Finally, by substituting 𝑠net and 𝑠total into Eq. \n(10), the damping parameter  𝛼FiM is obtained as shown in Fig. 4c. It can be clearly seen  that \n𝛼FiM(≈0.01) is insensitive to 𝑇, in sharp contrast to 𝛼FM which significantly increases as 𝑇 \napproaches 𝑇A. Note that Eq. (10) is valid only in the ferromagnetic limit and, therefore,  it \nis necessary to confirm that the measured temperature range ( 220 – 295 K) is in the deep \nferromagne tic regime.  This would be guaranteed by the facts that 1) Fig. 4b shows \n𝑇A~160  K , which is  far below the lowest 𝑇  in our measurement s (220 K ), and 2) the \nresonance frequency  at 𝑇A is expected to  be similar to or larger than about 50  GHz  under \n300 mT ,6) which is much larger than the experimentally obtained resonance frequency  at 220 \nK (12 GHz under 300 mT).  \nThe observation that 𝛼FiM is insensitive to 𝑇 indicates that the 𝑇 dependence of the \nspectral linewidth in FiMR is attributed to the 𝑇  dependence of the net spin density 𝑠net \ninstead of that of the effective Gilbert damping parameter. This conclusion is consistent with \nsome recent papers,16,17) but it is in sharp contrast to the interpretation of the previous \nreport s,6,7) where the 𝑇 dependence of the spectral linewidth in FiMR was attributed to the \nchange of the effective Gilbert damping parameter. Our results provide an additional and \nclear evidence that properly defined Gilbert damping parameter 𝛼FiM  of ferrimagnets  is \ninsensitive to temperature, supporting the validity of these papers .16,17) Here, we would like \nto mention that, e ven though Fig. 4c is an evidence for the temperature -insensitive 𝛼FiM of \nferrimagnets, it lacks the information of 𝛼FiM in the vicinity of 𝑇A. However, obtaining  𝛼FiM \nin the vicinity of 𝑇A based on FiMR  experiment s is challenging  because 1) experimental \nobservation of ferrima gnetic resonance at 𝑇A which was measured to be larger than 5 0 GHz 9 \n for certa in ferrimagnets6) is expected to be difficult with homodyne detection technique (40 \nGHz at maximum in our measurement system) and 2) obtaining the necessary parameter \n𝑠total  is difficult because the net spin density 𝑠net  cannot be obtained from the effective \ngyromagnetic ratio in the vicinity of 𝑇A. Therefore, we believe that Fig. 4c serves as a good \nexperimental evidence to conclude that 𝛼FiM of ferrimagnets is insensitive to temperature.  \nIn conclusion, we have provided the macroscopic theoretical description of \nferrimagnetic resonance and experimental results that support it. Our theory shows that the \nresonance frequency  and the spectral linewidth of ferrimagnetic resonance can be described  \nwell across the angular momentum compensation point , by adding the antiferromagnetic -\nlike inertial term to the equations of motion and by defining the Gilbert damping parameter \nproperly through the Rayleigh dissi pation function. Moreover, we performed the spin -torque \ninduced ferrimagnetic resonances at various temperatures and successfully observed that the \nresonance frequency  and the linewidth depend on temperature. By analyzing the spectrum \nbased on our theory, we found that the Gilbert damping parameter in ferrimagnets is \ninsensitive to temperature, which has been considered to be strongly temperature -dependent. \nOur work introduces a new framework for studying ferrimagnetic resonance that allows us \nto interpret the ferrimagnetic dynamics for a wide range of temperatures .  \n \nAcknowledgments  \nThis work was supported by the JSPS KAKENHI (Grants  No. 15H05702,  No. 17H04924 , \nNo. 17H05181 , No. 26103002, and No. 26103004),  Collaborative Research Program of the \nInstitute for  Chemical Research, Kyoto University, and R & D project  for ICT Key \nTechnology of MEXT from the Japan Society  for the Promotion of Science (JSPS). This \nwork was  partly supported by The Cooperative Research Project  Program of the Researc h \nInstitute of Electrical  Communication, Tohoku University. S. K. K. w as supported by the \nstartup fund at the University of Missouri . D. H. K. was supported  as an Overseas \nResearcher under the Postdoctoral Fellowship  of JSPS (Grant No. P16314). K. J. L. w as \nsupported by the National Research Foundation of Korea  (2017R1A2B2006119) . K. J. K.  \nwas supported by  the KAIST -funded Global Singularity Research Program for 2019 .  10 \n  \nReferences  \n1) T. Jungwirth , X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol.  11, 231 \n(2016).  \n2) V . Baltz, A. Manchon , M. Tsoi , T. Moriyama , T. Ono , and Y . Tserkovnyak , Rev. Mod. \nPhys.  90, 015005 (2018).  \n3) J. Železn ý, P. Wadley, K. Olejník, A. Hoffmann, and H. Ohno , Nat. Phys.  14, 220 \n(2018).  \n4) M. B.  Jungfleisch, W. Zhang,  and A.  Hoffmann, Phys. Lett. A  382, 865 (2018).  \n5) O. Gomonay, V . Baltz, A. Brataas,  and Y . Tserkovnyak, Nat. Phys . 14, 213 (2018).  \n6) C. D.  Stanciu, A. V . Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and Th. \nRasing , Phys. Rev. B  73, 220402(R) (2006).  \n7) M. Binder, A. Weber, O. Mosendz, G. 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B  97, \n220403(R) (2018).  \n \n  12 \n  \nFigure Captions  \nFig. 1.  The schematic illustration of the device and the measurement  setup . The direction \nof the external magnetic field  𝐻ext and the AC current  𝐼rf are indicated. 𝐻ext was applied  \nin-plane  45° away from the long axis of the strip . \n \nFig. 2.  (a) The ferrimagnetic resonance spectra as a function of the external magnetic field \n𝐻ext at several temperatures from 220 -295 K. The emerging peak at 𝐻ext≈50 mT below \n240 K is attributed to the out -of-plane resonan ce peak and are neglected in this study. (b) \nThe resonance frequency  𝑓res as a function of the resonance  magnetic field  𝐻res. The solid \nlines are the fitting results by Eq. (8). (c) The spectral linewidth ∆𝐻 as a function of 𝑓res. \nThe solid lines are the fitting results by Eq. (9).  \n \nFig. 3. Resonance parameters as a function of temperature extracted by the fitting in Fig. \n(2). (a) The effective Landé  g-factor  𝑔eff. (b) The effective  anisotropy field 𝐻ani. (c) The \neffective Gilbert damping parameter 𝛼FM. (d) The f requency -independent linewidth ∆𝐻0. \n \nFig. 4. (a) The net magnetization 𝑀net and the magnetizations  of two sublattices  𝑀FeCo and \n𝑀Gd as function s of temperature . (b) The net spin density  𝑠net, the spin densities of two \nsublattices 𝑠FeCo and 𝑠Gd, and the sum of the magnitudes of the two spin densit ies 𝑠total as \nfunction s of temperature . (c) The effective Gilbert damping parameter 𝛼FM and the \nproperly defined Gilbert damping parameter of ferrimagnets  𝛼FiM as function s of \ntemperature . \n \n \n \n  13 \n  \nFig. 1 . \n  \n14 \n Fig. 2.  \n  \n0 5 10 15 200204060 295 K\n 260 K\n 240 K\n 220 KDH [mT]\nfres [GHz]数式 y = a + b*x\n重み 機械的\n残差平方和 6.89093\nピ ア ソ ン の r 0.99995\n--\n補正Ｒ 二乗 0.99989\n値 標準誤差\nΔH切片 38.04028 0.57739\n傾き 18.31533 0.07106\n数式 y = a + b*x\n重み 機械的\n残差平方和 3.53715\nピアソンの r 0.99996\n--\n補正Ｒ 二乗 0.9999\n値 標準誤差\nΔH切片 47.96301 0.69263\n傾き 21.8284 0.08274\n数式 y = a + b*x\n重み 機械的\n残差平方和 2.37794\nピ ア ソ ン の r 0.99992\n--\n補正Ｒ 二乗 0.9998\n値 標準誤差\nΔH切片 53.44711 1.23634\n傾き 24.77597 0.13079\n0 100 200 300 400 50005101520\n 295 K\n 260 K\n 240 K\n 220 Kfres [GHz]\nHres [mT]\n050100150200\n f = 4 GHz\n 6 GHz\n 10 GHz\n 14 GHz\n 18 GHzT = 295 K\n050100150200\nT = 280 K\n050100150200\nT = 260 K\n050100150200\nT = 240 K\n050100150200\nT = 230 K\n0 100 200 300 400 500 600 700 800050100150200V [mV]\nHext [mT]T = 220 K\n(a) \n(c) \n(b) 15 \n Fig. 3.  \n \n  \n220 240 260 280 3004681012DH0 [mT]\nT [K]\n220 240 260 280 300-120-80-400Hani [mT]\nT [K]\n220 240 260 280 3000.050.100.150.20\naFM\nT [K]\n220 240 260 280 3002.02.53.03.54.04.5geff\nT [K]\n(a) \n(c) \n(b) \n(d) 16 \n Fig. 4.  \n \n \n \n100 150 200 250 30001234Spin density [ ´10-6 J s/m3]\nT [K] snet\n sFeCo\n sGd\n stotal\nTA~160K\n0 50 100 150 200 250 300-101234Magnetization [ ´105 A/m]\nT [K] Mnet (by SQUID)\n MFeCo\n MGd\nTM » 110 K\n150 200 250 3000.000.050.100.150.20\na\nT [K] aFM\n aFiM\nTA~160K\n(a) \n(c) \n(b) "    },    {        "title": "1104.2399v1.Optimized_Effective_Potential_Model_for_the_Double_Perovskites_Sr2_xYxVMoO6_and_Sr2_xYxVTcO6.pdf",        "content": "arXiv:1104.2399v1  [cond-mat.mtrl-sci]  13 Apr 2011Optimized Effective Potential Model for the Double\nPerovskites Sr 2−xYxVMoO 6and Sr 2−xYxVTcO 6\nI V Solovyev\nNational Institute for Materials Science, 1-2-1 Sengen, Tsukuba , Ibaraki 305-0047,\nJapan\nE-mail:solovyev.igor@nims.go.jp\nAbstract. In attempt to explore half-metallic properties of the double perovs kites\nSr2−xYxVMoO 6and Sr 2−xYxVTcO 6, we construct an effective low-energy model,\nwhich describes the behavior of the t2g-states of these compounds. All parameters\nof such model are derived rigorously on the basis of first-principles electronic structure\ncalculations. In order to solve this model we employ the optimized effe ctive potential\nmethod and treat the correlation interactions in the random phase approximation.\nAlthough correlation interactions considerably reduce the intraat omic exchange\nsplitting in comparison with the Hartree-Fock method, this splitting s till substantially\nexceeds the typical values obtained in the local-spin-density appro ximation (LSDA),\nwhich alters many predictions based on the LSDA. Our main results ar e summarized\nas follows: (i) all ferromagnetic states are expected to be half-me tallic. However, their\nenergies are generally higher than those of the ferrimagnetic orde ring between V- and\nMo/Tc-sites (except Sr 2VMoO 6); (ii) all ferrimagnetic states are metallic (except fully\ninsulating Y 2VTcO 6) and no half-metallic antiferromagnetism has been found; (iii)\nmoreover, many of the ferrimagnetic structures appear to be un stable with respect to\nthe spin-spiral alignment. Thus, the true magnetic ground state o f the most of these\nsystemsisexpected tobemorecomplex. Inaddition, wediscusssev eralmethodological\nissues related to the nonuniqueness of the effective potential for the magnetic half-\nmetallic and insulating states.\nPACS numbers: 75.25.-j, 72.80.Ga, 71.10.-w, 75.47.-mOptimized Effective Potential Model for Double Perovskites 2\n1. Introduction\nRecently double perovskites A2−xA′\nxBB′O6, where AandA′are the di- and three-\nvalent (typically alkali-earth or rare-earth) elements, respective ly, and B(B′) are\nthe transition-metal elements, have attracted considerable att ention [1], following\nexperimental discoveries of the room-temperature ferromagne tism and the colossal-\nmagnetoresistance effect in Sr 2FeMoO 6[2], which make these materials promising in\nthe field of spin electronics. The extraordinary properties of the d ouble perovskites are\nfrequently attributed to their half-metallic electronic structure, although the situation\ncan be rather subtle [3]. Nevertheless, the first discoveries spark the new research\nactivity aiming at the systematic search of half-metallic double-pero vskite materials.\nMany such predictions has been made on the basis of first-principles electronic structure\ncalculations. Besides ferromagnetic (FM) compounds, a particular attention was paid\nto the half-metallic antiferromagnets – the systems, where the ha lf-metallicity coexists\nwith the zero net magnetization [4, 5]. However, even apart from th e technological\nissues, related to the defects in the double-perovskite structur e, there is a number of\nfundamental problems, which call many predictions into question:\n(i) Most of the calculations are based on the local-spin-density appr oximation (LSDA)\nor its refinement – the generalized gradient approximation, which ha ve many\nlimitations for the transition-metal oxides. Particularly, all these t heories employ\nthe functional dependence of the exchange-correlation energy obtained in the limit\nof homogeneous electron gas. However, many unique properties o f the double\nperovskites are related to the alternation of two different transit ion-metal sites\nBandB′, and in this sense the systems are essentially inhomogeneous. Thus , even\nthough the double-perovskite compounds may be classified as itiner ant electron\nmagnets, it does not necessary mean that their properties are we ll described within\nLSDA;\n(ii) In the vast majority of calculations, it is believed that the LSDA+ Utechnique,\nwhich incorporates the physics of on-site Coulomb interactions [6, 7], can serve as\na reasonable alternative to LSDA and provide more superior descrip tion for the\nproperties of double perovskites. However, the LSDA+ Uscheme for the double\nperovskites is ill-defined: there are many uncertainties related to t he values of the\non-site Coulomb repulsion Uat the sites BandB′, which are frequently taken\nas adjustable parameters, the subspace of “localized orbitals”, a nd the double-\ncounting term [8]. Moreover, the total energy in LSDA+ Uis taken in the ad hoc\nform, which is valid only in the atomic limit and equivalent to some constra ined\ndensity-functional theory [7]. In such a situation, all predictions o f the half-metallic\nbehavior for the double perovskites should be taken very cautious ly.\nIn this paper we will illustrate how these problem could be resolved by a pplying\nthe optimized effective potential (OEP) method, and what the cons equences on the\nelectronic andmagnetic propertiesofthedoubleperovskites could be. The OEPmethodOptimized Effective Potential Model for Double Perovskites 3\nis formulated in the spirit of Kohn-Sham density functional theory [9 ], by introducing\nan auxiliary one-electron problem, which is specified by some local effe ctive potential\nand which is used for the search of the true spin density through th e minimization of the\ntotal energy of the system [10, 11, 12, 13]. In this work we will apply the OEP technique\nforthe solutionof thelow-energy model, where allthe parameters arederived rigorously,\non the basis of first-principles electronic structure calculations. B esides simplicity, such\na model analysis typically allows one to concentrate directly on the mic roscopic picture,\nunderlying the considered phenomena. For the first applications of the OEP model, it is\nmore convenient to consider the t2g-systems, whose electronic and magnetic properties\nare predetermined by the behavior of the t2g-states of the sites BandB′located near\nthe Fermi level. In this case, the effective potential is specified by o nly three parameters,\nwhich should be used for the minimization of the total energy.\nWeselect two prototypical examples Sr 2−xYxVMoO 6andSr 2−xYxVTcO 6, for which\nwe adopt the undistorted double perovskite structure with the lat tice parameters a=\n7.846 and 7.899 ˚A, respectively. In fact, a= 7.846 ˚A is the experimental lattice\nconstant for SrLaVMoO 6[14], which is close to theoretical values obtained in the first-\nprinciples calculations [15], and a= 7.899 ˚A is the theoretical lattice constant for\nLa2VTcO 6[16]. Deviations from the ideal double perovskite structure are exp ected to\nbe small [16] and were not considered here. The band-filling depende nce due to the\nsubstitution of Sr by Y was treated in the virtual-crystal approxim ation. According to\nthe electronic structure calculations in the local-density approxima tion (LDA), the basic\ndifference between Sr 2VMoO 6and Sr 2VTcO 6is in the relative position of the V- and\nMo(Tc)-states: in Sr 2VMoO 6, the V- and Mo-states are separated in energy and form\ntwo nonoverlapping bands, while in Sr 2VTcO 6these states form one common band\n(Fig. 1). Both Sr 2−xYxVMoO 6and Sr 2−xYxVTcO 6(and other isoelectronic to them\n/s49/s48 /s56 /s54 /s52 /s50 /s48/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s68/s101/s110/s115/s105/s116/s121 /s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41/s32\n/s32/s32\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\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s85/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s87/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s88/s32/s32/s75/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s88/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s76\n/s49/s48 /s56 /s54 /s52 /s50 /s48\n/s68/s101/s110/s115/s105/s116/s121 /s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41/s32\n/s32/s32\n/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\n/s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s85/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s87/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s88/s32/s32/s75/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s88/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s76\nFigure 1. Electronic structure near the Fermi level in the local-density appr oximation\n(total and partial densities of states, and the energy dispersion of thet2g-bands) for\nSr2VMoO 6(left) and Sr 2VTcO 6(right). The position of the Fermi level is shown by\nthe dot-dashed line.\ncompounds compounds) were intensively discussed in the literature in the context of the\nhalf-metallic applications. For instance, SrLaVRuO 6, SrLaVMoO 6, and La 2VTcO 6were\nproposed to be the half-metallic antiferromagnets [14, 16, 17, 18]. On the other hand,\nthe metallic ferrimagnetic (i.e., non spin-compensated) behavior was also suggested for\nSrLaVMoO 6[15, 19]. In general, such conclusions are sensitive to the values of t heOptimized Effective Potential Model for Double Perovskites 4\non-site Coulomb repulsion U, which are used in the calculations.\nThe rest of the paper is organized as follows. The method is discusse d in Sec. 2.\nParticularly, in Sec. 2.1 we briefly describe construction of the low-e nergy model for\nthe double perovskites and in Sec. 2.2 – the basic idea of the OEP meth od. Results of\nthe model OEP calculations are shown in Sec. 3. Particularly, the obt ained electronic\nband structure is discussed in Sec. 3.1, the screening of on-site int eractions due to\nthe correlation effects and its consequences on the properties of Sr2−xYxVMoO 6and\nSr2−xYxVTcO 6is considered in Sec. 3.2, and the behavior of the total energy and\nstability of the ferrimagnetic states are discussed in Sec. 3.3. Finally , brief summary of\nthe work is presented in Sec. 4.\n2. Method\n2.1. Construction and parameters of the low-energy model\nThe first step of our approach is the construction of the effective low-energy model for\nthet2g-states of the double perovskites:\nˆH=/summationdisplay\nij/summationdisplay\nσ/summationdisplay\nmm′tmm′\nijˆc†\nimσˆcjm′σ+1\n2/summationdisplay\ni/summationdisplay\nσσ′/summationdisplay\nmm′m′′m′′′Ui\nmm′m′′m′′′ˆc†\nimσˆc†\nim′′σ′ˆcim′σˆcim′′′σ′,(1)\nwhich is derived in the Wannier basis, by starting from the LDA band st ructure. The\nWannier functions are specified by the indices m=xy,yz, orzx, andσstands for the\nspin indices ↑or↓. The one-electron part tmm′\nijof the model Hamiltonian was derived\nby using downfolding method, and the Coulomb (and exchange) inter actionsUi\nmm′m′′m′′′\nwere obtained by combining the constrained density-functional th eory with the random-\nphase approximation (RPA). The method was discussed in the literat ure, and for details\nthe reader is referred to the review article [20]. There are three s ets of parameters in\nthe model Hamiltonian (1):\n(i) the “charge transfer” energies, or the energy splitting betwe en “atomic” t2g-levels of\nthe V and Mo (Tc) sites, which are described by the site-diagonal pa rt oftmm′\nij. The\nvalues of these parameters are −391 and 94 meV for the VMo and VTc compounds,\nrespectively. Thus, as expected from the LDA band structure (F ig. 1), for the VMo\ncompounds, the atomic V-states are located lower in energy and we ll separated\nfrom the Mo-states, while for the VTc compounds, the situation is r eversal and the\nsplitting is considerably smaller;\n(ii) the transfer integrals, which are described by the off-diagonal elements of tmm′\nijwith\nrespect to the site indices. As expected, the largest parameter is theddπ-transfer\nintegral between nearest-neighbor V- and Mo(Tc)-sites, which is 271 (245) meV.\nAnother important parameter is the ddσ-transfer integral between next nearest\nneighbors of the same atomic type. Its values in the V- and Mo-subla ttices of\nthe VMo compounds are −72 and−116 meV, respectively. Similar values were\nobtained forthe VTc compounds: −73 and−113 meV forthe V- andTc-sublattice,\nrespectively;Optimized Effective Potential Model for Double Perovskites 5\n(iii) intraatomic Coulomb and exchange interactions in the t2g-band, which are screened\nby all other states. At each site of the system, these interaction s are fully\nspecified by two Kanamori parameters [21]: the intraorbital Coulom b interaction\nU=Ummmmand the exchange interaction J=Umm′m′mform/negationslash=m′. In the perfect\ncubic environment, the third Kanamori parameter U′=Ummm′m′(the interorbital\nCoulomb interaction) can be related to UandJasU′=U−2J[20]. Other types\nof matrix elements of /bardblUmm′m′′m′′′/bardblare equal to zero. The band-filling dependence\nofU,U′, andJ, as obtained in the virtual crystal approximation, is explained in\nFig. 2. Generally, as the number of t2g-electrons increases, the effective Coulomb\n/s48/s49/s50/s51/s52\n/s74/s32/s77/s111/s85\n/s32/s39/s77/s111/s85/s32/s77/s111\n/s74/s32/s86/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s101/s86/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54/s85/s32/s86\n/s85\n/s32/s39/s86\n/s48/s49/s50/s51/s52\n/s74/s32/s84/s99/s85\n/s32/s39/s84/s99/s85/s32/s84/s99\n/s74/s32/s86/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s101/s86/s41\n/s32/s32\n/s32/s32/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s84/s99/s79\n/s54/s85/s32/s86\n/s85\n/s32/s39/s86\nFigure 2. Intraorbital Coulomb interaction U, interorbital Coulomb interaction U′,\nand the exchange interaction Jfor the Sr 2−xYxVMoO 6(left) and Sr 2−xYxVTcO 6\n(right) compounds.\ninteractions also increase, reflecting the tendencies of the RPA sc reening [20]. On\nthe other hand, the effective exchange interaction Jis less sensitive to the band\nfilling. The Coulomb repulsion UVis larger for the VMo-compounds, due to\nthe specific electronic structure of Sr 2−xYxVMoO 6(Fig. 1): since the V-band is\nseparated and located in the low-energy part of the spectrum, it w ill be populated\nin the process of the band-filling. It will effectively reduce the numbe r of channel\navailable for the RPA screening of UV, for example, due to the transitions from the\noccupied oxygen band to the unoccupied part of the V-band [20]. Th us,UVwill be\nless screened in the VMo-systems in comparison with the VTc-ones, where the V-\nand Tc-states form one common band (Fig. 1).\n2.2. Optimized effective potential method\nInorder tosolve themodel andfindthemagneticgroundstateoft hedoubleperovskites,\nwe employ the optimized effective potential method. The OEP model w orks in the spirit\nof the Kohn-Sham density functional theory [9] on the double-per ovskite lattice, where\nthe kinetic energy\nEkin=/summationdisplay\nσocc/summationdisplay\nn1\nΩBZ/integraldisplay\ndk/angbracketleftcσnk|ˆtk|cσnk/angbracketright,Optimized Effective Potential Model for Double Perovskites 6\nthe number of electrons\nnτ=/summationdisplay\nσ/summationdisplay\nmocc/summationdisplay\nn1\nΩBZ/integraldisplay\ndkcmτ†\nσnkcmτ\nσnk,\nand the spin magnetic moment\nmτ=/summationdisplay\nmocc/summationdisplay\nn1\nΩBZ/integraldisplay\ndk/parenleftBig\ncmτ†\n↑nkcmτ\n↑nk−cmτ†\n↓nkcmτ\n↓nk/parenrightBig\nat the site τare constructed from the one-electron orbitals cσnk, which are obtained\nfrom the solution of the auxiliary one-electron problem\n/parenleftbigˆtk+ ˆvσ\neff/parenrightbig\n|cσnk/angbracketright=εσnk|cσnk/angbracketright (2)\nwith the potential ˆ vσ\neff. In the above notations, ˆtkis the Fourier image of ˆtij=/bardbltmm′\nij/bardbl,\neach eigenvector |cσnk/angbracketrightis specified by its components cmτ\nσnkin the basis of t2g-orbitals\nm=xy,yz, orzxof the atomic type τ= V or Mo(Tc), nis the band index, and kis\nthe position in the first Brillouin zone with the volume Ω BZ.\nThe parameters of the potential ˆ vσ\neff, which is supposed to be “local” (or diagonal\nwith respect to the site indices), are obtained from the minimization o f the total energy\nE=Ekin+EC+EX+Ecorr, (3)\nwhere the Coulomb ( EC) and exchange ( EX) energies are given by\nEC=1\n2/summationdisplay\nτ/summationdisplay\nσσ′/summationdisplay\nmm′m′′m′′′Uτ\nmm′m′′m′′′nστ\nmm′nσ′τ\nm′′m′′′\nand\nEX=−1\n2/summationdisplay\nτ/summationdisplay\nσ/summationdisplay\nmm′m′′m′′′Uτ\nmm′′′m′′m′nστ\nmm′nστ\nm′′m′′′,\nrespectively, in terms of the density matrix:\nnστ\nmm′=occ/summationdisplay\nn1\nΩBZ/integraldisplay\ndkcmτ†\nσnkcm′τ\nσnk.\nThe expression for the correlation energy, which is treated in the r andom phase\napproximation, is given by [22, 23, 24]:\nEcorr=−1\n4ΩBZ/integraldisplay\ndqTr/braceleftBig\nln/bracketleftBig\n1−ˆP(iω,q)ˆU/bracketrightBig/bracketleftBig\n1−ˆUˆP(iω,q)/bracketrightBig\n+2ˆP(iω,q)ˆU/bracerightBig\n,(4)\nwhereˆUare the diagonal matrices /bardblUτ\nmm′m′′m′′′/bardblwith respect to the site indices τ,\nˆP=/bardblPττ′\nmm′m′′m′′′/bardblis the polarization in the imaginary frequency:\nPττ′\nmm′m′′m′′′(iω,q) =/summationdisplay\nσocc/summationdisplay\nnunocc/summationdisplay\nn′1\nΩBZ/integraldisplay\ndk2(εσnk−εσn′k+q)\nω2+(εσnk−εσn′k+q)2cmτ†\nσn′k+qcm′τ\nσnkcm′′τ′†\nσnkcm′′′τ′\nσn′k+q,(5)\nand matrix multiplication in (4) implies the convolution over two orbital in dices:\n(ˆUˆP)ττ′\nmm′m′′m′′′=/summationtext\nll′Uτ\nmm′ll′Pττ′\nll′m′′m′′′.Optimized Effective Potential Model for Double Perovskites 7\nFor the cubic systems (and neglecting the orbital polarization effec ts), the effective\npotential is specified by three parameters: ˆ v↑V\neff=−∆V\nex/2, ˆv↓V\neff= ∆V\nex/2, ˆv↑Mo(Tc)\neff=\n∆ct−∆Mo(Tc)\nex/2, and ˆv↓Mo(Tc)\neff= ∆ct+∆Mo(Tc)\nex/2. In the other words, ∆V\nexand ∆Mo(Tc)\nex\nis the exchange-correlation splitting between minority- and majorit y-spin states at the\nsites V and Mo(Tc), respectively, and ∆ ctis the “charge transfer energy” between sites\nV and Mo(Tc). All these parameters are obtained by minimizing the to tal energy (3).\nIn practice, first we calculate the total energy on some coarse me sh of these parameters,\nidentify the minimum, then continue the calculations around the minimu m by using\nfiner mesh, and so on until reaching necessary accuracy for the t otal energy minimum\n(less than 1 meV).\n3. Results and Discussions\n3.1. Electronic structure and arbitrariness of the OEP proc edure for half-metallic and\ninsulating states\nIn the following, we consider two magnetic solutions: ferromagnetic (∆V\nex>0 and\n∆Mo(Tc)\nex>0) and ferrimagnetic (∆V\nex>0, while ∆Mo(Tc)\nex<0), and find parameters of\nthe effective potential by minimizing the total energy for each case . Total and partial\ndensities of states for Sr 2−xYxVMoO 6, obtained from the solution of the Kohn-Sham\nequations (2) with the optimized potential, are shown in Fig. 3. All FM s olutions are\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114/s89/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s32/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s50/s48/s49/s48/s48/s49/s48/s50/s48/s89\n/s50/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48\n/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s83/s114\n/s50/s86/s77/s111/s79\n/s54\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114/s89/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s89\n/s50/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\nFigure 3. Total and partial densities of states as obtained in the optimized eff ective\npotential method for the ferromagnetic (upper panel) and ferrim agnetic (lower panel)\nconfigurations of Sr 2−xYxVMoO 6. The Fermi level is at zero energy.Optimized Effective Potential Model for Double Perovskites 8\nhalf-metallic: the Fermi levels crosses the majority-spin band, while the minority-spin\nband is completely empty. In the case of Sr 2VMoO 6and SrYVMoO 6, there is a strong\nhybridization between V- and Mo-states, which form the common ma jority-spin band.\nHowever, in Y 2VMoO 6the majority-spin band is split into the Mo- and V-subbands,\nwhich are separated by an energy gap. This behavior is related to th e large Coulomb\nrepulsion UVin Y2VMoO 6(Fig. 2), which additionally shifts the V-states to the higher-\nenergy part of spectrum. Any deviation from the half-metallic beha vior and partial\npopulation of the minority-spin band increases the total energy of Sr2−xYxVMoO 6and\nthus makes this state unstable.\nThe choice of the parameters of the effective potential in the half- metallic regime is\nnot unique. For example, any redistribution of states in the unoccu pied minority-spin\nband does not affect the ground-state properties. Moreover, a ny change of ∆V\nex, ∆Mo\nex,\nand ∆ ct, which does not deform the partially populated majority-spin band ( apart from\nthe constant shift), does not affect the ground-state propert ies either. Quantitatively,\nthis condition can be formulated as follows: the ground-state prop erties in the half-\nmetallic regime are controlled by the single parameter a= ∆ct+(∆V\nex−∆Mo\nex)/2. Any\nchange of ∆V\nex, ∆Mo\nex, and ∆ ct, which keeps a, does not affect the ground-state properties\nin the half-metallic regime.\nAll ferrimagnetic configurations of Sr 2−xYxVMoO 6are metallic. In this case all\nthree parameters of the effective potential are uniquely defined. The degree of the spin\npolarization ( P) in the ferrimagnetic state is sensitive to the concentration xand the\ndefinition of P[25]. For example, using the simplest “density of states” definition (s ee\nFig. 3), one obtains P= 0.3, 0.1, and 0.2 for x= 0, 1 and 2, respectively. On the other\nhand, the “Bloch-Bolzmann transport theory” definition yields P= 0.1, 0.1, and 0.5 for\nx= 0, 1 and 2, respectively. The experimental value of Pfor SrLaVMoO 6, measured\nby using the point-contact Andreev reflection technique, is about 0.5 [14]. Nevertheless,\nas we will see below, many of these ferrimagnetic states appear to b e unstable and\ncalculated spin polarization cannot be directly compared with the exp erimental data.\nSimilar behavior was obtained for Sr 2VTcO 6and SrYVTcO 6, where again half-\nmetallic FM state competes with the normal metallic ferrimagnetic sta te (Fig. 4)\nY2VTcO 6requires a special attention: it has six electrons in the t2g-band. The\nsystem is insulating and all metallic configurations have higher energie s. Therefore,\nin the FM state, the parameters ∆V\nex, ∆Tc\nex, and ∆ ctcan be chosen in an arbitrary\nway. Indeed, any combination of these parameters (provided tha t the system remains\ninsulating) yields the same number of electrons nτ= 3, spin magnetic moments\nmτ= 3µB, andthetotal energyof thesystem. Thecorrelationenergy is ex actly equal to\nzero for this particular case. The arbitrariness with the choice of t he effective potential\nremains also in the antiferromagnetic state of Y 2VTcO 6(Fig. 5), although the number\nof arbitrary parameters decreases. Indeed, the splitting betwe en V- and Mo-states with\nthe same spin affects the ground-state properties. However, th e bands with different\nspins can be rigidly shifted relative to each other and, as long as the s ystem remains\nin the insulating state, this shift does not affect the ground-state properties. Thus,Optimized Effective Potential Model for Double Perovskites 9\n/s51 /s50 /s49 /s48 /s49 /s50 /s51/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\n/s51 /s50 /s49 /s48 /s49 /s50 /s51/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114/s89/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s32/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114/s89/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\nFigure 4. Total and partial densities of states as obtained in the optimized eff ective\npotential method for the ferromagnetic (upper panel) and ferrim agnetic (lower panel)\nconfigurations of Sr 2VTcO 6and SrYVTcO 6. The Fermi level is at zero energy.\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s50/s48/s49/s48/s48/s49/s48/s50/s48/s89\n/s50/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\nFigure 5. Total and partial densities of states as obtained in the optimized eff ective\npotentialmethod forthe antiferromagneticconfigurationofY 2VTcO 6. The Fermi level\nlies in the band gap.\nthe properties of the antiferromagnetic Y 2VTcO 6are defined by only two parameters:\n∆ctand (∆V\nex−∆Tc\nex), while the third parameter (∆V\nex+ ∆Tc\nex) can be taken arbitrarily.\nFinally, wewouldlike toemphasize thatantiferromagneticY 2VTcO 6becomesinsulating\ndue to the large exchange splitting (both at the V- and Tc-sites) ob tained in the OEP\ncalculations. If the splitting were smaller, the majority-spin V- and T c-bands would\neventually merge, yielding the half-metallic antiferromagnetic state , which was obtained\nin the LSDA calculations for the isoelectronic La 2VTcO 6[16]. Thus, such half-metallicOptimized Effective Potential Model for Double Perovskites 10\nantiferromagnetic state is probably an artifact of LSDA, and more rigorous treatment\nof the correlation effects will open the band gap in the both spin chan nels.\n3.2. Screening and the effective Stoner model\nSince the effective potential is local (or diagonal with respect to th e site indices) and\ndoes not affect the orbital degrees of freedom, the OEP scheme h as many similarities\nwith the Stoner model of magnetism, where the strength of the int raatomic splitting ∆τ\nex\nbetween the minority- and majority-spin states is controlled by the effective parameter\nIτ: ∆τ\nex=Iτmτ. Thus, using the intraatomic exchange splitting, obtained from the\nminimizationofthetotalenergy, andcorrespondingvaluesofthes pinmagneticmoments\nin the ground state (Fig. 6), one can find the effective Stoner para meterIτ\nOEP. All such\n/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s77/s111/s77/s97/s103/s110/s101/s116/s105/s99/s32/s77/s111/s109/s101/s110/s116/s115/s32/s40\n/s66 /s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54/s86\n/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s84/s99/s77/s97/s103/s110/s101/s116/s105/s99/s32/s77/s111/s109/s101/s110/s116/s115/s32/s40\n/s66 /s41\n/s32/s32\n/s32/s32/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s84/s99/s79\n/s54/s86\nFigure 6. Local spin magnetic moments at the V- and Mo(Tc)-sites, as obtain ed in\ntheoptimizedeffectivepotentialmethodforSr 2−xYxVMoO 6(left)andSr 2−xYxVTcO 6\n(right).\nconsiderations are valid only for the ferrimagnetic metallic state, wh ere ∆τ\nexare uniquely\ndefined. The results for Sr 2−xYxVMoO 6are explained in Fig. 7. The effective Stoner\nparameters derived from the OEP approach are of the order of 1.3 -1.6 eV at the V-sites\nand 1.0-1.1 eV at the Mo-sites. Similar parameters were obtained for Sr2−xYxVTcO 6:\nIV\nOEP= 1.27 eV and ITc\nOEP= 1.06 eV in the case of Sr 2VTcO 6, andIV\nOEP= 1.28 eV\nandITc\nOEP= 1.18 eV in the case of SrYVTcO 6. All of them are systematically larger\nthat the typical values of Iτobtained in the local-spin-density approximation (typically,\nclose 1 eV for the 3 d-elements and even smaller for the 4 d- and 5d-elements [26]).\nThis example clearly shows the importance of the rigorous treatmen t of the exchange-\ncorrelation interactions (beyond LSDA) in the physics of double per ovskites.\nAnother limiting case is the Hartree-Fock (HF) approximation, which neglects the\ncorrelation interactions. Then, corresponding Stoner paramete r describing the splitting\nin the system of three t2g-orbitals, are given by Iτ\nHF= (Uτ+ 2Jτ)/3. They are also\nplotted in Fig. 7. The difference between Iτ\nHFandIτ\nOEPis the measure of correlation\ninteractions and their impact on the magnetic properties of the dou ble perovskites. One\ncan see that Iτ\nOEPis reduced by about 10-20% in comparison with Iτ\nHF, which indicates\nthe importance of correlation interactions.\nFinally, it is instructive to consider the effect of the static screening alone. TheOptimized Effective Potential Model for Double Perovskites 11\n/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s73/s32/s86\n/s82/s80/s65/s73/s32/s77/s111\n/s82/s80/s65/s73/s32/s77/s111\n/s79/s69/s80/s73/s32/s77/s111\n/s72/s70/s73/s32/s86\n/s79/s69/s80/s73/s32/s86\n/s72/s70/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s83/s116/s111/s110/s101/s114/s32/s80/s97/s114/s97/s109/s101/s116/s101/s114/s115/s32/s40/s101/s86/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54\nFigure 7. EffectiveStoner parametersforSr 2−xYxVMoO 6asderivedin the optimized\neffective potential (OEP) method for the ferrimagnetic state, in t he Hartree-Fock(HF)\napproximation, and using only static values of Coulomb and exchange interactions in\nthe random phase approximation (RPA).\ncorresponding on-site interaction parameters in RPA are given by\nˆURPA=1\nΩBZ/integraldisplay\ndq/bracketleftBig\n1−ˆUˆP(0,q)/bracketrightBig−1ˆU,\nwhich yields the Stoner parameter Iτ\nRPA= (Uτ\nRPA+ 2Jτ\nRPA)/3. Such a picture\ncan be regarded as the static limit of the GWapproximation [27, 28], where the\nfrequency-dependent screened Coulomb interaction ˆURPA(iω) is replaced by its static\ncounterpart ˆURPA≡ˆURPA(0)[29]. Typically, it issufficient toconsider onlysite-diagonal\ncontributions to ˆURPA, because if ˆUis site-diagonal, the off-diagonal contributions to\nˆURPAare expected to be small [30]. These results are also shown in Fig. 7. O ne can\nclearly see that the parameters Iτ\nRPAare overscreened and strongly underestimated in\ncomparison with Iτ\nOEP. This indicates the importance of dynamical screening effects in\nthe construction of the optimized effective potential.\n3.3. Total energy and stability of the ferrimagnetic state\nThe behavior of the total energy and of its partial contributions is explained in Fig. 8.\nFor all considered compounds, except Sr 2VMoO 6, the ferrimagnetic configuration has\nlower energy. The tendencies towards ferrimagnetism increase wit h the band-filling.\nThe main factor stabilizing the ferrimagnetic state is the gain of the k inetic energy. The\nmain factor destabilizing it is the loss of the exchange energy.\nIn order to investigate the local stability of the ferrimagnetic phas e, we calculate\nthe spin stiffness D, which is related to the second derivative of the total energy withOptimized Effective Potential Model for Double Perovskites 12\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s107/s105/s110/s101/s116/s105/s99\n/s32/s67/s111/s117/s108/s111/s109/s98\n/s32/s101/s120/s99/s104/s97/s110/s103/s101\n/s32/s99/s111/s114/s114/s101/s108/s97/s116/s105/s111/s110\n/s32/s116/s111/s116/s97/s108/s69/s110/s101/s114/s103/s121/s32/s100/s105/s102/s102/s101/s114/s101/s110/s99/s101/s32/s40/s101/s86/s47/s102/s111/s114/s109/s117/s108/s97/s32/s117/s110/s105/s116/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s107/s105/s110/s101/s116/s105/s99\n/s32/s67/s111/s117/s108/s111/s109/s98\n/s32/s101/s120/s99/s104/s97/s110/s103/s101\n/s32/s99/s111/s114/s114/s101/s108/s97/s116/s105/s111/s110\n/s32/s116/s111/s116/s97/s108/s69/s110/s101/s114/s103/s121/s32/s100/s105/s102/s102/s101/s114/s101/s110/s99/s101/s32/s40/s101/s86/s47/s102/s111/s114/s109/s117/s108/s97/s32/s117/s110/s105/s116/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s84/s99/s79\n/s54\nFigure 8. Total energy difference between ferrimagnetic and ferromagnet ic states and\npartial contributions to this energy difference, calculated for Sr 2−xYxVMoO 6(top)\nand Sr 2−xYxVTcO 6(bottom).\nrespect to the spin-spiral vector qin the ferrimagnetic state:\nE(q)≈E(q0)+D(q−q0)2,\nwhereq0= (π/a,π/a,π/a ) andais the cubic lattice parameter. More specifically,\nthe numerical calculations of E(q), were performed by employing the generalized Bloch\ntheorem [31] and the so-called local-force theorem for the infinites imal spin rotations.\nThe latter states that the total energy change for the small rot ations can be replaced by\nthe change of the one-electron energies, obtained from the Kohn -Sham equations (2) for\nthe rigid spin rotations of the effective potential ˆ vσ\neff[32]. The theorem is rather general\nand can be proven by considering the rotational invariance of the e xchange-correlation\nfunctions [8]. Thus, the range of the applicability of this theorem is no t limited by the\nlocal-spin-density approximation, for which it was originally proven, b ut can be also\nextended to the OEP approach [8], provided that the effective pote ntial ˆvσ\neffis uniquely\ndefined. Particularly, since the choice of ˆ vσ\neffis not unique in the case of insulating\nY2VTcO 6, this strategy cannot be directly applied to this system.\nThe behavior of Din the OEP model is explained in Fig. 9. Particularly, Dis\nnegative in the case of Sr 2VMoO 6, Sr2VTcO 6, and SrYVMoO 6. Thus, the ferrimagnetic\nstate in these compounds is unstable with respect to the incommens urate spin-spiral\nstate. Since FM state has lower energy in the case of Sr 2VMoO 6(Fig. 8), the spin-spiral\ninstability of the ferrimagnetic state may be finally resolved in the fav or of the collinearOptimized Effective Potential Model for Double Perovskites 13\n/s45/s50/s45/s49/s48/s49/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s84/s99/s79\n/s54/s83/s112/s105/s110/s32/s83/s116/s105/s102/s102/s110/s101/s115/s115/s32/s40/s101/s86/s32/s65/s50\n/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54/s111\nFigure 9. Spin stiffness in the ferrimagnetic state calculated using optimized eff ective\npotentials for Sr 2−xYxVMoO 6(filled circles) Sr 2−xYxVTcO 6(empty squares).\nFM arrangement, to make it the true magnetic ground state of Sr 2VMoO 6. However,\nin Sr2VTcO 6and SrYVTcO 6already ferrimagnetic state has lower energy. Moreover,\nthis ferrimagnetic state itself appears to be unstable with respect to the noncollinear\nspin-spiral alignment. Thus, the ground state in this case will be neit her ferro- nor\nferrimagnetic.\n4. Summary and Conclusions\nIn summary, using first-principles electronic structure calculation s, we have constructed\nthe effective low-energy model for the t2g-states of double perovskites Sr 2−xYxVMoO 6\nand Sr 2−xYxVTcO 6. Then, the ground-state properties of the obtained model have\nbeen investigated on the basis of the optimized effective potential m ethod, by treating\ncorrelation interactions in the random-phase approximation. The r eason for using\nsuch strategy was twofold. On the one hand, such approach allows one to improve\nthe commonly used local-spin-density approximation for the double p erovskites and\nincorporate the physics of electron correlations beyond the homo geneous electron gas\nlimit. On the other hand, since OEP is formulated as the ground-stat e method,\nwhere all the parameters are derived rigorously by minimizing the tot al energy of\nthe system, it gets rid of many ambiguities inherent to the LSDA+ Umethod.\nParticularly, although Sr 2−xYxVMoO 6, Sr2−xYxVTcO 6, and other isoelectronic to\nthem compounds are frequently proposed to exhibit half-metallic fe rromagnetic or\nantiferromagnetic properties, the situation is probably not such o ptimistic because of\nthe electron correlation effects, which affect the intraatomic exch ange splitting andOptimized Effective Potential Model for Double Perovskites 14\nmakes it substantially larger than in LSDA. Of course, this will revise m any predictions\nbased on the LSDA. For example, Y 2VTcO 6, which was expected to be half-metallic\nantiferromagnet, appears to be insulator. The behavior of other materials is even more\ncomplicated. For the most of them (except Sr 2VMoO 6), the half-metallic ferromagnetic\nstate is energetically less favorable than the (normal) metallic ferrim agnetic one.\nNevertheless, this ferrimagnetic state appears to be also unstab le with respect to the\nspin-spiral alignment. Thus, the true magnetic ground state of th ese double perovskites\nwill be the spin spiral or some more complicated non-collinear magnetic structure.\nFinally, we have argued that the effective potential (and therefor e the one-electron\nband structure) for the half-metallic and insulating ferro- or antif erromagnetic state is\nnot uniquely defined. Generally, there is a bunch of parameters, wh ich yield the same\ntotal energy, occupation numbers and spin magnetic moments at t he transition-metal\nsites.\nAcknowledgments\nI wish to thank the support from the Federal Agency for Science a nd Innovations,\ngrant No. 02.740.11.0217, during my stay at the Ural State Technic al University\n(Ekaterinburg, Russia).\nReferences\n[1] Serrate D, De Teresa J M and Ibarra M R 2007 J. Phys.: Condens. Matter 19023201\n[2] Kobayashi K-I, Kimura T, Sawada H, Terakura K and Tokura Y 19 98Nature395677\n[3] Solovyev I V 2002 Phys. Rev. B65144446\n[4] van Leuken H and de Groot R A 1995 Phys. Rev. Lett. 741171\n[5] Pickett W E 1998 Phys. Rev. B5710613\n[6] Anisimov V I, Zaanen J and Andersen O K 1991 Phys. Rev. B44943\n[7] Solovyev I V, Dederichs P H and Anisimov V I 1994 Phys. Rev. B5016861\n[8] Solovyev I V and Terakura K 1998 Phys. Rev. B5815496\n[9] Kohn W and Sham L J 1965 Phys. Rev. 140A1133\n[10] Talman J D and Shadwick W F 1976 Phys. Rev. A1436\n[11] Kotani T and Akai H 1996 Phys. Rev. B5416502\n[12] Kotani T 1998 J. Phys.: Condens. Matter 109241\n[13] Engel E and Schmid R N 2009 Phys. Rev. Lett. 103036404\n[14] Gotoh H, Takeda Y, Asano H, Zhong J, Rajanikanth A and Hono K 2009Applied Physics Express\n2013001\n[15] Park M S and Min B I 2005 Phys. Rev. B71052405\n[16] Wang Y K, Lee P H and Guo G Y 2009 Phys. Rev. B80224418\n[17] Park J H, Kwon S K and Min B I 2002 Phys. Rev. B65174401\n[18] Uehara M, Yamada M and Kimishima Y 2004 Solid State Commun. 129385\n[19] Song W, Wang J, Meng J and Wu Z 2010 J. Comp. Chem. 311795\n[20] Solovyev I V 2008 J. Phys.: Condens. Matter 20293201\n[21] Kanamori J 1963 Prog. Theor. Phys. 30275\n[22] Pines D 1999 Elementary Excitations in Solids (Oxford: Westview Press)\n[23] von Barth U and Hedin L 1972 J. Phys. C51629\n[24] Aryasetiawan F, Miyake T and Terakura K 2002 Phys. Rev. Lett. 88166401Optimized Effective Potential Model for Double Perovskites 15\n[25] Mazin I I 1999 Phys. Rev. Lett. 831427\n[26] Gunnarsson O 1976 J. Phys. F6587\n[27] Hedin L 1965 Phys. Rev. 139A796\n[28] Aryasetiawan F and Gunnarsson O 1998 Rep. Prog. Phys. 61237\n[29] Aryasetiawan F, Imada M, Georges A, Kotliar G, Biermann S and L ichtenstein A I 2004 Phys.\nRev.B70195104\n[30] Solovyev I V 2006 Phys. Rev. B73155117\n[31] Sandratskii L M 1998 Adv. Phys. 4781\n[32] Bruno P 2003 Phys. Rev. Lett. 90087205"    },    {        "title": "1101.5994v1.Ferrimagnetism_and_disorder_in_epitaxial_Mn_2_x_Co_x_VAl_thin_films.pdf",        "content": "arXiv:1101.5994v1  [cond-mat.mtrl-sci]  31 Jan 2011Ferrimagnetism and disorder in epitaxial\nMn2−xCoxVAl thin films\nMarkus Meinert, Jan-Michael Schmalhorst, and G¨ unter\nReiss\nDepartment of Physics, Bielefeld University, 33501 Bielef eld, Germany\nE-mail:meinert@physik.uni-bielefeld.de\nElke Arenholz\nAdvanced Light Source, Lawrence Berkeley National Laborat ory, CA 94720,\nUSA\nAbstract. The quaternary full Heusler compound Mn 2−xCoxVAl with x= 1\nis predicted to be a half-metallic antiferromagnet. Thin fil ms of the quaternary\ncompounds with x= 0...2were preparedby DCand RFmagnetron co-sputtering\non heated MgO (001) substrates. The magnetic structure was e xamined by x-ray\nmagnetic circular dichroism and the chemical disorder was c haracterized by x-ray\ndiffraction. Ferrimagnetic coupling of V to Mn was observed f or Mn 2VAl (x= 0).\nForx= 0.5, we also found ferrimagnetic order with V and Co antiparall el to Mn.\nThe observed reduced magnetic moments are interpreted with the help of band\nstructure calculations in the coherent potential approxim ation. Mn 2VAl is very\nsensitive to disorder involving Mn, because nearest-neigh bor Mn atoms couple\nanti-ferromagnetically. Co 2VAl has B2 order and has reduced magnetization. In\nthe cases with x≥0.9 conventional ferromagnetism was observed, closely relat ed\nto the atomic disorder in these compounds.\nSubmitted to: J. Phys. D: Appl. Phys.Ferrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 2\n1. Introduction\nHalf-metallic fully compensated ferrimagnets (HMFi), which are also k nown as half-\nmetallic antiferromagnets, attracted large interest during the pa st years. A material\nwith this property would exhibit full spin polarization at the Fermi leve l, but the\nmagnetization would be effectively zero. It was first predicted for M n and In doped\nFeVSb [1]. Among others, La 2VMnO 6and related double perovskites [2], and\ncertain diluted magnetic semiconductors have been later predicted to be half-metallic\nantiferromagnets as well [3]. This is interesting for technological ap plications, e.g.\nfor spin transfer torque switching, which depends on the magnetiz ation and the spin\npolarization of the material to be switched.\nGalanakis et al.pointed out that it may be possible to synthesize a HMFi by\nsubstituting CoforMnintheHeuslercompoundMn 2VAl[4]. Mn 2VAlisa(potentially\nhalf-metallic) ferrimagnet with antiparallel coupling of Mn and V momen ts and a\ntotal moment of -2 µBper formula unit. The high Curie temperature of 760K makes\nit interesting for practical applications. Numerous experimental [ 5, 6, 7, 9, 10] and\ntheoretical [11, 12, 13, 14, 15] studies are found in the literature . Following the Slater-\nPauling rule for Heusler compounds, m=NV−24 [16], the magnetic moment mis\nto be taken as negative, because the number of valence electrons NVis 22. Thus, by\nadding effectively two electrons per unit cell, the magnetization shou ld vanish. This\ncan be achieved by substituting one Mn with one Co atom, which has tw o additional\nelectrons. Ab initio simulations were carried out on this system in the L2 1structure\nwith Mn and Co randomly spread across the Wyckoff 8c sites and V and Al on the\n4a and 4b sites. Indeed, a HMFi is found with magnetic moments of: - 1.388 (Mn),\n0.586 (Co), 0.782 (V), 0.019 (Al) [4]. It was shown by Luo et al.that the site\noccupation preference in Mn 2YAl depends on the number of valence electrons of Y: if\nit is lower than the one of Mn, Ywould preferentially occupy the 4a/b sites, but if it is\nhigher,Ywould rather occupy the 8c sites together with Mn, changing the st ructure\nto the Hg 2CuTi type [17]. Accordingly, one can expect an occupation as propo sed by\nGalanakis et al.in Mn 2−xCoxVAl (MCVA).\nFor many practical applications it is necessary to prepare thin films o f the\nmagnetic materials. Therefore one has to find suitable deposition te chniques and\noptimize the parameters. The parent compounds Mn 2VAl and Co 2VAl [18, 19] have\nbeen successfully synthesized in the bulk and epitaxial growth of Mn 2VAl films with\nL21ordering on MgO (001) single crystals was also demonstrated [20, 2 1]. In this\npaper we present experimental results on the structural and ma gnetic properties of\nepitaxial Mn 2−xCoxVAl thin films.\n2. Methods\n2.1. Experimental details\nThe samples were deposited using a UHV co-sputtering system equip ped with five\nDC and two RF 3” magnetron sputtering sources, arranged in a con focal sputter-up\ngeometry. Up to four sources can be used simultaneously. The tar get-to-substrate\ndistance is 21cm and the inclination of the sources is 30◦. The substrate carrier can\nbe heated to 1000◦C by an infrared heater from the backside. An electron beam\nevaporator with one crucible is placed in the center of the chamber a t a distance of\n50cm to the sample. The base pressure of the system was typically 5 ·10−10mbar.Ferrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 3\nElemental targets of Mn, Co, V, and Al of 99.95% purity were used. The\nsputtering pressure was set to 2 ·10−3mbar. The correct sputter power ratios were\nset up using a combined x-ray reflectivity and x-ray fluorescence t echnique.\nAll samples used in this study had the following stack sequence: MgO ( 001)\nsingle crystal / Mn 2−xCoxVAl 18nm / Mg 0.5nm / MgO 1.5nm with x=\n0/0.5/0.9/1.0/1.1/1.5/2. The upper MgO was deposited by e-beam evaporation.\nThe protective Mg / MgO bilayer was deposited after cooling the samp les to prevent\noxidation and interdiffusion. Diffraction measurements on Mn 2VAl films deposited at\nvarious temperatures revealed that a substrate carrier temper ature of at least 600◦C\nwas necessary to obtain good order, but temperatures above 70 0◦C lead to strong\nMn sublimation, which can not be reliably compensated by higher sputt ering power\n(compare with [20]). Therefore all samples discussed in this paper we re deposited at\na carrier temperature of 700◦C.\nX-ray diffraction (XRD), reflectometry (XRR), and fluorescence (XRF) were\nperformed in a Philips X’Pert Pro MPD diffractometer with Cu anode, Br agg-\nBrentano and collimator point focus optics, an open Euler cradle and an Amptek\nfluorescence detector in a He enclosure.\nX-ray magnetic circular dichroism (XMCD) was measured at beamline 6 .3.1 of\nthe Advanced Light Source. A magnetic field of ±1.6T parallel to the incoming\nx-ray beam was applied, the sample surfaces were inclined by 30◦with respect to\nthe incoming beam. Element specific magnetic hysteresis loops were t aken with a\nmagnetic field of up to ±2T. The magnetic field was switched for every energy point\nto obtain the dichroic signal. Data were taken at 20K, 150K, 200K, a nd 300K.\nAll XMCD spectra were taken at least twice, with polarizations of +60 % and -60%.\nSystematic measurements were performed in the surface sensitiv e total electron yield\nmode. Additionally, following an idea by Kallmayer et al.[22], the visible light\nfluorescence of the MgO substrate was detected by a photo diode behind the sample.\nThus, bulk information of the films could be obtained in x-ray transmis sion.\n2.2. Band structure calculations\nBand structurecalculationsofdisorderedcompoundswere perfo rmedwith the Munich\nSPRKKR package, a spin-polarized relativistic Korringa-Kohn-Rost oker code [23].\nThe ground state self-consistent potential calculations were per formed on 834 kpoints\nin the irreducible wedge of the Brillouin zone. The exchange-correlat ion potential\nwas approximated by the Perdew-Burke-Ernzerhof implementatio n of the generalized\ngradient approximation [24], the Fermi energy was determined usin g Lloyd’s formula\n[25, 26]. The angular momentum expansion was taken up to lmax= 3. A scalar\nrelativistic representation of the valence states was used in all cas es, thus neglecting\nthe spin-orbit coupling. For Mn 2VAl the atomic spheres approximation was applied\nand Co 2VAl was treated with full potential calculations. Half-metallic groun d states\nwere obtained for Mn 2VAl and Co 2VAl with their respective bulk lattice parameters.\nTo account for disorder, the coherent potential approximation ( CPA) was used. In\nour calculations with the ideally ordered L2 1structure, Mn 2VAl has a total moment\nof 2.01µB/f.u., with 1.54 µBon Mn and -1.03 µBon V. Co 2VAl has a total moment of\n1.99µB/f.u., with0.87 µBonCoand0.28 µBonV. Thesevaluesarein goodagreement\nwith calculations presented by other authors [27].Ferrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 4\n/s51/s53\n/s51/s48\n/s50/s53\n/s50/s48\n/s49/s53\n/s49/s48\n/s68/s32/s40/s110/s109/s41\n/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48\n/s120/s48/s46/s54\n/s48/s46/s53\n/s48/s46/s52\n/s48/s46/s51\n/s48/s46/s50\n/s48/s46/s49\n/s48/s46/s48\n/s101/s91/s48/s48/s49/s93/s32/s40/s37/s41\n/s49/s46/s48\n/s48/s46/s56\n/s48/s46/s54\n/s48/s46/s52\n/s48/s46/s50\n/s48/s46/s48/s83\n/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48\n/s120/s32/s83/s66/s50\n/s32/s83/s76/s50/s49/s49/s49/s48/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s51/s52 /s51/s51 /s51/s50 /s51/s49 /s51/s48 /s50/s57 /s50/s56\n/s50/s113/s32/s40/s100/s101/s103/s41/s32/s120/s32/s61/s32/s48\n/s32/s120/s32/s61/s32/s49\n/s32/s120/s32/s61/s32/s50\n/s32\n/s53/s46/s57/s48\n/s53/s46/s56/s53\n/s53/s46/s56/s48\n/s53/s46/s55/s53\n/s53/s46/s55/s48\n/s53/s46/s54/s53/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s99/s32/s40/s197/s41/s40/s97/s41\n/s40/s98/s41 /s40/s100/s41\n/s40/s99/s41 /s40/s101/s41\nFigure 1. (a) :θ-2θscans of the (002) reflections of Mn 2VAl (x= 0) and Co 2VAl\n(x= 2). Clear Laue oscillations are visible in both cases. (b): out-of-plane lattice\nparameter cas function of x.(c): Order parameters SB2andSL21as functions\nofx.(d): Microstrain ε[001]and(e): coherence length Dand as functions of x.\nThe dashed line in (e)denotes the film thickness.\n3. Experimental results and discussion\n3.1. Lattice structure\nAll MCVA films were found to be highly epitaxial with MCVA [001] /bardblMgO [001],\nrockingcurvewidthsof0.6◦to 1.5◦, andanMCVA[100] /bardblMgO[110]in-planerelation.\nLaueoscillationsobservedatthe (002)reflectionsdemonstratet he latticeandinterface\ncoherence of the films in the two limiting cases of Mn 2VAl and Co 2VAl (Fig. 1(a)).\nForx= 1, however, the oscillations are less pronounced.\nFigure 1(b) displays the out-of-plane lattice parameter cas a function of x.\nAccordingtoVegard’slaw[8], alineardecreaseofthe latticeparamet erwithincreasing\nxcan be expected for a simple substitutional model. However, a signifi cant deviation\nfrom this law is observed at x= 1. This indicates, as we will see in detail later, a\nstructural and magnetic order-disorder transition. For Mn 2VAl,cis slightly lower\nthan the bulk value of 5.875 ˚A [9]; Co 2VAl has also a slightly reduced ccompared to\nthe bulk value of 5.77 ˚A [18]. This is compatible with a tetragonal distortion causedFerrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 5\nby the epitaxial matching with the substrate: the lattice is expande d in the plane\nand shrinks in the out-of-plane direction. For the case of Co 2TiSn we have recently\nperformed first principles calculations of the change in total energ y for this type of\nlattice distortion. In this case it is of the order of 25 −50meV, and is thus easily\nactivated during the film growth [28]. For the compounds presented here, we expect\na similar energy range.\nTakamura’s extended order model for Heusler compounds [29] was applied to\nobtain the order parameters SB2andSL21from the measured XRD peak intensities.\nUnlike Webster’s model [33], this model takes the dependence of SL21onSB2into\naccount. The structure factors were obtained from the measur ed intensities by\ncorrecting for the Lorentz-Polarization term and the temperatu re factor with an\neffective Debye-Waller factor of Beff= 0.4.SB2is calculated from the four structure\nfactor ratios of (002) and (222) versus (022) and (004), respe ctively.SL21is calculated\nas the average of the (111) structure factor versus (022) and (004). The full atomic\nscattering factors including angular dependence and anomalous co rrections were used\nin the numerical model calculations. As shown in Fig. 1(c), the Mn 2VAl films are\nordered in the L2 1structure with significant V-Al disorder ( SL21≈0.4). With\nincreasing Co content, the L2 1order disappears in the alloy system; Co 2VAl does not\nshow any sign of L2 1ordering. On the other hand, the degree of B2 order increases\nslightly with increasing Co content, from SB2= 0.7 toSB2= 0.8, i.e., 85% to 90% of\nthe Co atoms are on the 8c sites. However, we note here that disor der between Co,\nMn, and V can not be identified with this method, because the atomic f orm factors\nare too similar.\nA Williamson-Hall analysis [34] of the integral peak widths of the (002) , (004),\nand (006) reflections was performed. A Gaussian instrumental pe ak broadening\nand a Lorentzian convolution of grain size and strain effects were as sumed, i.e., the\ncontributions were separated with\nB2\nobs=B2\ninst+B2\nss (1)\nand\nBss·cosθ=kλ\nD+4ε[001]sinθ (2)\nwithBobsbeing the observed width, Binstthe instrumental width, Bssthe size-strain\nwidth, the shape factor k= 0.9, the coherence length (grain size) Dand the averaged\n[001] component of the strain tensor ε[001]. The analysis results are displayed in Fig.\n1(d) and (e). The measured coherence length matches the film thic knesses quite well\nwithin the accuracyofthe measuringandfitting procedure. Aclear trend ofincreasing\nstrain can be observed, from 0.18% to 0.47%. The lattice mismatch of Co2VAl (3.1%)\nis about 2.4 times as large as the mismatch of Mn 2VAl (1.3%) with MgO. The same\nfactor applies to the strain values, which verifies the high quality of t he epitaxy. The\nlower degree of film coherence, the deviation from Vegard’s law and t he rather low\nstrain in spite of the large lattice mismatch indicate an increased dens ity of lattice\ndefects in Mn 1Co1VAl. The defects allow for relaxation of the film, which can reduce\nthe microstrain at a loss of coherence.\nZiebeckandWebsterfoundthatCo 2VAlcrystallizesintheL2 1phase,but exhibits\nsome preferential V-Al disorder [18]. The samples measured by the m were annealed at\n800◦C for 24h. The samples by Kanomata et al.were annealed at up to 1200◦C, and\nstill exhibited a complex grain structure consisting of L2 1and B2 ordered fractions.\nDeposition at 700◦C may thus be insufficient to promote L2 1order in Co 2VAl.Ferrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 6\n/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s88/s77/s67/s68\n/s53/s51/s48 /s53/s50/s53 /s53/s50/s48 /s53/s49/s53 /s53/s49/s48 /s53/s48/s53\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s120/s32/s61/s32/s48\n/s120/s32/s61/s32/s48/s46/s53\n/s120/s32/s61/s32/s49\n/s120/s32/s61/s32/s49/s46/s53\n/s120/s32/s61/s32/s50/s45/s50/s46/s49/s45/s49/s46/s56/s45/s49/s46/s53/s45/s49/s46/s50/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51\n/s56/s48/s48 /s55/s57/s53 /s55/s57/s48 /s55/s56/s53 /s55/s56/s48 /s55/s55/s53 /s55/s55/s48\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s120/s32/s61/s32/s48/s46/s53\n/s120/s32/s61/s32/s49\n/s120/s32/s61/s32/s49/s46/s53\n/s120/s32/s61/s32/s50\n/s45/s50/s46/s55/s45/s50/s46/s52/s45/s50/s46/s49/s45/s49/s46/s56/s45/s49/s46/s53/s45/s49/s46/s50/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51\n/s54/s54/s48 /s54/s53/s53 /s54/s53/s48 /s54/s52/s53 /s54/s52/s48 /s54/s51/s53 /s54/s51/s48\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s120/s32/s61/s32/s48\n/s120/s32/s61/s32/s48/s46/s53\n/s120/s32/s61/s32/s49\n/s120/s32/s61/s32/s49/s46/s53\n/s86 /s77/s110 /s67/s111\nFigure 2. Experimental XMCD spectra for V, Mn, and Co at 20K. The\ncorresponding XAS spectra were normalized to a post-edge ju mp height of 1.\nThe spectra for x= 0.9,1.1 are similar to x= 1 and are omitted for clarity.\nTable 1. Experimental total magnetic moments at 20K (given in µB/f.u.) and\nestimated Curie temperatures.\nmtotTC\nMn2VAl 0.88 ≫RT\nMn1.5Co0.5VAl 0.1 -\nMn1.0Co1.0VAl 1.09 ≈350K\nMn0.5Co1.5VAl 2.29 -\nCo2VAl 1.66 ≈210K\nHowever, as stated initially, a higher deposition temperature was no t usable because\nof Mn sublimation.\n3.2. Magnetic and electronic structure\nWe begin with a discussion of the XMCD spectra in dependence on x, which are\nshown in Fig. 2 (a)-(c). For x= 0, i.e., for pure Mn 2VAl, we find an antiparallel\nalignment of the Mn and V moments. This is preserved up to x= 0.5, going along\nwith an antiparallel coupling of Co to Mn. Here, we find the predicted f errimagnetic\norder with the Co and V moments pointing opposite to the Mn moments . With\nfurther increasing x, all magnetic moments point in the same direction; the alloys\nbecome ferromagnets. This transition is closely related to chemical disorder which\nis indicated by the deviation of the lattice parameter from Vegard’s la w. Across the\nstoichiometry series the shape of the spectra changes significant ly. Most prominently,\nthe splitting of the V and Mn lines vanishes above x= 0.9 and above. The appearance\nof this splitting is directly correlated with the appearance of ferrima gnetism. The line\nshape of the Mn XMCD for x= 1.5 is very similar to the Mn line shape in Co 2MnAl\nor Co2MnSi [35]. For the ferrimagnetic coupling of Co and Mn, they haveto b e secondFerrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 7\n/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s88/s77/s67/s68\n/s53/s51/s48 /s53/s50/s53 /s53/s50/s48 /s53/s49/s53 /s53/s49/s48 /s53/s48/s53\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s101/s108/s101/s99/s116/s114/s111/s110/s32/s121/s105/s101/s108/s100\n/s32/s108/s117/s109/s105/s110/s101/s115/s99/s101/s110/s99/s101/s49/s46/s48\n/s48/s46/s56\n/s48/s46/s54\n/s48/s46/s52\n/s48/s46/s50\n/s48/s46/s48\n/s45/s48/s46/s50/s88/s77/s67/s68\n/s54/s54/s48 /s54/s53/s53 /s54/s53/s48 /s54/s52/s53 /s54/s52/s48 /s54/s51/s53 /s54/s51/s48\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s101/s108/s101/s99/s116/s114/s111/s110/s32/s121/s105/s101/s108/s100\n/s32/s108/s117/s109/s105/s110/s101/s115/s99/s101/s110/s99/s101\n/s77/s110\n/s86\nFigure 3. Normalized XMCD spectra of Mn and V in electron yield and\nluminescence detection.\nnearestneighborsonoctahedralpositions. Coand Mn on tetrahe dralnearest-neighbor\npositions couple ferromagnetically, as in Co 2MnGe [33] and the other Co 2Mn-based\nHeusler compounds.\nTo assert that the complex shape of the Mn and V spectra is not a su rfacial effect,\nwe have measured the transmitted x-ray intensity in luminescence d etection at room\ntemperature for Mn 2VAl. The XMCD spectra are almost equal in total electron yield\nand in transmission (see Figure 3), although in both cases the L3 pre -peak is more\npronounced in transmission. However, compared to the total are a of the peaks, this\ndeviation is small. The fine structure of the spectra is consequently related to the\nelectronic structure of the films rather than to a surface effect.\nUsing the sum rule analysis we extracted the spin and orbital magnet ic moments\nfrom the XMCD spectra [36]. Table 1 summarizes the total magnetic m oments\nobtained from sum rule analysis and provides estimates of the Curie t emperatures\nobtained from temperature dependent XMCD for x= 0,1,2. Figure 4 displays the\nelement specific total moments in dependence on x. Because of dynamical screening\neffects of the x-ray field, the sum rules fail for the early 3d transit ion metals [30]. To\ncompensate the resulting spectral mixing effects, the apparent s pin magnetic moments\ncan be multiplied with correction factors as suggested by D¨ urr et al.and Scherz et al.,\ni.e. 1.5 for Mn [31] and 5 for V [32]. Actually, the applied correctionfactors depend on\nthe actual electronic structure and can not be simply transferre d to different systems.\nHowever, we assume that this influence is rather small, so that quan titative results\ncan be obtained.Ferrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 8\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48\n/s45/s48/s46/s53\n/s45/s49/s46/s48\n/s45/s49/s46/s53/s109/s116/s111/s116/s32/s40/s109/s66/s32/s47/s32/s97/s116/s111/s109/s41\n/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48\n/s120/s70/s105/s77 /s70/s77\n/s32/s86\n/s32/s77/s110\n/s32/s67/s111\nFigure 4. Element specific magnetic moments as functions of x. Ferrimagnetic\n(FiM) order is observed for x≤0.5, ferromagnetic (FM) order is observed for\nx≥0.9.\nIn Mn 2VAl we find a lowered Mn moment (1 µB) and an enhanced V moment\n(−1.1µB), resultinginatotalmagnetizationof0 .88µB/f.u. Nochangeofthemagnetic\nmoments was observed at RT as compared to 20K, hence the Curie t emperature\nis much higher than RT. The film is not well described by a pure L2 1order\nmodel. As discussed earlier, the film has some disorder between Mn an d (V,Al). In\nthis case, Mn atoms reside on sites surrounded by other Mn atoms, which couple\nantiferromagnetically at short distance. Indeed, by calculating th e self-consistent\npotential in SPRKKR with 20% Mn-Al or Mn-V exchange, we find antipa rallel\ncoupling of the antisites. For Mn-Al exchange, the Mn(8a) moment is reduced to\n1.22µBand the Mn on the Al site has −2.48µB. The V moment is reduced to\n−0.83µB. This results in a total magnetization of 0 .85µB/f.u., and the average Mn\nmoment is consequently 0 .85µB. In the case of Mn-V exchange, the Mn(8a) moment\nremains at 1 .58µBand the Mn on the V site has −2.63µB. The V moment on the\n4b site is −0.87µBand +0.84µBon the 8a site. In this case the total moment is\n1.78µB/f.u., with an average Mn moment of 1 .16µB. Further, the case of Mn-Al\nexchange is energetically preferred with respect to the Mn-V exch ange. Seeing the\nlow total and Mn moments and the high V moment, a preferential Mn- Al exchange in\nMn2VAl is thus in good agreement with the structural and the magnetic data. Our\ncalculations show that the 20% Mn-Al disorder and B2 disorder bare ly influence the\nhalf-metallicgapofMn 2VAl. ForB2disorder,the totalmagneticmoment alsoremains\nunaffected. In contrast, 20% Mn-V disorder destroy the gap. Th is is in contrast to\nthe findings by Luo et al., obtained with a supercell approach in a pseudopotential\ncode. They state that the gap is preserved under 25% Mn-V disord er [17].\nCo2VAl has a reduced Co moment (0 .69µB) and a V moment of 0 .28µB, giving a\ntotal magnetization of 1.66 µB/f.u. The film has B2 order, which is expected to reduce\nthe magnetization from the highly ordered L2 1case. We find magnetic moments of\n0.75µBfor Co and 0.4 µBfor V in a B2 ordered SPRKKR calculation, with a total\nmoment of 1.86 µB/f.u., in good agreement with our measurements. Some additionalFerrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 9\ndisorder involving Co and V could explain the further reduced moment s. The Curie\ntemperatureisabout210K,whichissignificantlylowerthanthevalue forbulksamples\n(310K [18]). A calculation of the Curie temperature with SPRKKR within the mean\nfield approximation (see [37] for details of the procedure) yields 352 K in the L2 1\ncase and 165K in the B2 ordered case. The observed significant red uction of the\nCurie temperature in the disordered alloy is thus in agreement with th eory. The\nhalf-metallic gap of Co 2VAl vanishes in the B2 structure.\nAtx= 0.5, a nearly complete magnetic compensation with a total moment of\nonly 0.1µB/f.u. is observed. Remarkably, at x= 1.5 the total magnetic moment\nbecomes larger than 2 µB/f.u., caused by the high Mn moment of 1.67 µB. This is\nin agreement with the different Mn line shape: in, e.g., Co 2MnAl, in which Mn has\na similar line shape, Mn has a moment of about 3 µB[33]. Thus, the mechanism\nmainly responsible for the ferromagnetic coupling of all moments is th e preferentially\ntetrahedral (instead of octahedral) coordination of Mn atoms wit h Co.\n4. Conclusions\nEpitaxial thin films of Mn 2−xCoxVAl have been synthesized on MgO (001) substrates\nby DC and RF magnetron co-sputtering. It was intended to observ e a ferrimagnetic\ncompensation of the magnetization at x= 1. The films have significant chemical\ndisorder, depending on the degree of Mn-Co substitution. Mn 2VAl was found to\nbe L2 1ordered, with a preferential Mn-Al disorder and additional V-Al d isorder.\nThe Mn-Al disorder reduces the total moment considerably, beca use the nearest-\nneighbor Mn atoms couple antiferromagnetically in this configuration . Accordingly,\nthe magnetization of Mn 2VAl is very sensitive to disorder involving Mn. However, the\nband structure calculations suggest that only Mn-V disorder has a n influence on the\nhalf-metallic gap.\nBecause of the disorder, a nearly complete magnetic compensation was observed\nfor Mn 1.5Co0.5VAl. With further Co substitution, the electronic structure chang es\nconsiderably, and a parallel coupling of Co, Mn, and V was observed. We suppose\nthat Co and Mn become preferentially nearest neighbors, which lead s to a parallel\ncoupling of their magnetic moments.\nThe Co 2VAl films, being the second extremum of the substitutional series, had\nB2 order. The band structure calculations with B2 order suggest r educed moments,\nbut the experimentally determined moments are further reduced, which indicates\nadditional disorder involving Co. The Curie temperature was significa ntly reduced,\nwhich is in agreement with the trend observed in the mean field calculat ion. It\nis in principle possible to obtain a high degree of L2 1order in bulk Co 2VAl by\nappropriate thermal treatment, but our maximum substrate tem perature was limited\nby Mn evaporation. While it may be possible to obtain the correct occu pation for the\nferrimagnetic compensation in the bulk, it seems not possible to obta in films with a\nhigh degree of order.\nAcknowledgements\nThe authors gratefully acknowledge financial support by the Deut sche Forschungsge-\nmeinschaft (DFG) and the Bundesministerium f¨ ur Bildung und Forsc hung (BMBF).\nThey thank for the opportunity to work at BL 6.3.1 of the Advanced Light Source,\nBerkeley, USA, which is supported by the Director, Office of Science , Office of BasicFerrimagnetism and disorder in epitaxial Mn 2−xCoxVAl thin films 10\nEnergy Sciences, of the U.S. Department of Energy under Contra ct No. DE-AC02-\n05CH11231.\nReferences\n[1] van Leuken H and de Groot R A 1995 Phys. Rev. Lett. 741171\n[2] Pickett W E 1998 Phys. Rev. B 5710613\n[3] Akai H and Ogura M 2006 Phys. Rev. Lett. 97026401\n[4] Galanakis I, ¨Ozdo˜ gan K, S ¸a¸ sio˜ glu E and Akta¸ s B 2007 Phys. Rev. B 75092407\n[5] Kawakami M, Yoshida Y, Nakamichi T, Ishida S and Enokiya H 1981J. Phys. Soc. Jpn. 50\n1041\n[6] Yoshida Y, Kawakami M and Nakamichi T 1981 J. Phys. Soc. Japan 502203\n[7] Itoh H, Nakamichi T, Yamaguchi Y and Kazama N 1983 Trans. Japan Inst. Met. 24265\n[8] Vegard L 1921 Zeitschrift f¨ ur Physik 517\n[9] Nakamichi T and Stager C V 1983 J. Magn. Magn. Mater. 3185\n[10] Jiang C, Venkatesan M and Coey J M D 2001 Solid State Commun. 118513\n[11] Ishida S, Asano S and Ishida J 1984 J. Phys. Soc. Japan 532718\n[12] Weht R and Pickett W E 1999 Phys. Rev. 6013006\n[13] S ¸a¸ sio˜ glu E, Sandratskii L M and Bruno P 2005 J. Phys.: Condens. Matter 17995\n[14]¨Ozdo˜ gan K, Galanakis I, S ¸a¸ sioglu E and Akta¸ s B 2006 J. Phys.: Condens. Matter 182905\n[15] Chioncel L, Arrigoni E, Katsnelson M I and Lichtenstein A I 2009 Phys. Rev. B 79125123\n[16] Galanakis I, Dederichs P H and Papanikalou N 2002 Phys. Rev. B 66174429\n[17] Luo H, Zhu Z, Ma L, Xu S, Zhu X, Jiang C, Xu H and Wu G 2008 J. Phys. D: Appl. Phys. 41\n055010\n[18] Ziebeck K R A and Webster P J 1974 J. Phys. Chem. Solids 351\n[19] Kanomata T, Chieda Y, Endo K, Okada H, Nagasako M, Kobaya shi K, Kainuma R, Umetsu R\nY, Takahashi H, Furutani Y, Nishihara H, Abe K, Miura Y and Shi rai M 2010 Phys. Rev. B\n82144415\n[20] Kubota T, Kodama K, Nakamura T, Sakuraba Y, Oogane M, Tak anashi K and Ando Y 2009\nAppl. Phys. Lett 95222503\n[21] Klaer P, Jorge E A, Jourdan M, Wang W H, Sukegawa H, Inomat a K and Elmers H J 2010\nPhys. Rev. B 82024418\n[22] Kallmayer M, Schneider H, Jakob G, Elmers H J, Balke B and Cramm S 2007 J. Phys. D: Appl.\nPhys.401552\n[23] The Munich SPR-KKR package, version 5.4, H.Ebert et al,\nhttp://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR; H E bert, Fully relativistic band\nstructure calculations for magnetic solids: Formalism and Application Electronic Structure\nand Physical Properties of Solids (Lecture Notes in Physics vol 535) ed H. Dreyss (Berlin:\nSpringer) pp 191\n[24] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 773865\n[25] Lloyd P and Smith P V 1972 Adv. Phys. 2169\n[26] Zeller R 2008 J. Phys.: Condens. Matter 20035220\n[27] Kandpal H C, Fecher G H and Felser C 2007 J. Phys. D: Appl. Phys. 401507\n[28] Meinert M, Schmalhorst J-M and Reiss G 2010 Appl. Phys. Lett. 97012501\n[29] Takamura Y, Nakane R and Sugahara S 2009 J. Appl. Phys. 10507B109\n[30] Ankudinov A L, Nesvizhskii and Rehr J J 2003 Phys. Rev. B 76115120\n[31] D¨ urr H A, van der Laan G, Spanke D, Hillebrecht F U and Bro okes N B 1997 Phys. Rev. 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Matter 23036001"    },    {        "title": "1712.04624v1.Theoretical_Proposal_for_Determining_Angular_Momentum_Compensation_in_Ferrimagnets.pdf",        "content": " 1 Theoretical P roposal for Determining  Angular Moment um Compensation in \nFerrimagnets   \n \nZhifeng Zhua) Xuanyao Fong, and Gengchiau Liangb) \nDepartment of Electrical and Computer Engineering,  National University of Singapore, Singapore  \nThis work demonstrates that the magnetization and angular momentum compensation temperature ( TMC \nand TAMC) in ferrimagnets  (FiM) can be unambiguously determined by performing two sets of temperature \ndependent current switching , with the symmetry reverses at TMC and TAMC, respectively.  A theoretical model \nbased on the  modified Landau -Lifshitz -Bloch equation is developed to systematically study the spin torque  \neffect under different  temperatures, and  numerical simulations are performed to corroborate  our proposal. \nFurthermore,  we demonstrate that the recently reported linear relation between T AMC and TMC can be  explained  \nusing the Curie -Weiss theory .  \n \n \n    Magnetization dynamics of ferrimagnets (FiM) driven \nby spin- orbit torque (SOT) has attracted considerable \nattention, especially in material with antiferromagneti c \ncoupled transition -metal (TM) and rare- earth (RE) alloy \n(e.g. GdX(FeCo)1-X or Co 1-XTbX) [1-7]. The magnetization \nin FiM can be tuned through temperature ( T) [2, 3] or \nmaterial composition ( X) [4 -6], resulting in the  \ncompensation point (TMC or XMC) with zero net \nmagnetization ( mnet). The different g -factors of TM and \nRE induce a nother  compensation ( TAMC or XAMC) where \nthe net  angular momentum ( Snet) vanishes  [8]. \nFurthermore, FiM has faster dynamics than ferromagnet (FM) because of  the strong exchange coupling between \nsub-lattices , and in contrast to antiferromagnets (AFM), \nthe finite m\nnet enables the read out of FiM magnetic state \nusing tunnel magnetoresistance ( TMR ) effect.   \n    Recently, the domain wall dynamics near T AMC is \npredicted to be free from Walker breakdown [9] due to the decoupling of two collective coordinates , and the spin \ntorque is greatly enhanced at the vicinity of  X\nMC [5]. To \nexploit the rich physics near the two compensation points,  \none has to unambiguously determine the T MC and TAMC. \nTMC can be determined in many ways, including the direct \nmeasurement of  magnetization using vi brating sample \nmagnetometer  (VSM) [5] or an indirect measurement of \nthe anomalous Hall resista nce (RAHE) versus magnetic \nfield ( H) loops at different T with coercivity field ( HC) \npeak s at TMC [2]. However, it remains difficult to \nexperimentally determine the TAMC. In addition, the \ndifference between T AMC and TMC can vary  from a few K \nto several tens of K in different samples.  \n    In this L etter, we propose a device structure and  show \nthat the TAMC and TMC can be  unambiguously  determined \nby conducting two sets of current induced switching. \nFirst, we analytically exploit the symmetries  in both types \nof switching. Next, a theoretical  model based on  the \nmodified Landau- Lifshitz- Bloch (LLB) equation  is \ndeveloped to systematically describe the T  dependent FiM \ndynamics . Finally, numerical simulation s are performed to verify our proposal, and the relation between T AMC and \nTMC in different samples are studied  to show  the generality \nof our model .  \nThe device structure,  schematically depicted in Fig. 1 , \nconsists of a magnetic tunnel junction (MTJ) deposited on top of the HM layer , with the two sets of operations \ndefined in Fig. 1 (a) and (b), respectively. The MTJ \nincludes one perpendicular -FM pinned layer (PL) and one  \nperpendicular -FiM free layer (FL), sandwiched by a \nspacer layer  (e.g. MgO ). The FiM in this study is \nGd\nX(CoFe) 1-X where  Gd (Co Fe) dominates  at low  (high)  \n \n \nFIG. 1. Device structures with perpendicular FiM -MTJ  \ndeposited on the HM layer.  The red dash line  denotes t he \ncharge current , which  is controlled by the voltage s (VA,B,C). \nThe cross symbol denotes the blocking  of current path, \nwhich can be achieved by using a transistor. (a) The FL is \nswitched by SOT, assisted by the HX. (b) The FL is switched \nby STT. (c, d) Illustration of T  dependent switching \ncorresponding to (a ) and (b), respectively.   \n 2 T. Since CoFe has a larger g -factor, TAMC is higher than \nTMC. In addition, t he direction  of charge current ( JC) is \ncontrolled by the voltage s (i.e. VA,B,C). \nAs shown in Fig. 1 (a), a lateral JC flowing  through the \nHM layer  generates  spin orbit torque  (SOT) acting on the \nFiM layer due to  the spin Hall effect [10] and Rashba -\nEdelstein effect [11]. To achieve  deterministic switching \nin the perpendicular direction, an external magnetic field \n(HX) along the current direction is required  [1]. For \nsimplicity, we define it as type- I switching . As we will \ndiscuss later,  the switching direction  in this type is \ndetermined by mnet, and  a reversal in switching direction  \nwill be observed across TMC by plotting the T dependent \nmTM-J loop. In experiment, m TM can be obtained by \nmeasuring RAHE in a Hall bar structure, by noting that RAHE \nis mainly determined by the magnetic moment of TM \nelement [1]. The switching direction of this type can be  \nunderstood by analyzing two torques. The first one, τ  = \n∆m × HX, generated by H X only determines the switching \ndirection; the other one, ∆m = m × (m × σ) where σ is \nthe spin polarization, or iginates from the spin torque and  \nshould be  sufficient to overcome the energy barrier. It is \nclear  that the switching direction  is reversed by using  an \nopposite  HX or σ , which qualitatively agrees with  the \nexperimental result [12]. Fig. 1 (c) illustrates the type-I \nswitching in different T regions. For T  < TMC, a +y -\npolarized  σ generates ∆m in the − y-direction, resulting \nin τ = + z, i.e. mnet is switched from down to up.  For both \nTAMC > T > TMC and T > TAMC, a −y-polarized  σ generates \n∆m in the +y -direction, resulting in τ = − z, i.e. m net is \nswitched from up to down. Therefore , the mTM-J loop is \nreversed across  TMC. \nThe operation in  Fig. 1 (b) is defined as type -II. The \nelectrons flowing  through the MTJ structure are polarized \nby the PL, and exert spin transfer torque (STT) on the FL \nby transferring the ir angular momentum [13, 14] . There is \nalso finite SOT since  JC flows through the HM layer . We \nneglect it for two reasons : a) it  doesn ’t lead to \ndeterministic switching  due to the lack of  HX; b) since \nonly the horizontal component  of JC contributes to SOT , \nthe torque  is too small to disturb the equilibrium FiM state. \nAs discussed later, the switching symmetry in this type \nreverses at T AMC, which is determined by the nature of spin \ntorque . According to the formula of spin torque ( τST = s × \n(s × σ)), it aligns the angular momentum  antiparallel to \nσ. For both T < TMC and T > TAMC (see Fig. 1  (d)), mnet is \nopposite to s net, and the switching direction  follows  the \nconventional spin torque switching [15]. However, the \nswitching pattern is abnormal for TAMC  > T > TMC, where \nspin torque aligns  mnet antiparallel to σ . This \nphenomenon was first reported by Jiang et al. [8] in the  \nstudy of STT switching of CoGd . In this region,  mnet is \nparallel to snet. Due to the effect of STT , snet is aligned antiparallel to σ , resulting  in an  antiparallel \nconfiguration between  mnet and σ. By plotting the mTM-J \nloop, the switching direction  in type -II is reversed across \nTAMC. \nTo understand the se FiM dynamics, a model which can \nsystematically capture T  dependence is required. The \ncommonly used Landau -Lifshitz- Gilbert (LLG ) model  [8, \n16-19] is invalid for this purpose since it is based on the \nfixed magnetization length  assumption . To describe the T \ninduced magnetization length change , an additional \nlongitudinal relaxation term is introduced in to the LLB \nmodel . The model has been  widely used to describe the \nFM dynamics at elevated T  [20, 21] , and  recently  the LLB  \nof FiM  is also developed to simulate laser-induced \nswitching [22]. In this L etter, we derive  the FiM-LLB \nequation with the spin torque  contribution .  The dynamics  \nof spin angular momentum at each lattice site is  described \nby the atomistic LLG equation  \nˆ [ ( ( ) ( ( ))]I + )- H z γ ζλ= × ×× + ××s sH ss H ss ,    (1) \nwhere H is the effective field including  anisotropy field \nand exchange interaction s with other atoms, ζ  is the \nrandom thermal field , and HI is the spin torque  field.  \nBased on this atomistic equation, the collective behavior \nof sub -lattices is described by the correspon ding Fokker \nPlanck equation  \n0ˆ { ( ( ))\n()\n[ ( )]} 0IfHz\ntN\nTf\nNγγ λ\nγλ\nµ∂∂+ ×− ×× +\n∂∂\n∂+ ×× =\n∂NH N N H\nNN,     (2) \nwhere f is the spin distribution function, N is the vector on \na sphere with | N| = 1, and λ  is the damping constant . By \ntransforming the spin angular momentum to the \nmagnetization using 3( ,) dN f t≡< >=∫m s NN\ntogether with the use of  mean field approximation [22, 23] , \nthe final form  of the LLB equation is derived as  \n0,\n, 2\n0,\n, 2( ) (1 )\n()()vv MFA\nv vv v v v\nv\nvv v\nv\nvm\nmγ\n⊥⋅= × −Γ −\n××−Γmmm mH m\nm mm,     (3)  \nwhere the subscript v  denotes TM or RE element, the \n0, 0,\n,v vk MFA\nv ext A v v k\nvvJJ\nµµ=++ + H HH m m   is the mean \nfield with the exchange coupling coeffic ient, J0. Γ and \n⊥Γ  are the longitudinal and transverse damping \ncoefficients, respectively . m0,v is the equilibrium \nmagnetization obtained by solving two coupled  Curie -\nWeiss equation s   3 0,\n0, 0,\n0,()v\nvv\nvBξ\nξ= mξ\n,   4(a) \n0,\n0, 0,\n0,()k\nkk\nkBξ\nξ= mξ\n,   4(b) \nwith0,MFA\nv vvβµ= Hξ .  \n    Eq. 3  contains two coupled equations for TM and RE, \nwhich need to be solved simultaneously . Numerical \nintegration of Eq. 3  proceeds using a 4th order predictor -\ncorrector method, i.e.  the first 4 steps are obtained by  a \n4th order Runge -Kutta method , after which, the predictor \nis calculated using the 4th order multi- step Adams -\nBashforth method, and the corrector is computed using a \n4th order Adams -Moulton implicit method.  The model is \nverified by benchmarking with experimental M-H loop \nand M-J loop [1, 2] . Furthermore, the model can capture \nthe essential physics including the exchange interaction \nbetween TM and RE elements, T induced magnetization  \nreduction [23], transition from RE to TM dominant by \nchanging T or X [2, 4] , peak of  HC [2] and spin torque [6]  \nat TMC. The details of the model derivation and the ability \nto capture essential physics  will be described elsewhere.  \n    Based on the modified LLB model, we then perform \nnumerical simulations to  verify our  proposals . First of all,  \nthe effect of T  on magnetization is investigated. As shown \nin Fig. 2  (a), the magnetizations of TM and RE reduce \nwith T and vanish  at a common Curie temperature,  TC = \n315 K, which is a unique  property of FiM  since their FM \ncounterparts  have distinct Curie temperatures (TC_Fe  = \n1043 K, T C_Gd  = 292 K). This finding also highlights the \nstrong exchange coupling between sub -lattices . The use \nof different g factors and magnetic moments for TM and \nRE ( gTM = 2.05, gRE = 2, µTM = 2.217 µB, µRE = 7.63µ B) \nleads to  the separation  of mnet and snet as shown in Fig. 2 \n(b). The transition from RE  to TM dominant occurs at TMC \n= 165 K, and the angular momentum transition happens at T\nAMC  = 195 K.  In addition, we find that the separation \nbetween TMC and TAMC varies from several K to several \ntens of K, depending on the strength of exchange coupling \n \nFIG. 2. Effect of T on (a) sub- lattice magnetization, and (b) \nnet magnetization  (left y -axis)  and net angular momentum  \n(right y -axis) . The m = 1 is defined at T = 0. The intersections \nof the dash line with the magnetization and angular \nmomentum denote the TMC and TAMC, respectively.  between sub -lattices.  It is worth noting that the results in \nFig. 2 are obtained from equilibrium state calculation by \nsolving the coupled Curie- Weiss equations, which cannot \nbe experimentally verified due to the incapability of  \nmeasuring the angular momentum. As we have discu ssed, \nboth compensation points can be determined by \nexploiting the spin torque symmetry around T AMC and TMC. \nThe simulation results for T  dependent type- I and type -II \nswitching are shown in Fig. 3. In  type-I, mTM is switched  \nfrom up to down under positive current for T  < 165 K, \nwhereas the switching direction is reversed for higher T  \n(see Fig. 3  (a)). In contrast, the symmetry reversal in type -\nII occurs at T = 195 K  (see Fig. 3  (b)). According to the \ndiscussion in Fig. 1, TMC and TAMC are determined to be \n165 K and 195 K, respectively. The agreement between the equilibrium state calculation (Fig. 2) and the dynamic \nspin torque switching (Fig. 3) demonstrates the \neffectiveness of our model and provides an experimentally accessi ble method to determine T\nAMC.   \n    Finally, we study the relationship between T AMC and \nTMC in samples with different X. As shown in Fig. 4 (a), a \ngood linearity between TAMC and TMC is observed, which \ncan be explained using the T  dependent magnetization \n[24], i.e. the difference in sub -lattices magnetization leads \nto different compensation points. However, instead of  \nusing a simple power -law relation between MS and T [24], \nwe solve the coupled Curie- Weiss equation to \nsystematically capture the T  induced magnetization  \nchange. Similar to the atomistic modeling result in ref. \n[23], we show that both T MC and TAMC only exist in certain \nregion (0.23 < X < 0.26). For X < 0.23, due to small \namounts of RE, the samples are TM dominant for all  \n \nFIG. 3. Magnetization of the TM element versus J C under \ndifferent T in (a) type-I, and (b) type -II. The HX in (a) is 1 \nmT. The initial magnetization is m Z = 1 (−1) for the thick blue \n(thin red) lines. The two dash line s divide the diagram into \nthree  regions, i.e. T < TMC, TAMC > T > TMC, and T  > TAMC. \n 4  \nFIG. 4 . (a) Relation between TAMC and TMC. Blue dots are \nsamples with different X ranging from 0.23 to 0.26, which \nare fitted using a linear  function (red line) . (b) The effect of \nX on TC, TAMC, and TMC, where  TMC and TAMC only exist  for \n0.23 < X < 0.26.  \ntemperatures, whereas the coincidence of T MC and TC \nmakes RE dominant for X > 0.26. This explains the  \nexperimental observation that only certain samples have \nTMC [1]. The ability to capture T AMC, TMC and TC in \ndifferent samples demonstrates the generality of our model, and the knowledge of the relation between  T\nAMC, \nTMC and X would be helpful in designing room temperature \nFiM based electronic devices.           In conclusion, we proposed a device structure  to identify \nthe T\nAMC and TMC by conducting T  dependent current \nswitching.  To systematically describe the FiM dynamics at \ndifferent T, we have extended the LLB model to include \nthe spin torque effect.  We verify this  proposal by  \nnumerically  simulating two sets of T  dependent switching , \nand the results agree with the equilibrium state calculation \nusing Curie -Weiss equations. Furthermore, the recent \nreported linear relation between T AMC and TMC can be  well \nexplained using our model. The work should facilitate the \nstudy of current excited dynamics in FiM based devices.  \nWe acknowledge financial support from CRP award no. \nNRF -CRP12 -2013 -01 and MOE2013 -T2-2-125. \n  \na)a0132576@u.nus.edu, b)elelg@nus.edu.sg \n[1] N. Roschewsky, T. Matsumura, S. Cheema, F. Hellman, T. Kato, S. \nIwata, S. Salahuddin , Spin -orbit torques in ferrimagnetic GdFeCo alloys, \nApplied Physics Letters, 109 (2016) 112403.  \n[2] O. Takaya, K. Kab -Jin, T. Takayuki, K. Sanghoon, M. Takahiro, Y. \nHiroki, T. Arata, O. Teruo, Temperature dependence of magnetoresistance \nin GdFeCo/Pt heter ostructure, Applied Physics Express, 9 (2016) 073001.  \n[3] W. Seung Ham, S. Kim, D. -H. Kim, K. -J. Kim, T. Okuno, H. Yoshikawa, \nA. Tsukamoto, T. Moriyama, T. Ono, Temperature dependence of spin- orbit \neffective fields in Pt/GdFeCo bilayers, Applied Physics Letters, 110 (2017) \n242405.  \n[4] J. Finley, L. 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Chubykalo -Fesenko, Landau -Lifshitz- Bloch \nequation for ferrimagnetic materials, Physical Review B, 86 (2012) 104414.  \n[23] T.A. Ostler, R.F.L. Evans, R.W. Chantrell, U. Atxitia, O. Chubykalo -\nFesenko, I. Radu, R. Abrudan, F. Radu, A. Tsukamoto, A. Itoh, A. Kirilyuk, \nT. Rasing, A. Kimel, Crystallographically amorphous ferrimagnetic alloys: \nComparing a localized atomistic spin model with experiments, Physical Review B, 84 (2011) 024407.  \n[24] Y. Hirata, D.-H. Kim, T. Okuno, T. Nishimura, D.- Y. Kim, Y . Futakawa, \nH. Yoshikawa, A. Tsukamoto, K. -J. Kim, S. -B. Choe, T.  Ono, Correlation \nbetween Compensation Temperatures of Magnetization and Angular \nMomentum in GdFeCo Ferrimagnets, arXiv:1710.07779, (2017).\n \n \n"    },    {        "title": "1403.0614v1.CaBaCo__4_O__7____A_ferrimagnetic_pyroelectric.pdf",        "content": "arXiv:1403.0614v1  [cond-mat.str-el]  3 Mar 2014CaBaCo 4O7: A ferrimagnetic pyroelectric\nR. D. Johnson,1,∗K. Cao,2and P. G. Radaelli1\n1Clarendon Laboratory, Department of Physics, University o f Oxford, Oxford, OX1 3PU, United Kingdom\n2Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom\n(Dated: November 2, 2018)\nMagneto-electric coupling in pyroelectric CaBaCo 4O7is investigated using ab-intio calculations\nand Landau theory. The former shows that exchange-strictio n is strong enough to produce a giant\nchange in electric polarization upon ferrimagnetic orderi ng, comparable to the experimentally de-\ntermined value of ∼17mC/m2. Furthermore, Landau theory demonstrates that magneto-el astic\ncoupling in CaBaCo 4O7is responsible for the strong magneto-electric coupling ap pearing close to\nthe magnetic phase transition.\nPACS numbers: 75.85.+t, 75.50.Gg, 77.70.+a, 75.10.-b\nI. INTRODUCTION\nSolid-state materials that adopt a polar crystal struc-\nture have sustained interest in condensed matter physics,\nand have become key components in technology. For ex-\nample, the change in the intrinsic bulk electric polar-\nization of non-centrosymmetric pyroelectric compounds,\nwhich occurs upon varying the temperature of the ma-\nterial, forms the basis of infra-red sensing devices. Also,\nresearch into ferroelectric materials, in which inversion\nsymmetry is broken at a phase transition giving rise to\nswitchable ferroelectric states, has lead to the develop-\nment of electronic devices such as ferroelectric random\naccess memory (FeRAM). Multiferroic materials form a\nsubset of ferroelectrics, in which spontaneous electric\npolarization is coupled to long-range magnetic order.\nResearch in this field has recently undergone a renais-\nsance of interest, following the discovery of magnetic-\nfield-switchable electric polarization in the now canon-\nical systems TbMnO 31and TbMn 2O52– opening new\nroutes towards the development of novel multifunctional\ndevices.\nTo integrate multiferroic materials into technology it\nis necessary to identify systems that exhibit a very large\nmagnetically-induced ferroelectric polarization close to\nroom temperature. However, the largest magnetically-\ninduced ferroelectric polarization measured to date,\n(2870µCm−2observed in CaMn 7O12below 90 K3–5), is\ntwo orders of magnitude smaller than that of a good fer-\nroelectric. Recently, a much larger spin-assisted change\nin polarization of ∼17000µCm−2was measured in\nCaBaCo 4O7below 64 K6,7— a very significant obser-\nvation that, if confirmed, could pave the way for a new\ngeneration of magnetic ferroelectrics. In this paper, we\nperform first-principles calculations and a phenomeno-\nlogical analysis to study the magneto-electric coupling in\nCaBaCo 4O7. We demonstrate that all single crystal ex-\nperimental data are consistent with CaBaCo 4O7being\npyroelectric rather than ferroelectric in both paramag-\nnetic and magnetically ordered states. The large pyro-\nelectric currents observed near the magnetic phase tran-\nsition result from an exchange-striction-driven change of∆Pin the paramagnetic pyroelectric polarisation Ppyr.\nHowever, ∆ Pis always co-aligned in a fix relation with\nPpyr, either parallel or antiparallel depending on the sign\nof the magnetostrictive constant, and neither Ppyrnor\n∆Pare switchable.\nII. CRYSTAL AND MAGNETIC STRUCTURES\nAND ELECTRICAL PROPERTIES\nThe crystal structure of CaBaCo 4O7, shown in Figure\n1, was found to adopt the polar space group Pbn21at\nall temperatures below 400 K8. The structure comprises\ninterleaved kagom´ e and triangular layers of CoO 4tetra-\nhedra, which are buckled with respect to a high symme-\ntry, high temperature trigonal polar phase (space group\nP31c) common to other members of the RBaCo4O7\nseries9,10(R= rareearth, calcium or yttrium), but yet to\nbe observedin CaBaCo 4O7. In both spacegroupsymme-\ntries, CaBaCo 4O7is likely to be a non-switchable pyro-\nelectric material, since atoms in inversion-related struc-\ntures are separated by large distances. Furthermore, it\nfollowsthat ahightemperaturephasetransitionto acen-\ntrosymmetric group is extremely unlikely to occur below\nthe melting point.\nThegeometricfrustrationintrinsictoboth kagom´ eand\ntriangular lattices, well known to give rise to exotic frus-\ntrated magnetic ground states11,12, is lifted as a result\nof the CoO 4buckling. This structural distortion is re-\nported to be largest in CaBaCo 4O7[ 8], removing the\nfrustration, and promoting ferrimagnetic order develop-\ning atTc= 64 K. The magnetic structure8,13is shown in\nFigure 2. There are four symmetry inequivalent cobalt\nsites in the unit cell, labelled Co1, Co2, Co3, and Co4,\nand colored green, blue, red and pink, respectively, in\naccordance with the color scheme in reference 13. The\nmagneticmomentsofthe foursitesliewithin the abplane\nwithCo1andCo4momentsapproximatelyantiparallelto\nthose of Co2 and Co3. CaBaCo 4O7is a mixed valance\nsystem, with greater charge, and hence larger magnetic\nmoments, located on the Co1 and Co4 sites; a primary\ncomponent to the ferrimagnetism.\nMeasurements on a powder sample of CaBaCo 4O72\nFIG. 1. (Color online) Left: The crystal structure of\nCaBaCo 4O7. Calcium, barium, cobalt, and oxygen atoms\nare shown as black, green, blue, and red spheres, respective ly.\nThe CoO 4tetrahedra are shaded blue. Right: The triangular\nand kagom´ e CoO 4layers in the abplane.\n[ 6] showed that a pyrocurrent signal, corresponding\nto a change in electrical polarization, coincided with\nan anomaly in the dielectric constant at the ferrimag-\nnetic ordering transition Tc. The pyroelectric signal\nswitches sign with the external electric field, and the\nresults were therefore interpreted as evidence for ferro-\nelectricity and multiferroicity6,7. We note, however, that\nthe magnetically-induced change in polarisation (∆ P∼\n80µCm−2was found to be extremely small (approxi-\nmately 0.1 to 0.5%) with respect to the “pyroelectric\npolarization” Ppyr, with whatever definition might be\nadopted for it (see Appendix I for an extended dis-\ncussion); these results therefore have to be interpreted\nwith caution, as they could easily arise from an arte-\nfact. Similar measurements on single crystal samples\nshowed much larger pyroelectric currents developing at\nTc, consistent with a giant change in polarisation of\n∆P∼17000µCm−27. However, no switching behavior\nwas reported for the single crystal sample.\nIII. METHODS\nOur first-principles calculations were based on density-\nfunctional theory implemented in the Vienna ab-initio\nsimulations package (VASP)14,15. We used the spin-\npolarized generalized gradient approximation with on-\nsite Coulomb interactions, U, included for cobalt 3d or-\nbitals (GGA+U)16. By fixing the Hund coupling con-\nstantJ= 1 eV and testing several Uvalues we found\nthat the experimental ground state electronic structure\nbecomes metallic if U <3 eV. We therefore only present\nthe results for U= 4 eV, as CaBaCo 4O7is known to\nbe an insulator. We also performed calculations with\nU=6 eV, which produced very similar results. The pro-\njector augmented-wave (PAW)17method with a 500 eV\nplane-wave cutoff was used throughout, and a 6 ×4×4\nFIG. 2. (Color online) The groundstate magnetic structureo f\nCaBaCo 4O7. Cobalt ions are shown as spheres colored yellow\nor black if charge rich or charge poor, respectively. Magnet ic\nmoments are colored in accordance with the scheme adopted\nin reference 13. The twelve unique nearest neighbor exchang e\npaths are shown by black arrows.\nk-point mesh converges the calculation very well. In cal-\nculations of relaxed crystal structures, first atomic posi-\ntions, and then the lattice parameters, were left to vary\nuntil changes in total energy in the self-consistent calcu-\nlations were less than 10−7eV and the remnant forces\nwere less than 1 meV/ ˚A. The electric polarization was\ncalculated using the Berry phase method18.\nIV. MODELLING MAGNETISM AND\nPYROELECTRICITY\nWe turn to the symmetric Heisenberg model to de-\nscribe the magnetic interactions in CaBaCo 4O7. The ex-\npression for the magnetic Hamiltonian is simplified to\nHm=/summationtext\nijJijSi·Sj, whereJijare exchange integrals\nbetween cobalt spins SiandSj. For simplicity we only\nconsider nearest neighbor (NN) interactions of which 12\nare unique, labelled J1−J12in Fig. 2. Furthermore, it\nis reasonable to assume that the main exchange-striction\neffects primarily originate in NN interactions, and are\ntherefore proportional to Si·Sj.\nThe magnetic point group symmetry of CaBaCo 4O7\nbelowTc(m′m2′) only allows a change in electric polar-\nization parallel to the crystallographic caxis to be in-\nduced by exchange-striction. All twelve exchange inter-3\nTABLE I. Atom pair, bond length ( l), and exchange energies\nof a single unit cell for each spin configuration, given for al l\nNN exchange interactions. Note that in the calculations, th e\nmagnetic moments and exchange parameters are normalized\nto 1.\nJ Co pair l(˚A) FM AFM1 AFM2 AFM3\nJ1Co1-Co2 3.099 4 4 -4 -4\nJ2Co1-Co3 3.165 4 -4 4 -4\nJ3Co1-Co4 3.107 4 -4 -4 4\nJ4Co1-Co3 3.187 4 -4 4 -4\nJ5Co1-Co2 3.207 4 4 -4 -4\nJ6Co1-Co4 3.226 4 -4 -4 4\nJ7Co2-Co3 3.299 4 -4 -4 4\nJ8Co2-Co4 3.231 4 -4 4 -4\nJ9Co2-Co3 3.007 4 -4 -4 4\nJ10Co2-Co4 3.162 4 -4 4 -4\nJ11Co3-Co4 3.262 4 4 -4 -4\nJ12Co3-Co4 3.012 4 4 -4 -4\nactions can contribute to the change in electric polariza-\ntion in the experimental ferrimagnetic (EFM) structure,\nwhich can be calculated directly. However, the paramag-\nnetic (PM) phase presents a challenge, as spin polarized\ncalculations are required to give accurate results. Thus,\nto mimic the PM phase, we averaged several spin con-\nfigurations in such a way that individual NN exchange\ninteractions cancel, i.e.,/summationtext\nkSi(k)·Sj(k) = 0, where\nkdenotes the spin configuration — a method reliably\nused to calculate spin-phonon coupling in the PM phase\nof ZnCr 2O4[ 19]. The average electric polarization be-\ntween these spin configurations can then be considered\nan approximationof the polarization ofthe PM phase, as\nany magnetically-induced polarization will cancel, leav-\ning only that intrinsic to the crystal structure.\nIn CaBaCo 4O7the PM phase was modelled with four\ncollinear spin configurations that we label FM, AFM1,\nAFM2, AFM3, respectively. FMisaferromagneticstruc-\nture, while the other three are antiferromagnetic struc-\ntures of the type (- - + +), (- + - +), (- + + -) for\n(Co1,Co2,Co3,Co4),respectively. Thevaluesofexchange\ninteraction energies( Hm) for the four spin configurations\nare given in Table I, where each component exactly can-\ncels within the Heisenberg spin model approximation.\nHaving established a model for the PM phase, it is\nnow possible to calculate and compare the electric po-\nlarization of the PM and EFM structures. Note that in\nthe following the effect of spin-orbit coupling was found\nto be negligible, and so has been omitted from the final\nresults. Initially we performed calculations on a fixed\nexperimental atomic geometry13in order to isolate the\npure electronic contribution to the polarization. The re-\nsults, given in the first part of Table II, show only small\nvariations for all spin configurations, with ∆ Papproxi-\nmately 1 mC/m2– one order of magnitude smaller than\nthat measured by experiment7– indicating that the pureTABLE II. The calculated electric polarization in units\nmC/m2of the four spin configurations that average to give\nthe net PM polarization, and that calculated for the experi-\nmentally determined ferrimagnetic structure (EFM). For ea ch\nrow, the polarization in the FM state is taken as a reference.\n∆P=P(EFM)−P(PM).\nFM AFM1 AFM2 AFM3 PM EFM ∆ P\nElectronic only\n0 -0.16 0.36 1.06 0.31 0.74 0.43\nElectronic and ionic relaxation\n0 -14.0 -10.4 1.1 -5.8 -0.6 5.2\nElectronic, ionic, and lattice relaxation\n0 -17.71 -21.7 -3.1 -11.2 -5.3 5.9\nelectroniccontribution cannotfully explain the measured\nchange in polarization observed upon magnetic ordering.\nTherefore, an ionic contribution to the electric polariza-\ntion was calculated by fixing the experimental lattice pa-\nrameters and relaxing the internal ion positions. The\nresults are shown in the second part of Table II, where\nit can be seen that both the relative fluctuations and the\nfinal ∆Pare significantly enhanced. Finally, we allow for\nmagnetic strain coupling by performing full relaxations\nunder each magnetic configuration. The results, given\nin the third part of Table II, show significant differences\nwith respect to the electronic and ionic relaxation cal-\nculation, but give rise to a comparable ∆ P. Notably,\nthe sign of the polarization in the EFM phase changes\nsign upon relaxation, however the relative change, ∆ P,\nremains positive.\nThecorrectapproximationofthe PMphaseisessential\nin evaluating ∆ P. To double check our model, we also\ngenerated a model PM state with a set of noncollinear\nspin configurations that satisfy the same cancellation re-\nlation. In this case, fully relaxing the atomic geometry\ngives ∆P= 4.6 mC/m2with U=4 eV, which is in good\nagreement with that presented in Table II. All calcula-\ntions show that exchange-striction effects alone can give\nrise to a giant enhancement of the electric polarisation\nof the order 1-6 mC/m2, comparable to the experimental\nobservations. There remains a discrepancy between the\nactual magnitude of calculated and experimentally de-\ntermined change in electric polarization. However, this\neffect is typical in such calculations due to the limitation\nof current DFT in the treatment of the strongcorrelation\neffects of, in this case, cobalt 3 delectrons. Furthermore,\nthe approximation of the paramagnetic phase required\nforcalculationsmightreduce∆ PsincetherealPMphase\nis completely disordered, while our model is constructed\nfrom ordered states.4\nV.H−TPHASE DIAGRAM\nAs explained in detail in Appendix I, the electrical po-\nlarisation is an ill-defined concept in a pyroelectric, be-\ncause it depends upon the choice of a hypothetical cen-\ntrosymmetric phase that is not actually present in the\nphase diagram. However we chose to define it, the py-\nroelectric polarisation cannot be switched with an elec-\ntric field, although one can reverse it by simply rotat-\ning the crystal upside down. The key issue that re-\nmains to be addressed is whether ∆ Pis switchable,\nsince a positive answer would indicate a very unusual\nkind of spin-assisted ferroelectricity. In multiferroics,\nthe magnetically-induced polarisation changes sign upon\nreversing the polarity of the magnetic structure, which\nmust therefore be acentric. For CaBaCo 4O7, reproduc-\ning this mechanism is highly problematic: although the\ncrystalishighlypolar, thecobaltsublatticeisalmostcen-\ntrosymmetric and the cobalt magnetic structure is also\nquasi-centrosymmetric, the inverted EFM structure be-\ning essentially identical to the EFM structure. For this\nvery reason, ab-intiocalculations alone are of limited use\nin assessing whether ∆ Pcan switch, since there is no\nobvious way to construct a magnetic domain that would\nsupport a reversed ∆ P. We therefore adopt a different\napproachbaseduponaphenomenologicalLandautheory,\nand demonstrate that ∆ Pdoes not switch. Furthermore,\nthis minimal model is found to be sufficient to describe\nthe experimental behavior of CaBaCo 4O7when tuned\nby temperature and magnetic field close to Tc. We will\nemploy a scalar notation throughout. Extension to ten-\nsor notation is straightforwards but does not add to the\nessential physics of the problem.\nWe start by defining a simple Landau free energy for\nthe paramagnetic phase that accounts for its electrical\nproperties in the vicinity of the equilibrium pyroelectric\nstructure. We can write the free energy as:\nFp=−α\n2P2+β\n4P4−PE (1)\nwhereαandβare positive constants, which entirely\ndetermine the polarisability χ0an hyper-polarisability h0\noftheparamagneticphase—bothmeasurablequantities:\nχ0=∂P\n∂E/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nE=0=1\n2α\nh0=1\n2∂2P\n∂E2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nE=0=∓/radicalbigg\n9\n2χ5\n0β (2)\nSince we are only interested in the behaviour close to\nthe magnetic transition, we ignore the temperature de-\npendences of αandβ, which would give rise to conven-\ntional pyroelectric currents upon heating. From eq. 1 we\ncan also extract the equilibrium value of P:\nP0=±/radicalbiggα\nβ(3)We stress that P0isnota measurable quantity and\ndoes not correspond to a density of dipole moments Ppyr\nestimated using any of the reference structures proposed\nin Appendix I. In fact, this would entail extending eq. 1\nmuch beyond the range in which the quartic approxima-\ntion is valid. However, changes in Pin the vicinity of the\nequilibrium position are well defined and would result in\nameasurablecurrentinthe standardexperimentalsetup.\nBelow we present an estimation of P0fromab initio cal-\nculations.\nWe first consider how the Landau free energy in eq.\n1 can be modified in the magnetically ordered phase.\nThe point group symmetry of CaBaCo 4O7changes from\nmm2.1′tom′m2′at the magnetic ordering transition.\nOnly time reversal symmetry is broken, with all spa-\ntial symmetry operations preserved. The lowest order\nmagneto-electriccouplinginvariantin theLandauexpan-\nsion of the free energy is γPM2, whereγis a coupling\nconstant, P=P0+ ∆P, and ∆Pis the change in elec-\ntric polarization upon magnetic ordering as before. Mis\nthe magnetic order parameter, which, although primar-\nily antiferromagnetic, is coupled with the magnetic field\nthrough the ferrimagnetic component. γPM2is time re-\nversal even and parity odd, and as such is allowed prima\nfaciegiven the polar, paramagnetic parent phase point\ngroup. However, the Landau free energy must also be\ninvariant by any continuous or discrete free-space oper-\nator applied to the crystal as a whole. If one applies\nan inversion operator (not part of the crystal symme-\ntry) the magnetic structure remains almost invariant,\nas explained above, while Pchanges sign. Since this\nis an approximate relation, we can only conclude that\ntheγPM2term must be very small. An alternative, and\nperhaps more intuitive interpretation is to consider that\nCaBaCo 4O7undergoes a hypothetical non-polar to polar\nstructural phase transition to one of the reference struc-\ntures at very high temperatures. The term γPM2is not\ninvariant in any of the hypothetical parent phases, and\nis therefore rigorously excluded.\nWe proceed to demonstrate that a change in bulk elec-\ntric polarization can occur via a magneto-elastic contri-\nbution to the free energy described by the higher order\nterm−c\n2M2P2, which can result in a fractional change\n∆Pof the electric polarization.\nThe lowest order, stable Landau expansion of the free\nenergy may be written as:\nF(M,P,B) =F0−BM+a\n2(T−T∗)M2+b\n4M4\n−c\n2M2P2+d\n6M6−α\n2P2+β\n4P4−PE (4)\nwherea,banddandT∗are constants of the purely\nmagnetic part of the free energy. cis the magneto-elastic\ncoupling constant that may be positive or negative, and\nall other constants are defined such that positive values\nstabilize the free energy.\nIn zero applied electric field, two equilibrium condi-\ntions follow:\n∂F\n∂M=−B+a(T−T∗)M+bM3−cP2M+dM5= 0 (5)5\nand,\n∂F\n∂P=−αP+βP3−cM2P= 0 (6)\nSubstiting eq. 3 into equation 6 and rearranging the\nterms gives the solution\nP2=P02+c\nβM2(7)\nwhereMis found by solving equation 5, as described\nbelow. Equation 7 contains the essence of the physics\nof CaBaCo 4O7: at the magnetic ordering temperature,\nMbecomes non-zero, and an additional contribution to\nthe polarisation ∆ Pdevelops due to magneto-striction.\nThe sign of ∆ Pin relation to P0is fixed once and for all\nby the sign of the coupling constant cand can never be\nswitched.\nIn principle, it is possible to determine the values of\nαandβ, and therefore derive P0fromab initio calcula-\ntions of the paramagnetic phase in applied electric field.\nHowever, the aforementioned difficulty of performing ac-\ncurate calculations in the paramagnetic phase makes this\napproach impractical. We therefore chose a different\nmethod, exploiting the fact that P0enters as a param-\neter in the free energy expansion as a function of ∆ P\naround the magnetic ground state. As before, all calcu-\nlations here are performed with a Coulomb interaction\nenergy of U=4 eV. We first estimate the ab initio atomic\nstructure of the paramagnetic phase (corresponding to\n∆P= 0) by averaging the relaxed atomic structures in\nthe four spin configurations (FM, AFM1, AFM2, AFM3)\nwith fixed experimental lattice parameters. The ground-\nstate atomic structure (corresponding to ∆ P= ∆Pmax)\nwas previously determined by relaxing the atomic posi-\ntions with the ground-state magnetic structure (EFM).\nBy interpolating between these two extremes we can es-\ntimate atomic positions, and therefore calculate ground-\nstate energies, as a function of ∆ P. In these calcula-\ntions, it was assumed that Mdoes not vary, a reason-\nable approximation for values of ∆ Pclose to the ground\nstate. These energies were then fitted to the expression\nE=E0+a′(P0+ ∆P)2+b′(P0+ ∆P)4, which closely\nfollows the form of our Landau theory.\nTheresultsofthecalculationsandareplottedasclosed\ncircles in Figure 3 as a function of P/∆Pmax, together\nwith the fitted expression, plotted as a line. From the\nfit, we extract a value P0/∆Pmax= 26.51. Taking the\nreported value of ∆ Pmax= 17 mC/m2, this gives P0≃\n450 mC/m2We note that the calculated EvPcurve is\nsymmetrical about the origin (not shown in Figure 3).\nHence, there exists a single energy minima for each ±P0\ndomain, respectively. The energy needed to reverse ∆ P,\ni.e. switch between energetic minima, while preserving\nthe magnetization is about 411 meV, consistent with a\nvalue of 464 meV from direct ab-initio calculations This\nresult clearly indicates that ∆ Pis not switchable for a\ngivenP0domain.05101520253035\nP (∆P normalized to 1)-320-310-300Energy [eV]\n26.5 27 27.5 28 28.5\nP (∆P normalized to 1)Energy [eV]M = 1\nM = 0\n∆P\nFIG. 3. Fit of the polarization free energy to P, determined\nat points close to the energetic minimum by ab-initio calcu-\nlations. The energy for M= 0 and M= 1 is shown by red\nand black lines, respectively.\n10 20 30 40 50 60 70 80\nTemperature [K]051015M [emu g-1]\nFIG. 4. (Color online) The temperature dependence of the\nmagnetization in zero applied magnetic field. Data points\n(black circles) have been extracted from reference 7. The fit\nto the data, described in detail in the text, is shown as a red\nline.\nWe now turn to the magnetization. Substituting equa-\ntion 7 into equation 5 gives\n−B+a(T−Tc)M+˜bM3+dM5= 0 (8)\nwhereTc=T∗+c\naP02and˜b=b−c2\nβ. These ex-\npressions capture two further key aspects of the physics\nof CaBaCo 4O7. Coupling to the pyroelectric polariza-\ntion present in the paramagnetic phase will increase the\nmagnetic ordering temperature, and furthermore, may\ninduce a first-order (negative ˜b) magnetic phase transi-\ntion, as opposed to a second-order transition (positive\n˜b) expected in magnetic systems. Magnetization data\nmeasured parallel to the baxis forB≃0 has been repro-\nduced from reference 7 (Figure 4), which in the following\nwe take to represent the thermal evolution of the mag-\nnetic order parameter. Despite no evidence for magnetic\nhysteresis at the phase transition6, there occurs a sharp\njump in the magnetization at Tc— evidence for first-\norder behavior. This was confirmed by fitting equation\n8 to the data, in units of emu/g, and with B= 0. The6\n-10 -5 0 5 10\nB [T]450455460465P [mC m-2]62.5 K\n65 K\n68 K\n75 K60 K\nFIG. 5. (Color online) The magnetic field dependence on\nelectric polarization at four temperatures close to the pha se\ntransition. Data points (circles) have been extracted from\nreference 7. Simulations of the data are shown as solid lines ,\nwhich are colored according to the respective temperatures .\nNote that P0has been added throughout.\nconstants ˜banddwere allowed to vary freely in the fit,\nwithaset to unity having factored out a scaling param-\neter,ξ, to be determined later. The best fit is shown in\nFigure 4, with a=ξ,˜b=−0.112ξ,d= 0.00385ξ, and\nTc= 62.0 K. We note that ˜btakes a negative value, indi-\ncating that the magnetic phase transition is indeed first\norder due to coupling to P. This result might be verified\nexperimentally by investigating magnetic hysteresis.\nHaving established the temperature dependence of M,\nsetting the scaling parameter ξto 0.4 and the ratio of\nconstants candβthat couple MandPtoc\nβ= 210,\ngave the best qualitative agreement with the magnetic\nfield dependence of the electric polarization (Figure 5).\nThis result clearly demonstrates that the experimentally\ndetermined H-Tphase diagram close to Tccan be ex-\nplained by our phenomenological model based solely on\nmagneto-elastic coupling.\nFinally, weagainconsidertheenergybarrierassociated\nwith the hypothetical switching of ∆ P. The difference in\nenergy of two ±∆Pferroelectric domains can be written\nas:\n∆E=−c\n2M2(P0+∆P)2+c\n2M2(P0−∆P)2\n=−2cP0PM2(9)\ni.e.there is a large energy barrier to switching ∆ Pthat\nscales with P0, consistent with the results of the calcula-\ntions described in the above.\nVI. CONCLUSIONS\nTo summarize, we have demonstrated through ab-\ninitiocalculations that in CaBaCo 4O7, the giant changein electric polarization observed at the phase transition\nfrom pyroelectric paramagnetic to pyroelectric ferrimag-\nnetic can arise as a result of exchange-striction effects\nalone. The change in polarization was found to be an en-\nhancement, i.e.in the same direction as the polarization\nof the paramagnetic phase, and not switchable. Further-\nmore, such large changes are only predicted when one\nconsiders the relaxation of ionic positions. Our ab-initio\nresults are supported by Landau theory, which predicts\nthe correct magneto-electric behavior apparent close to\nTc.\nVII. APPENDIX I: ABSOLUTE VALUE OF\nELECTRIC POLARIZATION\nExperimentally, values of electric polarization may be\nmeasured in two ways. Firstly, relative changes in polar-\nization can be determined by integrating a pyroelectric\ncurrent measured upon tuning the system in question\nbetween two different structures along a given path, dis-\ncussed below. Secondly, the absolute polarization may\nbe measured if the system is ferroelectric, and domain\nswitching has been observed. Modern theories of electric\npolarization, where values can be quantum mechanically\ncalculatedusingthe Berryphasemethod18, closelyfollow\nthe pyroelectric experimental procedure. The theory is\ntherefore limited to calculating relative changes in elec-\ntric polarization as follows. The Berry phase method\nshows that the electric polarization is multivalued with a\nperiod of1\nΩeR, where Ω is the unit cell volume, eis the\nelectron charge, and Ris any lattice vector. Therefore,\nto define an absolute spontaneous polarization in a non-\ncentrosymmetric material, it is necessary to determine\nthe origin of the above periodic polarization. This may\nbe constrained by designating a ‘nearest’ (smallest total\ndisplacement of atoms) high temperature centrosymmet-\nric structure with zero net polarization, e.g. the cubic\nphase of BaTiO 3. In order to calculate an absolute value\nofPpyrin CaBaCo 4O7, it was therefore necessary to de-\nfine a hypothetical, high temperature crystal structure\nwith zero net polarization - a centrosymmetric super-\ngroup of Pbn21. To find such structures, we used the\nPSEUDO code of the Bilbao Crystallographic Server20,\nwhich allows one to determine the nearest supergroup\nstructure for an arbitrary subgroup structure. With the\nexperimental atomic structure as input13, two suitable\nsupergroup structures were identified with space groups\nPnnaandPccn, and the structural parameters are given\nin Table III. In terms of atomic displacements, the Pnna\nstructure is the nearest.\nUnfortunately, calculations showed that both the two\nsupergroup structures are metallic, preventing ab-initio\nBerry phase calculations. To obtain a qualitative esti-\nmation of Ppyr, we therefore assume the two reference\nstructures are insulating and use point charge models to\ncalculate the electric polarization, i.e. P=/summationtext\niZiri/Ω,\nwhereriis the position of the ith ion, and Ziis the corre-7\nTABLE III. Atomic positions of the PnnaandPccnreference\nstructures, given in fractional coordinates of the subgrou p\nsetting. ux,uy,uzare the displacement vectors that generate\nthe experimental Pbn21structure from the centrosymmetric\nstructures with the origin shifted by (0, 0, -0.2) and (0, 0,\n-0.44) for PnnaandPccn, respectively.\nPnna\nAtom Fractional Coordinates uxuyuz\nCa (0.0000, 0.67290, 0.7500) 0.0066 0.00000 -0.0792\nBa (0.0000, 0.66510, 0.2500) 0.0043 0.00000 0.0500\nCo1 (0.0000, 0.99970, 0.7500) 0.0155 0.00000 -0.0168\nCo2 (0.8703, 0.13215, 0.4999) 0.1230 0.03665 -0.0097\nCo3 (0.1297, 0.13215, 0.0001) 0.1230 -0.03665 -0.0097\nCo4 (0.2673, 0.00000, 0.5000) 0.0000 -0.07910 -0.0159\nO1 (0.9845, 0.00000, 0.0000) 0.0000 0.00440 0.0460\nO2 (0.9955, 0.50000, 0.0000) 0.0000 -0.00860 0.0285\nO3 (0.7500, 0.25000, 0.5000) 0.0317 0.01130 0.0815\nO4 (0.7500, 0.75000, 0.0000) -0.0181 -0.00820 0.0141\nO5 (0.0000, 0.15220, 0.2500) -0.0549 0.00000 0.0497\nO6 (0.2340, 0.08130, 0.7530) -0.0305 0.02840 0.0532\nO7 (0.2340, 0.91870, 0.2470) 0.0305 0.02840 0.0532\nPccn\nAtom Fractional Coordinates uxuyuz\nCa (0.0000, 0.75000, 0.50000) 0.0066 -0.07710 -0.06920\nBa (0.0000, 0.75000, 0.00000) 0.0043 -0.08490 0.06000\nCo1 (0.1241, 0.96030, 0.37455) -0.1086 0.03940 0.11865\nCo2 (0.8703, 0.28665, 0.24990) 0.1230 -0.11785 0.00030\nCo3 (0.1297, 0.21335, 0.75010) 0.1230 -0.11785 0.00030\nCo4 (0.3759, 0.96030, 0.12545) -0.1086 -0.03940 0.11865\nO1 (0.9900, 0.99790, 0.75875) -0.0055 0.00650 0.04725\nO2 (0.9900, 0.49790, 0.74125) 0.0055 -0.00650 0.04725\nO3 (0.7891, 0.32580, 0.38765) -0.0074 -0.06450 -0.04615\nO4 (0.7500, 0.74180, 0.75000) -0.0181 0.00000 0.02410\nO5 (0.8548, 0.10255, 0.99975) 0.0903 0.04965 0.05995\nO6 (0.2109, 0.17420, 0.61235) -0.0074 -0.06450 -0.04615\nO7 (0.3548, 0.89745, 0.00025) -0.0903 0.04965 0.05995\nsponding effective charge. Using simple valence charges\nwithZ(Ca)=Z(Ba)=+2, Z(Co)=+2.5, Z(O)=-2,theab-\nsolute value of polarization of the PM phase is estimated\nto be -775 mC/m2forPnnaand 263 mC/m2forPccn.\nAt first sight, it is surprising that two reference struc-\ntures give completely different values for Ppyr. However,\nthe displacement patterns shown in Table III provide an\nexplanation for this discrepancy. In the following we con-\nsider only the displacement along the cdirection, as theabplane components cancel exactly by symmetry. For\nPnna, almost all cations move in an opposite direction\nto the anions, giving rise to the large polarization differ-\nence along the + cdirection. By comparison, in the case\nofPccn, cations and anions are both displaced along the\nFIG. 6. (Color online) 2D illustration of the ambiguity as-\nsociated with choosing the correct path from centrosymmet-\nric reference structure to the polar phase. Here we fix the\nposition of the anions and use cations to represent different\nphases, where solid red represents the polar phase, dotted\nblue for the H 1reference and dotted red for the H 2reference.\nBlack arrows denote directions of polar distortion.\npositive and negative cdirections, resulting in polar con-\ntributions that partially cancel with each other, giving\nrise to the relatively small polarization difference with a\nsign opposite to that for Pnna.\nA simple analogy can be drawn if one considers a 2D\nlatticeofpositiveandnegativecharges,illustratedin Fig-\nure 6. Here, it is clear how two opposite displacements of\ncharge can result in different ‘nearest’ centrosymmetric\nstructures, giving rise to an ambiguity associated with\nthe choice of non-polar reference structures. Although\nH2appears energetically more favourable than H 1in the\n2D example, the case of CaBaCo 4O7is much more com-\nplex making it difficult to determine the correctreference\nstructure. In general, real ferroelectric distortions are\nmuch smaller than those considered in this paper, so it\nis often trivial to identify the nearest reference structure,\nand the polarization may be determined absolutely if the\nparent phase crystal structure is known. Despite both\nhypothetical centrosymmetric structures of CaBaCo 4O7\nhavingunphysicalstructuraldistortions, carefulvariable-\ntemperature experiments could be performed to identify\nthe direction of the pyroelectricpolarization with respect\nto the crystal structure, and hence remove the ambiguity\nin determining Ppyr.\nACKNOWLEDGMENTS\nThis work was funded by an EPSRC grant, number\nEP/J003557/1, entitled “New Concepts in Multiferroics\nand Magnetoelectrics”.\n∗r.johnson1@physics.ox.ac.uk\n1T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima,and Y. Tokura, Nature 426, 55 (2003).\n2N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha, and8\nS.-W. Cheong, Nature 429, 392 (2004).\n3G. Zhang, S. Dong, Z. Yan, Y. Guo, Q. Zhang, S. Yunoki,\nE. Dagotto, and J.-M. Liu, Phys. Rev. B 84, 174413\n(2011).\n4R. D. Johnson, L. C. Chapon, D. D. Khalyavin, P. Manuel,\nP. G. Radaelli, and C. Martin, Phys. Rev. Lett. 108,\n067201 (2012).\n5N. Perks, R. D. Johnson, C. Martin, L. C. Chapon, and\nP. G. Radaelli, Nature Communications 3, 1277 (2012).\n6K. Singh, V. Caignaert, L. C. Chapon, V. Pralong,\nB. Raveau, and A. Maignan, Phys. Rev. B 86, 024410\n(2012).\n7V.Caignaert, A.Maignan, K.Singh, C. Simon, V.Pralong,\nB. Raveau, J. F. Mitchell, H. Zheng, and L. Chapon,\nArXiv:1302.5541 [cond-mat.mtrl-sci].\n8V. Caignaert, V. Pralong, A. Maignan, and B. Raveau,\nSolid State Comm. 149, 453 (2009).\n9L.C.Chapon, P.G.Radaelli, H.Zheng, andJ.F.Mitchell,\nPhys. Rev. B 74, 172401 (2006).\n10A. Huq, J. Mitchell, H. Zheng, L. Chapon, P. Radaelli,\nK. Knight, and P. Stephens, Journal of Solid State Chem-istry179, 1136 (2006).\n11S. Nakatsuji, Y. Nambu, H. Tonomura, O. Sakai, S. Jonas,\nC. Broholm, H. Tsunetsugu, Y. Qiu, and Y. Maeno, Sci-\nence9, 1697 (2005).\n12S.-H. Lee, H. Kikuchi, Y. Qiu, B. Lake, Q. Huang,\nK. Habicht, and K.Kiefer, Nature Materials 6, 853(2007).\n13V. Caignaert, V. Pralong, V. Hardy, C. Ritter, and\nB. Raveau, Phys. Rev. B 81, 094417 (2010).\n14G. Kresse and J. Hafner, Phys. Rev. B 47, RC558 (1993).\n15G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169\n(1996).\n16A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.\nRev. B.52, R5467 (1995).\n17P. E. Blochl, Phys. Rev. B 50, 17953 (1994).\n18R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47,\n1651 (1993).\n19C. J. Fennie and K. M. Rabe, Phys. Rev. Lett. 96, 205505\n(2006).\n20C. Capilla, M. I. Aroyo, and J. M. Perez-Mato, Z. Krist\n226(2), 186 (2011)."    },    {        "title": "0907.1348v1.Magnetic_hysteresis_in_a_molecular_Ising_ferrimagnet__Glauber_dynamics_approach.pdf",        "content": "arXiv:0907.1348v1  [cond-mat.stat-mech]  8 Jul 2009Magnetic hysteresis in a molecular Ising ferrimagnet: Glau ber\ndynamics approach\nA. A. Bukharov1, A. S. Ovchinnikov1, N. V. Baranov1,2, K. Inoue3\n1Department of Physics, Ural State University, Ekaterinbur g, 620083 Russia\n2Institute of Metal Physics RAS and Ural State University, Eka terinburg, 620083, Russia\n3Institute for Advanced Materials Research,\nHiroshima University, Hiroshima, Japan\n(Dated: October 31, 2018)\nMotivated by recent experimental results reporting giant c oercive fields in Co(II)-\nbased molecular magnets we present a theory of hysteresis ph enomena based on the\nGlauber stochastic dynamics. Unusual form of hysteresis lo ops is similar to those\nof found in Co-based quasi-one-dimensional ferrimagnet Co PhOMe at low tempera-\ntures. Temperaturedependenceofthecoercive field hasacha racteristic formwithan\ninflection that may serve as an indicator of the Glauber dynam ics in real compounds.\nA relevance of the model for other Co-based molecular magnet s is discussed.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nOne of the remarkable phenomena found in molecular magnetic mater ials is the magnetic\nhysteresis similar to those observed in hard magnets.1,2In particularly, magnetic hysteresis\ncan be observed in the absence of a long-range magnetic order in ze ro-dimensional single-\nmolecule magnets (SMM)3and one-dimensional (1D) single chain magnets (SCM).4,5The\nhysteresis phenomena in SMM are affected by large easy-axis magne tic anisotropy and by\nthe weak intermolecular interactions.6Although a large easy-axis magnetic anisotropy can\nalso be of importance in quasi-1D magnets, an origin of magnetic hyst eresis in these systems\nis not yet well understood.7\nDiscovery of slow relaxation of magnetization in quasi-1D compound\nCo(hfac) 2[NIT(C 6H4p-OMe)] (or CoPhOMe)4evoked interest to the Glauber kinetic\nmodel suggested for relaxation dynamics in 1D Ising ferromagnet.8,9,10The source of a2\nstrong easy-axis anisotropy are Co(II) ions that enable to treat collective properties of this\ncompound by means of 1D Ising model. Recently, pronounced hyste resis loops with giant\ncoercivity were observed in another Co(II)-based materials, nam ely [Co(hfac) 2NIT-C 6H4-\nO-R]11, where R=(CH 2)3CH3, and [Co(hfac) 2]·BNO∗(BNO∗is chiral triplet bis(nitroxide)\nradical).12Despite the Glauber dynamics seems to be plausible, one should take in to\naccount that chains are packed in a 3D structure, and with a cooling 1D units undergo a\nphase transition into 3D magnetic array. It means that another ex planation of hysteresis\nphenomena related with domain wall (DW) dynamics comes into the play . High coercive\nfields reported in Refs.11,12at low temperatures, i.e. broad hysteresis loops, support the\nreasonings. Note that considerations of slow magnetic relaxation b elow 3D ordering in\nthe framework of the DW approach turns out to be fruitful for Mn (II)-based quasi-1D\nferrimagnets.13,14\nBeingguidedthismotivationwepresentatheoryofmagnetichyster esis forIsingferrimag-\nnetic chain calculated within the Glauber dynamics approach. An exte nsion of the Glauber\nmodel for higher spins (more than 1 /2) has been done in a modeling of photoinduced re-\nversible magnetization15,16. Despite equilibrium properties of mixed spin Ising systems have\nattracted a much of attention, a study of nonequilibrium aspects o f the model started only\nrecently.17In the paper we perform the mean field treatment of the Glauber-t ype stochastic\ndynamics of the mixed (3/2,1) ferrimagnetic chain by considering fluc tuations of local fields\nin the spirit of the generalized mean-field theory.18Our goal is to verify a validity of the\nmodel for real Co-based molecular ferrimagnetic chains. The choic e of the spin quantum\nnumbers is stipulated by a possible prototype of the model, the meta l-organic compound\n[Co(hfac) 2]·BNO∗studied recently by two of us, which embodies Co(II) ions with spin 3/ 2\nand the chiral triplet biradical ligands BNO∗with spin 1. However, the results we obtain\nmay be of interest for another Co-based quasi-1D magnets.\nThe paper is organized as follows. In Sec.II we obtain dynamic equatio ns of the Ising\nferrimagnetic ( S,σ) chain model derived through the Glauber stochastic dynamics in a p res-\nence of a time-varying magnetic field. In Sec. III we present result s of hysteresis behavior\nobtained via numerical calculations of these equations for the case S= 3/2,σ= 1. A depen-\ndence of shape of the hysteresis cycles on the ratio between the m agnetic field frequency and\nthe spin flip frequency is studied. In Sec. IV we confirm by a Monte Ca rlo (MC) method\nthat a hysteresis loops arise in a static magnetization process. Disc ussions are relegated to3\nthe Conclusion part.\nII. MODEL\nThe Hamiltonian of the Ising (3/2,1) ferrimagnetic chain is given by\nH=J/summationdisplay\ni(σiSi+σi+1Si)−H(t)/parenleftBigg/summationdisplay\niσi+/summationdisplay\niSi/parenrightBigg\n, (1)\nwhere the first term sums interactions between the nearest neigh bors with the spin variables\nσi= 0,±1 andSi=±1/2,±3/2, andJ >0 favors an antiferromagnetic coupling of the\nadjacent sites. The Zeeman term describes interaction of the spin s with an oscillating\nmagnetic field of the sinusoidal form\nH(t) =H0cos(ωt)\nwith a frequency ω. The system is in contact with an isothermal heat bath at given tem-\nperature T.\nThe relaxation of the interacting Ising system cannot be obtained f rom its Hamiltonian\nbecause it eliminates intrinsic spin dynamics (precession in the local fie ld). Nevertheless,\nrelaxation phenomena can be described by means of a phenomenolog ical equation which\nspecifies the transition rate from one spin configuration to anothe r. In general, we assume\nthat transition from one configuration to another involves changin g a single spin, i.e. ac-\ncording to Glauber dynamics,8,9,10at a rate of 1 /τtransitions per unit time.\nAccording to formalism of Ref.17the master equation for one sublattice can be written\nin suggestion that spins on another sublattice momentarily frozen, i.e.\nd\ndtP{S}(σ1,σ2,...,σ j,...,σ N)\n=/summationdisplay\nj/summationdisplay\nσ′\nj/ne}ationslash=σjωj(σ′\nj→σj)P{S}(σ1,σ2,...,σ′\nj,...,σ N)\n−/summationdisplay\nj\n/summationdisplay\nσ′\nj/ne}ationslash=σjωj(σj→σ′\nj)\nP{S}(σ1,σ2,...,σ j,...,σ N),(2)4\nd\ndtP{σ}(S1,S2,...,S j,...,S N)\n=/summationdisplay\nj/summationdisplay\nS′\nj/ne}ationslash=Sjωj(S′\nj→Sj)P{σ}(S1,S2,...,S′\nj,...,S N)\n−/summationdisplay\nj\n/summationdisplay\nS′\nj/ne}ationslash=Sjωj(Sj→S′\nj)\nP{σ}(S1,S2,...,S j,...,S N),(3)\nwhereP{S}(σ1,σ2,...σj,...,σ N)istheprobabilitythatthesystemhasthespinconfiguration\n{σ1, σ2, ... σ j, ..., σ N}in the first sublattice leaving the spins {S}of the second sublattice\nfixed,ωj(σ′\nj→σj) is the probability per unit time that the j-th spin changes from the value\nσ′\njtoσj. The similar notations hold for another sublattice.\nThe transition probabilities ωj(σ′\nj→σj) andωj(S′\nj→Sj) are imposed by the principle\nof detailed balance. Indeed, the principle requires the probabilities o f statesP{S}andP{σ}\nto be stationary at equilibrium\nωj(σj→σ′\nj)P{S}(σ1,σ2,...,σ j,...,σ N) =ωj(σ′\nj→σj)P{S}(σ1,σ2,...,σ′\nj,...,σ N),\nωj(Sj→S′\nj)P{σ}(S1,S2,...,S j,...,S N) =ωj(S′\nj→Sj)P{σ}(S1,S2,...,S′\nj,...,S N).\nProvided the probabilities P{S}andP{σ}are Boltzmann distributions, i.e.\nP{S}(σ1,σ2,...,σ′\nj,...,σ N)\nP{S}(σ1,σ2,...,σ j,...,σ N)=exp/bracketleftBig\n−β/braceleftBig\nJ/parenleftBig\nσ′\njSj−1+σ′\njSj/parenrightBig\n−Hσ′\nj/bracerightBig/bracketrightBig\nexp/bracketleftBig\n−β/braceleftBig\nJ/parenleftBig\nσjSj−1+σjSj/parenrightBig\n−Hσj/bracerightBig/bracketrightBig, (4)\nP{σ}(S1,S2,...,S′\nj,...,S N)\nP{σ}(S1,S2,...,S j,...,S N)=exp/bracketleftBig\n−β/braceleftBig\nJ/parenleftBig\nσjS′\nj+σj+1S′\nj/parenrightBig\n−HS′\nj/bracerightBig/bracketrightBig\nexp/bracketleftBig\n−β/braceleftBig\nJ/parenleftBig\nσjSj+σj+1Sj/parenrightBig\n−HSj/bracerightBig/bracketrightBig (5)\none get\nωj(σj→σ′\nj)\nωj(σ′\nj→σj)=exp[σ′\njyj]\nexp[σjyj],ωj(Sj→S′\nj)\nωj(S′\nj→Sj)=exp[ξjS′\nj]\nexp[ξjSj], (6)\nwhereyj=β/bracketleftbig\nH−J(Sj−1+Sj)/bracketrightbig\n, andξj=β/bracketleftbig\nH−J(σj+σj+1)/bracketrightbig\n,β= 1/T.\nIn Glauber dynamics the relationships (6) are resolved as follows\nωj(σj→σ′\nj) = Ωexp[σ′\njyj]/summationtext\nσ′′\njexp[σ′′\njyj], ω j(Sj→S′\nj) = Ωexp[ξjS′\nj]/summationtext\nS′′\njexp[ξjS′′\nj], (7)\nwhere Ω = 1 /τis a number of spin changes per unit time. Strictly speaking, Eqs.(7) a re\nrelevant for a equilibrium process with a constant magnetic field. Nev ertheless, it is reliable\nwhen the field sweep frequency is much less then that of spin transit ionsω≪Ω.5\nExpectation value of the σ-sublattice magnetization at the moment tis given by\n/an}bracketle{tσi/an}bracketri}ht=/summationdisplay\nσiσip(σi), (8)\nwhere\np(σi) =/summationdisplay\n{S}/summationdisplay\n{σ}′P{S}(σ1,...,σ i,...,σ N) (9)\nis the probability to find the σivalue. The sum with the prime runs over all σvariables\nexcept that of i-th site. Similar definitions are hold for another sublattice.\nThe probability satisfies the following dynamic equation\ndp(σi)\ndt=−Ωp(σi)+Ω/angbracketleftBigexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/angbracketrightBig\n, (10)\nwhich is obtained from Eq.(9) with an account of Eqs.(2-3) and (7). T he average in the\nright-hand side is determined as follows\n/angbracketleftBigexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/angbracketrightBig\n=/summationdisplay\n{S}/summationdisplay\n{σ}exp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]P{S}(σ1,...,σ i,...,σ N). (11)\nThe analogous equation is derived for another sublattice\ndp(Si)\ndt=−Ωp(Si)+Ω/angbracketleftBigexp[ξiSi]/summationtext\nS′\niexp[ξiS′\ni]/angbracketrightBig\n. (12)\nThe system of Eqs.(10,12) can be treated in the mean-field approxim ation (MFA)\n/angbracketleftBigexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/angbracketrightBig\n≈exp[σi/an}bracketle{tyi/an}bracketri}ht]/summationtext\nσ′\niexp[σ′\ni/an}bracketle{tyi/an}bracketri}ht], (13)\nthat yields\n\nd/an}bracketle{tσi/an}bracketri}ht\ndt=−Ω/bracketleftBig\n/an}bracketle{tσi/an}bracketri}ht−σBσ(σ/an}bracketle{tyi/an}bracketri}ht)/bracketrightBig\n,\nd/an}bracketle{tSi/an}bracketri}ht\ndt=−Ω/bracketleftBig\n/an}bracketle{tSi/an}bracketri}ht−SBS(S/an}bracketle{tξi/an}bracketri}ht)/bracketrightBig\n,(14)\nwhere the Brillouine function Bs(x) is introduced\nBs(x) =/parenleftBig\n1+1\n2s/parenrightBig\ncoth/bracketleftBig/parenleftBig\n1+1\n2s/parenrightBig\nx/bracketrightBig\n−1\n2scoth/bracketleftBigx\n2s/bracketrightBig\n.6\nFIG. 1: Hysteresis loops of magnetization per cell M=/an}bracketle{tσ/an}bracketri}ht+/an}bracketle{tS/an}bracketri}htat different temperatures θ.\nInset: the coercive field hcvs temperature.\nBy taking the uniform arrangement /an}bracketle{tσi/an}bracketri}ht=/an}bracketle{tσ/an}bracketri}ht,/an}bracketle{tSi/an}bracketri}ht=/an}bracketle{tS/an}bracketri}htand, as a consequence, /an}bracketle{tyj/an}bracketri}ht=\nβ[H(t)−2J/an}bracketle{tS/an}bracketri}ht],/an}bracketle{tξj/an}bracketri}ht=β[H(t)−2J/an}bracketle{tσ/an}bracketri}ht] one obtain eventually the dynamical equations\n\n\nd/an}bracketle{tσ/an}bracketri}ht\ndt=−Ω/bracketleftBig\n/an}bracketle{tσ/an}bracketri}ht−σBσ/parenleftbig\nσβ{H(t)−2J/an}bracketle{tS/an}bracketri}ht}/parenrightbig/bracketrightBig\n,\nd/an}bracketle{tS/an}bracketri}ht\ndt=−Ω/bracketleftBig\n/an}bracketle{tS/an}bracketri}ht−SBS/parenleftbig\nSβ{H(t)−2J/an}bracketle{tσ/an}bracketri}ht}/parenrightbig/bracketrightBig\n.(15)\nAt equilibrium ( d/dt→0) one recover the usual MFA equations from the system.\nIn one dimension the mean-field approximation is poor since the local fi eldsyiandξi\nfluctuate strongly from one site to another. To overcome partly t he drawback we use a\ngeneralization of the MFA approach like those used in Ref.18\nTake the following approximation for the averages (11)\n/angbracketleftBigexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/angbracketrightBig\n≈/summationdisplay\nSi−1,Siexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]p(Si−1)p(Si), (16)7\n/angbracketleftBigexp[ξiSi]/summationtext\nS′\niexp[ξiS′\ni]/angbracketrightBig\n≈/summationdisplay\nσi,σi+1exp[ξiSi]/summationtext\nS′\niexp[ξiS′\ni]p(σi)p(σi+1), (17)\ni.e. the averages depend on the spin variables at the adjacent sites of another sublattice.\nAfter substitution Eqs.(16-17) into Eqs.(10,12) one obtain the dyn amic equations of the\ngeneralized mean-field approximation written through the probabilities\nd\ndtp(σi) =−Ω/bracketleftbigg\np(σi)−/summationdisplay\nSi−1,Siexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nyi=β[H−J(Si−1+Si)]p(Si−1)p(Si)/bracketrightbigg\n,\nd\ndtp(Si) =−Ω/bracketleftbigg\np(Si)−/summationdisplay\nσi,σi+1exp[ξiSi]/summationtext\nS′\niexp[ξiS′\ni]/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nξi=β[H−J(σj+σj+1)]p(σi)p(σi+1)/bracketrightbigg\n.(18)\nBy determining a time evolution of these quantities one recover a time dependence of ex-\npectation values for the sublattice magnetizations according to Eq .(8).\nIII. NUMERICAL RESULTS\nLet us now discuss hysteresis phenomenon for (3 /2,1) ferrimagnetic Ising chain obtained\non the base of Eqs.(15) and (18). For numerical calculations it is con venient to rewrite the\nMFA equations in the form\nd/an}bracketle{tσ/an}bracketri}ht\nd˜τ=−/an}bracketle{tσ/an}bracketri}ht+B1([h−2/an}bracketle{tS/an}bracketri}ht]/θ),\nd/an}bracketle{tS/an}bracketri}ht\nd˜τ=−/an}bracketle{tS/an}bracketri}ht+(3/2)B3\n2(3[h−2/an}bracketle{tσ/an}bracketri}ht]/2θ).\nwhere ˜τ= Ωt, andh=H/J,θ=T/Jare reduced field and temperature, respectively.\nRegarding the system (18) there is a way to bring down a number of d ifferential equa-\ntions from 7 till 5 by using the normalizing condition for probabilities pand definition of\nobservables. Indeed, hold four differential equations for the pro babilities p(±1),p(±3/2)\nd\nd˜τp(±1) =−p(±1)+/summationdisplay\nSi−1,Siexp[±(h−Si−1−Si)/θ]/summationtext\nσ′\niexp[σ′\ni(h−Si−1−Si)/θ]p(Si−1)p(Si), (19)\nd\nd˜τp(±3/2) =−p(±3/2)+/summationdisplay\nσi,σi+1exp[±3(h−σj−σj+1)/(2θ)]/summationtext\nS′\niexp[S′\nih−σj−σj+1\nθ]p(σi)p(σi+1),(20)8\nand add the corresponding equation for the observable /an}bracketle{tS/an}bracketri}ht\nd/an}bracketle{tS/an}bracketri}ht\nd˜τ=−/an}bracketle{tS/an}bracketri}ht+/summationdisplay\nσi,σi+13\n2B3\n2/parenleftBig\n3[h−σj−σj+1]/2θ/parenrightBig\np(σi)p(σi+1),\nobtained from Eq.(18). The system is supplemented by the pure alge braic relations\np(0) = 1−p(−1)−p(1),\np(−1/2) =p(3/2)−2p(−3/2)−/an}bracketle{tS/an}bracketri}ht+1/2,\np(1/2) =/an}bracketle{tS/an}bracketri}ht−2p(3/2)+p(−3/2)+1/2\nthat yield the leaving probabilities.\nThe sums in the right-hand sides of Eqs.(19,20) are explicitly given by\n/summationdisplay\nSi−1,Siexp[σi(h−Si−1−Si)/θ]/summationtext\nσ′\niexp[σ′\ni(h−Si−1−Si)/θ]p(Si−1)p(Si) =exp[σi(h+3)/θ]\nZ1[(h+3)/θ]p(−3/2)2\n+2exp[σi(h+2)/θ]\nZ1[(h+2)/θ]p(−3/2)p(−1/2)+exp[σi(h+1)/θ]\nZ1[(h+1)/θ]/bracketleftbig\np(−1/2)2+2p(−3/2)p(1/2)/bracketrightbig\n+2exp[σih/θ]\nZ1[h/θ]/bracketleftbig\np(−3/2)p(3/2)+p(−1/2)p(1/2)/bracketrightbig\n+exp[σi(h−1)/θ]\nZ1[(h−1)/θ]/bracketleftbig\np(1/2)2+2p(−1/2)p(3/2)/bracketrightbig\n+2exp[σi(h−2)/θ]\nZ1[(h−2)/θ]p(1/2)p(3/2)+exp[σi(h−3)/θ]\nZ1[(h−3)/θ]p(3/2)2,\nfor the spin-1 sublattice and\n/summationdisplay\nσi,σi+1exp[Si(h−σj−σj+1)/θ]/summationtext\nS′\niexp[S′\ni(h−σj−σj+1)/θ]p(σi)p(σi+1) =exp[Si(h+2)/θ]\nZ3\n2[(h+2)/θ]p(−1)2\n+2exp[Si(h+1)/θ]\nZ3\n2[(h+1)/θ]p(−1)p(0)+exp[Sih/θ]\nZ3\n2[h/θ][p(0)2+2p(−1)p(1)]\n+2exp[Si(h−1)/θ]\nZ3\n2[(h−1)/θ]p(0)p(1)+exp[Si(h−2)/θ]\nZ3\n2[(h−2)/θ]p(1)2,\nfor the spin-3/2 one. Here,\nZ1(x) =/summationdisplay\nσ′\niexp[σ′\nix] = 2cosh( x)+1, Z 3\n2(x) =/summationdisplay\nS′\niexp[S′\nix] = 2cosh( x/2)+2cosh(3 x/2).\nThe results of such calculations performed within the Runge-Kutta method are presented\nin Figs.1-3 for the field frequency ω/Ω = 10−4(Ω = 1). Initial values were chosen to\ncorrespond either to total saturation or disorder in both sublatt ices. A stationary regime is9\napproximately reached after one full sweep. Fig.1 shows the evolut ion of hysteresis loops at\nvarioustemperatures. Itcanbeclearlyseenthattheareaofthe hysteresisloopmonotonically\ndecreases with increasing temperature. The hysteresis curves a re characterized by well-\ndefined steps, and for higher temperature ( θ >0.7) have the form with a loop in the middle\nclose to that of observed experimentally in Co(II)-based quasi-1D ferrimagnetic compound\nCoPhOMe at temperatures around /greaterorsimilar4.5 K.4The process of thermally activated spin flip\nformation reduces the effective intrinsic coercive field (see inset in F ig.1). Note that the\ntemperature dependence of the coercivity exhibits a behavior (a c urve with an inflection)\ndistinguished from that of predicted by a thermal activation theor y by Egami19used for\nhard magnetic materials.20WhenTtends to zero the value of the coercive field becomes\nequal to 2 J, i.e. it is determined by the exchange coupling (not by anisotropy) as expected\nfor Ising systems.\nThe both analytical approaches give qualitatively same results (see Fig.2). Note only\nthat a more detailed account of fluctuations within the generalized M FA squeezes the area\nof the hysteresis loops.\nTo elucidate physics standing for this hysteresis behavior, magnet izations of the both\nsublattices arepicked outin Fig.3. The results resemble a spin-orient ational phase transition\nin the Heisenberg antiferromagnet and are interpreted as follows. At strong fields both\nsublattices are polarized along the field. With its decreasing the bigge r spins retain their\ndirections while smaller spins change smoothly their alignment into oppo site to gain in an\nexchange energy. During further demagnetization process, whe nh <0, the bigger spins\nreorient to be again arranged along the field. This causes a sharp sp in reorientation in\nanother sublattice due to the exchange coupling which is stronger t han the corresponding\nZeeman interaction. The rapid change is accompanied by the side effe ct which stands out\nmore noticeably in the MFA calculations, namely, an appearance of ”w hiskers” in the entire\nhysteresis curve at low temperatures. The process is completed b y a gradual saturation\nof small-spin sublattice magnetization along the applied field. The back sweep goes on\nsimilarly.\nWealsofoundouthowthemagnetizationprocessdependsonthere altionshipbetween the\nfrequency Ω (a number of spin transitions per unit time) and the mag netic field frequency\nω. We plotted the corresponding curves in Fig.4 where two cases are p resented, namely,\nthe quasistatic regime ω/Ω = 10−4and the regime when the discrepancy between both the10\n-2-1 0 1 2\n-10 -5  0  5  10M\nhθ = 0.7\n-2-1 0 1 2\n-10 -5  0  5  10M\nhθ = 1\n-2-1 0 1 2\n-10 -5  0  5  10M\nhθ = 1.3\nFIG. 2: Comparison of hysteresis curves calculated within M FA (solid) and generalized MFA\n(dashed) approaches.\nfrequencies are not so drastic ω/Ω = 10−1. As well we show in the Figure the pure static\nmagnetization process which almost coincides with that of the quasis tatic regime. One see\nthat in the case ω/Ω = 10−1the hysteresis loop transforms into the narrow S-like form11\nFIG. 3: Hysteresis loops for the sublattice /an}bracketle{tσ/an}bracketri}ht(big dotted), /an}bracketle{tS/an}bracketri}ht(small dotted) and the entire M\n(solid) magnetizations ( θ= 0.4).\nsimilar to those found experimentally in the compound CoPhOMe at 2 .0 K and 3 .5 K.\nPhysically, the increasing the ratio ω/Ω at a fixed magnetic field frequency means that a\nspin dynamics governed by Ω slows down.\nIV. CONFIRMATION BY MONTE CARLO SIMULATIONS\nIn the previous section we have shown that in the quasistatic regime ω≪Ω the both\nanalytical models demonstrate an appearance of hysteresis loops of a peculiar form in the\nmiddle of magnetization curves. However, we note that spin fluctua tions taken into account\nin the framework the generalized MFA are strongly restricted since they hold a translational\nsymmetry of the chain, i.e. they are the same within an each elemenat ry cell. In order to\ncheck whether the hysteresis phenomena are stable against fluct uations in common case, we\nstudy the static magnetization process by a Monte Carlo method.\nWe apply standard importance sampling methods21,22to simulate the Hamiltonian given\nby Eq.(1). Periodic boundary conditions on N= 64, 256 chains were imposed and configu-12\n-2-1 0 1 2\n-10 -5  0  5  10M\nhθ = 0.2\na)\n!/Ω = 0\n!/Ω = 1/10000\n!/Ω = 1/10\n-2-1 0 1 2\n-10 -5  0  5  10M\nhθ = 0.2\nb)\n!/Ω = 0\n!/Ω = 1/10000\n!/Ω = 1/10\nFIG. 4: Evolution of magnetization curves for different frequ ency regimes ( θ= 0.2): (a) MFA; (b)\ngeneralized MFA13\nrations were generated by sequentially traversing the sublattices and making single-spin flip\nattempts. The flips are accepted or rejected according to a heat -bath algorithm. Our data\nwere generated with 104Monte Carlo steps per spin in the chain after 103warming steps\nper spin. We checked that the size effects do not substantially effec t the results (Fig. 5).\n-2-1 0 1 2\n-10 -5  0  5  10M\nhθ = 0.2\nN = 256\nN = 64\nFIG. 5: MC magnetization curves for different chain lengths.\nQualitatively, the same features of the hysteresis loops as obtaine d in MFA and the\ngeneralized MFA are also found in MC. Namely, the hysteresis loop of a n almost ideal\nrectangle form appears in the middle of the magnetization curve at lo w temperatues. The\ntemperature evolution of the hysteresis curves reproduces qua liatatively that of predicted\nby the analytical treatments, however, the MC hysteresis loop na rrows more rapidly with\nan increasing of temperature (Fig. 6).\nV. CONCLUSIONS\nWe analyse a magnetic hysteresis of Co-based quasi-1D ferrimagne tic magnets within\nthe model of the mixed spin Ising chain. By using a Glauber dynamics ap proach we build\nhysteresis loops that come up when a sinusoidal magnetic field is applie d. We found that the\nunusual shape of the calculated hysteresis cycles coincide with tho se found experimentally\nin the CoPhOMe ferrimagnet at low temperatures. However, anoth er Co-based molecular14\nchains11,12demonstrate a hysteresis behavior that is scarcely agreed with th e analytical\ntreatment, i.e. these materials behave similar to very hard magnets with a high coercive\nfield and broad hysteresis loops.\nIt is likely that a reason behind the hardness of the materials is interc hain interactions\nthat result in 3D ordering at low temperatures, and, as consequen ce, change a character\nof elementary excitations. Namely, random spin flips throughout th e Ising chain, that is a\nfeature of the Glauber dynamics, are substituted for spin flips in th e vicinity of domain walls\nseparating regions of opposite magnetizations that causes their d isplacements. Then the\ndemagnetization process is related with the thermally activated DW m otion. Similar ideas\nhave been recently argued in Ref.7, where the Mn(III)-based systems23,24were suggested to\nfind out a role of easy axis anisotropy in hysteresis phenomena.\n1R. Sessoli, D. Gateschi, A. Caneschi, M.A. Novak, Nature 365(1993) 365 .\n2L. Bogani, A. Vindigni, R. Sessoli, D. Gatteschi, J. Mater. C hem.18(2008) 4750.\n3R. Sessoli, H. L. Tsai, A. R. Schake, S. Wang, J. B. Vincent, K. Folting, D. Gatteschi, G.\nChristou and D. N. Hendrickson, J. Am. Chem. Soc. 115(1993) 1804.\n4A. Caneschi, D. Gatteschi, N. Lalioti, C. Sangregorio, R. Se ssoli, G. Venturi, A. Vindigni, A.\nRettori, M. G. Pini and M. A. Novak, Europhys. Lett. 58(2002) 771; Angew. Chem. Int. Ed.\n40(2001) 1760.\n5R. Clerac, H. Miyasaka, M. Yamashita and C. Coulon, J. Am. Che m. Soc.124(2002) 12837.\n6D. Gatteschi, R. Sessoli and J. Villain, Molecular Nanomagn ets, Oxford University Press, Ox-\nford, UK, 2006.\n7R. Sessoli, Angew. Chem. Int. Ed. 47(2008) 5508.\n8R. J. Glauber, J. Math. Phys. 4(1963) 294.\n9K. Kawasaki, Phys. Rev. B 145(1966) 224.\n10M. Suzuki, R. Kubo, J. Phys. Soc. Jpn 24(1968) 51.\n11N. Ishii, Y. Okamura, S. Chiba, T. Nogami, T. Ishida, J. Am. Ch em. Soc. 130(2008) 24.\n12Y. Numata, K. Inoue, N. Baranov, M. Kurmoo, and K. Kikuchi, J. Am. Chem. Soc. 129(2007)\n9902.\n13A.S. Ovchinnikov, I.G. Bostrem, V.E. Sinitsyn, A.S. Boyarc henkov, N.V. Baranov, K. Inoue,15\nPhys. Rev. B 74(2006) 174427.\n14E. Lhotel, D.B. Amabilino, C. Sporer, D. Caneau, J. Veciana, and C. Paulsen, Phys. Rev. B 77\n(2008) 064416.\n15M. Nishino, S. Miyashita, Phys. Rev. B 63(2001) 174404.\n16M. Nishino, K. Boukheddaden, S. Miyashita, F. Varret, Phys. Rev. B79(2005) 064452.\n17B. Deviren, M. Keskin, O. Canko, J. Mag. Magn. Mat. 321(2009) 458 and references therein.\n18E.Z. Meilikhov, JETP Letters 79(2004) 620.\n19T. Egami, phys. stat. sol. (a) 19(1973) 747.\n20F.T. Parker, Solid State Commun. 50(1984) 637; H. Kronm¨ uller, J. Mag. Magn. Mat. 7(1978)\n341.\n21G.M. Buend ´ia and R. Cardona, Phys. Rev. B 59, (1999) 6784.\n22M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statist ical Physics, OxfordUniversity\nPress, Oxford, UK, 1999.\n23K. Bernot, J. Luzon, R. Sessoli, A. Vindigni, J. Thion, S. Ric heter, D. Leclercq, J. Larionova,\nA. van der Lee, J. Am. Chem. Soc. 130(2008) 1619.\n24C. Coulon, H. Miyasaka, R. Cl´ erac, Struct. Bonding (Berlin )122(2006) 163.16\n-2-1 0 1 2\n-10 -5  0  5  10M\nhθ = 0.2\nMC\nGMFA \nMFA\n-2-1 0 1 2\n-10 -5  0  5  10M\nhθ = 0.8\nMC\nGMFA \nMFA\nFIG. 6: Temperature evolution of MC hysteresis loops in a com parison with the mean field (MF)\nand the generalized mean field (GMF) calculations ( N= 256)."    },    {        "title": "0705.3731v1.Ferrimagnetism_and_antiferromagnetism_in_half_metallic_Heusler_alloys.pdf",        "content": "arXiv:0705.3731v1  [cond-mat.mtrl-sci]  25 May 2007pss header willbe provided by the publisher\nFerrimagnetism and antiferromagnetism in half-metallic\nHeusler alloys\nIosifGalanakis∗1,Kemal ¨Ozdo˘gan2,ErsoyS ¸as ¸ıo ˘glu3,4,andBekir Aktas ¸2\n1Department of Materials Science, Universityof Patras,GR- 26504 Patra,Greece\n2Department of Physics,Gebze Instituteof Technology, Gebz e, 41400, Kocaeli, Turkey\n3Institutf¨ ur Festk¨ orperforschung, Forschungszentrum J ¨ ulich, D-52425 J¨ ulich, Germany\n4FatihUniversity, PhysicsDepartment, 34500, B¨ uy¨ ukc ¸ek mece,˙Istanbul,Turkey\nReceived 15November 2003, revised30 November 2003, accept ed 2December 2003\nPublishedonline 3December 2003\nPACS75.47.Np, 75.50.Cc, 75.30.Et\nHalf-metallic Heusler alloys are among the most promising m aterials for future applications in spintronic\ndevices. Although most Heusler alloys are ferromagnets, fe rrimagnetic or antiferromagnetic (also called\nfully-compensated ferrimagnetic) alloys would be more des irable for applications due tothe lower external\nfields. Ferrimagnetism can be either found in perfect Heusle r compounds or achieved through the creation\nof defects inferromagnetic Heusler alloys.\nCopyrightlinewillbeprovidedbythe publisher\nIntroduction : The family of the ferromagnetic Heusler alloys, e.g. NiMnSb or Co2MnSi, have been\nextensively studied during the last years due to their poten tial applications in magnetoelectronic devices\n[1]. Their main advantage with respect to other half-metall ic systems is their structural similarity with\nthe binary semiconductors and their high Curie temperature s. First principles calculations have been ex-\ntensively employed to study their electronic and magnetic p roperties (see Refs. [2, 3, 4] and references\ntherein). One of the most important features of these alloys is the Slater-Pauling behavior of their total\nspin magnetic moment which is given simply as a function of th e number of valence electrons in the unit\ncell [5, 6]. Authors have studied in the recent years several aspects of these half-metallic alloys like the\nproperties of surfaces [7, 8, 9] and interfaces with semicon ductors [10, 11], the quaternary [12, 13], the\norbitalmagnetism[14,15],theeffectofdopinganddisorde r[16,17],theexchangeconstants[18]andthe\nmagneto-opticalproperties[19].\nHalf-metallicferrimagneticHeusler alloyslike Mn 2VAl, where Mn andV atomshaveantiparallel spin\nmoments, are of particular interest since they create small er external magnetic fields and thus lead to\nsmaller energy losses [20, 21]. In the extreme case like Cr 2MnSb (alloys with 24 valence electrons) Cr\nandMnspinmomentscanceleachotherandthecompoundsarena medasfully-compensatedhalf-metallic\nferrimagnetsorsimplyhalf-metallicantiferromagnets[2 2]. Defectsinthesealloysshowaveryinteresting\nbehavior. When we substitute Co atomswith Cr(Mn)in the Co 2Cr(Mn)Al andCo 2Cr(Mn)Sicompounds,\nthe impurity atoms couple antiferromagnetically with the o ther transition metal atoms lowering the total\nspin moment [23, 24]. Co and Fe impurities in Mn 2VAl and Mn 2VSi ferrimagnetic alloys have spin\nmoments antiparallel to Mn and thus the total spin moment rea ches closer to the zero value [25]. In this\nmanuscript we will complete these studies presenting resul ts for the Cr(Mn) impurites in ferromagnetic\nCo2Mn(Cr)Al and Co 2Mn(Cr)Si alloys, the case of V and Cr impurities in ferrimagn etic Mn 2VAl and\nMn2VSi alloys and finally the 24-valence half-metallic antifer romagnetic alloys Cr 2FeZ (Z= Si, Ge, Sn)\nusingthefull–potentialnonorthogonallocal–orbitalban dstructurescheme(FPLO)[26].\n∗Corresponding author: e-mail: galanakis@upatras.gr , Phone+30 2610 969925, Fax +30 2610 969368\npss datawillbe providedby thepublisher2 I.Galanakis, K. ¨Ozdo˘ gan, E.S ¸as ¸ıo˘ glu, and B.Aktas ¸:\nTable 1 Total and atom-resolved spin magnetic moments in µBfor the [X 1−xX∗\nx]2YZ compounds. The atom-\nresolved spin moments have been scaled toone atom. We donot p resent the moments of the Z sp-atom since they are\nnegligible withrespect tothe X andY transition-metal atom s.\nCompound mComX∗mYmTotalCompound mMnmX∗mVmTotal\n[Co0.9Cr0.1]2MnAl 0.72 -2.57 2.79 3.43 [Mn0.9V0.1]2VAl -1.68 -1.00 1.03 -2.13\n[Co0.9Cr0.1]2MnSi 0.97 -1.55 3.03 4.40 [Mn0.9V0.1]2VSi -1.17 -1.10 0.92 -1.33\n[Co0.9Mn0.1]2CrAl 0.74 -1.75 1.70 2.60 [Mn0.9Cr0.1]2VAl -1.66 -1.8 1.12 -2.20\n[Co0.9Mn0.1]2CrSi 0.94 -0.95 2.14 3.60 [Mn0.9Cr0.1]2VSi -1.06 -1.58 0.96 -1.20\n[Co1−xCrx]2MnZ and [Co 1−xMnx]2CrZ (Z= Al, Si) alloys : In the first part of our study we will\nconcentrate on the case of defects in the strong ferromagnet ichalf-metallic Co 2CrZ and Co 2MnZ (Z= Al\norSi) Heusler alloys. InRef. [23]we havestudiedthe case of Cr defectsinCo 2CrAl andCo 2CrSi alloys;\nCr atoms substitute Co atoms at the perfect X sites. We expand ed this study in Ref. [24] studying the\ncase of Mn impurities in Co 2MnAl and Co 2MnSi compounds. In all cases the impurity atoms had spin\nmoments antiparallel to the other transition metal atoms lo wering the total spin moment while keeping\nthe half-metallic character of the parent compounds. We hav e expanded this study now considering Mn\nimpurities in Co 2CrAl and Co 2CrSi and Cr impurities in Co 2MnAl and Co 2MnSi and present both the\ntotalandatom-resolvedDOS’sinFig. 1. Aswasthecasealsoi nRefs. [23,24]allcompoundswithdefects\npresent the half-metallic behaviorand only in the case of [C o0.9Cr0.1]2MnSi alloy is the width of the gap\nslightly smaller with respect to the perfect Co 2MnSi alloy (not shown here). The total spin moments(see\nTable 1) are considerably smaller than the spin moments of th e perfect compounds: 5 µBfor Co2MnSi,\n4µBfor Co 2MnAl and Co 2CrSi, and 3 µBfor Co 2CrAl. Thus these kind of defected alloys could be\nvaluable for realistic applications. As for the compounds i n the previous paragraph we have performed\ncalculationsforseveralconcentrationsbut resultsaresi milar totheconcentrationwhichwepresent.\n[Mn1−xVx]2VZ and [Mn 1−xCrx]2VZ (Z= Al, Si) alloys : In Ref. [25] we have shown that when we\nsubstitute Co or Fe for Mn in the ferrimagnetic hafl-metallic Mn2VAl and Mn 2VSi alloys, the Co and\nFe impurity atoms have spin moments parallel to the V atoms at the Y site and antiparallel to the Mn\natoms. Thus the negative total spin moment of the alloys beco mes smaller in magnitude and for specific\nconcentrationofdefectswe geta half-metallicantiferrom agnet. We shownowresultswhenwe use V and\nCrasimpurityatomswhichhavelowervalencethanMn. Ascanb eseeninFig. 2,alloyswithCrimpurities\nalmost keep unaltered the gap in the spin-up band for both Al a nd Si-based compounds. Contrary to Cr\natoms, the lighter vanadium shows a smaller exchangesplitt ing and the Fermi level falls within a spin-up\npickof theV DOS completelydestroyingthe half-metallicit y. We shouldalso notethat we presentresults\nonly for a concentration of defects of x=0.1. We have performed calculations also for x=0.025,0.05 and\n0.2butthebehaviorofthecompoundsissimilar tothecase wh ichwepresent.\nEvenmoreinterestingisthebehaviorofthespinmomentspre sentedinTable1. InthecaseofCoandFe\nimpurities,duetotheirpositivespinmoment,thetotalspi nmomentwasapproachingzeroasweincreased\nthe concentration. Contraryto these chemical elements, V a nd Cr hybridizeless with the neighboringMn\natomswhichshowamoreatomiclike behaviorandthusincreas etheabsolutevalueoftheirspinmagnetic\nmoment. V or Cr atomshave smaller spin momentsthan Mn atomsb ut they are in low concentrationand\nas a result the total spin momentincreasesin magnitudefrom the -2µBof the perfectMn 2VAl and the -1\nµBof the perfect Mn 2VSi (note that the total spin momentsare negativesince thes e alloys have less than\n24 valence electrons according to the Slater-Pauling rule [ 6]). Thus such a doping with Cr and V would\nhavenoadvantagesovertheperfectMn 2VAlandMn 2VSi compoundsforrealisticapplications.\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheimpss header willbe provided by the publisher 3\n-6-3036[Co0.9Cr0.1]2MnAl\n[Co0.9Cr0.1]2MnSi\n-4 -2 0\nE (eV)-6-3036[Co0.9Mn0.1]2CrAl\n[Co0.9Mn0.1]2CrSi\n-202\n-4 -2 0\nE (eV)-202\n-4 -2 0-202\n-202DOS (states/eV)Co                                                 Cr\nCr                                                   Z  Co                                                 MnMn                                                  Z\nFig. 1(Color online) Total and atom-resolved DOS\nfor Cr impurities in Co 2MnAl and Co 2MnSi alloys\n(upper panel) and Mn impurities in Co 2CrAl and\nCo2CrSicompounds (lower panel).-6-3036[Mn0.9V0.1]2VAl\n[Mn0.9V0.1]2VSi\n-4 -2 0\nE (eV)-6-3036[Mn0.9Cr0.1]2VAl\n[Mn0.9Cr0.1]2VSi\n-202\n-4 -2 0\nE (eV)-202\n-4 -2 0-202\n-202DOS (states/eV)Mn                                              Vimp\nV                                                     ZMn                                                CrV                                                     Z\nFig. 2(Color online) Total and atom-resolved DOS\nfor V (upper panel) and Cr (lower panel) impurities\nin the case of Mn 2VAl and Mn 2VSi alloys. DOS’s\nhave beenscaled toone atom.\n-6-3036DOS (states/eV)Fe\nCr\ntotal\n-4 -2 0\nE (eV)-6-3036\n-4 -2 0\nE (eV)Cr2FeSi\nCr2FeGe                                    Cr2FeSn\nFig. 3Total, Cr- and Fe-resolved DOS for the\nCr2FeZalloyswhereZisSi,GeorSnandforalattice\nconstant of 6.2 ˚A.-6-3036DOS (states/eV)Fe\nCr\ntotal\n-4 -2 0\nE (eV)-4 -2 0\nE (eV)-6-3036a=6.0 Å                                                 a=6.2 Å                                                       \na=6.1 Å                                                 a=6.3 ÅCr2FeSi\nFig. 4(Color online) Total, Cr- and Fe-resolved\nDOS for the Cr 2FeSialloyand for four different val-\nues of the latticeconstant.\nCr2FeZ(Z=Si,Ge,Sn)alloys: MotivatedbyourresultsontheCr 2MnZalloys[22],wedecidedtostudy\nalso another family of 24-valence electrons compounds cont aining Fe instead of Mn: the Cr 2FeZ alloys\nwhereZ is Si, Ge orSn. In Fig. 3 we present the total andthe Cr a ndFe-resolveddensityof states (DOS)\nfor a lattice constant of 6.2 ˚A for all three alloys. For this lattice constant all the comp oundsunder study\npresent a region of low DOS in the spin-up band and the Fermi le vel falls within this region. Contrary\nto the compounds containing Mn, where the spin-up occupied s tates are mainly of Cr character and the\noccupied spin-downstates are of mainly Mn character, for th e Fe-based alloys both the spin-up and spin-\ndown occupied states exhibit a more mixed character. The ele ctronic structure is more complicated than\nthe Mn-based alloys and as a result when we slightly vary the l attice constant the band-structure changes\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheim4 I.Galanakis, K. ¨Ozdo˘ gan, E.S ¸as ¸ıo˘ glu, and B.Aktas ¸:\nin a way that the the region of low DOS is completely destroyed . This is illustrated in Fig. 4 where we\npresenttheDOSforCr 2FeSiforfourvaluesofthelatticeconstant. Crspin-upstat esbelowtheFermilevel\nand the Cr spin-up states just above the Fermi level move one t owards the other completely destroying\nthe regionof high spin-polarizationuponcompression. Lat tice constants largerthan 6.3 ˚A are unlikelyto\nbe achieved experimentally. The reason is the different hyb ridization between the transition-metal atoms\nwhenwesubstituteFeforMn. Thesmallerexchangesplitting oftheFeatomsclosesthegapandthisleads\nalsotoadifferentdistributionoftheCrchargewhichnomor eshowsalargeband-gapasfortheMn-based\nalloys. Finally we shouldalso discussthe spinmagneticmom ents. Fe atomsforthe lattice constantof6.2\n˚A possess a moment slightly smaller than -3 µBand each Cr atom has a spin moment of around 1.5 µB\nresultingin the almost vanishingtotal spin momentrequest edby theSlater=Paulingbehaviorforthe half-\nmetalswith 24valenceelectrons[6]. Overallthe Cr 2FeZ alloyswith 24-valenceelectronsarenot suitable\nfor realistic application, contrary to the Cr 2MnZ compoundssince half-metallicity is very fragile in the ir\ncase (e.g. calculations suggest that the region of low spin- up DOS persists for Cr 2FeSi only between 6.2\nand6.3˚A).\nSummary : We havepresentedfirst-principlescalculationsonseveral Heusleralloys. Cr(Mn)impurites\noccupying Co sites in Co 2Mn(Cr)Al and Co 2Mn(Cr)Si alloys couple antiferromagnetically to the other\ntransition metal atoms resulting in ferrimagnetic compoun ds with lower total spin moments similarly to\nour previous studies on these alloys [23, 24]. V and Cr impuri ties occupying Mn sites in ferrimagnetic\nMn2VAl and Mn 2VSi alloys present spin moments parallel to the Mn atoms cont rary to the behavior of\nthe Fe and Co impurities in these alloys and they lead to large r absolute values of the total spin moments\n[25]. Finally,rhe24-valenceelectronalloysCr 2FeZ(Z=Si,Ge,Sn)showaregionoflowdensityofstates\ninsteadoftherealgappresentedbytheisovalentCr 2MnZ(Z=P,As, Sb,Bi)[22]whichpersistsonlyfora\nverynarrowrangeoflattice constants.\nReferences\n[1] Half-metallic alloys: fundamentals and applications, Eds.: I. Galanakis and P. H. Dederichs, Lecture notes in\nPhysics vol.676 (BerlinHeidelberg: Springer 2005).\n[2] I.Galanakis, Ph. Mavropoulos, and P.H. Dederichs, J.Ph ys. D:Appl. Phys. 39, 765 (2006).\n[3] I.Galanakis andPh. Mavropoulos, J.Phys.: Condens. Mat ter inpress [preprint: cond-mat/0610827].\n[4] P.H.Dederichs, I. Galanakis, andPh. Mavropoulos, J.El ectronMicroscopy 54, i53(2005).\n[5] I.Galanakis, P.H.Dederichs, andN. Papanikolaou, Phys .Rev. B66, 134428 (2002).\n[6] I.Galanakis, P.H.Dederichs, andN. Papanikolaou, Phys .Rev. B66, 174429 (2002).\n[7] I.Galanakis, J. Phys.: Condens. Matter 14, 6329 (2002).\n[8] I.Galanakis, J. Magn. Magn. Mater. 288, 411 (2005).\n[9] M. Leˇ zai´ c,I. 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KGaA, Weinheim"    },    {        "title": "2211.15074v1.Anomalous_Nernst_effect_in_compensated_ferrimagnetic_CoxGd1_x_films.pdf",        "content": " 1 Anomalous Nernst effect in compensated ferrimagnetic CoxGd1-x films Ruihao Liu1,2, Li Cai1,2, Teng Xu1,2, Jiahao Liu1,2, Yang Cheng1,2 and Wanjun Jiang1,2,a)  1State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China 2Frontier Science Center for Quantum Information, Tsinghua University, Beijing 100084, China a)Author to whom correspondence should be addressed: jiang_lab@tsinghua.edu.cn  Abstract: The anomalous Nernst effect (ANE) is one of the most intriguing thermoelectric phenomena which has attracted growing interest both for its underlying physics and potential applications. Typically, a large ANE response is observed in magnets with pronounced magnetizations or nontrivial Berry curvature. Here, we report a significant ANE signal in compensated ferrimagnetic CoxGd1-x alloy films, which exhibit vanishingly small magnetization. In particular, we found that the polarity of ANE signal is dominated by the magnetization orientation of the transition metal Co sublattices, rather than the net magnetization of CoxGd1-x films. This observation is not expected from the conventional understanding of ANE but is analogous to the anomalous Hall effect in compensated ferrimagnets. We attribute the origin of ANE and its Co-dominant property to the Co-dominant Berry curvature. Our work could trigger a more comprehensive understanding of ANE and may be useful for building energy-harvesting devices by employing ANE in compensated ferrimagnets.      Recent studies on thermoelectric effects have stimulated renewed understandings of their physical origins and also triggered increasing application opportunities1–22. The anomalous Nernst effect (ANE) is one of the representative examples2-15. When a longitudinal temperature gradient (∇𝑇) is applied to a slab of magnetic material, the induced transverse electric field (𝑬) can be empirically formulated as23,24: 𝑬=𝑄!(𝑯𝒛×∇𝑇)+𝑄#(𝜇!𝑴𝒛×∇𝑇)\t\t\t       (1) where\t𝜇! is the vacuum permeability, 𝑄!\tand 𝑄# is the (ordinary/anomalous) Nernst coefficients, and 𝑯𝒛, 𝑴𝒛 are the applied perpendicular magnetic field, and the spontaneous net perpendicular magnetization, respectively. The two terms on the right side of Eq. (1) represent the ordinary Nernst effect (ONE) and ANE, respectively. Note that the contribution from ONE is typically smaller than that of ANE, which is thus often neglected25. From Eq. (1), it is clear that a large ANE is generally expected in ferromagnets with appreciable spontaneous magnetization3,4,6,7,10,11,13. Very recently, it is discovered that Eq. (1) becomes invalid in several antiferromagnets with no net magnetizations, where a large ANE could occur as a result of the nontrivial Berry curvature9,15.   2 Compensated ferrimagnets (FIMs) are currently attracting considerable attention from the spintronics community for enabling the efficient spin-orbit torque switching26-32 and the ultrafast domain wall motion33–38. Compensated FIMs exhibit minimal net magnetizations as a consequence of the antiparallel configuration of the involved magnetic sublattices39,40. Thus, it is not expected to observe pronounced ANE signals, as directly guided by Eq. (1). This intriguing aspect will be experimentally examined in the present study. Rare-earth-transition-metal (RE-TM) CoxGd1-x amorphous films are typical compensated FIMs, in which the unequal magnetic moments of the Co and the Gd elements are antiferromagnetically coupled39,40. This results in a vanishingly small net magnetization (𝑴𝒛=𝑴𝑪𝒐−𝑴𝑮𝒅), where 𝑴𝑪𝒐 and 𝑴𝑮𝒅 represent for the antiparallel magnetization of the Co and the Gd elements, as shown in Fig. 1(a). By changing the relative atomic composition ratio (x) between the Co and the Gd elements, or the external temperature (T), the contribution from each sublattice and the resultant magnetization could be effectively tuned. For example, crossing the critical composition compensation point (𝑥()*≈0.78), or the temperature compensation point 𝑇()*, the dominant contribution to the net magnetization is either from the RE or TM elements, respectively27,41–47.     In the present work, through utilizing an integrated on-chip method, we observe significant ANE signals in CoxGd1-x films. Furthermore, the polarity of ANE signals in the RE-dominant sample is opposite to that of the TM-dominant sample. This observation is very similar to the behavior of the electrical counterpart of ANE, the anomalous Hall effect (AHE). This indicates that ANE could also be dominated by the magnetization of the TM sublattice, rather than the net magnetization. The origin of ANE and its TM-dominant property could be jointly attributed to the TM-dominant Berry curvature of the RE-TM compounds.      Magnetic multilayers of stacking order from the bottom to the top Ta(1)/Pt(3)/CoxGd1-x(𝑡+),-)/Ta(3) (numbers in parentheses are thicknesses in nanometers) are deposited on thermally oxidized silicon substrates using an AJA magnetron sputtering system (Orion 8). The base pressure of the main chamber is better than 1 × 10−8 torr and the Ar pressure is 3.0 mtorr. The Co-dominant and Gd-dominant samples (Co0.84Gd0.16 and Co0.74Gd0.26, respectively) are synthesized through co-sputtering of the Co and Gd targets with a fixed sputtering power of Co (50 W, deposition rate 0.25 Å/s), while varying the power of Gd at 25 W (deposition rate 0.14 Å/s) and at 40 W (deposition rate 0.27 Å/s), respectively. Based on the deposition rate, the relative composition is determined (See Part 1 in Supplementary Materials). The thickness of the amorphous CoxGd1-x layer is about 4.7 nm for Co0.84Gd0.16 and 6.2 nm for Co0.74Gd0.26. Through applying out-of-plane magnetic fields (𝐻.), magnetic properties (𝑀.−𝐻. loop) are measured using a Superconducting Quantum Interference Device (SQUID, Quantum Design MPMS-3). Perpendicular magnetic anisotropy (PMA) is obtained in both samples, as shown in Fig. 1(b). It is displayed that the coercive field 𝐻/ of these two samples are about 0.08 and 0.10 kOe, and the saturation net magnetization 𝑀0 are about 260 emu/cc and 100 emu/cc, respectively. Magnitudes of 𝑀0 in both samples are relatively small, which is consistent with the  3 expectation of compensated ferrimagnetism48. \n FIG. 1. (a): Schematic illustration of the collinear spin configurations in compensated ferrimagnet CoxGd1-x with PMA. Showing in the upper panel is the spin configuration of the Co-dominant film, while the lower panel is the Gd-dominant film, respectively. The brown and green arrows represent the antiparallel and unequal magnetization of the Co and Gd elements, respectively. The black arrow represents the out-of-plane magnetic field. (b): The hysteresis loops (𝑀.−𝐻.) of the Co0.84Gd0.16 (red lines) and Co0.74Gd0.26 (blue lines) samples, respectively. (c): An optical image of the on-chip device that was made on a 100 nm thick Si3N4 insulating substrate. In this device, the AHE measurements are conducted by injecting an electric current from terminal 3 (or 4) to terminal 7 (or 8) and the AHE voltage can be obtained from terminals 5 and 6. The ANE measurements are conducted by injecting an electric current into the zigzag heater (from terminal 1 to terminal 2) and a transverse Nernst voltage can be obtained from terminals 3 and 4 (or 5 and 6 or 7 and 8, which are located at different positions from the on-chip heater). (d): The AHE loops (𝑅12−𝐻.) of Co0.84Gd0.16 (red lines) and Co0.74Gd0.26 (blue lines), respectively.  These compensated FIMs are also deposited on the 100 nm thick Si3N4 membranes. Note that the Si3N4 membrane is electrically insulating, which, however, exhibits a relatively high in-plane thermal conductivity49. Devices are patterned by utilizing standard photolithography and lift-off techniques, as shown in Fig. 1(c). The AHE and ANE measurements are conducted on the same device by using a triple-axis superconducting magnet under the He atmosphere of 300 torr13. The AHE loops (𝑅12−𝐻.) are displayed in Fig. 1(d) (See Supplementary Fig. S1 for the sign convention) and from which a PMA is further verified. Note that coercive fields (𝐻/) from the AHE measurements are larger than those from SQUID measurements (about 0.08 kOe and 0.10 kOe from SQUID and about 0.2 kOe and 2.2 kOe form the AHE measurement, \n 4 respectively). This could be attributed to the additional edge-pinning effect that was introduced during the lithography process50,51. The AHE resistance 𝑅12 can be formulated from an empirical relation: 𝑅12=𝑅!𝑯𝒛+4𝜋𝑅#𝑴𝒛       (2) where 𝑅!, 𝑅# are the ordinary and anomalous Hall coefficients. The contribution from the ordinary Hall effect (OHE) is negligible, as evident from the minimal change of 𝑅12 above saturation52. The SQUID magnetometry measures the net magnetization, which is parallel to the applied magnetic field above saturation53. Thus, magnetic hysteresis loops of these two samples should exhibit the same polarity. The AHE measurements in compensated FIMs, however, probe primarily the responses from the 3d shell of Co, due to the weak coupling between conduction electron spins with the inner 4f shell of the Gd magnetic moments54-57. Hence, the AHE responses are dominated by the magnetization configuration of the Co element in this FIM system54-57. This can be confirmed from the same polarity of magnetic hysteresis loops (𝑀.−𝐻.), as shown in Fig. 1(b), and the opposite polarity of AHE loops (𝑅12−𝐻.) due to the opposite magnetization direction of the Co element, as shown in Fig. 1(d).  \n FIG. 2. (a): Heat distribution map of the device. This simulation is conducted by using the COMSOL software and incorporating the material-specific parameters. During the simulation, an electric current 𝐼 of 3.00 mA is injected into the heater. (b): Resistance of the heater as a function of temperature 𝑇 (red lines, lower horizontal axis) and as a function of 𝐼3 (blue lines, upper horizontal axis). (c): Simulated temperature 𝑇 and (d): Simulated temperature gradient ∇𝑇 profile at different positions when injecting 𝐼 of 1.00 mA (green dots), 2.00 mA (blue dots), and 3.00 mA (red dots), respectively.  \n 5 As shown in Fig. 1(c), the ANE measurements are conducted by first injecting an electric current 𝐼 into the on-chip heater. To identify whether the dissipation of Joule heating from the heater generates a temperature gradient along the 𝑋 direction, a finite element simulation using the COMSOL software is performed. By using 𝐼 = 3.00 mA, the simulated heat distribution map is shown in Fig. 2(a). By separately measuring the temperature-dependent resistance changes of the heater, and the resistance changes as a function of 𝐼3 at (nominal) room temperature, the temperature increase due to Joule heating can thus be estimated, as shown in Fig. 2(b). The evolution of the temperature 𝑇 and temperature gradient ∇𝑇 along the 𝑋 axis, is simulated and shown in Figs. 2(c) and 2(d), respectively. It is demonstrated that, within the range of 𝑋>20\tµm, both the temperature and temperature gradient decrease monotonously as the distance from the heater increases. Similarly, a larger electrical current corresponds to a larger temperature gradient at a fixed position. These simulation results suggest that an appreciable and evolving temperature gradient ∇𝑇 could be established in the current device geometry, which ensures the ANE measurements. As a result, a transverse Nernst voltage can be observed from three different channels (3 µm (X) ´ 27 µm (Y), at 𝑋 of 26.5 µm, 50.5 µm, and 62.5 µm), respectively. During the ANE measurements, the second harmonic signal is detected, since the current-induced Joule heating is proportional to 𝐼!.  \n FIG. 3. The ANE voltages (𝑉\"#$)\tas a function of 𝐻% when injecting a current of 1.00 mA (green lines, the estimated temperature gradients: 0.09 K/µm, 0.07 K/µm and 0.05 K/µm from the left to the right column), 2.00 mA (blue lines, temperature gradients: 0.21 K/µm, 0.15 K/µm and 0.12 K/µm) and 3.00 mA (red lines, temperature gradients: 0.48 K/µm, 0.34 K/µm and 0.28 K/µm) into the heater, measured from three different channels (three columns), in Co0.84Gd0.16 (upper row) and Co0.74Gd0.26 (lower row), respectively. (g): The optical image of the on-chip ANE device.   \n 6 The 𝑉\"#$ vs. 𝐻% loops are shown in Figs. 3(a)-(c) for Co0.84Gd0.16 and Figs. 3(d)-(f) for Co0.74Gd0.26, respectively (See Supplementary Fig. S1 for the sign convention). More raw data are presented in Fig. S2 through injecting various currents into the heater. Similar to AHE, slopes of the ANE loops in the saturation regime only exhibit minor variations, as expected from the negligible contribution of the ONE signal. Its contribution is thus neglected in the following discussion. The observed ANE loops in the Co0.84Gd0.16 and Co0.74Gd0.26 films exhibit an opposite sign, which is analogous to the AHE behavior. Furthermore, the coercive fields 𝐻& are nearly the same from the AHE and ANE measurements. These data demonstrate that ANE acts similarly to AHE, indicating that the ANE response could also be dominated by the magnetization of the TM element (Co) in RE-TM (CoxGd1-x) compensated FIMs.  Phenomenologically, the analogy between the AHE and ANE behaviors could be explained as follows: the dominant contribution to thermal transport is from the conduction electrons in the present metallic system2-5,22, rather than bosonic phonons or magnons, as indicated by the Wiedemann-Franz Law58,59, and the elcetrons driven by the tempareture gradient directly leads to ANE. This is similar to the electrical transport phenomena, such as AHE52. Furthermore, several early works have already suggested that these two effects are intrinsically connected, as implied by the Mott relation2-5,22. Thus, due to similar reasons elucidating AHE in RE-TM materials, the TM-dominant property of ANE could be qualitatively understood.  \n FIG. 4. The obtained ANE voltages 𝑉\"#$ as a function of injected current 𝐼! and the simulated temperature gradient ∇𝑇, at different channel positions close to the heater (red lines), intermediate (blue lines), and far from the heater (green lines), for Co0.84Gd0.16: (a)/(b) and Co0.74Gd0.76: (c)/(d). Linear fitting lines (gray lines) and corresponding functions are also shown in (b) and (d). \n 7 As suggested by Eq. (1), the linear dependence of ANE on ∇𝑇 is testified by the evolution of 𝑉\"#$ as a function of 𝐼!, and the extracted ∇𝑇 from the simulations, as shown in Figs. 4(a)-4(b) for Co0.84Gd0.16 and Figs. 4(c)-4(d) for Co0.74Gd0.26, respectively. Furthermore, by linearly fitting the ∇𝑇\tdependence of 𝑉\"#$, as shown in Figs. 4(b) and 4(d), the thermopower of ANE can be estimated as 𝑆'(≈0.15𝜇𝑉/𝐾 for Co0.84Gd0.16 and 𝑆'(≈−0.13𝜇𝑉/𝐾 for Co0.74Gd0.26, respectively.  TABLE I. The estimated tangent of the anomalous Hall angle (𝑡𝑎𝑛𝜃)*+), thermopower of the SE/ANE (𝑆((/𝑆'(), the first/second term of the linear response theory (𝑆,/𝑆,,) and the transverse thermoelectric conductivity (𝛼'() of Co0.84Gd0.16 and Co0.74Gd0.26. Sample 𝑡𝑎𝑛𝜃!\"# 𝑆$$a (µV K-1) 𝑆%$ (µV K-1)  𝑆& (µV K-1) 𝑆&& (µV K-1) 𝛼%$ (A m-1 K-1) Co0.84Gd0.16 -0.003 -8 0.15 0.126 0.024 0.071 Co0.74Gd0.26 0.002 -4 -0.13 -0.122 -0.008 -0.044 a Data are collected from Ref. 17. To study the origin and quantitatively demonstrate the Co-dominant property of ANE in CoxGd1-x films, we utilize the linear response theory60-62: 𝑆'(=𝜌((𝛼'(+𝜌'(𝛼((, (3) where 𝜌((/𝜌'( and 𝛼((/𝛼'( are the longitudinal/transverse electrical resistivity and longitudinal/transverse thermoelectric conductivity, respectively. We denote the first and the second term on the right-hand side of Eq. (3) as 𝑆,=𝜌((𝛼'( and 𝑆,,=𝜌'(𝛼((. Note that 𝑆,, can be converted into 𝑆,,=𝑡𝑎𝑛𝜃)*+\t𝑆((, where 𝜃)*+=𝑎𝑟𝑐𝑡𝑎𝑛\t(𝜌'(𝜌((⁄) is the anomalous Hall angle, and 𝑆(( is the thermopower of the Seebeck effect (SE). Here, 𝑆, represents the contribution from the intrinsic ANE to the total ANE signal, while 𝑆,, appears as a consequence of the product of the SE and the AHE.      These parameters are summarized in Table I. Since |𝑆,|\t≫|𝑆,,|, 𝑆, (or 𝛼'() can be regarded as the dominant contribution to the 𝑆'( (ANE). As 𝛼'( stems primarily from the Berry curvature of the electron bands near the Fermi level11,62, the origin of ANE could thus be interpreted. Furthermore, the sign of 𝛼'( is opposite in the two samples, which plays the dominant role in the different ANE polarities. As the electrons near the Fermi level are dominated by the 3d band of Co sublattices56 and the spin orientations of which are opposite in the Co/Gd-dominant samples, this could lead to distinctively different contribution from Berry curvatures, the sign change of 𝛼'( and consequently the sign change of ANE could occur. This thus justifies the TM-dominant contribution of ANE in RE-TM FIMs.  8     In conclusion, we have synthesized Co-dominant (Co0.84Gd0.16) and Gd-dominant (Co0.74Gd0.26) compensated ferrimagnetic films with perpendicular magnetic anisotropy, in which the magnetic properties and transport properties are systematically examined. Appreciable ANE signals are demonstrated in compensated ferrimagnets. Furthermore, it is found that the polarities of ANE are opposite in the two samples, which is analogous to the AHE behavior. These observations demonstrate that the ANE responses could also be intrinsically connected with the magnetism of the transition metal element, rather than the net magnetization in the RE-TM compensated ferrimagnets. The origin and the transition metal-dominant property of ANE could be understood by invoking the Berry curvature of the TM component. More importantly, it seems quite promising to make transverse thermoelectric devices, thermal piles for example, based on the opposite signs of ANE in the compensated ferrimagnets, which could output large thermoelectric voltages while exhibiting moderate stray fields.  See the Supplementary Materials for the composition characterizations of the CoxGd1-x films, sign conventions used in AHE and ANE measurements, raw data of ANE loops, X-ray diffraction measurements for evidence of the amorphous CoxGd1-x films, and the schematic device structure and parameters used in the COMSOL simulations.       This work was supported by the general program of NSFC (Grant Nos. 52271181, 51831005), the National Natural Science Foundation of China (NSFC) under the distinguished Young Scholar program (Grant No. 12225409), the Beijing Natural Science Foundation (Grant No. Z190009), the Tsinghua University Initiative Scientific Research Program and the Beijing Advanced Innovation Center for Future Chip (ICFC).  Data Availability The data that support the findings of this study are available from the corresponding author upon reasonable request.  References: 1 K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 2 D. Xiao, Y . Yao, Z. Fang, and Q. Niu, Phys. Rev. Lett. 97, 026603 (2006). 3 T. Miyasato, N. Abe, T. Fujii, A. Asamitsu, S. Onoda, Y . Onose, N. Nagaosa, and Y . Tokura, Phys. Rev. Lett. 99, 086602 (2007). 4 Y . Pu, D. Chiba, F. Matsukura, H. Ohno, and J. Shi, Phys. Rev. Lett. 101, 117208 (2008). 5 S. Onoda, N. 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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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Alff1,∗\n1Darmstadt University of Technology, Petersenstr. 23, 6428 7 Darmstadt, Germany\n2Max-Planck-Institute for Solid State Research, Heisenber gstr. 1, 70569 Stuttgart, Germany\n3Hahn-Meitner-Institut (HMI), 14109 Berlin, Germany\n4Department of Materials Science and Engineering,\nRoyal Institute of Technology (KTH), 10044 Stockholm, Swed en\n5Department of Physics, University of Uppsala Box 530, 75121 Uppsala, Sweden\n6European Synchrotron Radiation Facility (ESRF),\n6 Rue Jules Horowitz, BP 220, 38043 Grenoble Cedex 9, France\n(Dated: Received 13 December 2006)\nIn the search for new spintronic materials with high spin-po larization at room-temperature, we\nhave synthesized an osmium based double perovskite with a Cu rie-temperature of 725K. Our com-\nbined experimental results confirm the existence of a sizabl e induced magnetic moment at the Os\nsite, supported by band-structure calculations in agreeme nt with a proposed kinetic energy driven\nmechanism of ferrimagnetism in these compounds. The intrig uing property of Sr 2CrOsO 6is that\nit is at the endpoint of a metal-insulator transition due to 5 dband filling, and at the same time\nferrimagnetism and high-spin polarization is preserved.\nPACS numbers: 61.12.Ld 75.50.Gg 75.50.Pp 75.50.Vv 81.05.Z x\nA so-called half-metal is a highly desired material for\nspintronics, as only charge carriers having one of the\ntwo possible polarization states contribute to conduc-\ntion. In the class of the ferrimagnetic double perovskites\nsuch half-metals are well known, e. g. Sr 2FeMoO 6[1].\nThe here described compound Sr 2CrOsO 6is special, as\nit has a completely filled 5 d t2gminority spin orbital,\nwhile the majority spin channel is still gapped. It is\nthus at the endpoint of an ideally fully spin-polarized\nmetal-insulator transition. At the metallic side of this\ntransition we have the half-metallic materials Sr 2CrWO 6\n[2] and Sr 2CrReO 6[3, 4]. Within the unique materials\nclass of double perovskites, therefore, one can find high\nCurie-temperature ferrimagnets with spin-polarized con-\nductivity ranging over several orders of magnitude from\nferrimagnetic metallic to ferrimagnetic insulating tunable\nby electron doping. Note that Sr 2CrOsO 6, where a reg-\nular spin polarized 5 dband is shifted below the Fermi\nlevel, is fundamentally different from a diluted magnetic\nsemiconductor, where spin-polarized charge carriers de-\nrive from impurity states.\nWhile for simple perovskites as the half-metallic fer-\nromagnetic manganites the Curie-temperature, TC, is in\nthe highest case still close to room-temperature, half-\nmetallic ferrimagnetic double perovskitescan have a con-\nsiderably higher TC[5]. It has been suggested that fer-\nrimagnetism in the double perovskites is kinetic energy\ndriven [6, 7, 8]. In short, due to the hybridization of\nthe exchange split 3 d-orbitals of Fe3+(3d5, spin major-\nity orbitals fully occupied) or Cr3+(3d3, onlyt2gare\nfully occupied), and the non-magnetic 4 d/5d-orbitals of\n∗Electronic address: alff@oxide.tu-darmstadt.deMo, W, Re or Os ( N-sites), a kinetic energy gain is only\npossible for the minority spin carriers. This will lead\nto a corresponding shift of the bare energy levels at the\nnon-magnetic site, and a strong tendency to half-metallic\nbehavior. This mechanism is operative for the Fe3+and\nCr3+(Msites) compounds [2], where all 3 dmajority\nspin states resp. all t2gmajority spin sates are fully oc-\ncupied and represent localized spins. In agreement with\nband-structure calculations [1, 2, 6, 9, 10, 11] this mech-\nanism is naturallyassociated with half-metallic behavior,\nas the spin-polarized conduction electrons mediate anti-\nferromagnetic order between MandNions, and thus,\nferromagnetic order between the Msites. In addition,\nthis mechanism will lead to an inducedmagnetic mo-\nment at the non-magnetic sites as Mo, W, Re, or Os,\nin convincing quantitative agreement with recent band-\nstructure calculations. It has been suggested and verified\nby observation of the corresponding x-ray magnetic cir-\ncular dichroism (XMCD) that the induced magnetic mo-\nment at the non-magnetic site scales with TCof the com-\npound [12, 13, 14], or – in other words – the band-width\nin the conducting spin minority channel scales with the\nmagnetic transition temperature. This behavior has also\nbeen confirmed qualitatively by nuclear magnetic reso-\nnance (NMR) experiments for FeRe-based double per-\novskites [15]. Quantitatively, however, Re shows an un-\nusually enhanced induced magnetic moment in the dou-\nble perovskite structure, as compared to Mo, W, and Os.\nAn obvious way to change the band-filling is by dop-\ning at the alkaline earth site. Substituting for example\nSr2+by La3+results in electron doping. For the fer-\nrimagnetic double perovskites it has been shown that\nincreasing the band-width by La doping generically in-\ncreasesTC[16, 17, 19]. The increase is substantial,\ni.e. more than 100K per one doped electron per unit2\n/s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48 /s49/s48/s53 /s49/s50/s48 /s49/s51/s53 /s49/s53/s48/s48/s46/s48/s50/s46/s48/s120/s49/s48/s51/s52/s46/s48/s120/s49/s48/s51/s54/s46/s48/s120/s49/s48/s51/s56/s46/s48/s120/s49/s48/s51\n/s49/s48/s52 /s49/s48/s53 /s49/s48/s54 /s49/s48/s55 /s49/s48/s56/s49/s48/s48/s48/s50/s48/s48/s48/s51/s48/s48/s48/s52/s48/s48/s48/s40/s52/s32/s48/s32/s52/s32/s47/s32/s48/s32/s48/s32/s49/s50/s41/s32/s104/s101/s120\n/s40/s52/s32/s52/s32/s52/s41/s32/s99/s117/s98\n/s50/s57/s56/s32/s75\n/s50/s48/s48/s32/s75\n/s49/s48/s48/s32/s75\n/s50/s32/s75/s105/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\n/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s99/s97/s108/s99/s117/s108/s97/s116/s101/s100\n/s32/s100/s105/s102/s102/s101/s114/s101/s110/s99/s101/s105/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/s84/s32 /s61/s32/s53/s53/s51/s32/s75/s83/s114\n/s50/s67/s114/s79/s115/s79\n/s54\nFigure 1: (Color online) Rietveld refinements of the neutron\npowder data of Sr 2CrOsO 6taken at 553K. The observed and\ncalculated diffraction intensities as well as the difference pat-\ntern are shown. The upper left inset shows a view along the\n[111]-direction revealing the rhombohedral structure of t he\ndouble perovskite. The upper right inset shows the temper-\nature dependence of the Sr 2CrOsO 6(444) diffraction peak\n(cubic) splitting into a (404) and (0 0 12) peak (hexagonal).\ncell. However, there is an alternative way of doping\nby using the Nsite itself [20]. Within this line of\nthinking there is a natural explanation why Sr 2CrReO 6\nwas up to now the ferrimagnetic double perovskite with\nthe highest Curie-temperature, 635K [3]. Compared to\nSr2CrWO 6(TC≈450−500K[2, 18, 19])withW5+(5d1),\nSr2CrReO 6with Re5+(5d2) possesses just one electron\nmore in the spin minority channel. Therefore, it cor-\nresponds to the compound LaSrCrWO 6. The increase\nofTCby more than 100K in both cases (as estimated\nfrom the caseofLaSrFeMoO 6[16] and LaCaCrWO 6[19])\nunderlines the justification of this comparison. Follow-\ning this argument, Sr 2CrOsO 6with Os5+(5d3) becomes\na natural candidate for an ordered ferrimagnetic dou-\nble perovskite with even higher TC. After an early at-\ntempt to synthesize this compound, it has been (erro-\nneously) concluded that it is notferromagnetic at room-\ntemperature [21]. The CrTa, CrW, CrRe, CrOs series\nis an ideal system to study the effect of 5 dband filling.\nIn order to investigate these ideas we have synthesized\nphase pure polycrystalline Sr 2CrOsO 6samples and mea-\nsured the net magnetization by SQUID magnetometry.\nThe (magnetic) crystal structure has been extensively\nstudied by neutron scattering and XMCD.\nSingle phase polycrystalline samples of Sr 2CrOsO 6\nhavebeenpreparedfromstoichiometricamountsofSrO 2,\nCrO2(both Sigma-Aldrich, 99.999%), and Os metal\n(Chempur 99.99%) at 1273K in evacuated quartz tubes.\nThe mixture was grinded thoroughly, pressed into pel-\nlets and put into a corundum crucible. The crucible was\nplaced in a quartz tube which was further evacuated and\nsealed under high vacuum. The quartz tube was heated\nat 1273K for 2 days. The obtained product was further\nheated under oxygen flow at 673K overnight to remove\nexcess Os metal as an impurity. Extensive x-ray and/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s48/s46/s48/s49/s48/s45/s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s109/s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s66/s32/s61/s32/s49/s48/s48/s48/s32/s79/s101\n/s83/s114\n/s50/s67/s114/s79/s115/s79\n/s54/s32\n/s32/s32 /s51/s48/s48/s32/s75\n/s32/s32 /s53/s32/s75\n/s32/s32/s109 /s32/s40/s101/s109 /s117/s41\n/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\nFigure 2: (Color online) Magnetization vs temperature and\nhysteresis (at 5K and 300K) of Sr 2CrOsO 6(measured in\na superconducting quantum interference device (SQUID)).\nThe sample has been cooled and field-oriented in diamagnetic\nLiNO 3(melting point 480K) because of the high anisotropy\nfield.\nneutron diffraction analysis confirms the high phase pu-\nrity of the samples (see Fig. 1). The antisite disorder is\nnegligible below 0.1%.\nIn Fig. 1 neutron powder diffraction data collected\nwiththe instrumentE9(Ge monochromatorselectingthe\nneutron wavelength λ= 1.7971˚A) at the reactor BER II\nof the Hahn-Meitner-Institut in Berlin is shown. The de-\ntailed refinement results are presented in Table II. From\ndata refinement one can say that the crystal structure is\nclearly cubic above the magnetic transition temperature.\nAt 550K the crystal structure can still be refined by as-\nsuming a cubic structure. At 300K the sample is already\nrhombohedral. The upper right inset of Fig. 1 shows the\ntemperatureevolutionofthe (444)diffractionpeak in the\ncubic notation. At low temperatures the splitting into\ntwo peaks ((404) and (0 0 12) in the hexagonal notation)\nwith an observed intensity ratio of about 3 clearly evi-\ndences the rhombohedral structure. The tolerance factor\nat low temperatures is still close to one ( ∼0.992).\nFigure 2 shows the net magnetization and hysteresis\nTableI: Measuredandcalculated (usingthegeneralized gra di-\nent approximation including spin-orbit coupling (GGA+SO) )\nmagnetic moments of Sr 2CrOsO 6inµB/f.u. For a detailed\ndiscussion of the applied band-structure calculation see e .g.\nRef. [10, 11]. Note that the neutron measurements are total\nmoments per site. Calculated number of d-holes at Os: 4 .83\nmethod CrmSCrmLOsmS OsmL\nXMCD (300K) -0.17 0.015\nneutrons (298K) 1.92 -0.12\nneutrons (2K) 2.03 -0.73\nGGA+SO (0K) 2.216 -0.024 -1.214 0.1223\n/s49/s48/s56/s53/s48 /s49/s48/s57/s48/s48 /s49/s50/s51/s53/s48 /s49/s50/s52/s48/s48 /s49/s50/s52/s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s49/s48/s56/s53/s48 /s49/s48/s57/s48/s48 /s49/s50/s51/s53/s48 /s49/s50/s52/s48/s48 /s49/s50/s52/s53/s48\n/s45/s48/s46/s48/s57/s45/s48/s46/s48/s56/s45/s48/s46/s48/s55/s45/s48/s46/s48/s54/s45/s48/s46/s48/s53/s45/s48/s46/s48/s52/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s88/s65/s78/s69/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s112/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s88/s65/s78/s69/s83/s76\n/s50/s76\n/s51\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48/s46/s57/s57/s54/s48/s46/s57/s57/s56/s49/s46/s48/s48/s48\n/s40/s79/s115/s41/s40/s67/s114/s41\n/s116/s111/s116/s109/s111/s109/s101/s110/s116/s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s100/s105/s115/s116/s111/s114/s116/s105/s111/s110/s88/s77/s67/s68/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s88/s77/s67/s68\n/s32\nFigure 3: (Color online) XANES (left axis) and XMCD signal\n(right axis) at the Os L 2and L 3edges in Sr 2CrOsO 6. The\ninset shows the element resolved magnetic moments of Cr\n(µCr), Os (µOs), the total magnetic moment ( µtot) and the\nlattice distortion as determined from neutron scattering d ata\nrefinement (see Table II).\nbehavior of Sr 2CrOsO 6as measured in a superconduct-\ning quantum interference device (SQUID) magnetome-\nter.TCis estimated to be about 725K. The magnetiza-\ntion maximum at 550K is a consequence of the different\ntemperature evolution of the Cr magnetic moments and\nthe induced moments at the Os site (see inset of Fig. 3).\nThe drop of local magnetic spin moment at the Os site\nwithincreasingtemperatureisclearlynotassociatedwith\na change of the insulating sample behavior. It is most\nlikely, that the unusual magnetic behavior is coupled to\nthe distortion of the lattice, as can be motivated from\nTable II. In the same temperature range where the Os\nmoment changes, the Os and Cr oxygen octahedra are\nmarkedly twisted against each other (see topview of the\ncrystal structure in the inset of Fig. 1, and the combined\nplot of lattice distortion and magnetic moments vs. tem-\nperature in the inset of Fig. 2). Alternatively, one could\ninvokeadifficult tounderstandverystrongferromagnetic\ncoupling between the Cr ions along the chain Cr-O-Os-\nO-Cr. However, the finite induced magnetic moment at\nthe Os site, although reduced to -0.05 µBat 540K (for\ncomparison: in Sr 2FeReO 6the total Re moment is only\n-0.16µBat 5K [4]), survives up to TCindicating the va-\nlidity of the kinetic energy gain model.\nThe XMCD measurements on the Os L 2,3edges were\nperformed at the European Synchrotron Radiation Facil-\nity (ESRF) at beam line ID12 [22]. As shown in Fig. 3\na very clear XMCD signal is observed, signaling a stronglocal magnetic moment at the Os-site opposite to the net\nmagnetization. The experimentally determined value is\nin good agreementwith the neutronscatteringresult (see\ntable I).\nFig. 4 (right side) shows the site-resolved den-\nsity of states for the d-states obtained from band-\nstructure calculations, in the experimentally observed\nlow-temperature rhombohedral structure, using the gen-\neralized gradient approximation (GGA) including spin-\norbit coupling. The key result is that Sr 2CrOsO 6is, due\nto the completey filled Os t2gspin down orbitals, indeed\nat the end of spin-polarized metal-insulator transition\n(see sketch in Fig. 4, left side). This is also in agreement\nwith conductivity measurements on the powder samples,\nshowing about 10Ωcm at room-temperature. Our com-\nbined experimental and theoretical results show, that the\nkinetic-energy driven exchange mechanism [6, 7, 8] is not\nonly operable in the itinerant range but in a wide range\nof conductivities.\nIn summary, we have unambiguously demonstrated,\nthat an increase of the number of 5 delectrons leads to an\nincrease of the transition temperature in ferrimagnetic\ndouble perovskites irrespective of the changes in con-\nductivity. Fully spin-polarized conductivity as indicated\nby the large induced magnetic moments and supported\nfurther by band-structure calculations, can be changed\nfrom half-metallic behavior (known in Sr 2CrWO 6and\nSr2CrReO 6) to insulating behavior in the high-Curie\ntemperature ferrimagnetic insulator Sr 2CrOsO 6.\nThe authors acknowledge the support by the DFG\n(Project Ns. Al 560/4and Ul 146/4), SSF and VR. SNIC\nis acknowledged for providing computer facilities.\n-3 -2 -1 0 1\nenergy (eV)-4-2024DOS (states/eV/f.u.)\n2 3E\nCrM\nCrM Os (FM-I)W, Re (FM-M )t    -5d 2g\nTa (AFM-I )\nCr 3 d\nOs 5 d\nFigure 4: (Color online) Left: Band structure sketch for dif -\nferent fillings of the 5 dspin-down band (as realized in the\nmaterials Sr 2CrMO6withM= Ta, W, Re, Os). Right:\nBand-structure calculation for Sr 2CrOsO 6using the gener-\nalized gradient approximation including spin-orbit coupl ing\n(GGA+SO).\n[1] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura,\nand Y. Tokura, Nature (London) 395, 677 (1998).\n[2] J. B. Philipp, P. Majewski, L. Alff, A. Erb, R. Gross,T. Graf, M. S. Brandt, J. Simon, T. Walther, W. Mader,\nD. Topwal, and D. D. Sarma, Phys. Rev. B. 68, 144431\n(2003).4\n[3] H. Kato, T. Okuda, Y. Okimoto, Y. Tomioka, Y.\nTakenoya, A. Ohkubo, M. Kawasaki, and Y. Tokura,\nAppl. Phys. Lett. 81, 328 (2002).\n[4] J. M. De Teresa, D. Serrate, C. Ritter, J. Blasco, M. R.\nIbarra, L. Morellon, and W. Tokarz, Phys. Rev. B 71,\n092408 (2005).\n[5] For a recent review see: D. Serrate, J. M. De Teresa,\nand M. R. Ibarra, J. Phys.: Condens. Matter 19, 023201\n(2007).\n[6] D. D. Sarma, P. Mahadevan, T. Saha-Dasgupta, S. Ray,\nand A. Kumar, Phys. Rev. Lett. 85, 2549 (2000); see also\nCurr. Opinion in Solid State Mat. Sci. 5, 261 (2001).\n[7] J. Kanamori, and K. Terakura, J. Phys. Soc. Jpn. 70,\n1433 (2001).\n[8] Z. Fang, K. Terakura, andJ. Kanamori, Phys.Rev. B 63,\n180407(R) (2001).\n[9] Horng-Tay Jeng and G. Y. Guo, Phys. Rev. B 67, 094438\n(2003).\n[10] G. Vaitheeswaran, V. Kanchana, and A. Delin,\nAppl. Phys. Lett. 86, 032513 (2005).\n[11] G. Vaitheeswaran, V. Kanchana, and A. Delin, J. Phys.:\nConf. Ser. 29, 50 (2006).\n[12] P. Majewski, S. Gepr¨ ags, O. Sanganas, M. Opel,\nR. Gross, F. Wilhelm, A. Rogalev, and L. Alff,\nAppl. Phys. Lett. 87, 202503 (2005).\n[13] P. Majewski, S. Gepr¨ ags, A. Boger, M. Opel, A. Erb, R.\nGross, G. Vaitheeswaran, V. Kanchana, A. Delin, F. Wil-\nhelm, A. Rogalev, and L. Alff, Phys. Rev. B 72, 132402\n(2005).[14] M. Sikora, Cz. Kapusta, M. Borowiec, C. J Oates, V.\nProchazka, D. Rybicki, D. Zajac, J. M. De Teresa, C.\nMarquina, andM. RIbarra, Appl.Phys. Lett. 89, 062509\n(2006).\n[15] M.Wojcik, E.Jedryka, S.Nadolski, D.Rubi, C.Frontera ,\nJ. Fontcuberta, B. Jurca, N. Dragoe, and P. Berthet,\nPhys. Rev. B 71, 104410 (2005).\n[16] J. Navarro, C. Frontera, Ll. Balcells, B. Mart´ ınez, an d J.\nFontcuberta, Phys. Rev. B 64, 092411 (2001).\n[17] C. Frontera, D. Rubi, J. Navarro, J. L. Garc´ ıa-Mu˜ noz,\nJ. Fontcuberta, and C. Ritter, Phys. Rev. B 68, 012412\n(2003).\n[18] J. B. Philipp, D. Reisinger, M. Schonecke,\nA. Marx, A. Erb, L. Alff, R. Gross, and J. Klein,\nAppl. Phys. Lett. 79, 3654 (2002).\n[19] S. Gepr¨ ags, P. Majewski, R. Gross, C. Ritter, and L. Alff ,\nJ. Appl. Phys. 99, 08J102 (2006).\n[20] L. Alff, in Electron Correlation in New Materials and\nNanosystems , NATO Science Series II 241, K. Scharn-\nberg and S. Kruchinin (Eds.), Springer 2007.\n[21] A. W. Sleight, J. Longo, and R. Ward, Inorg. Chem. 1,\n245 (1962).\n[22] A. Rogalev, J. Goulon, Ch. Goulon-Ginet, and C. Mal-\ngrange, in Magnetism and Synchrotron Radiation , E.\nBeaurepaire et al.(Eds.), LNP vol. 565(Springer, 2001).\n[23] Y. Krockenberger, M. Reehuis, M. Tovar, K. Mogare, M.\nJansen, and L. Alff, J. Magn. Magn. Mater. 310, 1854\n(2007).5\nTable II: Results of the structure refinements of Sr 2CrOsO 6. For the neutron powder data collected at 2K, 100K, 200K\nand 298K the crystal structure was refined in the trigonal spa ce group R3. At the higher temperatures 540K and 770K the\nrefinements were carried out in the cubic space group Fm3m. The residuals of the crystal and magnetic structures are de fined\nasRN=P||Fo| −|Fc||/|Fo|andRMasRM=P||Io| − |Ic||/|Io|, respectively. For the refinements several parameters were\nconstrained to be equal. In these cases the standard deviati on is listed only for one of the equal parameters. As distorti on we\ndefinec/(√\n6a), i.e. the ratio c/anormalized to the value of a cubic structure in hexagonal not ation (√\n6). See also [23].\nT[K] 2 100 200 298 540 770\nspace group R3 R3 R3 R3 Fm3m Fm3m\na[˚A] 5.5176(3) 5.5170(4) 5.5178(3) 5.5181(6) 7.8243(2) 7.8455(3)\nc[˚A] 13.445(1) 13.454(1) 13.470(1) 13.500(1) – –\nV[˚A3] 354.47(4) 354.64(4) 355.16(4) 355.98(8) 479.00(4) 482.90(5)\nc/a 2.4367(4) 2.4386(3) 2.4412(4) 2.4464(8) – –\ndistortion 0.9948 0.9956 0.9966 0.9988 1 1\nxO,yO,zO, site0.335(1), 0.335(2), 0.336(2), 0.335(4), 0.2511(5), 0, 0, 0.2511(5), 0, 0,\n0.1908(7), 0.1900(7), 0.1876(8), 0.183(1), 24e 24e\n0.4169(6), 18 f0.4171(7), 18 f0.4171(8), 18 f0.417(2), 18 f\nxSr,ySr,zSr, site0, 0 , 0.251(1), 6 c0, 0 , 0.251(2), 6 c0, 0 , 0.251(2), 6 c0, 0 , 0.252(4), 6 c1\n4,1\n4,1\n4, 8c1\n4,1\n4,1\n4, 8c\nxOs,yOs,zOs, site0, 0 ,1\n2, 3b 0, 0 ,1\n2, 3b 0, 0 ,1\n2, 3b 0, 0 ,1\n2, 3b 0, 0, 0, 4 a0, 0, 0, 4 a\nxCr,yCr,zCr, site0, 0 , 0, 3 a 0, 0 , 0, 3 a 0, 0 , 0, 3 a 0, 0 , 0, 3 a1\n2,1\n2,1\n2, 4b1\n2,1\n2,1\n2, 4b\nB(Sr)[˚A2] 0.36(4) 0.40(4) 0.48(4) 0.63(4) 1.09(4) 1.47(5)\nB(Cr,Os)[ ˚A2]0.21(2) 0.22(2) 0.25(2) 0.30(3) 0.42(3) 0.51(4)\nB(O)[˚A2] 0.45(3) 0.47(3) 0.50(3) 0.67(3) 1.41(3) 1.72(4)\nd(Cr-O)[˚A] 1.947(7) 1.948(8) 1.946(9) 1.95(2) 1.948(4) 1.953(4)\nd(Os-O)[˚A] 1.957(7) 1.955(8) 1.957(9) 1.96(2) 1.965(4) 1.970(4)\nRN 0.039 0.040 0.040 0.037 0.034 0.039\nµCr[µB] 2.0(3) 1.9(3) 2.0(3) 1.9(2) 1.8(2) 0\nµOs[µB] -0.7(3) -0.6(3) -0.3(2) -0.1(2) -0.05 0\nRM 0.070 0.082 0.065 0.078 0.103 –"    },    {        "title": "1506.07028v1.Design_of_compensated_ferrimagnetic_Heusler_alloys_for_giant_tunable_exchange_bias.pdf",        "content": "arXiv:1506.07028v1  [cond-mat.str-el]  23 Jun 2015Design of compensated ferrimagnetic Heusler alloys for\ngianttunableexchangebias\nAjayaK.Nayak1∗,MichaelNicklas1,StanislavChadov1,PanchananaKhuntia1,ChandraShekhar1,\nAdel Kalache,1, Michael Baenitz1, YuriiSkourski2, Veerendra K. Guduru3, Alessandro Puri3, Uli\nZeitler3, J.M. D. Coey4&ClaudiaFelser1∗\n1Max Planck Institutefor Chemical Physicsof Solids, N¨ othn itzerStr. 40, D-01187 Dresden, Ger-\nmany\n2Dresden High Magnetic Field Laboratory (HLD), Helmholtz-Z entrum Dresden-Rossendorf, D-\n01328Dresden, Germany\n3High Field Magnet Laboratory, Institute for Molecules and M aterials, Radboud University Ni-\njmegen,Toernooiveld7, NL-6525ED Nijmegen,TheNetherlan ds\n4School ofPhysicsand CRANN, TrinityCollege, Dublin2, Irel and\nThe discovery ofmaterials withimprovedfunctionality can beaccelerated by rational mate-\nrial design.1Heusler compounds with tunable magnetic sublattices allow to implement this\nconcepttoachievenovelmagneticproperties.2Here,wehavedesignedafamilyofHeusleral-\nloyswithacompensatedferrimagneticstate. Inthevicinit yofthecompensationcomposition\nin Mn-Pt-Ga, a giant exchange bias (EB) of more than 3 T and a si milarly large coerciv-\nity are established. The large exchange anisotropy origina tes from the exchange interaction\nbetween the compensated host and ferrimagnetic clusters th at arise from intrinsic anti-site\ndisorder. We demonstrate the applicability of our design co ncept on a second material, Mn-\n1Fe-Ga, witha magnetic transition aboveroom temperature, e xemplifying the universality of\nthe concept and the feasibility of room-temperature applic ations. Our study points to a new\ndirection for novel magneto-electronic devices. At the sam e time it suggests a new route for\nrealizing rare-earth free exchange-biased hard magnets, w here the second quadrant magne-\ntizationcanbe stabilizedby the exchangebias.\nExchange bias corresponds to a shift of the hysteresisloop o f a ferromagnet along the mag-\nnetic field axis due to interfacial exchange coupling with an adjacent antiferromagnet.3,4It is em-\nployed for a variety of technological applications3–10, including magnetoresistive read heads and\nsensors. Oneofthemostimportantusesofexchangebiasis in spin-valvedevices,where an artifi-\ncialantiferromagnetexchange-biasedbyanormalantiferr omagnetallowsmagnetoresistancetobe\ndeveloped in a small field, stabilizing the magnetic state ag ainst repeated cycling.9Although the\noriginofexchangebiasisstillasubjectofdiscussion,its phenomenologyhasvariouslybeeninter-\npreted in terms of rough ferromagnetic (FM)- antiferromagn etic (AFM) interfaces11,12, a domain\nstatemodel13anduncompensatedinterfacialspins14,15. AlthoughbothFM andAFM subsystems\nare inseparable parts of an EB system, it is the latter that de termines the magnitude of the EB in\nthe system.15–17Therefore, it is important to search for an appropriate magn etically compensated\nmaterial to observe a maximum effect. Besides the EB phenome non, magnetically compensated\nmaterials are proposed to play a vital role in antiferromagn etic spintronics18,19and all-optical\nswitchingdevices20. Singlespinsuperconductivityhasevenbeenpredictedina spin-compensated\nhalf-metal.21Inthisletterwepresentadesignschemeforamagnetically- compensatedferrimagnet\nand demonstrate the relevance of this compensated magnetic state in realizing a giant exchange\n2biasand alarge coercivity.\nHeuslercompounds,whichprovidethematerialbaseforourd esignscheme,arewellknown\nfor their multi-functional properties, such as the giant ma gnetocaloric effect22, the field-induced\nshape-memory effect23and the topological insulating property24. The story of their success is\nbased onthefact thatnew materialscan bedesignedintheflex iblestructureoftheHeuslerfamily\non the basis of simple rules taking into account the position of the atoms, the number of valence\nelectrons,thedegreeofatomicdisorderandthestrengthof theexchangeinteractions2. Oneofthe\nfamous examples of valence electron count is the robustness of the Slater-Pauling rule, typically\nconnectedtohalf-metallicity,intheL2 1cubicHeuslercompounds.25Togetherwiththefactthatin\nHeusler compounds Mn develops a strong localized magnetic m oment, it can be used as a guide\nto search for compensated ferrimagnets in the Heusler famil y with 24 valence electrons. On the\nother hand, this provides us only a preliminary guide, since the compensated ferrimagnetic state\ntypically cannot be combined with half-metallicity. The cl assical example is Mn 3Ga26, which\nis predicted to exhibit a half-metallic state in the L2 1cubic phase, but the cubic phase does not\nform in the bulk due to a strong electronic instability leadi ng to a tetragonal strain breaking both\nthe compensated and half metallic states. However, either a half-metallic or a compensated state\ncan berecoveredbychemicalsubstitution.27Here, wefocusonthecompensatedmagneticstatein\nHeuslermaterials.\nCombining Mn 3Ga with another material of opposite net magnetization, Mn 2PtGa, forms\nan excellent starting point to design a zero-magnetization structure based on a compensated fer-\n3rimagnetic state. Mn 2PtGa consists of two non-equivalent types of Mn, one in the Mn -Ga and\nanother in the Mn-Pt planes of the inverse tetragonal struct ure (space group I-4m2)28. The Mn\nsitting in Mn-Ga planes possesses a higher magnetic moment d ue to its more localized nature.\nIt couples antiferromagnetically to the Mn in Mn-Pt planes. This configuration results in a net\nuncompensated magnetization of about 0.5 µB/f.u.arising from the larger moment of the Mn in\nMn-Ga planes.28On the other hand, the complete replacement of Pt by Mn to form Mn3Ga(space\ngroupI4/mmm)resultsinoneMnintheMn-GaandtwoMnintheMn-Mnplanes(f ormerMn-Pt\nplane). Thus, Mn 3Ga displaysa net uncompensated magnetizationof about 1 µB/f.u.of opposite\nsign to that in Mn 2PtGa.26,29Combining these two materials then suggests that we can crea te a\nfully compensated magnet with a compensation point for a par ticular Mn/Pt ratio. This design\nscheme is schematically depicted in Fig. 1. From first-princ iples calculations, it follows that the\ncritical composition with the zero magnetization is achiev ed in the solid solution Mn 3−xPtxGa at\na Pt content of about x0= 0.59, which is in good agreement with the experimental findings. O n\noptimizing the Mn/Pt ratio in Mn 3−xPtxGa, we always find a small lack of compensation in the\nmaterial, due to the formation of FM clusters by anti-site di sorder. This leads to an exceptionally\nlarge bulk EB and a large coercivity. In contrast to an artific ial antiferromagnet, which is a thin\nfilmstructurecomposedoftwoferromagneticlayersseparat edbyacouplinglayer9,herewecom-\nbine two isostructural ferrimagnetic compounds Mn 3Ga and Mn 2PtGa to obtain an intrinsically\nanisotropiccompensatedmagneticstateon an atomicscalei na bulkmaterial.\nIn order to characterize the magneticproperties of Mn 3−xPtxGa we have measured the tem-\nperature dependence of the low field magnetization, M(T). We find a systematic increase in the\n4ferrimagnetic N´ eel temperature ( TN) with increasing Mn content as shown in Fig. 2. The irre-\nversibility between the ZFC and FC curves reflects the appear ance of coercivity. We suggest that\nFM clusters embedded in the compensated host are the source o f this irreversibility. NMR mea-\nsurementsconfirmthattheseclustersoriginatefromrandom swapsbetweenPtintheMn-Ptplanes\nand Mn in the Mn-Ga planes (Supplementary Fig. 2). The irreve rsibility between ZFC and FC\nM(T)curves increases with increasing magnetic field demonstrat ing that cooling in higher fields\nhelpstheFM clusterstogrow insize(SupplementaryFig. 3).\nTheZFCmagnetizationat5K showsan unsaturatedbehaviorfo rmagneticfieldsupto14T\n(inset of Fig. 2), therefore, higherfields are required in or der to gain a better understandingof the\nmagnetic state of the system. Pulsed-field magnetization me asurements at 4.2 K up to a field of\n60TareshowninFig. 3a. For x= 0.5,0.6,and0.7thehysteresisloopscloseatafieldof 35−40T.\nThe non-saturating magnetization up to 60 T reflects the stre ngth of the dominant intersublattice\nMn-Mn exchange in the polycrystalline compensated ferrima gnetic host. The hysteresis reveals\nthat there is a FM component of 0.1−0.3µBin the compensated host, which may be compared\nwith its ferromagnetic collinear saturation of about 6µB/f.u.. We further obtain coercive fields of\nµ0HC= 2.2, 3.6, and 3.0 T at 4.2 K for x= 0.5, 0.6 and 0.7, respectively. These values are\ncomparable to the maximum HCobserved in rare-earth-based hard magnetic bulk materials . To\ninvestigatetheexchangeinteractionbetweentheFMcluste rsandAFMhost,wehavemeasuredFC\nhysteresisloops. TheFChysteresisloopstakenat4.2Kinad c-fieldof ±32TaredepictedinFig.\n3b. Theloopsfor x= 0.5,0.6,and0.7displayalargeshiftinthenegativefielddirec tionindicating\nthe existence of a large unidirectional anisotropy and a lar ge exchange bias. The hysteresis loops\n5close in a field of only 20−25T for all samples. Interestingly, the FC hysteresis loops re corded\nafter field cooling in 15 T and 25 T follow a similar path. This i mplies that a cooling field of\nµ0HCF= 15T is sufficientto saturatetheFM momentsand alarger cooling field does notfurther\nchange the magnetic state of the sample (inset of Fig. 3). For all samples the FC M(H)loops\npassalmostthroughtheorigininthethirdquadrant,result ingnearlysamecoercivityandexchange\nbias field. The EB increases monotonically for µ0HCFup to 10 T and then saturates for further\nincreasing coolingfields. It may beworthwhileto mentionth atfield of10 T is nearly sufficientto\nsaturatetheexchangebias,but thefull32 T isrequired tofu lly closetheloop.\nInMn3−xPtxGa,weobtainamaximumEBof HEB= 3.3Tforx= 0.6. Thisvalueisamong\nthe largest EB reported so far. The dependence of the EB and co ercivity on the Pt concentration\nmeasured after field cooling the sample in 15 T is displayed in the inset of Fig. 4a. Both, the EB\nfield (HEB) and the coercive field ( HC) exhibit a clear maximum around x= 0.6. We observe\na small increase in HCfor the FC loops in comparison to the ZFC loops, as generally f ound for\nmaterials showing an EB effect.4,5,30To demonstrate the temperature dependence of the EB we\nhavemeasuredhysteresisloopsatdifferenttemperaturesa fterfieldcoolingin5TforMn 2.4Pt0.6Ga\nand Mn 2.5Pt0.5Ga. The EBshowsa monotonicdecrease with increasing temper aturethatvanishes\naroundTN, asshowninFig. 4a.\nTheextremelylargeEBatlowtemperaturesintheMn-Pt-GaHe usleralloysisagoodstarting\npoint for achieving large EB in compensated materials. Ther efore, to establish a comprehensive\npicture we applied our design concept to another Heusler sys tem in the Mn-Ga family, Mn-Fe-\n6Ga, where the ordering temperature of the system can be varie d over a wide range to above room\ntemperature by tuning the composition (Supplementary Fig. 7-Fig. 14). Furthermore, Mn 2FeGa\nis nearly in a compensated magnetic state, whereas, Fe 2MnGa is ferromagnetic. By adjusting\nthe composition we can achieve a material with ferromagneti c clusters in an almost compensated\nferrimagnetic host and a N´ eel temperature aboveroom tempe rature. The temperature dependence\nofthe EBfor Mn 1.5Fe1.5Ga and Mn 1.8FeGa are shownin Fig. 4b. Mn 1.5Fe1.5Gaexhibitsan EB of\n1.1 T at 2 K, which is observableup to room temperature. ForMn 1.8FeGa we find an EB of 1.2 T\nand 0.04T at 2K and 300K, respectively.\nOurexperimentsshowthatmagneticcompensationofthehost isanimportantrequirementto\nachievealargeexchangebias. Wenowintroduceamodeltoexp laintheseobservations. Assuming\nferromagnetic clusters in the compensated ferrimagnetic h ost that are coherent with the host and\ncoupledtoitviaexchangebondsattheinterface, wecanelab oratetheconditionsfortheexchange\nbias. We assume that the FM clusters and their surroundings a re part of a single crystalline grain\nwith a uniform orientation of the uniaxial magnetocrystall ineanisotropy. By applying an external\nmagnetic field which is large enough to rotate the magnetizat ion in the ferromagnetic clusters,\nbut small enough not to rotate the compensated ferrimagneti c host, we find that the EB is well\ndescribed by\nH≈12JABSASB[1−cos(θK−θB)]·L2n2/3+KBsin2(θK−θB)·L3n\nδSA·cosθK·NA+SBcosθB·L3n. (1)\n7whereSA(B)are the amplitudes of the atomic magnetic moments in the host (cluster), δSA\naccounts for the imperfect magnetic compensation in the hos t;NA(B),ABare the number of atoms\ninside the host (cluster) and at the cluster/host interface , respectively; JABis the exchange cou-\npling constant at the cluster/host interface. KA,B>0are the anisotropy constants for the host\nand the cluster, θK,A,Bare the polar angles referring to the orientation of the anis otropy axis and\nmagnetizationoftheclusterand ofthehost,respectively.\nIn case of small FM clusters ( L3n≪NA) the key role is played by the compensationof the\nhost. TheEBfield can beexpressedas,\nH(M)≈α\nM+β, (2)\nwhereMis the total magnetization; αandβare parameters depending on the interface\nexchange-coupling,theparticularformoftheinterfaces, andthenumber,distributionandmagnetic\nmoments of the embedded ferromagnetic clusters. The theore tical estimate agrees well with the\nexperimentaldatathatshowanearlyinverserelationshipb etweenthe HEBandmagnetization(see\nFig. 1a).\nTo estimatethemaximumexchange-bias field Eq. 1 can be expre ssed in terms ofthe sizeof\ntheferromagneticcluster( L),\n8Hmax[T] = 0.21·(1/L[˚A])·105+0.54, (3)\nwhere the first term appears from the interface exchange coup ling and the second, from the\nmagnetic anisotropy (Supplementary). In order to achieve t he exchange bias fields of 3T, the\ncharacteristicFM clustersizemustnotexceed L∼104˚A.\nA key experimental observation is that at low temperature th e exchange bias is equal to the\ncoercivityin all samples. This can be explainedby consider ing additionalconditionsfor themag-\nnetization switching. For simplicity, we consider only tho se grains which are oriented differently\nfrom the polar angle, θK=π/2. Such grains exhibit a sharp rectangular-like hysteresis, in which\nthe coercivity corresponds to a regime of fast domain-wall m otion. This rapid process is the tran-\nsition between two states: one, comprising the domain wall ( induced by the competition between\nthe strong negative magnetic field and the positive exchange pinning at the host/cluster interface)\nand the new one, which is magnetically uniform. The eliminat ion of the domain wall occurs for\nthesmallexternalfieldvalues( H <0)at whichtheenergies ofthesetwostatesbecomecompara-\nble. Thedetailedtheoreticalinterpretationofourfinding scanbefoundinsupplementarymaterial.\nFrom this model we estimate that H+\nCis only about -1 mT, where the cluster size Lis the lead-\ning contribution. Taking into account that the measured hys teresis is the statistical sum over all\norientations, H+\nCwillhavean evensmalleramplitude.\nTo conclude, we have established a new approach to design com pensated ferrimagnetic\n9Heusler alloys. A small lack of compensation allows us to ach ieve an extremely large EB and\nmatching coercivity. A simple model allows us to quantify th e experimental results that show\nan inverserelationship between the total magnetization an d magnitudeof the exchange-bias field.\nThe present finding provides new insight into the mechanism o f EB and suggests exploring sim-\nilar coherent materials that can be implemented in thin film. Furthermore, the achievement of a\ncoercivity of morethan 3 T based on exchange anisotropy prop oses a new approach to permanent\nmagnet design. Permanent magnets are expected to deliver a l arge flux density. Their working\npoint,stabilizedbycoercivity,fallsnaturallyinthesec ondquadrantofthehysteresisloop,because\nofthedemagnetizingfield. Thepresentworkoffersanaltern ativeapproach-tostabilizethesecond\nquadrant magnetization by exchange bias. However, to reali zable in practical applications it must\nbecoupledwithalargermagnetizationinordertoproduceau sefulfluxdensity. Anon-percolating\nferromagnetic fraction approaching 30%, rather than aroun d 3% in Mn-Pt-Ga and less than 12%\nin Mn-Fe-Ga, is needed. The balance between the volume of FM c lusters and compensated host\nneeds to be optimized to achieve a large exchange bias as well as a large magnetization. Hence,\nthe present finding is a good starting point for designing rar e-earth free hard magnets comprising\nof two hard magnetic phases, which is different form the conv entional exchage spring approach.\nOur design concept together with the flexibility and huge num ber of materials crystallizing in the\nHeusler structure indicate the high potential in designing new materials with a large EB and a\nlarge coercivity at room temperature. Finally, the large EB and coercivity in the nearly compen-\nsatedHeusleralloysmaysignificantlyexpandtheimportanc eofthesematerialsinthephysicsand\nmaterialsciencecommunityduetohigh potentialforantife rromagneticspintronics.\n101. Eberhart, M. E. &Clougherty, D. P. Looking for design in materials design. Na t. Mater. 3,\n659-661(2004).\n2. Graf,T.,Felser,C. &Parkin,S.S.P.SimplerulesfortheunderstandingofHeusle rcompounds.\nProg. Solid StateChem. 39,1 (2011).\n3. Meiklejohn, W. H. &Bean, C. P. New Magnetic Anisotropy. Phys. 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Valence electron rules for prediction\nof half-metallic compensated-ferrimagnetic behaviour of Heusler compounds with complete\nspinpolarization.J. Phys.: Condens. Matter 18, 6171-6181(2006).\n26. Chadov, S., Kiss, J., &Felser, C. Improving Spin-Transport by Disorder. Adv. Func . Mater.\n23, 832-838(2013).\n27. Kurt, H. et al. Cubic Mn 2Ga Thin Films: Crossing the Spin Gap with Ruthenium. Phys. Re v.\nLett.112,027201(2014).\n28. Nayak, A. K., Nicklas, M., Chadov, S., Shekhar, C., Skour ski, Y., Winterlik, J. &Felser,\nC. Large Zero-Field Cooled Exchange-Bias in Bulk Mn 2PtGa. Phys. Rev. Lett. 110, 127204\n(2013).\n1329. Rode, K. et al. Site-specific order and magnetism in tetra gonal Mn 3Ga thin films. Phys. Rev.\nB87184429(2013).\n30. Leighton, C., Nogu´ es, J., J¨ onsson- ˚Akerman, B. J. &Schuller I. K. Coercivity Enhancement\nin Exchange Biased Systems Driven by Interfacial Magnetic F rustration. Phys. Rev. Lett. 84,\n3466-3469(2000).\nAcknowledgements: We thank J.A.Mydoshand Erik Kampert for valuablediscussio nson\nthe present work. This work was financially supported by the D eutsche Forschungsgemeinschaft\nDFG (Projects No. TP 1.2-A and No. 2.3-A of Research Unit FOR 1 464 ASPIMATT) and by\nthe ERC Advanced Grant No. (291472) ”Idea Heusler”. The expe riments at the High Magnetic\nFieldLaboratoryDresden(HLD)andHighFieldMagnetLabora toryNijmegenweresponsoredby\nEuro-MagNETIIundertheEuropeanUnionContract No. 228043 .\nAuthor Contributions Allauthorscontributedsubstantiallyto thiswork.\nAuthor Informations The authors declare no competing financial interests. Corre spon-\ndence and requests for materials should be addressed to C.F. (felser@cpfs.mpg.de) or A.K.N.\n(nayak@cpfs.mpg.de).\n14Figure 1 Design of a compensated magnetic state. a , A theoretical modeling showing\nthat thePt content controls theamount of Mn-spins pointing down by varyingtheuncom-\npensated magnetization, M, from1.1µBfor Mn 3Ga to0.5µB, in the opposite direction,\nfor Mn 2PtGa, thereby, going through the compensation point ( M= 0) at a Pt content of\nx= 0.59(upper panel). Approaching compensation maximizes HEB(lower panel): solid\ncurve corresponds to the model calculation deduced from the ab-initiocalculated mag-\nnetic moments, closed circles corresponds to the experimen t.b, Magnetic moments of\nmanganese in two different sublatices for the tetragonal Mn 3Ga (see the crystal struc-\nture), one Mn in Mn-Ga plane with a localized moment ≈3.1µB/f.u.(spin up, red arrow)\nand two Mn in Mn-Mn planes with a larger sum moment ≈4.2µB/f.u.(spin down, blue\narrow). TheMnintheMn-Gaplanecouplesantiferromagnetic ally totheMnintheMn-Mn\nplanes resulting in a net moment of 1.1µB/f.u.with spin down in the present illustration.\nc, Perfect magnetic compensation is expected in Mn 2.41Pt0.59Ga.d, Magnetic moment of\nmanganese in two sublatices for Mn 2PtGa (see the crystal structure), one Mn in Mn-Ga\nplanewithalargerlocalizedmoment(spinup,redarrow)and oneMninMn-Ptplanewith\na smaller moment (spin down, blue arrow). This configuration displays a net moment of\naround0.5µB/f.u.with spin up. Therefore, on going from Mn 3Ga to Mn 2PtGa, a mag-\nneticcompensation isencountered withequal andoppositel y aligned Mnmomentsinthe\nMn-Ga and Mn-Pt planes. e, A small anti-site disorder can bring about the formation FM\nclusters inside the compensated host. The exchange interac tion between the FM clus-\nters and the compensated host establishes a strong exchange anisotropy in the system\n15leading totheobservation of anexchange bias.\nFigure 2 Compensated ferrimagnetic state in Mn-Pt-Ga. M(T)for Mn 3−xPtxGa (x=0.2,\n0.5 and 0.6) measured in 0.01 T. Open symbols represent the ze ro-field cooled (ZFC)\ncurves and closed symbols the field cooled (FC) curves. In the ZFC mode, the sample\nwas initially cooled to 2 K and the data were taken upon increa sing the temperature in\nappliedmagneticfield. IntheFCmode,thedatawerecollecte dwhilecoolinginfield. The\ninset shows the zero field cooled M(H)hysteresis loop for Mn 2.4P0.6Ga measured at 2 K\ninafield of ±14 T.\nFigure 3 Hysteresis loops in Mn-Pt-Ga. a ,M(H)isotherms measured up to 60 T at\n4.2K.Thedataforsuccessivesamplesareshiftedby 0.3µBalongthemagnetizationaxis\nforbetter clarity. b,Fieldcooled(FC) M(H)loops measuredupto ±32Tforx= 0.5,0.6,\nand 0.7 after field cooling the samples in µ0HCF= 15T (closed symbols) and 25 T (open\nsymbols). The loop for x= 0.7is shifted by −0.3µBand forx= 0.5by+0.3µBalong the\nmagnetizationaxis. Theinsetshowsthedependence ofexcha nge-bias field( HEB)onthe\ncoolingfield( HCF).HEBiscalculatedusing HEB=−(H+\nC+H−\nC)/2,whereH+\nCandH−\nCare\nthelower and upper cut-offfields at which themagnetization becomes zero.\nFigure 4 Exchange bias and coercive fields for Mn-Pt-Ga and Mn-Fe-Ga. a, Tempera-\nture dependence of the EB for Mn 2.4Pt0.6Ga and Mn 2.5Pt0.5Ga. The inset of ashows the\ncoercive field HC(circles) and HEB(squares) as a function of the Pt concentration xin\nMn3−xPtxGa.b, Temperature dependence of theEB for Mn 1.5Fe1.5GaandMn 1.8FeGa.\n16/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s45/s49/s50 /s45/s56 /s45/s52 /s48 /s52 /s56 /s49/s50/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s72 /s61/s48/s46/s48/s49/s32/s84\n/s32/s120 /s61/s48/s46/s54\n/s32/s120 /s61/s48/s46/s53\n/s32/s120 /s61/s48/s46/s50/s77/s110\n/s51 /s45/s120/s80/s116\n/s120/s71/s97\n/s215/s48/s46/s54/s54/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41 /s215/s49/s48/s45/s52\n/s32\n/s84 /s32/s40/s75/s41/s215/s49/s46/s51/s77/s110\n/s50/s46/s52/s80/s116\n/s48/s46/s54/s71/s97\n/s72 /s32/s40/s84/s41/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s32/s32\n/s84 /s61/s53/s32/s75/s90/s70/s67/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50\n/s45/s51/s48 /s45/s49/s53 /s48 /s49/s53 /s51/s48/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s48/s46/s51/s48/s46/s54\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s49/s50/s51/s120 /s61/s48/s46/s53\n/s120 /s61/s48/s46/s54\n/s120 /s61/s48/s46/s55/s32\n/s32\n/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41/s84 /s61/s52/s46/s50/s32/s75/s97\n/s32/s48/s72 /s32/s40/s84/s41\n/s32/s98\n/s77/s110\n/s50/s46/s52/s80/s116\n/s48/s46/s54/s71/s97\n/s32/s32/s72\n/s69/s66/s32/s40/s84/s41\n/s72\n/s67/s70/s32/s40/s84/s41/s84/s61/s52/s46/s50/s32/s75/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s48/s49/s50/s51/s52/s48/s72\n/s69/s66/s40/s84/s41\n/s32/s77/s110\n/s50/s46/s53/s80/s116\n/s48/s46/s53/s71/s97\n/s32/s77/s110\n/s50/s46/s52/s80/s116\n/s48/s46/s54/s71/s97\n/s32\n/s84 /s32/s40/s75/s41/s48/s72\n/s67/s70/s61/s53/s32/s84/s97\n/s48/s72\n/s67/s70/s61/s53/s32/s84/s32/s77/s110\n/s49/s46/s53/s70/s101\n/s49/s46/s53/s71/s97\n/s32/s77/s110\n/s49/s46/s56/s70/s101/s71/s97\n/s32\n/s32/s98\n/s48/s72\n/s67/s70/s61/s49/s53/s32/s84/s84 /s61/s52/s46/s50/s32/s75\n/s77/s110\n/s51/s45 /s120/s80/s116\n/s120/s71/s97/s32/s72\n/s69/s66\n/s32/s72\n/s67/s48/s72 /s40/s84/s41\n/s32/s32\n/s120"    },    {        "title": "2102.11143v1.The_role_of_density_dependent_magnon_hopping_and_magnon_magnon_repulsion_in_ferrimagnetic_spin__1_2___S___chains_in_a_magnetic_field.pdf",        "content": "arXiv:2102.11143v1  [cond-mat.str-el]  22 Feb 2021The role of density-dependent magnon hopping and magnon-ma gnon repulsion in\nferrimagnetic spin-(1/2, S) chains in a magnetic field\nW. M. da Silva1,2and R. R. Montenegro-Filho2\n1Secretaria de Educa¸ c˜ ao da Para´ ıba, 58015-900 Jo˜ ao Pess oa-PB, Brasil\n2Laborat´ orio de F´ ısica Te´ orica e Computacional, Departa mento de F´ ısica,\nUniversidade Federal de Pernambuco, 50760-901 Recife-PE, Brasil\nWe compare the ground-state features of alternating ferrim agnetic chains (1 /2,S) withS=\n1,3/2,2,5/2 in a magnetic field and the corresponding Holstein-Primako ff bosonic models up to order/radicalbig\ns/S, withs= 1/2, considering the fully polarized magnetization as the bos on vacuum. The single-\nparticle Hamiltonian is a Rice-Mele model with uniform hopp ing and modified boundaries, while\nthe interactions have a correlated (density-dependent) ho pping term and magnon-magnon repul-\nsion. The magnon-magnon repulsion increases the many-magn on energy and the density-dependent\nhopping decreases the kinetic energy. We use density matrix renormalization group calculations to\ninvestigate the effects of these two interaction terms in the bosonic model, and display the quanti-\ntative agreement between the results from the spin model and the full bosonic approximation. In\nparticular, we verify the good accordance in the behavior of the edge states, associated with the\nferrimagnetic plateau, from the spin and from the bosonic mo dels. Furthermore, we show that the\nboundary magnon density strongly depends on the interactio ns and particle statistics..\nI. INTRODUCTION\nThe gapped phases of magnetic insulators are respon-\nsible for magnetization ( m) plateaus in the mvs. mag-\nnetic fieldcurves[1]. In onedimension, theseincompress-\nible phases satisfy the topological Oshikawa-Yamanaka-\nAffleck [2] criteria, and exhibit associated edge states in\nopen spin chains[3]. The gappedphasesareseparatedby\ngapless phases that have a low-energy physics described\nby the Tomonoga-Luttinger-liquid theory [4]. Thus, the\nmagneticfield hinducesquantumphasetransitionsinthe\nspin chain, with quantum critical points at the plateau\nextremes [5, 6]. In the vicinity of the quantum critical\npoints, the magnons are in a high-dilute regime and can\nbe treated as a gas of hard-core bosons [7, 8] or fermions\n[9]. In this approximation, the energy is comprised only\nby asimple hopping term and the uniform Zeeman term,\nwhich plays the role of chemical potential in the effective\nmodel. Following this approach, we can show that the\nm(h) curve (magnon density) presents a square-root sin-\ngularity in the gapless side of the transition. The first\ncorrection to this law is linear and obtained by taking\ninto account magnon-magnon interaction through a phe-\nnomenological scattering length [10–14].\nAnother kind of hopping term is known as corre-\nlated(ordensity-dependent )hopping and is essential\nin the modeling of a variety of quantum systems [15].\nOne of these terms, the bond-charge interaction [16],\nis used to model electrons in strongly correlated ma-\nterials [17–21] and was particularly investigated in the\ncontext of high- Tcsuperconductivity. Besides, the Hub-\nbard model with this term has an exact solution in a\nspecial point of the parameter space [22–25]. On the\nother hand, the extended boson Hubbard model with a\ndensity-dependent hopping is an effective Hamiltonian\nfor bosonic molecules, typically polar species [26–28], in\noptical lattices [29–36]. Further, general correlated hop-ping hard-core bosonic Hamiltonians are investigated to\nunderstand the physics of frustrated insulating magnetic\nmaterials [29–31, 37].\nThe ground-state of ferrimagnetic chains satisfy the\nLieb-Mattis theorem [38] and exhibit ferromagnetic and\nantiferromagnetic long-range orders [39]. and was inves-\ntigatedintheisotropic[40–42]andanisotropic[41,43,44]\ncases. Ferrimagnetic systems ˙In particular, the behavior\nof the edge states associated with the 1/3 magnetization\nplateau of the AB 2anisotropic chain was recently inves-\ntigated [45]. Furthermore, rich phase diagrams are ob-\nserved through doping [46–50] or adding geometric frus-\ntration [51–58] to the ferrimagnetic models. In partic-\nular, ferrimagnetic spin-(1/2, S) chains under an ap-\nplied magnetic field present magnetization plateaus at\nm=S−1/2 (ferrimagnetic plateau) and m=S+ 1/2\n(saturation plateau), where mis the magnetization per\nunit cell [59–64]. On the experimental side, the one-\ndimensional magnetic phase of a variety of bimetallic\ncompounds was shown to be modeled by spin-(1/2, S)\nferrimagnetic chains [65–70]. Recently, the full magne-\ntization curve of the charge-transfer salt (4-Br- o-MePy-\nV)FeCl 4was experimentally investigated [71] and shown\nto be a spin-(1/2, 5/2) chainabove the three-dimensional\nordering temperature.\nSince spin-( s,S) ferrimagnetic chains have a long-\nranged ordered ground state, linear and interacting spin-\nwave theory [72] from the classical ferrimagnetic state\nwas used to characterize their low-energy magnetic ex-\ncitations, mainly through the Holstein-Primakoff formal-\nism [59, 60, 73–78]. Furthermore, linear spin-wavetheory\nfrom the fully polarized state [64] gives good results for\nthe gapped and gapless phases of spin-(1/2, S) chains in\na magnetic field.\nHereweshowthattheHolstein-PrimakoffHamiltonian\nup to order/radicalbig\ns/Sgives almost exactly results to the\nground-state phase diagram of ferrimagnetic spin-(1/2,2\nS) chains in a magnetic field. We use the density matrix\nrenormalization group (DMRG) [79, 80] to obtain the\nmagnetization curves, besides local properties, from the\nHolstein-Primakoff Hamiltonian and the spin Hamilto-\nnian. In addition to the simple hopping and thenearest-\nneighbor interacting terms, a correlated hopping term is\nessential to obtain a good accord between the numerical\nresults from the spin and bosonic models. In Sec. II we\npresent the Holstein-Primakoff Hamiltonian and the an-\nalytical formulas for the hard-core boson approximation.\nIn Sec. III we compare the DMRG results for the magne-\ntization and local properties from the Holstein-Primakoff\nHamiltonian and the spin model. In Sec. IV we present\na summary of the main results of the manuscript.\nII. HOLSTEIN-PRIMAKOFF BOSONIC\nHAMILTONIAN FROM THE FULLY\nPOLARIZED VACUUM\nThe alternating mixed-spin ( s= 1/2,S) chain with L\nunit cells has the Hamiltonian\nHspin=JL/summationdisplay\nj=1(sj·Sj+sj·Sj−1)−BSz\ntot,(1)\nwhere\nSz\ntot=/summationdisplay\nj(sz\nj+Sz\nj) (2)\nis thez-component of the total spin, and we consider\nthe magnetic field Bin thezdirection, with gµB≡1,\nwhereµBis the Bohr magneton and gis theg-factor.\nThe spin −1/2 are attached to asites, while spin −Sto\nbsites along the chain, and we study chains for which\nS= 1,3/2,2,5/2, as schematically shown in Fig. 1(a).\nThe ground-state total spin for B= 0 isS−sand has\na copy in each sector in the range −(S−s)≤Sz\ntot≤\n(S−s), as expected from the Lieb-Mattis theorem [38],\nwith a ferrimagnetic (FRI) long-range ordered state. If\na little magnetic field is applied, the ground state with\nSz\ntot= (S−s) is chosen. Further, the ground state has\na finite gap to spin excitations carrying a spin ∆ Sz=\n+1, which induces a magnetization plateau at mFRI=\n(S−s). Also, a second magnetization plateau is the fully\npolarized plateau at mFP=S+s.\nThefully polarized state is an exact ground state of\nthe spin Hamiltonian, and we build the spin-wave the-\nory considering it as the magnon vacuum. Making the\nHolstein-Primakoff mapping on asites\nsz\nj=s−na\nj,\ns−\nj=√\n2sa†\nj/parenleftBig\n1−na\nj\n2s/parenrightBig1/2\n≈√\n2sa†\nj/parenleftBig\n1−na\nj\n4s/parenrightBig\n,(3)and onbsites:\nSz\nj=S−nb\nj;\nS−\nj=√\n2Sb†\nj/parenleftBig\n1−nb\nj\n2S/parenrightBig1/2\n≈√\n2Sb†\nj/parenleftBig\n1−nb\nj\n4S/parenrightBig\n,(4)\nFIG. 1. (a) Schematic representation of the alternating spi n\nmodel, with spin-1 /2 atasites, and spin- Satbsites, with\nS= 1,3/2,2,and 5/2. Thez-direction is the direction of an\napplied magnetic field Band the superexchange coupling is J.\n(b) Holstein-Primakoff bosonic Hamiltonian up to O(S−1/2),\nwith a hopping term t=J√\nsS, local potentials εa=−2SJ\nandεb=−2sJ, nearest-neighbor interaction V=J, and\ndensity dependent correlated hopping process X=J/radicalbig\ns/S.\nTheasites have a hard-core constraint, na≤1, while the\nconstraint nb≤2Sis imposed on bsites, with na(nb) as the\nnumber of bosons in one a(b) site. The magnetic field Bacts\nas a chemical potential µin the bosonic model: µ=−B.\n(c) The bosonic approximations investigated: t,t−V, and\nt−X−V. In thetandt−Vapproximations, there is a hard-\ncore constraint on all sites; while in the t−X−Vmodel, the\nbsites can accommodate up to two magnons.\nwherena\nj=a†\njajandnb\nj=b†\njbj, wearrivein the following\nspin-wave Hamiltonian3\nHSW=J/summationdisplay\nj/braceleftBigg/parenleftBig\nS−nb\nj/parenrightBig/parenleftBig\ns−na\nj/parenrightBig\n+√\nsS/bracketleftBigg/parenleftBig\n1−nb\nj\n4S/parenrightBig\nbja†\nj/parenleftBig\n1−na\nj\n4s/parenrightBig\n+\n+b†\nj/parenleftBig\n1−nb\nj\n4S/parenrightBig/parenleftBig\n1−na\nj\n4s/parenrightBig\naj/bracketrightBigg\n+√\nsS/bracketleftBigg/parenleftBig\n1−na\nj\n4s/parenrightBig\najb†\nj+1/parenleftBig\n1−nb\nj+1\n4S/parenrightBig\n+\n+a†\nj/parenleftBig\n1−na\nj\n4s/parenrightBig/parenleftBig\n1−nb\nj+1\n4S/parenrightBig\nbj+1/bracketrightBigg\n+/parenleftBig\ns−na\nj/parenrightBig/parenleftBig\nS−nb\nj+1/parenrightBig/bracerightBigg\n+\n−B/summationdisplay\nj/parenleftBig\nS+s−na\nj−nb\nj/parenrightBig\n+O(S−1). (5)\nDropping the classical energy of the ferromagnetic state\nEclass= 2JLsS−B/parenleftbig\nS+s/parenrightbig\nL,therelevantmagnonHamil-\ntonian is:\nHSW=Ht+HX+HV+O(S−1),(6)\nwithHSW=HSW−Eclass.\nAs sketched in Fig. 1(b), the Htterm comprises a\nhopping process and distinct local potentials on aandb\nsites:\nHt=t/summationdisplay\nj/parenleftBig\nb†\njaj+1+b†\nj+1aj+1+H. c./parenrightBig\n+/summationdisplay\nj[(εa−µ)na\nj+(εb−µ)nb\nj] (7)\nwith\n\n\nt=J√\nsS, hopping parameter;\nεa=−2SJ, local potential on asites;\nεb=−2sJ, local potential on bsites;\nµ=−B.(8)\nIn an open chain, if aorbis a boundary site, the local\npotential is half of the above value:\n/braceleftBigg\nε(boundary)\na =−SJ\nε(boundary)\nb=−sJ.(9)\nConsidering the local potentials, we see that the magnon\nhas a higher probability to be found on asites, and this\nprobability increases with S. However, since the hopping\nparameter tincreaseswith√\nS, quantum fluctuations are\nrelevant for moderate values of S, and the magnons can\novercome the potential barrier between aandbsites.\nWe observe that the bulk Hamiltonian Htis a partic-\nular case of the Rice-Mele model [81]:\nHRice-Mele=/summationdisplay\nj/parenleftBig\nt2b†\njaj+1+t1b†\nj+1aj+1+H. c./parenrightBig\n+/summationdisplay\nj(εana\nj+εbnb\nj), (10)\nputtingµ= 0 and considering the general case of al-\nternating hopping: t1(t2) for intra-cell (inter-cell) hop-\nping. The Rice-Mele Hamiltonian was originally pro-\nposed to model the physics of electrons in polymers, butis a paradigmatic model to the understanding of topolog-\nical insulators, and can be realized by atoms in optical\nlattices [82]. The model presents the bulk-boundary cor-\nrespondence [83], and an interacting version was recently\ninvestigated [83] to probe the connection between topol-\nogyandparticle-particleinteractions[83]. TheRice-Mele\nmodel recovers the bulk Hamiltonian Htfort1=t2.\nHowever, for an open chain, the mapping of the spin\nmodel requires local potentials on the boundary sites,\nEq. (9).\nThe second term in the bosonic Hamiltonian (6) is a\ndensity-dependent or correlated hopping term given by\nHX=−X/summationdisplay\nj/bracketleftbig\n(a†\nj+a†\nj+1)nb\njbj+H. c./bracketrightbig\n,(11)\nwith\nX=J\n4/radicalbiggs\nS. (12)\nSinces= 1/2 andna\nj≤1, a hard-core constraint must\nbe imposed on asites for all models considered. Hence,\nwe discard a term similar to the Xterm but with aandb\nvariablesexchanged, with X′=J\n4/radicalbig\nS/sas the correlated\nhopping amplitude. As sketched in Fig. 1(b), the energy\nof the system is lowered by the hopping of a magnon to a\nsite that is alreadyoccupied. In other words, the magnon\nprobability to overcome the potential barrier between a\nandbsites increases if there is one magnon on the bsite.\nThis term becomes relevant for higher magnon densities,\nsince it is an interaction term, and X→0 asSincreases.\nThe last term in the Hamiltonian (6), see Fig. 1(b),\nis a repulsive term between magnons in nearest neighbor\nsites:\nHV=V/summationdisplay\njna\nj+1(nb\nj+nb\nj+1), (13)\nwithV=J, and increases the energy for higher magnon\ndensities. For one magnon per unit cell, this term favors\nthe magnon localization on alternating asites, since the\nlocalpotentials( ε)favorsthe occupationof asites. Thus,\nε(anydensity) and V(higher densities) favormagnonlo-\ncalization on asites, while quantum fluctuations (tunnel-\ning between aandbsites) are favored by t(any density)\nandX(higher densities).4\nIn this work, we compare data from three approxima-\ntions of Hamiltonian (6), as summarized in Fig. 1(c),\nand from the spin Hamiltonian (1). The first, which we\nidentify as the tapproximation, is an analytical solution\nto the free hard-core model. In this approximation, all\nsites have a hard-core constraint and can be occupied by\nonly one magnon: na\nj≤1 andnb\nj≤1, for any j. This\nconstraint implies that there is not an energy contribu-\ntion from the Xterm, and we drop the nearest-neighbor\ninteraction V. In the second approximation, t−V, we\nkeep the hard-coreconstraint but add the Vcontribution\ntoHt. The last approximation, t−X−V, has the three\ntermsHt,HX, andHV. We consider a hard-core con-\nstraint on asites and a constraint nb\nj≤2 onbsites. As\nwe present below, the relaxation of the hard-core con-\nstraint on bsites and the consequent activation of the\ncorrelated hopping term implies an excellent agreement\nbetween the results of this approximation and the spin\nmodel.\nA. Free hard-core magnons [ t-approximation]: Ht\nandL→ ∞\nThe first approximation to the magnetization curve, a\nmany-magnon state, is to consider the magnons as free\nhard-core bosons or free fermions. This mapping is exact\nin the high-dilute regime ofmagnons, near the saturation\nfield. Here, we extend this approach for the full range of\nthe magnetization, ( S−s)≤m≤(S+s), and compare\ntheir results to more precise calculations considering the\ninteraction terms.\nThe single-magnon energies are given by the term Ht,\nEq. (7), in Hamiltonian (6). Using the following Fourier\ntransforms:\n/braceleftBigg\naj=1√\nL/summationtext\nke−ik/4eikjak;\nbj=1√\nL/summationtext\nke+ik/4eikjbk,(14)\nwhere a phase e±ik/4is included to ease the calculation,\nHtbecomes:\nHt=/summationdisplay\nk/parenleftbig\na†\nkb†\nk/parenrightbig/parenleftbigg\nεaγk\nγkεb/parenrightbigg/parenleftbigg\nak\nbk/parenrightbigg\n,(15)\nwhere\nγk= 2tcos(k/2). (16)\nAfter diagonalization, the Hamiltonian is written as\nHt=/summationdisplay\nkω(−)\nkn(−)\nk+ω(+)\nkn(+)\nk, (17)\nwhere the dispersion relations are\nω(±)\nk=εa+εb\n2±ωk=−J(s+S)±ωk,(18)with\nωk=/radicalBigg/parenleftbiggεb−εa\n2/parenrightbigg2\n+γ2\nk\n=J/radicalbig\n(S−s)2+4sScos2(k/2),(19)\nandn(±)\nk=α(±)†\nkα(±)\nk, where\n/parenleftbigg\nα(−)\nk\nα(+)\nk/parenrightbigg\n=/parenleftbigg\ncosθk−sinθk\nsinθkcosθk/parenrightbigg/parenleftbigg\nak\nbk/parenrightbigg\n,(20)\nwith\n/braceleftBigg\ncos2θk=1\n2+S−s\n2ωk;\nsin2θk=1\n2−S−s\n2ωk.(21)\nA magnetic field (or chemical potential) adds a + Ben-\nergy term to the two bands.\n-1 0 1\nk /  π-2(S+s)-2S-2s = -10Energy / Jinactive\none - magnon\n       states-1-0.500.51\nk /  π02s = 1\n-1-0.500.51\nk /  π-2s = -10 B=2(s+S)=−µ\nB=2SFP\nFRIB=0vacuum\none magnon\nper unit cell\nFIG. 2. One-magnon states from the fully polarized (FP)\nvacuum and B= 0 as a function of lattice wave-vector k.\nIn thet-approximation, the high energy band is inactive. For\nB= 2(s+S) =−µ, the exact ground state is the FP vacuum;\nwhile there is one magnon per unit cell in the system for\nB= 2S, and the ground state is ferrimagnetic (FRI).\nThe magnon bands are shown in Fig. 2 for (i) B= 0;\n(ii)B= 2(s+S), the saturation field, for which m=\nmFP= (S+s); and (iii) B= 2S, the critical field of the\nferrimagnetic plateau, for which m=mFRI= (S−s), or\n1 magnon per unit cell: mFP−mFRI= (S+1/2)−(S−\n1/2) = 1. As expected from the Lieb-Mattis theorem,\nthe magnetization in the null field is S−s. In the free\nhard-corebosonorfreefermionapproximation,wefillthe\nsingle-particle states following a Fermi distribution up to\nthe effective Fermi wave-vector kF. As shown in Fig. 2,\nif we follow this procedure the two bands should be filled\nforB= 0(twomagnonsperunitcell)andthemagnetiza-\ntion curves of the spin model would not be reproduced.\nWe have shown in Ref. [64] that this problem can be\novercome, even for finite T, by introducing an effective\nchemical potential µto the upper band, in a way similar\nto Takahashi’s solution to the ferromagnetic linear chain\n[84]. In particular, µ→ −J=−2sJasT→0, such\nthat the overall effect of µatT= 0 is the suppression\nof the upper band. Hence, in the free hard-core approx-\nimation, we must consider only the lower band in the5\ncalculations, as schematically indicated in Fig. 2. Thus,\nfor example, the energy per unit cell in the free hard-core\napproximation is written as\nE\nL=1\nLkF/summationdisplay\nk=−kFω(−)\nk(22)\nfor a magnon density per unit cell n, where\nkF=πn, (23)\nandn=mFP−m.\n1. Average local magnetizations\nIf the chain has a magnon density per unit cell n\nand considering the hard-core approximation, the aver-\nage magnetizations of aandbsites are given by\n/braceleftBigg\n/angbracketleftsz/angbracketright=s−1\nL/summationtextkF\nk=−kF/angbracketleftna\nk/angbracketright(asites);\n/angbracketleftSz/angbracketright=S−1\nL/summationtextkF\nk=−kF/angbracketleftnb\nk/angbracketright(bsites).(24)\nUsingEqs. (20)anddiscardingtermsinvolvingtheupper\nband, we obtain\n/braceleftBigg\n/angbracketleftna\nk/angbracketright=n(−)\nkcos2θk;\n/angbracketleftnb\nk/angbracketright=n(−)\nksin2θk.(25)\nWe, thus, have\n/braceleftBigg/angbracketleftbig\nsz/angbracketrightbig\n=s−1\n2−S−s\n2/summationtextkF\nk=−kF1\nωk;/angbracketleftbig\nSz/angbracketrightbig\n=S−1\n2+S−s\n2/summationtextkF\nk=−kF1\nωk,(26)\nafter the substitution of Eqs. (21) in Eqs. (25) and the\nresults in Eqs. (24).\nIII. DMRG RESULTS FOR THE SPIN MODEL\nAND THE BOSONIC HAMILTONIANS\nA. Methodology\nWe use the density matrix renormalization group to\nobtain the magnetization curves and local properties of\nthe spin, t−V, andt−X−Vmodels; and also com-\npare this data with the analytical results from the free\nhard-core model ( t-model) in the thermodynamic limit:\nL→ ∞. These approximations are summarized in Fig.\n1(c). All DMRG results (boson and spin models) are ob-\ntained through the Algorithms and Libraries for Physics\nSimulations (ALPS) project [85] for chains with L= 128\nunit cells, with one asite at one extreme and a bsite\nat the other. If the system has an asite at the left\nextreme and a bin the right ( a-bboundaries), the renor-\nmalization process for the magnetization step inside the\nferrimagnetic plateau becomes trapped in a metastablestate for S= 3/2,2,and 5/2. In these cases, the global\nenergyminimum is reachedbythe algorithmif the chains\nhave absite at the left extreme and an asite at the right\nextreme ( b-aboundaries). In the other magnetizations,\nthis is irrelevant, i. e., the same state is calculated for\nthea-bor theb-aboundaries. We retain a maximum of\n243 states per block and the maximum discarded weight\nless than 10−9.\nFor the spin model, the magnetization curves are cal-\nculated from the lowest energy state for a fixed Szand\nB= 0:E(Sz); since for B/negationslash= 0 we need only to add\nthe Zeeman term, such that EB(Sz) =E(Sz)−BSz. In\na gapless (non-plateau) phase, the magnetization curves\nare made of finite steps in a finite-size system. Defining\nthe extreme points of these finite steps as B−andB+,\nwe have\nB±=±[E(Sz±1)−E(Sz)] (27)\nfor the step at Sz. In a gapless phase, B−→B+as\nL→ ∞, while in a thermodynamic-limit plateau state\nB−/negationslash=B+forL→ ∞. In the last case, B±are quantum\ncritical fields separatingthe plateau insulating state from\na gapless critical Luttinger liquid phase. For the bosonic\nmodels, the magnon density per unit cell nas a function\nof chemical potential µis obtained with a similar pro-\ncedure. We calculate the lowest energy state for a fixed\nnumber of bosons N, withN=nLandµ= 0. The value\nof the chemical potential at the extremes ( µ−andµ+) of\nthe step at Nare given by\nµ±=±[E(N±1)−E(N)]. (28)\nA gapless phase has µ+→µ−asL→ ∞, while in a\nplateau insulating phase µ+/negationslash=µ−in the thermodynamic\nlimit. The transformation from the boson to the spin\nlanguage is performed through the following equations:\n/braceleftBigg\nn=m−mFP,and\nB=−µ.(29)\nB. Magnetization curves and local magnetizations\nThe magnetization curves from the spin model and the\nboson Hamiltonians are shown in Fig. 3(a). The models\npresent two magnetization plateaus, one at the ferrimag-\nnetic magnetization mFRI=S−1/2 and the fully polar-\nized state mFP=S+1/2. The saturation field BFP, end\npoint of the FP plateau, is obtained through the closing\nof the single-particle magnon gap with B, as sketched in\nFig. 2, at B=BFP= 2(S+s). The saturation field\nBFPfrom any bosonic approximation is rigorously equal\nto its exact value since the fully polarized state, an ex-\nact eigenstate of the spin model, is the vacuum for the\nbosonic models.\nThe Oshikawa-Yamanaka-Affleck topological criteria\n[2] states that a plateau can appear in the magnetiza-\ntion curve of spin systems if\nSu−m=integer, (30)6\n0 1 2 3 4 5 6\nB / J0.30.40.50.60.70.80.91m / ms(1/2, 5/2)\n(1/2, 2)\n(1/2, 3/2)\n(1/2, 1)Spin, DMRG (L=128)\nt, L →∞\nt - X - V, DMRG (L=128)(a)\nFRI plateau\nFRI plateau\nFRI plateau\nFRI\nplateau arrowsedge states\n0 1 2 3 4 5 6\nB / J00.51n(1/2, 5/2)\n(1/2, 2)\n(1/2, 3/2)\n(1/2, 1)Spin, DMRG (L=128)\nt, L →∞\nt - X - V, DMRG (L=128)(b)\nt - V, DMRG (L=128)13/22 5/2\nGapless\nLuttinger\nliquidFRI plateau\n     (n = 1)\nFIG. 3. (a) Magnetization per unit cell ( m) of (1/2,S) chains,\nwithS= 1,3/2,2,and 5/2, normalized by its saturation\nvalue (ms) and (b) magnon densities per unit cell nas func-\ntions ofBin units of J. DMRG data for (full lines) the spin\nmodel and for ( •) thet−X−Vapproximation. We present\nin both figures the results for the t-model (dashed lines) and\nL→ ∞ . In (b), we also show the magnon density as a func-\ntion ofBunits ofJfor (•) thet−Vapproximation, calculated\nwith DMRG for a system with L= 128. In the thermody-\nnamic limit, the ferrimagnetic plateau is observed for n= 1,\nwhile the gapless Luttinger liquid phase for 0 < n < 1. On\nboth figures, we indicate (arrows) the magnetic field at which\nthe edge states are occupied by one magnon.\nwhereSuis the maximum spin of a unit cell, unless the\nground state spontaneously break translation symmetry.\nThis corresponds to a number of magnons per unit cell\nn= 0,1,2,...,S u(Su−1/2) for integer (half-integer) Su,\nfrom the fully polarized state. Since Su=S+1/2 (sat-\nurated magnetization) for spin-(1/2, S) chains, plateaus\ncould appear at m=S+1/2,(S−1/2),...,0(1/2), for\nhalf-integer (integer) S. In the spin-(1/2, S) chains, the\ndata shows that there are two magnetization plateaus,\none at the fully polarized ( n= 0) magnetization and the\nother at one magnon per unit cell ( n= 1), the ferrimag-\nnetic plateau. The other possible magnetization plateaus\nare inhibited by the magnon-magnon interaction term V\n[Eq. (13)]. The magnetization curves of ferrimagnetic\nspin chains with 1 /2< s < S can exhibit other plateaus\nbetween the ferrimagnetic and the fully polatized ones.\nThis is observed, for example, in spin-(1,2) and spin-\n(1,3/2) chains [86]. For these chains, the ferrimagnetic\nplateau state has two magnons per unit cell, n= 2, andthemagnetizationcurvesalsoexhibitsaplateauat n= 1.\nWe also observe the occupancy of the edge states of the\nferrimagnetic plateau by one magnon at the indicated\nmagnetic fields. These edge states appears in finite-size\nopen systems associated with topological aspects [3, 45]\nof the ferrimagnetic state.\nThe non-interacting Rice-Mele model (10) does not\npresent edge states for uniform hopping. However, re-\ncently [83], it was shown that an interacting fermionic\nsystem, with a local Coulomb interaction, presents ef-\nfective edge states, and that a fraction of the boundary\ncharge is, in fact, related to the bulk properties. Our\nnon-interacting Hamiltonian (7) has the modified local\npotentials in the boundaries (9), required from the spin\nmapping, and thus localized edge states. Furthermore,\nour interacting model is a bosonic system having the cor-\nrelated hopping and nearest-neighbor coulomb repulsion.\nIn the Sec. IV, we study the boundary magnon den-\nsity and compare it from relevant interacting and non-\ninteracting bosonic models.\nThe gapless phase between the magnetization plateaus\nis a Luttinger liquid phase with power-law decay of the\ntransverse spin correlation functions [4] and has a dy-\nnamical exponent z= 1.\nWe notice in Fig. 3(a) that as Bdecreases from\nBFP, the magnetization from the free hard-core model,\nt-approximation, departs from that of the spin model\nat roughly half filling of the lower magnon band ω(−),\nEq. (18). At this filling, the interaction effects start to\nbecome relevant. The critical field of the ferrimagnetic\nplateau from the free hard-core model: 2 S, see Fig. 2,\nbecomes more near its value for the spin model as Sin-\ncreases, as also confirmed in Table I. This effect can be\nattributed to the local energy of the asites that becomes\ndeeper in comparison with the local energy of the bsites,\nas can be seen in the energy term (7) and the sketch in\nFig. 1. Thus, the magnons become more localized on\nasites asSincreases, and the XandVenergy terms,\nEqs. (11) and (13), respectively, become lesser relevant.\nTheXterm, due to the low occupancy probability of\nonebsite by two magnons, and also because X→0\nasS→ ∞, see Eq. (12). The Vterm, on the other\nhand, because the probability of finding two magnons\nin nearest neighbor sites is also very low. Further, the\nt−X−V-approximation, which has all energy terms in\nHamiltonian(6)active, is inexcellentagreementwith the\nresults for the spin model. Even the location of the edge\nstates is well reproducedby the t−X−V-approximation.\nIn Fig. 3(b) we show the average magnon density per\nunit cell/angbracketleftn/angbracketright. Besides the models presented in (a), we add\nthet−V-approximation. The presence of the magnon-\nrepulsionmakestheaccordancewith thespin modelgood\nfor/angbracketleftn/angbracketright>1/2, while in the t-approximation this agree-\nment is good up to a value of /angbracketleftn/angbracketrightless than 1/2. As\n/angbracketleftn/angbracketright →1, reaching the ferrimagnetic plateau, the repul-\nsionVincreases much the energy of the system, and,\nthus, implies a lower value of the critical magnetic field\nBFRI, compared to the spin model. The location of the7\nTABLE I. Critical field of the ferrimagnetic plateau in units\nofJfor the spin-(1/2, S) chains in the free hard-core boson\napproximation t-approximation and the spin model.\nS t -approx.: 2 S spin model: β2S−β\nβ\n1 2 1.76 0.13\n3/2 3 2.85 0.05\n2 4 3.88 0.03\n5/2 5 4.91 0.02\nTABLE II. Average spins at a(/angbracketleftsz/angbracketright) andb(/angbracketleftSz/angbracketright) sites for\nthe spin-(1/2, S) chains in the free hard-core boson approxi-\nmationtand the spin model: ( /angbracketleftsz/angbracketright,/angbracketleftSz/angbracketright), atm=S−s, the\nferrimagnetic magnetization.\nS t -approx. spin model\n1 ( −0.27,0.77) ( −0.29,0.79)\n3/2 ( −0.34,1.34) ( −0.36,1.36)\n2 ( −0.38,1.88) ( −0.39,1.89)\n5/2 ( −0.40,2.40) ( −0.41,2.41)\nedge states in the finite-size system is also different be-\ntween the spin and the t−V-approximation. Further,\ntheX-term, the density-dependent hopping term, lowers\nthe energy, and the full /angbracketleftn/angbracketright-curve of the spin model is\nwell reproduced by the t−X−Vapproximation. We\nnotice, however, that only for the finite size spin-(1/2, 1)\nchain studied, the lower critical field of the ferrimagnetic\nplateau is B≈0.06J. In Sec. IV, we present the magnon\ncurve for N >128.\nThe average spin and magnon density at aandbsites\nare shown in Fig. 4. Notice that the average spin at\nbsites, Fig. 4(b), is normalized by S. Also shown are\nthe probability of occupancy of aandbsites. We use\nthe expressions (26) to obtain the average spins from the\nt-approximation, while we calculate the average magnon\ndensity from the spin model with\n/braceleftBigg\n/angbracketleftna/angbracketright=1\n2−1\nL/summationtextL\nl=1/angbracketleftsz\nl/angbracketright,\n/angbracketleftnb/angbracketright=S−1\nL/summationtextL\nl=1/angbracketleftSz\nl/angbracketright,(31)\nfor a finite spin chain of size Land open boundaries.\nThe value of the average spin at asites from any of the\nconsidered bosonic approximations is in good agreement\nwith its value from the spin model. A relevant departure\nbetween the approximations and the spin model occurs\nin the average spin on bsites as the ferrimagnetic mag-\nnetization plateau is approached. However, even the t-\napproximationprovidesgoodvaluesfortheaveragespins,\nas shown in Table II for the ferrimagnetic magnetiza-\ntion. Also, the results become indistinguishable, even on\nbsites, as Sincreases or as the fully polarized plateau\nis approached. Furthermore, as in the magnetization re-\nsults, the t−X−Vmodel is an excellent approximation\nto the spin Hamiltonian. The data in Figs. 4(c) and (d)\nconfirm that due to the higher value ofthe localpotential\nonasites, theprobabilityofoccupancyof asitesishigher0.40.60.8 1\nm / ms-0.2500.250.5 <Sz\na>\n(1/2, 1)\n(1/2, 3/2)\n(1/2, 2)(1/2, 5/2)\n0.40.60.8 1\nm / ms0.80.91\n<Sz\nb> / S\n(1/2, 1)(1/2, 3/2)(1/2, 2)(1/2, 5/2)(a) (b) Spin, DMRG(L=128)\nt, L→∞\nt - X - V, DMRG (L=128)\n0.40.60.8\nm / ms00.20.40.60.81<na>(1/2, 1)(1/2, 3/2)(1/2, 2)\n(1/2, 5/2)\n0.40.60.81\nm / ms00.10.2\n<nb>(1/2, 1)\n(1/2, 3/2)\n(1/2, 2)\n(1/2, 5/2)\n(c) (d)\nFIG. 4. Average spin at (a) a,/angbracketleftSz\na/angbracketright, and (b) b,/angbracketleftSz\nb/angbracketright,\nsites, in this case normalized by S, for (1/2, S) chains with\nS= 1,3/2,2,and 5/2, as a function of the normalized mag-\nnetization m/ms, wheremsis the saturation magnetization\nof each chain. The hard-core boson tapproximation in the\nthermodynamic limit (dashed lines), DMRG results for the\nspin model (full lines) and the t−X−Vapproximation ( •),\nboth for systems with L= 128. Average magnon densities at\n(c)a,/angbracketleftna/angbracketright, and (d) b,/angbracketleftnb/angbracketright, sites as a function of m/msand\nthe same legend of (a) and (b).\nthan that on bsites, as discussed in the context of the\nmagnetization curves in Fig. 3. In particular, quantum\nfluctuations are reduced as Sincreases, since /angbracketleftnb/angbracketright →0 in\nthis limit, as the data in Fig. 4(d) suggests.\nIn Figs. 5(a) and (b) we present the local magnon\ndensities in finite chains with open boundaries for the\nspin and the t−X−Vmodels. The magnon densities\nfrom the spin model are calculated through\n/braceleftBigg\n/angbracketleftna\nl/angbracketright=1\n2−/angbracketleftsz\nl/angbracketright;\n/angbracketleftnb\nl/angbracketright=S−/angbracketleftSz\nl/angbracketright,(32)\nand/angbracketleftncell\nl/angbracketright=/angbracketleftna\nl/angbracketright+/angbracketleftnb\nl/angbracketright. We show the data for the spin-\n(1/2, 1) and spin-(1/2,5/2) chains, for two low magnon\ndensities ( n= 8/128,n= 16/128), near the fully polar-\nized magnetization plateau, and two high magnon densi-\nties, more near the ferrimagnetic magnetization plateau\n(n= 72/128,n= 116/128), in a chain with L= 128 unit\ncells. For the lower magnon densities (hard-core limit)\nthe results for t−X−V-approximation departs from the\nspin model data near the boundaries for the spin-(1/2,\n1). However, the accordance between the two models is\nexcellent in the case of the spin-(1/2, 5/2), even near the\nboundaries, for the two lower magnon densities shown.8\n00.020.040.060.08\n00.050.10.15\n00.5\n0 64 12800.20.40.60.8n = 8/128 = 0.0625\nn = 16/128 = 0.12500\nn = 72/128 = 0.56250spin: a site\nspin: b sitespin: cell\nt - X - VAverage magnon densities along the chain\nln = 116/128 = 0.90625(a) (1/2, 1) chain\n00.020.040.060.08\n00.050.10.15\n00.5\n0 64 12800.20.40.60.8n = 8/128 = 0.0625\nn = 16/128 = 0.12500\nn = 72/128 = 0.56250spin: b sitespin: cell\nt - X - V\nln = 116/128 = 0.90625(b) (1/2, 5/2) chain\nspin: a site\n00.20.40.60.81\n00.20.40.60.81\n00.20.40.60.81\n0 1000.20.40.60.81(1/2, 1)\n(1/2, 3/2)\n(1/2, 2)\nl(1/2, 5/2)(c) edge statesDensity change along the chain δ<n>FRI+1→FRIcell: spin\na site: spin\nb site: spin\nFIG. 5. Average magnon densities along (a) the (1 /2,S= 1) chain and (b) the (1 /2,S= 5/2) withL= 128 unit cells. Magnon\ndensities at ( •)asites,/angbracketleftna,l/angbracketright, at (•)bsites,/angbracketleftnb,l/angbracketright, and (•) total magnon density, /angbracketleftncell,l/angbracketright, for unit cell lfrom the spin model\nand (/trianglesolid) thet−X−Vapproximation, both calculated with DMRG for systems with L= 128. The following total average\nmagnon densities are shown: n= 8/128,16/128,72/128,and 116 /128, from top to bottom, corresponding, respectively, to\nthe magnetizations (a) m= 184/128,176/128,120/128,and 76/128; and (b) m= 376/128,368/128,312/128,and 268/128.\n(c) Average magnon distribution change δ/angbracketleftn/angbracketrightFRI+1→FRI,l =/angbracketleftn/angbracketrightN=L,l− /angbracketleftn/angbracketrightN=L−1,lalong the chain as a function of unit cell\nlshows the presence of the edge states for the four chains stud ied. (d) The average magnon probability density /angbracketleftn/angbracketrightalong a\nchain with L= 128 at (circles) aand (diamonds) bsites for (white symbols) N=L−1 and (red symbols) N=Lbosons from\nthet−X−Vapproximation of the (1/2,1) chain. (e) Sketch of the ground states for N=L−1 andN=Lbosons. Average\nmagnon densities are shown as grey circles, fluctuations bet weenaandbsites are indicated by a blue stripe, and we use a red\nfilled curve to represent the localized orbital in the bounda ry unit cell.\nFor the two higher magnon densities, there is an excel-\nlent agreement between the spin model and the t−X−V\napproximation for the two chains.\nIn Fig. 5(c), we show the magnon distribution in\nthe edge localized state occupied by one magnon for the\nfour chains studied. To calculate it, we notice that the\nedge state appears between the two magnetization steps:\nSz=L(S−s) andSz=L(S−s) + 1, which will join\nin the thermodynamic limit, and make the S−sferri-\nmagnetic plateau, see Fig. 3(a) and (b). Thus, to visu-alize the spatial extent of the edge state, we consider the\nmagnon distribution change, δ/angbracketleftn/angbracketrightFRI+1→FRI, between a\ntotal number of magnons N=L−1 [Sz=L(S−s)+1]\nandN=L[Sz=L(S−s)]:\nδ/angbracketleftn/angbracketrightFRI+1→FRI,l=/angbracketleftn/angbracketrightN=L,l−/angbracketleftn/angbracketrightN=L−1,l.(33)\nWe notice in the data shown in Fig. 5(c) that a tiny dis-\ncrepancyisobservedbetweentheresultsforthe t−X−V-\napproximationand the spinmodel inthe caseofthe (1/2,\n1)chain,whilefortheotherchainstheagreementisexcel-9\nlent. Theholeaddedtothemany-magnonstateat N=L\nbecomes well localized in the boundary cell, mainly on\nthe boundary asite, thus implying the presence of this\nempty orbital in the N=L−1 [Sz=L(S−s)] state.\nThe localization in the boundary increases with Sas ex-\npected from the increasing of the bulk gap. In fact, the\nfluctuations of the magnon density in the boundary be-\ncome neglegible as S→ ∞, and the boundary hosts one\nmagnon in this limit. We can attibute this behavior to\nthe increasing potential barrier between bulk and edge,\nEqs. (8) and (9), respectively, as S→ ∞.\nAs an example of the data used to make the results\nshown in Fig. 5(d), we exhibit in Fig. 5(d) the average\nmagnon density along the chain, /angbracketleftn/angbracketright, foraandbsites,\nin thet−X−Vapproximation of the (1/2,1) chain.\nWe notice that the asite at the left boundary is almost\nempty for N=L−1, while its ≈0.6 forN=L, implying\nthe value shown in (c). We also notice a sizable change\nin the occupation of the bsite at the left boundary for\nthe two fillings, while /angbracketleftn/angbracketrightdoes not change at the right\nboundary. A sketch of the two ground states is shown in\nFig. 5(e). For N=L−1, we have an insulating state\nwith themagnonaveragehigheron asitesthan on bsites.\nQuantum fluctuations between aandbsites in nearby\nunit cells are expected, given the excellent accordance\nbetween the results from the t−X−V-approximation\nandthe spinmodel in the high-densitylimit, m→mFRI.\nAdding one magnon to the ( L−1)-magnon state, we\nobtain the magnetization at m=S−sin the finite-\nsize system, which is also an insulating state. The added\nmagnon occupies the empty localized orbital state in the\nunit cell of the boundary asite. The penetration of this\nedge state in the gapped bulk is very tiny and decreases\nwith increasing S, see Fig. 5(c), as can be estimated\nthrough the average magnon distribution.\nIV. THE RELEVANCE OF THE BOUNDARY\nPOTENTIALS AND INTERACTIONS IN THE\nBOUNDARY MAGNON DENSITY OF THE\nBOSONIC MODEL\nWith the help of Fig. 6, we discuss the relevance of the\ndifference between the local potential in the boundaries\nand the bulk sites on the boundary magnon densities,\nas well as the importance of interactions in it. We use\nthe parameters of the spin-(1/2, 1) chain, but focus on\nthe bosonic Hamiltonian, for three models in finite-size\nchains: (1) the interacting model considering the map-\nping to the spin system ( X/negationslash= 0,V/negationslash= 0) - the t−X−V\nmodel; (2)thecorrespondingnon-interactingmodel,i. e.,\nthet−X−Vmodel with X= 0 and V= 0, which has\na hard-core constraint on asites and the bsites can host\nup to 2 bosons; and (3), the tapproximation, which has\na hard-core constraint on aandbsites. One of the sets,\nFigs. 6(a) and (c), shows data for the boundary poten-\ntials distinct from the bulk potentials, with the values in\nEqs. (9) and (8), respectively. The other set, Figs. 6(b)00.511.522.5\nB / J122124126128130132134136138\n00.511.522.5\nB / J122124126128130132134136138 N(b) Bound. pot. = Bulk pot.\nt - X - V (a) Bound. pot. ≠ Bulk pot. \nB = 0.06 J     with\n X=0, V=0t - X - V t - X - V \nt, finite L t, finite Lt - X - V \n     with\n X=0, V=0\n0 10\nl00.20.40.60.81\n0 10\nl00.20.40.60.81δ <n>Ni→Nf, l\n050100\nl00.1\n120\nl(d) Bound. = Bulk (c) Bound. ≠ Bulk \nFIG. 6. DMRG results for finite-size modified Rice-Mele\nmodels with L= 128. In (a) and (c), the boundary potentials\nhave the values in Eq. (9); while in (b) and (d), the boundary\npotentials have the same values of the bulk local potentials\nonaandbsites,εaandεbin Eq. (8). Taking the parameters\nfrom the spin-(1/2, 1) chain, we show data for the full inter-\nacting model, t−X−V, thet−X−Vmodel with X= 0 =V,\nand the hard-core approximation t. In (a) and (b), we present\nthe number of magnons Nas a function of magnetic field (in\nunits of J)B=−µ, where µis the chemical potential in\nthe bosonic system. In (c) and (d), we show the change in\nmagnon density per unit cell δ/angbracketleftn/angbracketrightNi→Nf,l=/angbracketleftn/angbracketrightNf,l−/angbracketleftn/angbracketrightNi,l\nbetween states N=NiandN=Nfas a function of unit cell\nl, where the arrows in (a) and (b) identify NiandNffor each\nmodel.\nand (d), takes the value of the boundary potentials equal\nto the bulk ones, Eq. (8). The arrows in Figs. 6(a) and\n(b) indicate the edge states whose boson density change\nδ/angbracketleftn/angbracketrightNi→Nf,l=/angbracketleftn/angbracketrightNf,l−/angbracketleftn/angbracketrightNi,l (34)\nis shown in Figs. 6(c) and (d), respectively, where Nf\n(Ni) is the highest (lowest) value of Nbetween the two\nmagnon density steps separated by the occupancy of the\nedge state.\nFor the bulk different from the boundaries, Fig. 6(a),\nwe see that the edge state is very robust, appearing even\nin thetapproximation. This does not occur for the t\napproximation if the boundary is equal to the bulk, Fig.\n6(b), which is an expected result for the Rice-Mele model\n[83]. However, we notice in Fig. 6(a) that, in comparison\nwith the spin model, the plateau width is very underesti-\nmatedinthe tapproximation,sincewearenotdiscarding\nthe single-particle states in the upper band shown in Fig.\n2. We observethat this upper band wasdiscarded for the\nL→ ∞results shown in Figs. 3 and 4, due to the in-\ntroduction of the effective chemical potential connected\nwith the Takhashi’s constraint [64]. Considering also the10\nspin model, the error in the plateau width for t−X−V\ninteracting case is very tiny, with a lower critical field\n≈0.06J. Also in Fig. 6(a), bulk and boundaries dis-\ntinct, we note that the edge state in the non-interacting\nt−X−Vmodel appears only for two bosons over the\nN= 127 state, Fig. 6(a), with one of the bosons oc-\ncupying an extended state and the other an edge state,\nas shown in Fig. 6(c). This shows that the distinction\nbetween boundaries and bulk is not sufficient to a good\nrepresentation of the edge state in the spin model, the\ninteractions are essential for it. In the case of the tap-\nproximation, the particle statistics provides a change in\nthe single particle quantum states that mimic the effect\nof theXandVinteractions in the t−X−Vmodel. In\nFig. 6 (b), where the boundary is equal to the bulk, we\nshow that edge states appear in the t−X−Vmodel in\nthe interacting and non-interacting cases. As mentioned\nabove, the edge state of the non-interacting case does\nnot appear in the tapproximation. Further, the bound-\nary density, Fig. 6 (d), in the interacting t−X−Vcase\nappears in the right extreme and has a value approxi-\nmately equal to that in Fig. 6 (c). From these results,\nwecanassertthat theinteractions XandV, and the par-\nticle statistics, are essential to understand the boundary\nmagnon density.\nV. SUMMARY AND DISCUSSION\nWe have investigated the Holstein-Primakoff bosonic\nHamiltonian, up to order/radicalbig\ns/S, of the alternating spin-\n(s= 1/2,S) chains in a magnetic field, and considering\nthe fully polarized as the vacuum. Three bosonic Hamil-\ntonians were considered: the first has only a hopping\nterm (t-approximation) and distinct local potentials ( ε)\non the spin-1 /2 (asite) and the spin- S(bsite) sites , this\napproximation is a Rice-Mele Hamiltonian with bound-\naries different from the bulk and uniform hopping term.\nThesecondonehasthetermsofthe t-approximationplus\namagnon-magnonrepulsion V(t-V-approximation); and\nthe last approximation shows the tandVterms, and a\ndensity-dependent correlatedhopping term X(t−X−V-\napproximation). In the t−X−Vapproximation, bsites\ncan accommodate up to two magnons, while in the oth-\ners a hard-core constraint is imposed on aandbsites.\nThe local potentials, at any density, and Vfor higherdensities favor magnon localization on the asites. On\nthe other hand, quantum fluctuations, magnon tunneling\nbetween aandbsites, are favored by t, at any density,\nandXfor higher densities. We use the density matrix\nrenormalization group to investigate the spin model and\nthe bosonic Hamiltonians t−Vandt−X−Vin finite-\nsize open systems, while for the t-approximation we have\nconsidered its analytical solution. We compare the mag-\nnetization and magnon densities per unit cell as a func-\ntion of a magnetic field, average bulk density and local\ndensities along the chains from the spin model and the\nbosonic approximations. From the ferrimagnetic plateau\n(one magnon per unit cell) to saturation (empty chain),\nthet−X−V-approximation is in excellent agreement\nwith the spin model, while the tandt−Vresults de-\npart from that of the spin model as the magnon density\nincreases. This, thus, showstherelevanceofbothinterac-\ntion terms, magnon-magnonrepulsion and the correlated\nhopping term X, and its associated particle fluctuations,\nto describe the many-body system near the ferrimagnetic\nplateau. The edge state associated with the insulating\nferrimagnetic plateau is well reproduced by the bosonic\nmodel. In particular, we have shown that the magnon\nboundary densities are strongly dependent on the inter-\nactions and the particle statistics.\nThe use of the fully polarized state as the vacuum en-\nables a better understanding of the underlying quantum\nprocesses in the spin Hamiltonian, compared to a ferri-\nmagnetic vacuum, which is not an exact eigenstate of the\nspin Hamiltonian. 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The maximum | ∆S| and adiabatic temperature change ( ΔTad) at T N \nare 11 J kg-1 K-1 and 1.9 K, respectively, for a magnetic fi eld change of 20 kOe. Moreover, it is \nfound that the complex magnetic phase transitions taking place below T N produce additional \n⎯but smaller ⎯ features on ∆S(T).  \n \n   PACS number(s): 75.50.Gg, 75.25.+z, 75.30.Kz, 75.60.Ej \n \n*Corresponding author. Email: wilfrid.prellier@ensicaen.fr  \n \n I. Introduction: \nResearch activity in the area of magneto-thermodynamics (MT) has received rapid \nimpetus in the recent past, mainly because of th e discovery of giant magnetic entropy changes in \nthe Gd 5(SixGe1−x)4 compounds.1, 2 Many efforts are underway to disc over materials showing large \nmagneto-caloric-effect (MCE) under moderate applied magnetic fields3, so that magnetic cooling \ntechnology may become a real ity in the near future.4,5 MCE is an importan t phenomena of MT \nwhich manifests itself as an is othermal magnetic entropy change ( ∆S) or an adiabatic \ntemperature change ( ∆Tad) when the magnetic material is exposed to a varying magnetic field.6 \nRefrigeration based on the MCE is advantageous in that it is an environmentally friendly and \nenergy efficient alternative to the co mmonly used vapor-cycle refrigeration.6 It was found that \ngiant MCE is exhibited by various in termetallic materials such as Gd 5(Si,Ge) 4 1,2, Mn(As,Sb) 7, \nLa(Fe,Si) 13 8,9, MnFe(P,As)10 , Heusler alloys 11,12 and some other compounds (see reviews Ref. \n3, 13-15 and references therein). In the case of tr ansition metal oxides, most of the MCE studies \nso far were focused on mixed-valenc y rare-earth based manganese oxides3,16, since some of them \nexhibit large | ∆S| values close to room temperat ure which are comparable with Gd.17  \nUp to now, no studies were devoted to the magnetocaloric properties of binary \nmanganese oxides (like MnO 2, Mn 2O3,, Mn 3O4 and Mn 5O8). This family of compounds combines \nseveral features favorable for applications; th ey are cheap, harmless and have a high chemical \nstability.18 Also, these oxides are potenti al candidates for i) rechargeable lithium batteries, ii) \ncatalysts, and iii) soft magnetic materials for transformers cores.18 Among all these \nantiferromagnetic oxides, the spinel Mn 3O4 is the only one which exhibits a  long-range \nferrimagnetic ordering19, i.e. a transition at wh ich significant MCE can be expected. It belongs to \nthe class of normal-spinel (AB 2O4) structured compounds, in which the tetrahedral A sites are occupied by a divalent cation (Mn2+) while all the octahedral B si tes are occupied by trivalent \ncations (Mn3+). At 1445 K, this compound undergoes a tetragonal to cubic phase transition (c/a \n~1.16) resulting from a collective Ja hn-Teller distortion within the e g orbitals of Mn3+ (t2g3 eg1).20 \nBelow 42 K~T N, a complex succession of rearrangements takes place in the magnetic structure. \nDespite the longstanding history19-24, this issue is still the subject  of intense research activity.25-27 \nAt the present time, the most widely accep ted picture of its magnetic ordering is 25-27: (i) At T N, \nMn 3O4 exhibits a  transition from the paramagne tic state towards a non-collinear ferrimagnetism \nof Yafet-Kittel type.28 That means a triangular structure in which the Mn2+ spins order \nferromagnetically along [110] (cubic settings) while the Mn3+ are symmetrically canted around [-\n1 -1 0 ], leading to a net mome nt antiparallel to that of Mn2+; (ii) At T 1∼39 K this coplanar spin \nstructure lying in (1 -1 0) evol ves to a spiral-type ordering relate d to a conical di stribution of the \nMn3+ spins around [110] (this type of magnetic struct ure is incommensurate  with the chemical \nlattice); (iii) Finally, at T 2∼33 K, the spin order rec overs a planar type stru cture. However, there \nis a doubling of the magnetic cell with respect to the chemical one, induced by a subdivision of \nthe Mn3+ into two sets. While the Mn2+ spins still order ferroma gnetically along [110], the Mn3+ \nspins exhibit a canting angle of ≈ 69° with respect to [-1 -1 0]. Very recently, Kim et al.  reported \nthat this magnetic transition is accompanied by a structural transition from tetragonal to \nmonoclinic symmetry, which is highly sensitive to the strength and orientation of the external \nmagnetic field.27  \nIn this context, our motivation to investigate the magnetocaloric properties of Mn 3O4 is \ntwofold. First, from a fundamental point of view , this compound offers an opportunity to study \nthe MCE associated with a wide variety of ma gnetic or magnetostructural transitions. Second, \nfrom an application point of view, the existen ce of such a sequence of transitions could be expected to generate MCE over a wide temperatur e range. This aspect is a favorable condition to \nobtain large refrigeration capacit y. Furthermore, our investigati ons also provide a comparative \nstudy on the reliability of the calorimetric and magnetic methods which are commonly used to \nderive the isothermal entropy changes. \n \nII. Experimental details: \nOxalate based sol-gel process has been chosen to synthesize Mn 3O4 polycrystalline \nsample. The initial product is manganese oxalate dihydrate (MnC 2O4·2H 2O), prepared using \nmanganese acetate tetrahydrate [Mn(CH 3COO) 2·4H 2O] and oxalic acid [C 2H2O4·2H 2O] as \nprecursors with ethanol as a solvent. Thermal decomposition of MnC 2O4·2H 2O at 500°C for 2 h \nduration yields Mn 2O3 as a major product; further annealing of this sample at 1100°C for 4 h in \nair leads to the formation of Mn 3O4 powder. This powder is grounded in an agate mortar, pressed \ninto bars and finally sintered at 1100°C for 12h. The cr ystallinity and struct ure has been studied \nusing a Panalytical X’Pert Pr o diffractometer using Cu K α radiation, which indicate monophasic \nnature of this compound consis tent with the hausmannite Mn 3O4 reported in the literature. A \ndetailed study of the formation me chanism has been reported elsewhere.18, 29 A superconducting \nquantum interference device (SQUID) based ma gnetometer (Quantum Design, MPMS-XL5) has \nbeen used for the temperature dependent magnetization measurements M H(T) in low-field, while \nmagnetic isotherms M T(H) up to 90 kOe were recorded by m eans of an extraction technique in a \n“physical property measurement system” (PPMS,  Quantum design). The heat-capacity curves \nC(T)  (in H=0 and H=20 kOe) were recorded from 2 to 100 K by using a semi-adiabatic relaxation technique in the PPMS. To stress the comparison between the calorimetric and magnetic methods, the paper is focused on the data  corresponding to a field change from zero to 20 kOe. Note also that H = 20 kOe is the typical value to be considered for potential applications \nusing permanent magnets. \nIII. Results and discussion: \nThe location of the above mentioned transitions (T N,T1,T2) on the magnetization and heat \ncapacity data are reported in the insets of Figure 1 and 2a , respectively. Apart from T N which has \na clear signature in both cases, the features associated to T 1 and T 2 are faint. To reveal them \nclearly, we display the temperature dependence of  the first derivative of the magnetization curve \n(–dM/dT) recorded upon cooling in a small magne tic field of 200 Oe (inset of Figure 1) while \ninset of Figure 2a shows the temperature depe ndence of C/T in zero field. There is a good \nconsistency between these two results, both leading to T N ≈ 42.75 K, T 1 ≈ 39.75 K and T 2 ≈  \n34.25 K. The positions of these transitions are in line with those previously reported in the \nliterature.23,25-27 \nIn order to derive the magnetic entropy ch ange, we first used the common method based \non the Maxwell equation applied to a series of isothermal magnetization curves. These \nisothermal magnetization curves were recorded as follows: (i) Each of th ese curves was preceded \nby a zero-field-cooling, ensuring to start from a virgin magneti c state; (ii) The temperature \nspacing between two successive M T(H) curves was quite large out of  the region of the transitions \n(δT=5K), and it was decreased to a small value inside it ( δT=2K); (iii)  The data were recorded \nupon increasing the magnetic fiel d from 0 to a value of H Max. Two series were recorded, with \nHMax = 20 kOe and H Max = 90 kOe. The results of ΔS(T) for magnetic field changes equal or \nlower than 20 kOe were checked to be the sa me in both cases. Making use of the Maxwell \nequation with such data, the temperature dependence of ΔS associated to a magnetic field change \nfrom 0 to H is usually derived from the formula:6 ∫ ⎥⎦⎤\n⎢⎣⎡\n∂∂= ΔH\nHdHTHTMHTS\n0'')',(),(     (1a) \nIt must be emphasized that this  relationship assumes that M(T,H)  can be regarded as a state \nfunction, i.e. taking a single value independent of the magneto thermal history. Accordingly, the \nMaxwell equation can be safely used around a sec ond-order transition, whereas  it is in principle \ninapplicable to a first-order transition.30-42 In the latter case, the analysis of the magnetic data that \nis required to estimate ΔS is much more complex. In the recent  years, various models have been \nproposed to address this issue wh ich is still intensively debated.31-33,35,37,38,40,42 \nIn the present case of Mn 3O4, the MCE primarily takes place around T N which is a second-order \ntransition. We thus derived ΔS(T) from the Maxwell equation, by using a slightly modified form \nof this equation which is more tractable in practice:15,41,43,44 \n∫∂∂= ΔH\ndHHTMTHTS\n0')',( ),(     (1b). \nThe main panel of Figure 1 displays the - ΔS(T) curve obtained at H = 20 kOe, when considering \nthe M T(H) curves recorded upon field increasing. As  expected for a ferrimagnetic transition, one \nobserves a large peak centered around T N, which reaches a maximum value of ∼ 7.5 Jkg-1K-1. \nThe overall shape of - ΔS(T) was found to be the same for larg e values of H, the main difference \nbeing the height of the peak which reaches ∼ 13 Jkg-1K-1 and ~ 18 Jkg-1K-1 for H = 50 and 90 \nkOe, respectively. Beside this main peak, the second prominent feature of - ΔS(T) in Figure 1 is \nthe presence of a crossing from positive to negative values as temperature is decreased below 20 \nK. Similar features were reported in  various types of manganese oxides.45 However, it must kept \nin mind that artifacts can easily be ge nerated when using the Maxwell equation.30-42 Before addressing this issue more precisel y, let us consider the results of ΔS(T) obtained from the \ncalorimetric method. \nThis approach is based on the recording of heat capacity curves, C(T) measured in zero-\nfield and in the external magnetic  field H. Assuming that the basi c relationship dS = (CdT)/T is \nalways obeyed (i.e, dealing with reversible tran sformations), the isothermal entropy change is \ndirectly computed from the following equation,46 \n∫ ⎥⎦⎤\n⎢⎣⎡− ∫ ⎥⎦⎤\n⎢⎣⎡= − = ΔT T\nH dTTTCdTTHTCTS TS HTS\n0 00)0,( ),()( )( ),(    (2)  \nThe C(T) curves in H = 0 and H = 20 kOe were measured upon warming after a zero-\nfield cooling down to the lowest reachable temperature, i.e. 2 K in the present case. Below 2K, \nthe C/T values are linearly extrapolated to zero at T = 0 K. The - ΔS(T) derived using the above \nprocess is depicted in Figure 2a, where the main feature is again the presence of a large peak \ncentered at T N. However two main differences emerge when comparing to the magnetic method. \nFirst, the height of the peak is substantia lly higher, reaching a maximum value of 11 J K-1 kg-1 \ninstead of 7.5 J K-1 kg-1. Second, the - ΔS(T) at low-temperatures monotonically reaches zero as \nthe temperature is decreased wit hout exhibiting any crossover to negative values. The first issue \nis a direct consequence of the limited reso lution of the magnetization method. Indeed, the ΔS \nvalue derived from Eq. (1) actually reflects an average over the temperature range T ± δT, where \nδT is the spacing betw een two consecutive M T(H) curves. In particular, one can check that the \nmaximum value derived from magnetization, ΔS (44K) ≈ -7.5 J kg-1 K-1, well corresponds to \n44.7 2/ )(44\n44−=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛Δ∫+\n−T dTTST\nTC δδ\nδ J K-1 kg-1, where ΔSC(T) is the curve derived from heat capacity. \nOf course, reducing δT minimizes this effect, but this is limited by a growing uncertainty in the derivative of Eq. (1) when the M H(T) become too close to each othe r. In reality, the intrinsically \nhigher resolution of the calorimetric method makes it more suitable to derive the peak of ΔS(T) \nin  case of a sharp transition, as presently found around T N. \nIn other respects, the absen ce of positive values in the ΔS(T) curves derived from C(T) at \nlow-temperatures led us to reconsider the results obtained from the magnetization method. \nActually, the question is not about the quality of magnetic data nor the Maxwell equation in \nitself, but it deals with a po ssible inappropriate use of it.6,35,36,46 For instance, it is now widely \nrecognized that such a type of problem  is at the origin of a lot of huge ΔS values reported in the \nliterature, which are in fact artifacts.30-42 An important point sometimes forgotten is that the \nMaxwell equation is supposed to hold only in presence of a reversible magnetic behavior.6,35,36,46 \nHence, a first check about the legitimacy of usi ng the Maxwell equation is to compare the M(H) \ncurves recorded either upon fi eld-increasing (“Up” curve) or field-decreasing (“Down” curve). \nFigure 3 shows some of such magnetic cycles at various temperatures. The M(H) curves \nrecorded at temperatures higher th an ~ 30 K (see Figure 3a), i.e. in the temperature range of the \ntransitions, are found to be reversible (or almost  reversible), authorizing the use of the Maxwell \nequation. In contrast, at lower temperature (see Figure 3b), a noticeable hysteresis is observed between the field increasing and field decreasing branches of M(H) curves. This behavior \nbecomes much more prominent at low temperatures (below ∼ 15 K). Therefore, the Maxwell \nequation is simply not valid to estimate the ΔS using Equation (1) at low-temperatures.\n  Looking \nat Figure 3b, one can notice that the apparent “positive” values of ΔS(T) reflect a shift of the \nM(H) “Up”  curves towards higher fields as  the temperature is decreased. Meanwhile the \napplication of the Maxwell equation on the “Down”  series of M(H) curves would just yield \nvanishing positive values of ΔS(T) in the same low-temperature regime. In fact, such a low-temperature behavior of the M(H) curves is  mainly driven by the pronounced temperature \ndependence of the coercive field in Mn 3O4,19 a phenomenon which should not be confused with a \ngenuine MCE. \nFinally, a closer look at the - ΔS(T) curve derived from heat-capacity reveals two \nadditional features which are more clearly ev idenced in the logarithmic scale (Figure 2b). \nAround 40 K (close to T 1), one can observe a weak “kink”, while a small “dip” appears around \n33 K (near to T 2). The kink at T 1 can be regarded as an addi tional negative contribution of \nentropy change superimposed onto the low-temp erature side of the main peak around T N. Such a \nbehavior is well corresponding to a shoulder observed around T 1 in –dM/dT curve (see inset \nFigure 1). In terms of heat capacity , the above feature is also consistent with the fact that around \nT1, C(T, H=20kOe) lies below C(T, H=0) and has its maximum slightly shifted towards higher \ntemperature. At T 2, the anomaly looks different, since it ra ther appears as a “dip” superimposed \nonto the low-temperature wing of the main peak. It turns out that this feature comes from a \npeculiar influence of the magnetic  field on the C(T) curve around T 2. As compared to zero-field, \nthe peak present on the C(T) curve in 20 kOe starts developing at a lower temperature ( ≈ 32.5 K \ninstead of ≈ 33.0 K) while its maximum is slightly shifted to higher temperature ( ≈ 34.3 K \ninstead of ≈ 34.0 K). The crossing between the C(T) cu rves in H=0 and H=20 kOe is close to ≈ \n33.6 K. In terms of MCE, such a field-induced modification in the prof ile of the transition \ninduces a dip in - ΔS(T), that is centered at  the crossing temperature.  Moreover, the height of \nsuch a peak is expected to be only a fraction of the entropy jump associated with the first-order \ntransition at T 2. By integrating the anomaly on the C/T(T) curves in zero or 2T, this entropy \njump is found to be ∼ 0.5 J K-1 kg-1, while the height of the dip-like anomaly on the ΔS(T) curve \nis indeed smaller, i.e. ∼ 0.1 J K-1 kg-1.  With the calorimetric method, one can also obta in the second characte ristic quantity of \nthe MCE, i.e. the adiaba tic temperature change ΔTad. These values were derived from the \nentropy curves using the equation,46 \n[])( 00)( )( ) 0,(TS H ad STST H TT − = → Δ      ( 3 )  \nThe resulting ΔTad(T) curve in the case of field change from 0 to 20 kOe is shown on \nFigure 4. Similarly to the ΔS(T) curve, the main feature in ΔTad(T) is a prominent peak located \nclose to T N. One can also observe a clear signature of the anomaly at T 2 in the ΔTad(T) plot. It \ncan be noted that the maximum ΔTad of 1.9 K for a moderate applied magnetic field of 20 kOe, is \nlarge enough to regard Mn 3O4 as a potential refrigerant materi al. Nevertheless, the width of the \nΔS(T) and ΔTad(T) curves are quite small, which woul d yield only a modest relative cooling \npower. The best magnetocaloric materials reported  so far in the temperature range around 50 K \nbelong to the family of the Laves phases.3,15 Even though the performances of Mn 3O4 well \ncompare to those of these compounds showing second-order ferromagnetic transitions, slightly \nbetter results are observed in the case of first-order transitions. As a matter of fact, the best Laves \nphase ErCo 2 (TC = 37 K) yields a maximum ΔTad = 3 K for a field change of 20 kOe.3,15,47,48 \n \nIV. Conclusions: \nIn conclusion Mn 3O4 is found to exhibit a substant ial MCE around its ferrimagnetic \ntransition at T N (42.75 K). For a field change of  20 kOe, the peak values of ⎥ΔS⎥ and ⎥ΔTad⎥ are \n11 J K-1 kg-1 and 1.9 K, respectively. The vari ous transitions taking place from T N down to ∼ 33 \nK are also found to have signature s on the isothermal entropy cha nge. However, these effects are \nsmaller and do not really contribute to widen th e operating temperature range for applications in \nmagnetic refrigeration.  This study also well illustrates the advantages  of the calorimetric method as compared to \nthe widely used magnetic technique based on the Ma xwell equation: First, it is less prone to the \ngeneration of spurious features, as  those we presently observed in Mn 3O4 owing to the \nappearance of magnetic irreversib ility at low temperatures ; S econd, the calorimetric method \nfacilitates a better temperature resolution which allows to detect the small anomalies and avoid \nunderestimating the maximum ΔS value in case of sharp peaks. \n \n \nV. Acknowledgements: \nResearch at Laboratoire CRI SMAT is supported by Indo -French Centre for the \nPromotion of Advance Research /Centre Franco-Indien pour la Promotion de la Recherche \nAvancée (IFCPAR /CEFIPRA), the CNRS Energy Progr am (Research Project “Froid \nMagnétique”) and ENERMAT. \n \nReferences  \n 1 V. K. Pecharsky and K. A. Gschneider, Jr., Phys. Rev. Lett. 78, 4494 (1997). \n 2 A. O. Pecharsky, K. A. Gschneidner, Jr., and V. K. Pecharsky , J. Appl. Phys. 93, 4722 (2003).\n 3 K. A Gschneidner Jr., V. K. Pechars ky and A. O. Tsokol, Rep. Prog. Phys. 68, 1479 (2005), \nand references therein. \n 4 K. A. Gschneidner, Jr., V. K.  Pecharsky, Int. J. Refrig. 31, 945 (2008), and references therein. \n 5 B. Yu, M. Liu, P. W. Egolf, A. Kitanovski, Int. J. Refrig. 33, 1029 (2010), and references \ntherein. \n 6 A. M. Tishin and Y. I. Spichkin, The Magnetocaloric Effe ct and Its Applications . (Institute of \nPhysics, Bristol, Philadelphia, 2003), and references therein. \n 7 H. 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Shiga, Cryogenics 39, 915 (1999). \n Figure captions  \nFigure 1. Temperature dependence of the isothermal entropy change derived from the magnetic \nmethod, using the field-increasing branches of M( H) for a field variation from 0 to 20 kOe. The \ninset displays the derivative of the magnetizat ion curve measured in 200 Oe. The arrows show \nthe transitions taking place, from right to left, at T N, T1, and T 2.  \n \nFigure 2. (a): Main panel: Temperature dependen ce of the isothermal entropy change for a field \nvariation of 20 kOe, where the “magnetic” ΔS(T) (open squares) is  derived from the field \nincreasing branches of M(H), while the “calorimetric” ΔS(T) (filled circles) is derived from C(T) \ncurves recorded upon warming. The inset shows the temperature dependence of C/T in zero-field. (b): Zoomed view of the isothermal entropy change estimated from the heat capacity data, represented in a semi-logarithmic scale. The arro ws highlight the anomalies present, from right \nto left, at T\n1 and T 2. \n \nFigure 3. Isothermal magnetization curves reco rded either upon increasing (filled symbols) or \ndecreasing (empty symbols) the ma gnetic field. (a): High-T regime  with curves at 30, 35, 40 and \n45 K; (b): Low-T regime with curves at 5, 10, 15 and 20 K.  Figure 4. Temperature dependence of the adiabatic temperature change in Mn\n3O4, measured for a \nmagnetic field change of 20 kOe. "    },    {        "title": "2308.04274v2.Ionically_Driven_Synthesis_and_Exchange_Bias_in_Mn___4__N_MnN___x___Heterostructures.pdf",        "content": "1 \n Ionically-Driven Synthesis and Exchange Bias in Mn 4N/MnN x Heterostructures \n \nZhijie Chen,1 Christopher J. Jensen,1 Chen Liu,2 Xixiang Zhang,2 and Kai Liu1,* \n1Physics Department, Georgetown University, Washington, DC 20057, USA \n2King Abdullah University of Science & Technology, Thuwal 23955-6900, Saudi Arabia \n \nAbstract \nFerrimagnets have received renewed attention as a promising platform for spintronic \napplications. Of particular interest is the Mn 4N from the ε-phase of the manganese nitride as an \nemergent rare-earth-free spintronic material due to its perpendicular magnetic anisotropy, small \nsaturation magnetization, high thermal stability, and large domain wall velocity. We have achieved \nhigh-quality (001)-ordered Mn 4N thin film by sputtering Mn onto η-phase Mn 3N2 seed layers on \nSi substrates. As the deposited Mn thickness varies, nitrogen ion migration across the Mn 3N2/Mn \nlayers leads to a continuous evolution of the layers to Mn 3N2/Mn2N/Mn4N, Mn 2N/Mn4N, and \neventually Mn 4N alone. The ferrimagnetic Mn 4N indeed exhibits perpendicular magnetic \nanisotropy, and forms via a nucleation-and-growth mechanism. The nitrogen ion migration is also \nmanifested in a significant exchange bias, up to 0.3 T at 5 K, due to the interactions between \nferrimagnetic Mn 4N and antiferromagnetic Mn 3N2 and Mn 2N. These results demonstrate a \npromising all-nitride magneto-ionic platform with remarkable tunability for device applications. \n \n*Email: kai.liu@georgetown.edu  2 \n Ferrimagnets (FiMs) are characterized by antiparallel-coupled magnetic sublattices with \ndifferent magnetic moments. They have gained tremendous interests recently as a promising \nplatform for spintronic applications as they offer the combined benefits of ferromagnets (FMs) and \nantiferromagnets (AFs),1-4 such as easy manipulation with an external magnetic field and fast spin \ndynamics.5 Moreover, some FiMs such as rare-earth iron garnets,6 rare earth-transition metal \nfilms,1 and Mn 4N7 also possess perpendicular magnetic anisotropy (PMA) under appropriate \ngrowth conditions, a desirable functionality for device applications. Among them, Mn 4N stands \nout due to its high Curie temperature (T C = 745K),8 PMA,7, 9-11 small saturation magnetization (up \nto 145 emu/cm3),9 large domain wall velocity,12 and possibility of hosting non-trivial spin \ntextures.13, 14 It can also be doped with other elements to change its magnetic properties and \ncompensation point.15-17 As the only ferrimagnetic phase among the four stable manganese \nnitrides,18 namely θ-MnN,19 η-Mn 3N2,20, 21 ζ-Mn2N,22 and ε-Mn 4N,8 Mn4N has an anti-perovskite \nstructure with two inequivalent and anti-aligned Mn sublattices at the corners and face centers, and \na nitrogen atom at the body center.  \nFurthermore, there have been emerging interests in magneto-ionics, where energy-efficient \ncontrol of magnetic properties can be achieved via ionic migration.23-40 Nitrogen-based magneto-\nionics has been shown to exhibit good cyclabilities and fast ionic migration.34, 41-43 Recently, \nnitride-based exchange bias has been reported in MnN/CoFe heterostructures, which can be \nmanipulated via nitrogen ion migration in and out of the MnN layer using an electric field.44 This \nmotivates us to build an all-nitride magneto-ionic system taking advantage of the different \nmagnetic phases in Mn nitrides.  \nMn4N thin films are typically grown onto SrTiO 3 or MgO substrates at elevated \ntemperatures through molecular beam epitaxy, pulsed laser deposition, or reactive sputtering in a 3 \n nitrogen environment.7, 9-12, 14 The film quality is susceptible to the nitrogen flow rate or partial \npressure, and the optimum growth conditions vary from study to study.7, 11 It is challenging to grow \nhigh quality thin films of Mn 4N directly on Si substrates,45, 46 which are CMOS compatible. In this \nwork, we demonstrate that high-quality (001)-ordered Mn 4N thin films can be grown on Si \nsubstrates by directly sputtering pure Mn onto an Mn 3N2 seed layer at elevated temperatures, \nresulted from the chemical reaction between Mn and the nitrogen in the Mn 3N2 seed layer. In a \nnominally Mn 3N2 (20nm)/Mn ( 𝑡ெ) series of samples, by changing the deposited Mn thickness \n𝑡ெ  from 0 to 50 nm, nitrogen ion migration gradually transforms the layers into \nMn3N2/Mn2N/Mn4N, Mn 2N/Mn4N and eventually Mn 4N, confirmed by X-ray diffraction (XRD), \ntransmission electron microscopy (TEM), and electron energy loss spectroscopy (EELS). The \nMn4N films are found to exhibit PMA. First-order reversal curve (FORC) measurements reveal \nthat the Mn 4N forms with a nucleation-and-growth process. The nitrogen ion migration is also \nmanifested in a significant exchange bias, up to 0.3 T at 5 K, due to the interaction between \nferrimagnetic Mn 4N and antiferromagnetic Mn 3N2 and Mn 2N.  \nSeed layers of 20 nm Mn 3N2 were first reactive-sputtered onto Si substrate with 285nm \nthermally oxidized SiO 2 layer from a Mn target using direct current (dc) in an ultrahigh vacuum \nchamber with a base pressure better than 5×10ି଼ torr. The substrate temperature was kept at \n450 °C, and the Ar : N 2 ratio was held at 1 : 1 with a 5 mtorr sputtering gas pressure. These Mn 3N2 \nfilms were then left in vacuum for 30 min at the same substrate temperature to promote nitrogen \nreordering. Subsequently, 0 - 50 nm of Mn was deposited onto the Mn 3N2 layer at the same 450 °C \nsubstrate temperature in an Ar-only environment. After deposition, substrate heating was \nimmediately turned off, and the samples were cooled to room temperature before depositing a 5 \nnm Ti capping layer.  4 \n Structural characterizations were performed using XRD on a Panalytical X'Pert³ MRD with \nsymmetric 2θ-ω and grazing incidence geometries. Sample microstructures and composition \nanalysis were done using an FEI Titan Themes Cubed G2 300 (Cs Probe) TEM at KAUST.  \nMagnetic measurements were carried out using a Quantum Design superconducting \nquantum interference device (SQUID) magnetometer. Exchange bias was measured at 5 K by first \nfield-cooling the sample from 300 K in a 2 T magnetic field, all in the out-of-plane (OP) geometry. \nFORC measurements47-51 were done in a vibrating sample magnetometer at room temperature by \nsaturating samples in a 1.5 T OP magnetic field and then measuring from a reversal field ( 𝐻ோ) back \nto saturation. This process was repeated at different 𝐻ோ to fill the interior of the hysteresis loop, \ncreating a family of FORCs. A FORC distribution was then calculated using Eqn. (1), \n𝜌 (𝐻,𝐻ோ)≡ −ଵ\nଶெೄபమெ(ு,ுೃ)\nడு డுೃ.                                                   (1) \nwhere 𝑀ௌ is the saturation magnetization, and 𝑀(𝐻,𝐻ோ) is the magnetization at the applied field \n𝐻 with reversal field 𝐻ோ. The FORC distribution can also be represented in terms of local coercive \nfield and bias field ( 𝐻, 𝐻) defined by 𝐻=ଵ\nଶ(𝐻−𝐻ோ) and 𝐻=ଵ\nଶ(𝐻+𝐻ோ).                                                      \nThe η-phase Mn 3N2 is chosen as the seed layer for Mn 4N growth because it provides the \ncrystalline texture and nitrogen needed for the Mn 4N growth.52 As shown in Fig. 1(a) and (b), \nMn3N2 is antiferromagnetic (T N ~925 K), and its c-axis is about three times that of Mn 4N.21, 53 \nXRD reveals the growth of the Mn 3N2 phase, shown in Fig. 1(c), with a preferred orientation along \nthe (010) direction and the c-axis in the film plane. Furthermore, grazing incidence XRD confirms \nthat all peaks are from Mn 3N2 (Supplementary Material Fig. S1). Upon depositing 40 nm of Mn, \nXRD shows that the film is primarily the Mn 4N phase oriented along (001), with no appreciable \nMn3N2 phase left, as shown in Fig. 1(d) and Supplementary Material Fig. S1. Along with the 5 \n structure changes, there is also a drastic change in the film magnetic properties. As shown in Fig. \n1(e), the initial Mn 3N2 layer does not exhibit any magnetic signal, consistent with its \nantiferromagnetic nature ; interestingly, the sample deposited with Mn exhibits a square loop with \na large coercivity (0.27 T) and a small 𝑀ௌ (85 emu/cm3) that are typical of Mn 4N films.7, 9, 10 \nTo understand how the film transforms from Mn 3N2 to Mn 4N by only depositing Mn, we \nhave investigated a series of samples starting with 20 nm Mn 3N2 seed layer, but the deposited Mn \nnominal thickness varied from 0 to 50 nm with a 5 nm step size. From now on, each sample is \nreferred by its deposited Mn thickness ( 𝑡ெ) unless otherwise stated. Fig. 2(a) reveals how the Mn \nnitride phase evolves across the samples. Starting from 𝑡ெ = 0 nm, which is the Mn 3N2 layer, the \nonly peak is the Mn 3N2 (020). As 𝑡ெ  increases, a prominent peak emerges around 47.2 , \ncorresponding to the Mn 4N (002), indicating Mn 4N formation as Mn is deposited. On the other \nhand, the Mn 3N2 (020) peak diminishes and shifts to higher angles before eventually vanishing in \nthe 𝑡ெ = 30 nm sample. This trend indicates that the Mn 3N2 phase is fading and is not as stable \nas Mn4N at high temperatures, consistent with prior studies.19, 54 Interestingly, as the Mn 3N2 peak \ngets smaller, another peak emerges near 42.2  in the 𝑡ெ = 20 nm sample and grows larger before \ndisappearing in the 𝑡ெ  = 40 nm sample. This peak is the (111) diffraction from the ζ-phase \nMn2N0.86, which has a thermal stability and nitrogen content between η-Mn 3N2 and ε-Mn 4N.18, 19, \n54 In the 𝑡ெ = 40 and 45 nm samples, both Mn 3N2 and Mn 2N peaks have vanished, while the \nMn4N (002) peak gets even larger and closer to its expected location. Eventually, when 𝑡ெ \nreaches 50 nm, the α-Mn (221) peak shows up near 43 , indicating that some deposited Mn remain \nunreacted as the entire nitride film is now Mn 4N. Grazing incidence scans are included in the \nSupplementary Material Fig. S1, consistent with Fig. 2(a), along with full range 2 - scans (Fig. \nS2). 6 \n   The Mn 4N crystallite size has been estimated from the full-width-at-half-maximum \n(FWHM) of the (002) peak, after instrument broadening correction, using the Scherrer equation.55 \nThe Mn 4N crystallite size nearly doubles as more Mn is deposited, reaching a plateau after 𝑡ெ = \n40 nm, as shown in Fig. 2(b). This is consistent with the trend in Fig. 2(a), where the Mn 4N peak \nbecomes sharper and more prominent, and Mn 4N is the only phase after 𝑡ெ reaches 40 nm. This \ncrystallite size estimation is rather simplified, as it ignores peak width contribution from other \nfactors such as inhomogeneities in d-spacing. As the film stoichiometry changes due to the nitrogen \nmigration, any spread in N-content and the lattice parameters would lead to a broadening of the \npeak width. Interestingly, the overall narrowing trend of the Mn 4N peak width with increasing \n𝑡ெ suggests that the stoichiometry variation is suppressed at high 𝑡ெ, which is consistent with \nthe fact that when 𝑡ெ reaches 40 nm, only a single phase Mn 4N is observed.  Moreover, the peak \nlocations shift to higher angles as 𝑡ெ  increases, as shown in Fig 2(c). Nitride phases’ lattice \nconstants are known to be very sensitive to nitrogen content,18 as interstitial nitrogen usually causes \nthe lattices to expand. As Mn 3N2 loses nitrogen to the deposited Mn, its lattice contracts, causing \nthe Mn 3N2 peak to shift to a higher angle until this phase is gone. The Mn 4N peak location, on the \nother hand, stays relatively constant before changing rapidly beyond 𝑡ெ = 40 nm, likely caused \nby the nitrogen redistribution within the Mn 4N phase once the nitrogen from Mn 3N2 and Mn 2N \nhas been depleted. Thus, we postulate that as Mn is deposited onto the Mn 3N2 layer at elevated \ntemperatures, it reacts with the nitrogen coming from Mn 3N2 and forms Mn 4N. While Mn 3N2 loses \nnitrogen, it first turns into Mn 2N and eventually becomes Mn 4N. These reactions are summarized \nin Eqn. 2a, and they can be combined into one chemical reaction  (Eqn. 2b) since they are multistep \nreactions. \n2𝑀𝑛ଷ𝑁ଶ→ 3𝑀𝑛ଶ𝑁+𝑁; 4𝑀𝑛+𝑁→𝑀𝑛ସ𝑁; 2𝑀𝑛ଶ𝑁→𝑀𝑛ସ𝑁+𝑁            (2a) 7 \n 5𝑀𝑛+𝑀𝑛ଷ𝑁ଶ→ 2𝑀𝑛ସ𝑁;                                              (2b) \nWe also calculated the enthalpy of formation for the reaction shown in Eqn. 2b to be -110 kJ/mol \nusing the standard enthalpy of formation for Mn 4N and Mn 3N2,54 indicating that this reaction is \nthermodynamically favorable. \nCross-sectional TEM and EELS lines scans were performed to explore the microstructure \nand nitrogen content evolution as 𝑡ெ varies. Fig. 2(d) shows the EELS line scan along the film \nthickness (green line in the inset) for 𝑡ெ = 0 nm. The Mn:N composition ratio is determined to be \n58:42, consistent with the nominal atomic ratio of Mn 3N2. As 𝑡ெ increases to 40 nm, the EELS \nscan shown in Fig. 2(e) illustrates that the Mn:N ratio is now 84:16, also consistent with the \nnominal atomic ratio of Mn 4N. Interestingly, most of the nitride layers appear homogenous with \nconstant Mn:N ratio from both the cross-sectional TEM and EELS, indicating that nitrogen in the \nMn3N2 seed layer has redistributed to maintain a constant nitrogen concentration within the Mn \nnitrides after the Mn is deposited. These results further corroborate our postulation that nitrogen \nmoves from the Mn 3N2 seed layer into the Mn layer to form more stable Mn 4N.  \nWe then investigate the magnetic properties of this series of samples. Fig. 3(a) shows the \nroom temperature hysteresis loops with in-plane (IP) and out-of-plane (OP) magnetic fields. The \nOP loops get more square and broader as 𝑡ெ increases from 10 to 50 nm while the IP loops stay \nrelatively constant. These trends are further revealed by plotting the squareness, or ratio of \nremanence magnetization ( 𝑀) over 𝑀ௌ, for each sample, Fig. 3 (b). The OP and IP remanence are \nsmall and stay relatively constant for 𝑡ெ < 20 nm, indicating the lack of a clear magnetic easy \naxis. For 20 nm < 𝑡ெ < 35 nm, a sharp jump in OP remanence is observed, along with a drop in 8 \n IP remanence, indicating a clear easy axis has been established in the OP direction. At 𝑡ெ > 35 \nnm, the easy axis remains OP, while IP remanence increases slightly but remains low.  \nWe have also calculated the uniaxial magnetic anisotropy constant ( 𝐾௨, see Supplementary \nMaterial). As shown in Fig. 3(c), 𝐾௨ starts out to be negative for 𝑡ெ = 5 nm and shows a clear \nswitching from negative to positive, especially when 𝑡ெ > 20 nm, further confirming the magnetic \neasy axis switching to OP as more Mn 4N is formed. Note that 𝐾௨ exhibits the largest value (0.03 \nMJ/m3), or the film has the largest PMA when 35 nm < 𝑡ெ < 45 nm. This is also consistent with \nthe XRD result, which shows that Mn 4N is the only phase for this 𝑡ெ range. This 𝐾௨ value is \nsmaller than other reported values which range from 0.05 to 0.2 MJ/m3.7, 9-11, 56, 57 The uniaxial \nanisotropy has been attributed to the tetragonal distortion caused by in-plane tensile strains.7, 9, 10 \nHowever, in this study, the films that show the largest PMA has c/a close to 1 (Table 1 in \nSupplementary Material). The lack of in-plane strain may be the reason for our smaller 𝐾௨ values \nsince the films are deposited onto an amorphous SiO 2 layer. Additionally, the sizeable PMA in our \nfilms may have come from other contributions such as the shape anisotropy of the Mn 4N grains.58 \nTo investigate how the Mn 4N phase evolves with 𝑡ெ and the corresponding magnetization \nreversal, we have carried out FORC studies in the OP geometry at room temperature, as shown in \nFig. 4. For the 𝑡ெ = 10 and 20 nm samples, individual FORCs fill the major loops in a slanted \nfashion, Fig. 4a and 4c, respectively. The corresponding FORC distributions exhibit a prominent \nvertical ridge centered around 𝐻  = 0, which corresponds to reversible switching,51, 59 and a \nsmaller horizontal feature centered at 𝜇𝐻 = 120 mT and 150 mT, respectively (Fig. 4b, 4d). This \nindicates that the Mn 4N film is mainly reversible and magnetically soft. Likely for this 𝑡ெ range, \nthe Mn 4N phase is just emerging in small clusters scattered in an antiferromagnetic matrix of 9 \n Mn3N2 and Mn 2N. As more Mn 4N is formed, families of FORCs for 𝑡ெ = 30, 40, and 50 nm are \nconsiderably different, as individual FORCs return to positive saturation in a more horizontal \nfashion, consistent with the establishment of a magnetic easy axis (Figs. 4e, 4g, 4i). Their FORC \ndistributions are also strikingly different. The previous large vertical reversible ridge at 𝜇𝐻 = 0 \nbecomes smaller and eventually vanishes in the 𝑡ெ = 50 nm sample. The horizontal feature along \nthe 𝐻  axis now becomes prominent and shifts to higher 𝜇𝐻  of 460, 310, and 390 mT, \nrespectively (Fig. 4f, 4h, 4j). The change in relative intensity of the horizontal and vertical FORC \nfeatures likely indicate that Mn 4N forms via a nucleation-and-growth mechanism, similar to that \nreported previously in the ordering of L10 FeCuPt.59  \nIn this nominally Mn 3N2 (20nm) /Mn ( 𝑡ெ) series of samples, the evolution of the AF phase \nand the emergence of the FiM phase are also manifested in the exchange bias effect, which was \nstudied at 5 K after cooling the samples from room temperature in a positive 2 T OP magnetic field. \nA significant horizontal shift to the negative field direction, up to 300 mT, and a coercivity \nenhancement can be seen, Fig. 5(a), typical of exchange bias systems.44, 60-62 The 𝑡ெ dependence \nof coercivity ( 𝐻) and exchange field ( 𝐻ா) both exhibit non-monotonic trends, with an intriguing \npeak around 20 nm < 𝑡ெ < 30 nm, as shown in Fig. 5(b). These trends are likely a combined effect \nfrom the AF phase evolution as well as the FiM thickness and 𝑀ௌ variations. To further explore \nthe exchange anisotropy independent of the FiM, we have evaluated the interfacial exchange \nenergy (𝐽௧) per unit area using Eqn. 3:60, 62 \n𝐽௧=𝑀ிெ𝑡ிெ𝐻ா=𝑚ிெ𝐻ா/𝐴                                                       (3) \nwhere 𝑀ிெ, 𝑚ிெ, and 𝑡ிெ are the FiM saturation magnetization, saturation magnetic moment, \nand layer thickness, respectively, 𝐻ா is the exchange field, and A is the sample area. As shown in \nFig. 5(c), the dependence of 𝐽௧ on  𝑡ெ exhibits a bell-shaped plot that peaks around 20 to 30 10 \n nm. 𝐽௧ is small and increases continuously for 𝑡ெ of 5 - 15 nm, where the dominating AF phase \nis Mn3N2, as observed by XRD. Due to the high T N of Mn 3N2, only a small fraction of the AF \nspins is aligned to pin Mn 4N by field cooling from room temperature, resulting in a small 𝐽௧. \nHowever, for 20 nm < 𝑡ெ <35 nm samples, another AF phase, Mn 2N, starts to dominate. By \nfield cooling from 300 K, the Mn 2N is effectively coupled with Mn 4N, resulting in a significant \nexchange bias at 5 K. Exchange energy then quickly decreases as Mn 2N is turned into Mn 4N. By \n𝑡ெ = 40 nm, no AF phases can be identified from XRD and 𝐽௧ mostly vanishes. \nIn summary, we have achieved high-quality Mn 4N films growth by depositing pure Mn \nonto an Mn 3N2 seed layer. By varying the Mn thickness 𝑡ெ, the nitrogen concentration in the \nstarting Mn 3N2/Mn bilayers can be continuously tuned to be Mn 3N2/Mn2N/Mn4N, Mn 2N/Mn4N, \nand eventually to Mn 4N alone, as observed by XRD and TEM/EELS. With increasing 𝑡ெ, more \nMn4N is formed, with an increasing PMA reaching 0.03 MJ/m3. FORC measurements further \nreveal that Mn 4N forms via a nucleation-and-growth mechanism. A large exchange bias up to 0.3 \nT is found at 5 K in this all-nitride system. The variation of the exchange anisotropy is further \nattributed to the phase change of the antiferromagnets caused by nitrogen redistribution. These \nresults demonstrate an effective all-nitride magneto-ionic platform for studying the properties of \nthe emergent ferrimagnetic Mn 4N and its potential applications in spintronics. \nSupplementary Material \nGrazing incidence and full range 2 - X-ray diffraction patterns, Mn nitride phase \nevolution with layer thickness, cross-sectional TEM and EELS line scans, uniaxial magnetic \nanisotropy calculation, and additional sample information (PDF). \n 11 \n Acknowledgement \nThis work has been supported in part by the NSF (DMR-2005108, ECCS-2151809), by \nSMART (2018-NE-2861), one of seven centers of nCORE, a Semiconductor Research Corporation \nprogram, sponsored by National Institute of Standards and Technology (NIST), and by KAUST \n(OSR-2019-CRG8-4081). The acquisition of a Magnetic Property Measurements System \n(MPMS3), which was used in this investigation was supported by the NSF-MRI program (DMR-\n1828420). 12 \n Figures \n \n \nFigure 1. Schematics showing the lattice structure and spin orientation of (a) Mn 3N2 and (b) Mn 4N. \nXRD 2θ-ω scans of (c) 20 nm Mn 3N2 seed layer and (d) Mn 4N sample fabricated by depositing 40 \nnm Mn on top of 20 nm Mn 3N2. (e) Room temperature out-of-plane hysteresis loops for the same \nMn3N2 and Mn 4N samples.  \n \n13 \n  \n  \nFigure 2. (a) XRD 2θ-ω scans for nominally Mn 3N2 (20nm) / Mn ( 𝑡ெ) films, 𝑡ெ is the deposited \nMn thickness listed next to each curve. Vertical lines show the expected peak locations of Mn 4N \n(002) (red), Mn 3N2 (020) (blue), and Mn 2N0.86 (111) (black). Trends showing (b) Mn 4N crystallite \nsize and (c) Mn 4N (002) (red) and Mn 3N2 (020) (blue) peak location variations. Solid lines are \nguides to the eye. Dotted horizontal lines are the expected peak locations for Mn 4N (002) (red) and \nMn3N2 (020) (blue). EELS scans along the green line in the TEM insets showing Mn:N ratio across \nthe sample for (d) 𝑡ெ = 0 nm and (e) 𝑡ெ = 40 nm, where 𝑍 = 0 is the starting point of the interface \nbetween the substrate and MnN x layers. \n \n14 \n   \nFigure 3.  (a) Room temperature hysteresis loop with out-of-plane (left) and in-plane (right) \nmagnetic fields for nominally Mn 3N2 (20nm) / Mn ( 𝑡ெ) films, where 𝑡ெ is the deposited Mn \nthickness listed next to each curve. Trends for the (b) in-plane and out-of-plane remanence and (c) \n𝐾௨ as 𝑡ெ increases. Solid lines in (b) and (c) are guides to the eye.  \n \n15 \n  \nFigure 4. Families of FORCs (top row) and FORC distributions (bottom row) for (a,b) 𝑡ெ = 10 \nnm, (c,d) 20 nm, (e,f) 30 nm, (g,h) 40 nm, and (i,j) 50 nm. 𝑡ெ is the deposited Mn thickness on \ntop of 20 nm Mn 3N2 seed layer.  \n \n16 \n  \nFigure 5. (a) Hysteresis loops of the nominally Mn 3N2 (20 nm)/Mn ( 𝑡ெ ) series of samples \nmeasured at 5 K after 2 T field-cooling from 300 K, with the 𝑡ெ  listed for each sample. (b) \nDependence of coercivity (black) and exchange field (blue) on 𝑡ெ at 5 K. (c) Dependence of \ninterfacial exchange energy on 𝑡ெ  at 5 K where the color represents the different dominating \nantiferromagnetic phases: Mn 3N2 (blue), Mn 2N (red), and no antiferromagnets (black). Solid lines \nin (b) and (c) are guides to the eye.  \n \n  \n17 \n References \n1 L. Caretta, M. Mann, F. Buttner, K. Ueda, B. Pfau, C. M. Gunther, P. Hessing, A. Churikova, C. \nKlose, M. Schneider, D. Engel, C. Marcus, D. Bono, K. Bagschik, S. Eisebitt, and G. S. D. Beach, \nNat. Nanotechnol. 13, 1154 (2018). \n2 R. Blasing, T. P. Ma, S. H. Yang, C. Garg, F. K. Dejene, A. T. N'Diaye, G. Chen, K. Liu, and S. S. P. \nParkin, Nat. Commun. 9, 4984 (2018). \n3 J. Finley and L. Liu, Appl. Phys. Lett. 116, 110501 (2020). \n4 S. K. Kim, G. S. 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Phys. 33, R247 (2000). \n Supplementary Material  \n1 \n Ionically-Driven Synthesis and Exchange Bias in Mn 4N/MnN x Heterostructures \nZhijie Chen,1 Christopher J. Jensen,1 Chen Liu,2 Xixiang Zhang,2 and Kai Liu1,* \n1Physics Department, Georgetown University, Washington, DC 20057, USA \n2King Abdullah University of Science & Technology, Thuwal 23955-6900, Saudi Arabia \n \n1. Grazing incidence X-ray diffraction  \n \nFigure S1. Grazing incidence X-ray diffraction on samples with different 𝑡ெ showing the \nevolution of Mn 3N2 (black star), Mn 2N (red triangle), Mn 4N (blue circle), and Mn (green square) \nphases. 𝑡ெ = 0 nm is the 20nm Mn 3N2 seed layer.   \nThese scans were done on samples with deposited Mn thickness ( 𝑡ெ) from 0 nm to 50 nm.  \nStarting with 𝑡ெ = 0 nm, which is the 20 nm Mn 3N2 seed layer (black line), all the peaks (black \nSupplementary Material  \n2 \n star) are from the Mn 3N2 phase. As 𝑡ெ increases to 10 nm (red line), Mn 4N peaks (blue circle) \nemerge because of the reaction between Mn and nitrogen from Mn 3N2. Mn2N peaks (red triangle) \nalso show up as Mn 3N2 loses nitrogen. At 𝑡ெ = 20 nm, Mn 3N2 peaks are all gone after losing too \nmuch nitrogen while the Mn 4N peaks grow. Mn 2N peaks, on the other hand, persist until 𝑡ெ = 40 \nnm as the Mn 2N phase loses nitrogen and turns into Mn 4N. In the meantime, Mn 4N peaks grow \ntaller and sharper. At 𝑡ெ = 50 nm, there is no nitrogen available for Mn to react with and form \nMn4N. Thus, -Mn peaks show up. These results are consistent with the interpretation of the 2 -\n X-ray diffraction shown in Fig. 2(a) of the main text. \n2. Full range 2 - X-ray diffraction   \n \nFigure S2. Full range 2 - X-ray diffraction on samples with different 𝑡ெ showing the evolution \nof Mn3N2 (black star), Mn 2N (red triangle), Mn 4N (blue circle), and Mn (green square) phases. \n𝑡ெ = 0 nm is the 20nm Mn 3N2 seed layer.   \nSupplementary Material  \n3 \n These scans are similar to the one shown in Fig. 2(a). These are symmetric scans with a 1   offset \nto suppress the substrate peak. The peaks for the different phases are mainly located between 40  \n- 50 as shown in Fig. 2(a) of the main text. \n3. Phase evolution \nWe have included the expected nitrogen atomic percent and Mn nitride phases identified in each \nsample in the histogram below (Fig. S3). The expected nitrogen atomic percent is calculated using \nthe nominal Mn 3N2 seed layer thickness and the Mn thickness deposited on top ( t୬). The Mn \nphases in each sample are identified from the XRD peaks in Fig. S1 and S2.  \n \nFigure S3. Histogram showing Mn nitride phases and expected nitrogen atomic percentages in \neach sample. 𝑡ெ is the deposited Mn thickness on top of the 20 nm Mn 3N2. Colors represent \ndifferent Mn nitride phases.  \nSupplementary Material \n4 \n 4. Cross-sectional TEM and EELS line scans   \nThe cross-sectional TEM lamellas were fabricated using the Helios G4 UX FIB system (Thermo \nFisher Scientific) with a Ga+ beam source. Low-energy (2-5 kV) final polishing was employed to \nminimize the irradiation damage. The composition ratio of Mn and N was determined by EELS \nline-scan analysis using FEI Titan Themes Cubed G2 300 (Cs Probe) TEM at 300 kV.     \nEELS line scans are done on samples with 𝑡ெ = 0, 20, 40, 50 nm. Those for 𝑡ெ = 0 and 40 nm \nsamples are also included in Figs. 2(d) and 2(e) of the main text, and they are shown here for \ncompleteness (Fig. S4). The TEM images (left column) indicate the Mn nitride layers are mostly \nhomogenous without distinctive interfaces for all four samples. EELS scans measured across the \ngreen lines in the TEM images show that the composition ratio of Mn : N is continuously varying \ndue to the ionic motion of nitrogen within the nitride layers. At 𝑡ெ = 0 nm, atomic ratio of Mn:N \nis 58:42, consistent with the atomic ratio of Mn 3N2. As 𝑡ெ increases, nitrogen redistribute within \nthe Mn nitride layers and the Mn:N ratio changes to 66:34 for 𝑡ெ = 20 nm and 84:16 for 𝑡ெ = \n40 nm. This is also consistent with XRD results (Fig. S1 and S2) that indicate Mn 4N is the only \nnitride phase at 𝑡ெ = 40 nm. When 𝑡ெ increases to 50 nm, atomic percent ratio further increases \nto 92:8, likely because of the existence of pure Mn as shown in XRD (Fig. S1 and S2). Note that \nthere is some non-uniformity near the interfacial region between the substrate/capping layer and \nnitride layers, likely caused by interfacial mixing effect,1 as nitrogen tends to go into the substrate \nand capping layer more than Mn. Supplementary Material  \n5 \n  \nFigure S4. TEM (left column) and corresponding EELS line scan (right column) along the green \nline shown in TEM for (a,b) 𝑡ெ = 0 nm (Mn 3N2 seed layer), (c,d) 𝑡ெ = 20 nm, (e,f) 𝑡ெ = 40 \nnm, and (g,h) 𝑡ெ = 50 nm.  \nWe have also estimated the theoretical Mn/N ratio as 𝑡ெ changes. Fig. S5 shows the experimental \n(EELS) and calculated Mn/N ratio variation as 𝑡ெ changes. The theoretical Mn/N ratio has a \nlinear relationship with 𝑡ெ since the total nitrogen in the system is fixed, and the experimental \nratio largely agrees with the theoretical curve except for the 𝑡ெ = 50 nm samples, which has \nunreacted Mn remaining. \nSupplementary Material \n6 \n  \nFigure S5. Experimental (black) and theoretical (red) Mn/N ratio as a function of 𝑡ெ. \nExperimental data are determined from EELS results. Theoretical data are determined by fixing \nthe N atoms in the system and change Mn atoms as 𝑡ெ changes. \n5. Uniaxial magnetic anisotropy calculation  \nWe have also calculated the uniaxial magnetic anisotropy constant ( 𝐾௨) using2   \n𝐾௨=𝐾௨+ఓబ\nଶ𝑀ௌଶ,       (S1) \nwhere 𝐾௨ is the effective anisotropy estimated from the area difference between the IP and OP \nhysteresis loops and ఓబ\nଶ𝑀ௌଶ is the thin film demagnetization energy.  \n6. Sample information \nThe table below lists detailed sample information, including the nominal Mn 3N2 seed layer \nthickness, deposited Mn thickness ( 𝑡ெ), total thickness determined by X-ray reflectivity, \nsaturation magnetization ( 𝑀௦), Mn4N in-plane ( a-axis) and out-of-plane lattice constants ( c-axis). \nSupplementary Material \n7 \n Table 1. Sample layer thickness information, saturation magnetization, and lattice constants.  \nSample Nominal \nMn3N2 \nseed \nlayer \nthickness \n(nm) Nominal \nDeposited \nMn \nthickness \n(nm) Total \nthickness \n(nm) MS (RT) \n(emu/cm3) Ku \n(MJ/m3) \n Mn4N      \na-axis \n(nm) Mn4N      \nc-axis \n(nm) \n#1 20 0 --- --- --- --- --- \n#2 20 5 27.9 25.2 -0.007 --- --- \n#3 20 10 31.3 67.6 0.001 0.3872 0.3859 \n#4 20 15 36.3 78.3 0.004 0.3873 0.3861 \n#5 20 20 41.4 86.3 0.002 0.3866 0.3861 \n#6 20 25 45.0 80.2 0.010 0.3860 0.3862 \n#7 20 30 50.0 67.1 0.024 0.3860 0.3861 \n#8 20 35 54.4 79.1 0.030 0.3859 0.3860 \n#9 20 40 60.2 85.5 0.027 0.3855 0.3856 \n#10 20 45 65.4 71.2 0.028 0.3846 0.3848 \n#11 20 50 70.5 63.5 0.020 0.3843 0.3848 \n \nReferences \n1. C. T. Ma, W. Zhou, B. J. Kirby and S. J. Poon, AIP Adv. 12, 025117 (2022). \n2. T. Hirose, T. Komori, T. Gushi, A. Anzai, K. Toko and T. Suemasu, AIP Adv. 10, 025117 \n(2020). "    },    {        "title": "1504.04547v1.Model_for_the_FC_and_ZFC_Ferrimagnetic_Spinel.pdf",        "content": "arXiv:1504.04547v1  [cond-mat.str-el]  17 Apr 2015Model for the FC and ZFC Ferrimagnetic Spinel\nN. Karchev\nDepartment of Physics, Sofia University, James Bourchier 5 b lvd., 1164 Sofia, Bulgaria\n(Dated: October 20, 2021)\nThere are two methods of preparation of ferrimagnetic spine l. If, during the preparation, an\nexternal magnetic field as high as 300 O¨ e is applied upon cool ing the material is named field-cooled\n(FC).Iftheappliedfieldisabout1O¨ ethematerial iszero-fi eldcooled (ZFC).Toexplorethemagnetic\nand thermodynamic properties of these materials we conside r two-sublattice spin system, defined\non the bcc lattice, with spin- sAoperators SA\niat the sublattice Asite and spin- sBoperators SB\niat\nthe sublattice Bsite, where sA> sB. The subtle point is the exchange between sublattice A and\nB spins, which is antiferromanetic. Applying magnetic field along the sublattice A magnetization,\nduring preparation of the material, one compensates the Zee man splitting, due to the exchange, of\nsublattice B electrons. This effectively leads to a decrease of thesBspin. We consider a model with\nsBvarying parameter which accounts for the applied, during th e preparation, magnetic field.\nIt is shown that the model agrees well with the observed magne tization-temperature curves of\nzero field cooled (ZFC) and non-zero field cooled (FC) spinel f errimagnetic spinel and explains the\nanomalous temperature dependence of the specific heat.\nPACS numbers: 75.50.Gg,71.70.Ej,75.10.Dg,75.10.Lp\nI. INTRODUCTION\nThe magnetization-temperature and magnetic suscep-\ntibility curves for zero field cooled (ZFC) and non-zero\nfield cooled (FC) ferrimagnetic spinel display a notable\ndifference below N´ eel TNtemperature [1–13]. The (ZFC)\ncurve exhibits a maximum and then a monotonic de-\ncrease upon cooling from TN, while the (FC) curve in-\ncreases steeply, shows a dip near the temperature at\nwhich the (ZFC) curve has a maximum and finally in-\ncreases monotonically. There is also difference between\nso-called field-cooled-cooling and field-cooled-warming\nprocedures [14].\nThe specific heat curves for spinel ferrimagnetics show\nsharp peak at N´ eel temperature, which indicates fer-\nrimagnetic to paramagnetic transition, and tiny peak\nat temperature below N´ eel’s where the magnetization-\ntemperature curve has a maximum [3, 6, 9].\nAlthough the FC and ZFC spinel ferrimagnetics have\nintensively been studied, their magnetic and thermody-\nnamic properties have not been understood. There is\nnot an effective model which in unified way to explain\nthe experimental results.\nIn the present paper we consider two-sublattice spin\nsystem, defined on the bcc lattice, with spin- sAopera-\ntorsSA\niat the sublattice Asite and spin- sBoperators\nSB\niat the sublattice Bsite, where sA> sB. The subtle\npoint is the exchange between sublattice A and B spins,\nwhich is antiferromanetic. Applying magnetic field along\nthesublatticeAmagnetization, duringpreparationofthe\nmaterial, one compensates the Zeeman splitting of sub-\nlattice B electrons due to the exchange. This effectively\nleads to decrease of the sBspin. One can obtain an intu-\nition for this from spin-fermion model of spinel ferrimag-\nnetic with spin-1 /2itinerantelectronsat the sublattice B\nsite and spin- slocalizedelectronsat the sublattice Asite.\nAn applied, alongthe magnetizationof the localized elec-trons, external magnetic field compensates the Zeeman\nsplitting due to the spin-fermion exchange. Integrating\nout the fermions we obtain an effective spin model with\neffective spin sBwhich depends on the external magnetic\nfield (see Appendix A).\nWe consider a model with sBvarying parameter which\naccounts for the applied, during the preparation, mag-\nnetic field. First we present the method of calcula-\ntion exploring ZFC spinel with sA= 1.5 andsB= 1.\nWe obtained that the system has two phases. At low\ntemperature (0 ,T∗) the magnetic orders of the Aand\nBspins contribute to the magnetization of the sys-\ntem, while at the high temperature ( T∗,TN), the mag-\nnetic order of the sublattice B with smaller spin sBand\nwith a weaker intra-sublattice exchange is suppressed by\nmagnon fluctuations. Only the sublattice A spins, with\nstronger intra-sublattice exchange, have non-zero spon-\ntaneous magnetization. There is no additional symmetry\nbreaking, and the Goldstone boson has a ferromagnetic\ndispersion in both phases, but partial-order transition\ndemonstrates itself through tiny peak of specific heat as\na function of temperature.\nPartial order is well known phenomenon and has been\nsubject to extensive studies. Frustrated antiferromag-\nnetic systems has been studied by means of Green func-\ntion formalism. Partial order and anomalous temper-\nature dependence of specific heat have been predicted\n[15]. Experimentally the partial order has been observed\ninGd2Ti2O7[16]. Monte Carlo method has been uti-\nlized to study the nature of partial order in Ising model\nonkagom´elattice [17]. There are exact results for the\npartially ordered systems which precede the above stud-\nies [17–19]. The advantage of the present method of cal-\nculation is that it permits to consider the sublattice B\nspin as a varying parameter sB<1. We calculate the\nmagnetization-temperature curves and the specific heat\nas a function of temperature for sB= 0.7,sB= 0.4,2\nsB= 0.2 and fixed sA= 1.5, thus accounting for the in-\ncreasing of the applied, during preparation of FC spinel,\nmagnetic field.\nII. METHOD OF CALCULATION\nThe Hamiltonian of the ZFC ferrimagnetic spinel is\nH=−JA/summationdisplay\n≪ij≫ASA\ni·SA\nj−JB/summationdisplay\n≪ij≫BSB\ni·SB\nj\n+J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj (1)\nwhere the sums are over all sites of a body-centered cu-\nbic (bcc) lattice: ∝angbracketlefti,j∝angbracketrightdenotes the sum over the nearest\nneighbors, ≪i,j≫Adenotes the sum over the sites of\nthe A sublattice, ≪i,j≫Bdenotes the sum over the\nsites of the B sublattice. The first two terms describe\nthe ferromagnetic Heisenberg intra-sublattice exchange\nJA>0,JB>0, while the third term describes the inter-\nsublattice exchange which is antiferromagnetic J >0.\nTo study a theory with the Hamiltonian Eq.(1) it\nis convenient to introduce Holstein-Primakoff represen-\ntation for the spin operators SA\ni(a+,a) andSB\ni(b+,b).\nRewriting the effective Hamiltonian in terms of the Bose\noperators ( a+,a,b+,b) we keep only the quadratic and\nquartic terms. The next step is to represent the Hamil-\ntonian in the Hartree-Fock approximation:\nH≈HHF=Hcl+Hq (2)\nwith\nHcl= 6NJA(sA)2(uA−1)2+6NJB(sB)2(uB−1)2\n+ 8NJsAsB(u−1)2, (3)\nand\nHq=/summationdisplay\nk∈Br/bracketleftbig\nεa\nka+\nkak+εb\nkb+\nkbk−γk/parenleftbig\na+\nkb+\nk+bkak/parenrightbig/bracketrightbig\n,\n(4)\nwhereN=NA=NBis the number of sites on a sublat-\ntice. The two equivalent sublattices A and B of the bcc\nlattice are simple cubic lattices. The wave vector kruns\nover the reduced first Brillouin zone Brof a bcc lattice\nwhich is the first Brillouin zone of a simple cubic lattice.\nThe dispersions are given by equalities\nεa\nk= 4sAJAuA(3−coskx−cosky−coskz) + 8sBJu\nεb\nk= 4sBJBuB(3−coskx−cosky−coskz) + 8sAJu\nγk= 8J u√\nsAsBcoskx\n2cosky\n2coskz\n2(5)\nThe equations (5) show that Hartree-Fock parameters\n(uA,uB,u) renormalize the intra and inter-sublattice ex-\nchange constants ( JA,JB,J) respectively.To diagonalize the Hamiltonian one introduces new\nBose fields αk, α+\nk, βk, β+\nkby means of the transforma-\ntion\nak=ukαk+vkβ+\nka+\nk=ukα+\nk+vkβk\n(6)\nbk=ukβk+vkα+\nkb+\nk=ukβ+\nk+vkαk,\nwherethe coefficients ofthe transformation ukandvkare\nrealfunctionsofthewavevector k(B7). Thetransformed\nHamiltonian adopts the form\nHq=/summationdisplay\nk∈Br/parenleftBig\nEα\nkα+\nkαk+Eβ\nkβ+\nkβk+E0\nk/parenrightBig\n,(7)\nwith new dispersions\nEα\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk+εa\nk/bracketrightbigg\n(8)\nEβ\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+εb\nk−εa\nk/bracketrightbigg\nand vacuum energy\nE0\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk−εa\nk/bracketrightbigg\n(9)\nFor positive values of the Hartree-Fock parameters and\nall values of k∈Br, the dispersions are nonnegative\nEα\nk≥0, Eβ\nk≥0. When sA> sBtheαkboson is\nthe long-range (magnon) excitation in the system with\nEα\nk∝ρk2, near the zero wavevector, while the βkbo-\nson is a gapped excitation, with gap proportional to the\ninter-sublattice exchange constant Eβ\n0= 8Ju(sA−sB) .\nThe free energy of a system with Hamiltonian HHF\nequations (2), (3) and (4) is\nF= 6NJA(sA)2(uA−1)2+6NJB(sB)2(uB−1)2\n+ 8NJsAsB(u−1)2+1\nN/summationdisplay\nk∈BrE0\nk (10)\n+1\nβN/summationdisplay\nk∈Br/bracketleftBig\nln/parenleftBig\n1−e−βEα\nk/parenrightBig\n+ ln/parenleftBig\n1−e−βEβ\nk/parenrightBig/bracketrightBig\n,\nwhereβ= 1/Tis the inverse temperature. Then, the\nsystem of equations for the Hartree-Fock parameters is\n∂F/∂uA= 0, ∂F/∂uB= 0, ∂F/∂u= 0.(11)\nThe Hartree-Fock parameters are positive functions\nofT/J, solution of the system of equations (11) (see\nEqs.(B8)). Utilizing these functions, one can calculate\nthe spontaneous magnetization on the two sublattices\nMA=< S3\n1j> j is from sublattice A\n(12)\nMB=< S3\n2j> j is from sublattice B3\nandM=MA+MB, the spontaneous magnetization of\nthe system. In terms of the Bose functions of the αand\nβexcitations they adopt the form\nMA=sA−1\nN/summationdisplay\nk∈Br/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\n(13)\nMB=−sB+1\nN/summationdisplay\nk∈Br/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\n,\nwhereukandvkare functions of the wavevector k, coef-\nficients in the transformation (6).\nThemagnonexcitation- αkisacomplicatedmixtureof\nthe transversal fluctuations of the AandBspins (6). As\na result the magnons’ fluctuations suppress in a different\nwaythemagnetizationonsublattices AandB. Quantita-\ntivelythis dependsonthe coefficients ukandvk. At char-\nacteristic temperature T∗spontaneous magnetization on\nsublattice Bbecomes equal to zero, while spontaneous\nmagnetization on sublattice Ais still nonzero (see Fig.1).\nTo study the magnetic properties of the system above\nT∗we first consider the paramagnetic phase. To this end\nwe make use of the Takahashi modified spin-wave theory\n[20] and introduce two parameters λAandλBto enforce\nthe magnetization on the two sublattices to be equal to\nzero. The new Hamiltonian is obtained from the old one\nequation (1) by adding two new terms:\nˆH=H−/summationdisplay\ni∈AλASA3\ni+/summationdisplay\ni∈BλBSB3\ni (14)\nIn momentum space the new Hamiltonian ˆHadopts the\nform Eq.(4) with new dispersions\nˆεa\nk=εa\nk+λA,ˆεb\nk=εb\nk+λB.(15)\nUtilizing the same transformation (6) one obtains the\nHamiltonian ˆHin diagonal form (7) with dispersions ˆEα\nandˆEβobtained from Eqs.(8) replacing εa\nkandεb\nkwith\nˆεa\nkand ˆεb\nkrespectively. It is convenient to represent the\nparameters λAandλBin the form\nλA= 6JusB(µA−1), λB= 6JusA(µB−1).(16)\nThe dispersions ˆ εa\nkand ˆεb\nkarepositivefor all valuesofthe\nwavevector k, if the parameters µAandµBare positive.\nThe dispersions ˆEα\nkandˆEβ\nkare well defined if\nµAµB≥1. (17)\nTheβkexcitation is gapped ( ˆEβ\nk>0) for all values of\nparameters µAandµBwhich satisfy equation (17). The\nαexcitation is gapped if µAµB>1, but in the particular\ncase\nµAµB= 1 (18)\nˆEα\n0= 0, and near the zero wavevector ˆEα\nk≈ˆρk2. There-\nfor, in the particular case Eq. (18) αkboson is the long-\nrange excitation (magnon) in the system./s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48\n/s45/s49/s44/s48/s45/s48/s44/s53/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s83/s80/s79/s78/s84/s65/s78/s69/s79/s85/s83/s32/s32/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78/s84/s47/s74/s77/s65 \n/s77/s65 \n/s43/s77/s66\n/s77/s66/s115/s65 \n/s61/s49/s46/s53\n/s115/s66\n/s61/s49\nFIG. 1: (Color online) Spontaneous magnetization MA,MB\nandMA+MBas a function of the temperature in units of\nexchange constant T/Jfor ferrimagnetic spinel on bcc lattice\nwithsA= 1.5,sB= 1,JA/J= 2 and JB/J= 0.002. The\npartial order transition temperature T∗is the temperature\nabove which MB= 0.\nAboveN´ eeltemperature we introduced the parameters\nµAandµB(λA,λB) to enforce the sublattice AandB\nspontaneous magnetizations to be equal to zero. We find\nout these and Hartree-Fock parameters, as functions of\ntemperature, solving the system of five equations, equa-\ntions (11) and the equations MA=MB= 0, where the\nspontaneous magnetization has the same representation\nasequations(13)butwithcoefficients ˆ uk,ˆvk, anddisper-\nsionsˆEα\nk,ˆEβ\nkin the expressions for the Bose functions.\nThe numerical calculations show that above N´ eel tem-\nperature µAµB>1. When the temperature decreases\nthe product µAµBdecreases, remaining larger than one.\nThe temperature at which the product becomes equal to\none (µAµB= 1) is the N´ eel temperature Fig.(6).\nBelowTN, the spectrum contains long-range(magnon)\nexcitations. Thereupon, µAµB= 1 and one can use the\nrepresentation µB= 1/µA. Then,µAand Hartree-Fock\nparameters are solution of a system of four equations,\nequations (11) and the equation MB= 0.\nWe utilize the obtained functions µA(T),µB(T)uA(T),\nuB(T),u(T) (see Appendix C) to calculate the sponta-\nneous magnetization as a function of the temperature.\nFor a system with sA= 1.5,sB= 1,JA/J= 2 and\nJB/J= 0.002 the functions MA(T/J),MB(T/J) and\nMA(T/J) +MB(T/J) are depicted in figure (1). The\npartial order transition temperature T∗is the tempera-\nture above which MB= 0.\nThe customary formula for the entropy of a Bose sys-\ntem with Hamiltonian (7) is\nS=1\nN/summationdisplay\nk,δ/bracketleftbig\n(1+nδ\nk)ln(1+nδ\nk)−nδ\nklnnδ\nk/bracketrightbig\n,(19)\nwhereδstays for αandβ. The dispersions Eα\nkandEβ\nk\n(8) areused to define the Bosefunctions nα\nkandnβ\nkbelow\nT∗, dispersions ˆEα\nkandˆEβ\nkwithµB= 1/µAare used for4\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48 /s50/s50 /s50/s52 /s50/s54 /s50/s56 /s51/s48 /s51/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s67\n/s84/s47/s74/s115/s65\n/s61/s49/s46/s53\n/s115/s66\n/s61/s49\nFIG. 2: (Color online) Specific heat Cvs temperature, in\nunits of the exchange constant T/J, forJA/J= 2,JB/J=\n0.002,sA= 1.5 andsB= 1. The high temperature (green)\nvertical line marks N´ eel TNtemperature, while the low tem-\nperature (red) line marks partial order transition tempera ture\nT∗.\npartial order phase T∗< T < T N, and with µAµB>1\nfor paramagnetic phase above N´ eel temperature. With\nentropy, as a function of temperature in mind, one can\ncalculatethecontributionofmagnonstothespecificheat:\nC=TdS\ndT(20)\nThe resultant curve C(T/J), for a system with the same\nparameters, as above, is depicted in figure (4).\nThefigures(1)and(2)showthatthepresentmethodof\ncalculation describes correctlythe features of the system,\npartial order transition and anomalous behavior of the\nspecific heat at the temperature of this transition T∗.\nIII. FC SPINEL FERRIMAGNETICS\nFC spinel ferrimagnetics are prepared applying mag-\nnetic field along the sublattice A magnetization upon\ncoolingthematerial. ThiscompensatestheZeemansplit-\nting of sublattice B electrons which effectively leads to\ndecrease of the sBspin. The Hamiltonian of the FC\nspinel ferrimagnet is given by Eq.(1) with sBa varying\nparameter. The advantage of the method of calculation,\npresented above, is that we can use the same systems of\nequations for the three phases, 0 < T < T∗,T∗< T <\nTN,T > T N, and different values of the parameter sB.\nWe calculate the magnetization-temperature curves and\nthespecificheatasafunctionoftemperaturefor sA= 1.5\nand three different values of sB(sB= 0.7,sB= 0.4 and\nsB= 0.2), thus accounting for the increasing of the ap-\nplied, during preparation of FC spinel, magnetic field.\nThe resultant magnetization-temperature curves are\ndepicted in figure (3). The curve ”a” corresponds to\nZFC spinel (see figure (1)). The increasing of the ap-\nplied, during the preparation, magnetic field is mod-\neled by decreasing of sB. The curves ”b”, ”c” and ”d”/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52/s83/s80/s79/s78/s84/s65/s78/s69/s79/s85/s83/s32/s32/s77 /s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78\n/s84/s47/s74/s97/s98/s99/s100\nFIG. 3: Color online) Spontaneous magnetization MA+MB\nas a function of the temperature in units of the exchange\nconstant T/Jfor ferrimagnetic spinel on bcc lattice with\nJA/J= 2,JB= 0.002,sA= 1.5 and :a)(black) sB= 1,\nb)(red)sB= 0.7, c)(blue) sB= 0.4 and d)(green) sB= 0.2.\nThe dash (magenta) line shows the spontaneous magnetiza-\ntionMAof sublattice AforsA= 1.5 andsB= 0.2.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s67/s32/s115/s66\n/s61/s49\n/s32/s115/s66\n/s61/s48/s46/s55\n/s32/s115/s66\n/s61/s48/s46/s50/s115/s65\n/s61/s49/s46/s53\n/s84/s47/s74\nFIG. 4: (Color online) Specific heat Cvs temperature, in\nunits of the exchange constant T/J, forsA= 1.5 and different\nvalues of the effective spin sB: black curve sB= 1, red curve\nsB= 0.7 and blue curve sB= 0.2 .\nare magnetization-temperature curves for FC spinel fer-\nrimagnetics with sB= 0.7,sB= 0.4 andsB= 0.2. The\ncurve ”c” agrees well with the observed magnetization-\ntemperature curve of FC spinel MnV2O4with an exter-\nnal magnetic field as high as 300 O¨ e applied upon cool-\ning the material [4, 5]. The dash line shows the spon-\ntaneous magnetization MAof sublattice AforsA= 1.5\nandsB= 0.2. Comparing with line ”d”, which shows\nthe spontaneous magnetization MA+MBof the same\nsystem one can conclude that contribution of the sublat-\ntice B magnetization is small and Zeeman splitting, of\nsublattice B electrons, is approximately compensated.\nThe dependence of specific heat on temperature for\ndifferent values of sBis shown in figure (4). The curve\nwith higher maximumcorrespondsto ZFC spinel (see fig-\nure (2)) With increasing the applied, during the prepara-\ntion, magnetic field (with decreasing sB) the anomalous\ntemperature behavior of specific heat, at temperature of\npartialordertransition, remainswell defined but the pick5\ndecreases. The anomalous pick is well observed experi-\nmentally [3, 6, 9], and one can use it to determine the T∗\ntemperature of the partial order transition.\nIV. SUMMARY\nIn this paper is pointed out that a model which agrees\nwell with the observed magnetic and thermodynamic\nproperties of ZFC and FC spinel ferrimagnets is a two-\nsublattice Heisenberg model with spin- sAoperators at\nthe sublattice Asite, spin- sBoperators at the sublat-\nticeBsite, ferromagnetic intra-sublattice exchange and\nantiferromagnetic inter-sublattice exchange. The ap-\nplied magnetic field, along the sublattice A magnetiza-\ntion upon cooling the material, is accounted for varying\nonly one parameter sB.\nThe subtle point is that MnV2O4spinel hasan obvious\nanomalousmagnetic behavior,but the appearanceofthis\nanomaly is attributed to the strong orbital-spin coupling\n[4] which is not discussed in the model (1). The spinel\nMnV2O4is a two-sublattice ferrimagnet, with site Aoc-\ncupied by the Mn2+ion, which is in the 3 d5high-spin\nconfiguration with quenched orbital angular momentum,\nwhich can be regarded as a s= 5/2 spin. The B site is\noccupied by the V3+ion, which takes the 3 d2high-spin\nconfiguration in the triply degenerate t2gorbital and has\norbital degrees of freedom. Because of the strong spin-\norbitalinteractionit isconvenientto consider jjcoupling\nwithJA=SAandJB=LB+SB. The sublattice A\ntotal angular momentum is jA=sA= 5/2, while the\nsublattice Btotal angular momentum is jB=lB+sB,\nwithlB= 3, and sB= 1 [1]. Then the g-factor for\nthe sublattice AisgA= 2, and for the sublattice B\ngB=5\n4. The sublattice Amagnetic order is antiparallel\nto the sublattice Bone and the saturated magnetization\nisσ= 25\n2−5\n44 = 0, in agreement with the experimen-\ntal finding for ZFC spinel that the magnetization goes to\nzero when the temperature approaches zero. The Hamil-\ntonian of the system is\nH=−κA/summationdisplay\n≪ij≫AJA\ni·JA\nj−κB/summationdisplay\n≪ij≫BJB\ni·JB\nj\n+κ/summationdisplay\n/angbracketleftij/angbracketrightJA\ni·JB\nj (21)\nThe first two terms describe the ferromagnetic Heisen-\nberg intra-sublattice exchange κA>0,κB>0, while the\nthird term describes the inter-sublattice exchange which\nisantiferromagnetic κ >0. Theoperators JA\njandJB\njsat-\nisfy theSU(2) algebra, therefor we can use the Holstein-\nPrimakoff representation of the total angular momen-\ntum vectors JA\nj(a+\nj,aj)andJB\nj(b+\nj, bj), wherea+\nj, ajand\nb+\nj, bjare Bose fields. Farther on, we repeat the calcula-\ntions from sections II and III. The only difference is the\nexpression for the magnetization gAMA+gBMB. The\njjmodel shows that the anomalous magnetic and ther-\nmodynamic behavior of MnV2O4is a consequence of theferrimagnetic nature of the system.\nAppendix A: Spin-fermion model of spinel\nferrimagnet\nThe Hamiltonian of the spin-fermion model of ferri-\nmagnetic spinel defined on a body centered cubic lattice\nis\nH=−t/summationdisplay\n≪ij≫B/parenleftbig\nc+\niσcjσ+h.c./parenrightbig\n−µ/summationdisplay\ni∈Bni\n−JB1/summationdisplay\n≪ij≫BSB\ni·SB\nj+J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj(A1)\n−JA/summationdisplay\n≪ij≫ASA\ni·SA\nj−H/summationdisplay\ni∈ASzA\ni−H/summationdisplay\ni∈BSzB\ni,\nwhereSνB\ni=1\n2/summationtext\nσσ′c+\niστν\nσσ′ciσ′, with the Pauli matrices\n(τx,τy,τz), is the spin of the itinerant electrons at the\nsublattice Bsite ,SA\niis the spin ofthe localizedelectrons\nat the sublattice Asite,µis the chemical potential, and\nni=c+\niσciσ. The sums are over all sites of a body cen-\nteredcubic lattice, ∝angbracketlefti,j∝angbracketrightdenotesthe sum overthe nearest\nneighbors, while ≪ij≫Aand≪ij≫Bare sums over\nall sites of sublattice AandBrespectively. The Heisen-\nberg term ( JA>0) describes ferromagnetic Heisenberg\nexchangebetweenlocalizedelectronsand J >0is the an-\ntiferromagnetic exchange constant between localized and\nitinerant electrons. H >0 is the Zeeman splitting en-\nergy due to the external magnetic field (magnetic field\nin units of energy). We represent the Fermi operators,\nthe spin of the itinerant electrons and the density oper-\natorsniσin terms of the Schwinger bosons ( ϕi,σ,ϕ+\ni,σ)\nand slave fermions ( hi,h+\ni,di,d+\ni). The Bose fields are\ndoublets ( σ= 1,2) without charge, while fermions are\nspinless with charges 1 ( di) and -1 ( hi):\nci↑=h+\niϕi1+ϕ+\ni2di, c i↓=h+\niϕi2−ϕ+\ni1di,\nni= 1−h+\nihi+d+\nidi, sν\ni=1\n2/summationdisplay\nσσ′ϕ+\niστν\nσσ′ϕiσ′,\nc+\ni↑ci↑c+\ni↓ci↓=d+\nidi (A2)\nϕ+\ni1ϕi1+ϕ+\ni2ϕi2+d+\nidi+h+\nihi= 1 (A3)\nTo solve the constraint (Eq.A3), one makes a change of\nvariables, introducing Bose doublets ζiσandζ+\niσ[21]\nζiσ=ϕiσ/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig−1\n2,\nζ+\niσ=ϕ+\niσ/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig−1\n2,(A4)\nwhere the new fields satisfy the constraint ζ+\niσζiσ= 1. In\nterms of the new fields the spin vectors of the itinerant\nelectrons have the form\nSνB\ni=1\n2/summationdisplay\nσσ′ζ+\niστν\nσσ′ζiσ′/bracketleftbig\n1−h+\nihi−d+\nidi/bracketrightbig\n(A5)6\nWhen, in the ground state, the lattice site is empty, the\noperator identity h+\nihi= 1 is true. When the lattice\nsite is doubly occupied, d+\nidi= 1. Hence, when the lat-\ntice site is empty or doubly occupied the spin on this\nsite is zero. When the lattice site is neither empty nor\ndoubly occupied ( h+\nihi=d+\nidi= 0), the spin equals\nsi= 1/2ni,where the unit vector\nnν\ni=/summationdisplay\nσσ′ζ+\niστν\nσσ′ζiσ′(n2\ni= 1) (A6)\nidentifies the local orientation of the spin of the itinerant\nelectron.\nThe part of Hamiltonian Eq.(A1) with itinerant\nfermions can be rewritten in terms of Bose fields Eq.(A4)\nand slave fermions\nHB=−t/summationdisplay\n≪ij≫B/bracketleftbig/parenleftbig\nd+\njdi−h+\njhi/parenrightbig\nζ+\niσζjσ\n+/parenleftbig\nd+\njh+\ni−d+\nih+\nj/parenrightbig\n(ζi1ζj2−ζi2ζj1)+h.c./bracketrightbig\n×/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig1\n2/parenleftbig\n1−h+\njhj−d+\njdj/parenrightbig1\n2\n−JB1/summationdisplay\n≪ij≫Bmimjni·nj\n+U/summationdisplay\ni∈Bd+\nidi−µ/summationdisplay\ni∈B/parenleftbig\n1−h+\nihi+d+\nidi/parenrightbig\n(A7)\n−H/summationdisplay\ni∈B1\n2nz\ni[1−h+\nihi−d+\nidi].\nwhere\nmi=1\n2[1−h+\nihi−d+\nidi]. (A8)\nThe mixed, spin-fermion term, adopts the form\nHAB=J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·njmj (A9)\nThe Hamiltonian (A1) can be rewritten in the form\nH=HA+HB+HAB(A10)\nwhereHAis the contribution of sublattice A spins SA\ni.\nAn important advantage of working with Schwinger\nbosons and slave fermions is the fact that Hubbard term\nis in a diagonal form. The fermion-fermion and fermion-\nboson interactions are included in the hopping term.\nOne treats them as a perturbation. To proceed we ap-\nproximate the hopping term of the Hamiltonian Eq.(A7)\nsetting/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig1\n2∼1 and keeping only the\nquadratic, with respect to fermions, terms. This means\nthat the averagingin the subspace of the fermions is per-\nformed in one fermion-loop approximation. Further, we\nrepresenttheresulting hBandhABHamiltonianasasum\nof two terms\nHf=Hf\n0+Hfb\nint, (A11)where\nHf\n0=−t/summationdisplay\n≪ij≫B/parenleftbig\nd+\njdi−h+\njhi+h.c./parenrightbig\n+U/summationdisplay\ni∈Bd+\nidi\n−µ/summationdisplay\ni∈B/parenleftbig\n1−h+\nihi+d+\nidi/parenrightbig\n(A12)\n+4sJ−H\n2/summationdisplay\ni∈B/parenleftbig\nh+\nihi+d+\nidi/parenrightbig\n,\nis the Hamiltonian of the free dandhfermions, and\nHfb\nint=−t/summationdisplay\n≪ij≫B/bracketleftbig/parenleftbig\nd+\njdi−h+\njhi/parenrightbig/parenleftbig\nζ+\niσζjσ−1/parenrightbig\n(A13)\n+/parenleftbig\nd+\njh+\ni−d+\nih+\nj/parenrightbig\n(ζi1ζj2−ζi2ζj1)+h.c./bracketrightbig\n−JB1/summationdisplay\n≪ij≫Bmimjni·nj\nis the Hamiltonian of boson-fermion interaction.\nThegroundstateofthe system, without accountingfor\nthe spin fluctuations, is determined by the free-fermion\nHamiltonian h0and is labeled by the density of electrons\nn= 1−< h+\nihi>+< d+\nidi> (A14)\n(see equation (A2)) and the ”effective spin” of the sub-\nlattice B electron\nsB=1\n2/parenleftbig\n1−< h+\nihi>−< d+\nidi>/parenrightbig\n.(A15)\nAt half-filling\n< h+\nihi>=< d+\nidi> . (A16)\nTo solve this equation one sets the chemical potential\nµ=U/2. Utilizing this representation of µwe calculate\nthe effective spin sBas a function of applied magnetic\nfieldhfor parameters 4 t/U= 2.2 and 4sAJ/U= 4. The\nresult is depicted in figure (5).\nLet us introduce the vector,\nMν\ni=sB/summationdisplay\nσσ′ζ+\niστν\nσσ′ζiσ′M2\ni= (sB)2.(A17)\nThen, the spin-vector of itinerant electrons Eq.(A5) can\nbe written in the form\nSB\ni=1\n2mMi/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig\n,(A18)\nwhere the vector Miidentifies the localorientationof the\nspin of the sublattice B itinerant electrons.\nThe Hamiltonian is quadratic with respect to the\nfermions di,d+\niandhi,h+\ni, and one can average in the\nsubspace of these fermions (to integrate them out in the\npath integral approach). As a result, accounting for the\ndefinition of sB(see Eq.A15), one obtains SBi=Miand\nan effective model for spin vectors SBiwith Hamiltonian\nHeff=−JB2/summationdisplay\n≪ij≫BSB\ni·SB\nj,(A19)7\n/s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52/s48/s44/s53\n/s52/s116/s47/s85/s61/s50/s46/s50\n/s52/s115/s65\n/s74/s47/s85/s61/s52/s115/s66\n/s104\nFIG. 5: (Color online) Sublattice B effective spin sBas a\nfunctionofapplied magnetic field hfor parameters 4 t/U= 2.2\nand 4sAJ/U= 4.\nwhere the effective exchange constant JB2is calculated\nin the one loop approximation. The term with JB1ex-\nchange constant in equation (A7) and the mixed, spin-\nfermion term (A9), adopt the form\n−JB1/summationdisplay\n≪ij≫BSBi·SBj (A20)\nand\nJ/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj (A21)\nrespectively. Collecting all terms one obtains the Hamil-\ntonian of the FC ferrimagnetic spinel\nH=−JA/summationdisplay\n≪ij≫ASA\ni·SA\nj−JB/summationdisplay\n≪ij≫BSB\ni·SB\nj\n+J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj,\nwhereJB=JB1+JB2andsBdepending on the field\napplied during preparation Fig.(5). After the process of\npreparation the external magnetic field is set equal to\nzero. For systems with sB= 1, discussed in the pa-\nper, one has to consider two-band model for sublattice\nB fermions. The result remains the same even for this\nmore complicate system: the effective spin sBdecreases\nwhen the applied, under the preparation, magnetic field\nincreases.\nAppendix B: Hartree-Fock approximation\nLet us consider a theory with Hamiltonian (1). We\nintroduce Holstein-Primakoff representation for the spinoperators\nSA+\nj=SA1\nj+iSA2\nj=/radicalBig\n2sA−a+\njajaj\nSA−\nj=SA1\nj−iSA2\nj=a+\nj/radicalBig\n2sA−a+\njaj(B1)\nSA3\nj=sA−a+\njaj\nwhen the sites jare from sublattice Aand\nSB+\nj=SB1\nj+iSB2\nj=−b+\nj/radicalBig\n2sB−b+\njbj\nSB−\nj=SB1\nj−iSB2\nj=−/radicalBig\n2sB−b+\njbjbj(B2)\nSB3\nj=−sB+b+\njbj\nwhen the sites jare from sublattice B. The operators\na+\nj, ajandb+\nj, bjsatisfythe Bosecommutationrelations.\nIn terms of the Bose operators and keeping only the\nquadratic and quartic terms, the effective Hamiltonian\nEq.(1) adopts the form\nH=H2+H4 (B3)\nwhere\nH2=sAJA/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+sBJB/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(B4)\n+J/summationdisplay\n/angbracketleftij/angbracketright/bracketleftBig\nsAb+\njbj+sBa+\niai−√\nsAsB/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightBig\nH4=1\n4JA/summationdisplay\n≪ij≫A/bracketleftbig\na+\nia+\nj(ai−aj)2+(a+\ni−a+\nj)2aiaj/bracketrightbig\n+1\n4JB/summationdisplay\n≪ij≫B/bracketleftbig\nb+\nib+\nj(bi−bj)2+(b+\ni−b+\nj)2bibj/bracketrightbig\n+1\n4J/summationdisplay\n/angbracketleftij/angbracketright/bracketleftBigg/radicalbigg\nsA\nsB/parenleftbig\naib+\njbjbj+a+\nib+\njb+\njbj/parenrightbig\n(B5)\n+/radicalbigg\nsB\nsA/parenleftbig\na+\niaiaibj+a+\nia+\niaib+\nj/parenrightbig\n−4a+\niaib+\njbj/bracketrightBigg\nThe next step is to represent the Hamiltonian in the\nHartree-Fock approximation (2,3) and\nHq=sAJAuA/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+sBJBuB/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(B6)\n+Ju/summationdisplay\n/angbracketleftij/angbracketright/bracketleftBig\nsAb+\njbj+sBa+\niai−√\nsAsB/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightBig\nIt is convenient to rewrite the Hamiltonian in momen-\ntum space representation Eq.(4) The two equivalent sub-\nlattices A and B of the bcc lattice are simple cubic lat-\ntices. The wave vector kruns over the reduced first Bril-\nlouin zone Brof a bcc lattice which is the first Brillouin8\nzone of a simple cubic lattice. The dispersions are given\nby equalities Eqs. (5)\nTo diagonalize the Hamiltonian one introduces new\nBose fields αk, α+\nk, βk, β+\nkby means of the transforma-\ntion (6) where the coefficients of the transformation uk\nandvkare real function of the wave vector k\nuk=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+ 1\n\n(B7)\nvk=sign(γk)/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−1\n.\nThe transformed Hamiltonian adopts the form Eqs.(7)\nwith dispersionsEqs.(8) The free energy ofa system with\nHamiltonian HHF(7,8) is Eq.(10) The system of equa-\ntions for the Hartree-Fock parameters Eqs.(11), in terms\nof the Bose functions of the αandβexcitations, adopt\nthe form\nuA= 1−1\n3sA1\nN/summationdisplay\nk∈Brεk/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\nuB= 1−1\n3sB1\nN/summationdisplay\nk∈Brεk/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\nu= 1−1\nN/summationdisplay\nk∈Br/bracketleftbigg1\n2sA/parenleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/parenrightBig\n(B8)\n+1\n2sB/parenleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/parenrightBig\n−8Ju/parenleftBig\n1+nα\nk+nβ\nk/parenrightBig/parenleftBig\ncoskx\n2cosky\n2coskz\n2/parenrightBig2\n/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk\n\nwherenα\nkandnβ\nkare the Bose functions of αandβex-\ncitations.\nAppendix C: Takahashi modified spin-wave theory\n[20]\nTo study the paramagnetic phase of the system we\nmakeuseofthe Takahashimodified spin-wavetheory[20]\nand introduce two parameters λAandλBto enforce the\nmagnetization on the two sublattices to be equal to zero.\nIt is convenientto representthe parameters λAandλBin\nthe form Eqs.(16). We find out these and Hartree-Fock\nparameters, as functions of temperature, solving the sys-\ntem of five equations, equations (B8) and the equations\nMA=MB= 0. The numerical calculations show thatabove N´ eel temperature µAµB>1 (blue line in Fig.(6)).\nWhen the temperature decreases the product µAµBde-\ncreases, remaining larger than one. The temperature at\nwhich the product becomes equal to one ( µAµB= 1) is\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s49/s50/s51/s52/s53/s54\n/s84/s47/s74/s32\n/s32\n/s32\nFIG. 6: (Color online) Parameters µA,µBandµAµBas a\nfunction of the temperature, in units of the exchange con-\nstantT/J, for ferrimagnet with sublattice A spin sA= 1.5\nand sublattice B spin sB= 1. The high temperature vertical\n(cyan) line corresponds to the N´ eel temperature of ferrima g-\nnet to paramagnet transition. The low temperature vertical\n(green) line corresponds to partial order transition tempe ra-\nture.\nthe N´ eel temperature, marked by vertical cyan line in\nFig.(6).\nAt low temperature µA= 1 and µB= 1 (λA=λB=\n0). The Hartree-Fock parameters are positive functions\nofT/J, solution of the system of equations (B8). Uti-\nlizing these functions, one can calculate the spontaneous\nmagnetizationon the twosublattices at low temperature.\nAt characteristic temperature T∗spontaneous magneti-\nzationon sublattice Bbecomes equalto zero, while spon-\ntaneous magnetization on sublattice Ais still nonzero\n(see Fig.1 in the paper).\nAboveT∗the spectrum contains long-range (magnon)\nexcitations. Thereupon, µAµB= 1 and one can use the\nrepresentation µB= 1/µA. Then,µAand Hartree-Fock\nparameters are solution of a system of four equations,\nequations (B8) and the equation MB= 0.\nThe functions µA(T/J),µB(T/J) and\nµA(T/J)µB(T/J) are depicted in figure (6) for a\nsystem with sA= 1.5,sB= 1,JA/J= 2 and\nJB/J= 0.002.\nFor the same system the Hartree-Fock parameters\nuA(T/J),uB(T/J)u(T/J) are depicted in figure (7).\n[1] K. Adachi, T. Suzuki, K. Kato, K. Osaka, M. Takata and\nT. Katsufuji, Phys. Rev. Lett. 95, 197202 (2005).[2] Zhaorong Yang, Shun Tan, Zhiwen Chen, and Yuheng9\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52/s49/s44/s54/s49/s44/s56/s50/s44/s48/s50/s44/s50\n/s84/s47/s74/s117\n/s117/s65\n/s117/s66/s72/s65/s82/s84/s82/s69/s69/s45/s70/s79/s67/s75/s32/s80/s65/s82/s65/s77 /s69/s84/s69/s82/s83\nFIG. 7: (Color online) Hartree-Fock parameters as a functio n\nof the temperature, in units of the exchange constant T/J, for\nferrimagnet with sublattice A spin sA= 1.5 and sublattice\nB spinsB= 1. The high temperature vertical (cyan) line\nmarks the N´ eel temperature of ferrimagnet to paramagnet\ntransition. The low temperature vertical (blue) line marks\nthe partial order transition temperature.\nZhang, Phys. Rev. B 62, 13872 (2000).\n[3] H. D. Zhou, J. Lu, and C. R. Wiebe, Phys. Rev. B 76,\n174403 (2007).\n[4] V. O. Garlea, R. Jin, D. Mandrus, B. Roessli, Q. 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P.Sun, Solid State Communications 159, 88\n(2013).\n[12] Z. H. Huang, X. Luo, L. Hu, S. G. Tan, Y. Liu, B. Yuan,\nJ. Chen, W. H. Song, and Y. P. Sun, Journal of Applied\nPhysics115, 034903 (2014).\n[13] Dina Tobia, Juli´ an Milano, Maria Teresa Causa and\nElin L. Winkler, J. Phys.: Condens. Matter 27, 016003\n(2015).\n[14] Vincent Hardy, Yohann Br´ eard, and Christine Martin,\nPhys. Rev. B 78, 024406 (2008).\n[15] R. Quartuand H. T. Diep, Phys.Rev. B 55, 2975 (1997).\n[16] J. R. Stewart, G. Ehlers, A. S. Wills, S. T. Bramwell,\nand J. S. Gardner, J. Phys.: Condens. Matter 16, L321\n(2004).\n[17] P. Azaria, H. T. Diep, and H. Giacomini, Phys. Rev.\nLett.59, 1629 (1987).\n[18] V. G. Vaks, A. I. Larkin, and Y. N. Ovchinnikov, JETP\nLetters. 22, 820 (1966).\n[19] H. T. Diep, Ed., Frustrated Spin Systems , World Scien-\ntific (2004)\n[20] M. Takahashi, Phys. Rev. Lett. 58, 168 (1987).\n[21] D. Schmeltzer, Phys. Rev. B 43, 8650 (1991).\n[*] Electronic address: naoum@phys.uni-sofia.bg"    },    {        "title": "2210.14119v1.Ultrafast_Switching_in_Synthetic_Antiferromagnet_with_Bilayer_Rare_Earth_Transition_Metal_Ferrimagnets.pdf",        "content": "Ultrafast Switching in Synthetic Antiferromagnet\nwith Bilayer Rare-Earth Transition-Metal\nFerrimagnets\nChung Ting Ma1,*, Wei Zhou1, and S. Joseph Poon1,2\n1University of Virginia, Department of Physics, Charlottesville, Virginia, 22904, USA\n2University of Virginia, Department of Materials Science and Engineering, Charlottesville, Virginia, 22904, USA\n*ctm7sf@virginia.edu\nABSTRACT\nIn spintronics, it is important to be able to manipulate magnetization rapidly and reliably. Several methods can control\nmagnetization, such as by applying current pulses or magnetic fields. An applied current can reverse magnetization with\nnanosecond speed through the spin torque effect. For faster switching, subpicosecond switching with femtoseconds laser pulse\nhas been achieved in amorphous rare-earth transition-metal ferrimagnets. In this study, we employed atomistic simulations\nto investigate ultrafast switching in a synthetic antiferromagnet with bilayer amorphous FeGd ferrimagnets. Using a two-\ntemperature model, we demonstrated ultrafast switching in this synthetic antiferromagnet without external magnetic fields.\nFurthermore, we showed that if we initially stabilize a skyrmion in this heterostructure, the ultrafast laser can switch the skyrmion\nstate using the same mechanism. Furthermore, this bilayer design allows the control of each ferrimagnetic layer individually\nand opens the possibility for a magnetic tunnel junction.\nIntroduction\nThe ability to control magnetization is a critical component of designing memory and logical devices. In thin films, mag-\nnetizations are commonly manipulated through current or external fields. In spintronic devices, currents are often used\nto induce spin-transfer torque and spin-orbit torque to reliably switch magnetizations without magnetic fields1–5. Besides\nelectrical current, a laser pulse can also induce changes in magnetization. Subpicosecond demagnetization with femtosecond\nlaser pulse was first observed in ferromagnetic nickel film6. Since then, ultrafast manipulation of magnetization has drawn\nconsiderable interest for its potential applications. In ferromagnets, a multistep procedure has been demonstrated to switch\nmagnetization7–10. For example, in FePt nanoparticles, magnetizations are first thermally demagnetized, then re-magnetized\nthrough the laser-induced inverse Faraday effect9. In antiferromagnets, optical switching of antiferromagnetic order is observed\nin multiferroic TbMnO 3at 18 K11. Furthermore, reliable all-optical switching of magnetization in easy-plane CrPt has been\nproposed by unitizing the inverse Faraday effect12. Nonetheless, one-shot all-optical subpicosecond switching has only been\nobserved in ferrimagnets, such as rare-earth transition metal (RE-TM) ferrimagnets13–20and recently, Mn-based crystalline\nalloys21.\nAmorphous RE-TM ferrimagnetic films are one of the more appealing materials for applications. They consist of two\nantiferromagnetically coupled RE-TM sublattices, which align in an antiparallel direction. There exists a compensation\ntemperature (T Comp ) where the magnetic moment of the two sublattices cancel each other and magnetization goes to zero22, 23.\nRE-TM films contain several attractive properties, including perpendicular magnetic anisotropy (PMA)24, 25and high domain\nwall velocity26, 27. Furthermore, they are deposited at room temperature28and their composition can be tuned to adjust\nmagnetization and coercivity23, 28. Recent experiments also observed skyrmions in RE-TM thin films27, 29–31. One of the\nmost intriguing properties of RE-TM ferrimagnet is the access to one-shot all-optical ultrafast switching13–20. In previous\nstudies, it is revealed that angular momentum exchange between the two different sublattices is a key ingredient in all-optical\nswitching15, 32, 33. The requirement of having two different sublattices makes ferrimagnets, such as RE-TM, one of the few\nPMA materials to have this capability.\nIn this study, we investigate laser-induced ultrafast switching in a synthetic antiferromagnet (SAF) formed from a bilayer\nRE-TM ferrimagnet . A schematic diagram of this heterostructure is shown in Figure 1. In this heterostructure, two different\ncompositions of 5 nm thick FeGd combine to form a 10 nm thick SAF, with one layer having T Comp above room temperature\nand the other having T Comp below room temperature. Such control of T Comp in RE/TM films has been achieved experimentally\nby tuning composition of Fe and Gd23, where a higher T Comp was achieved by increasing rare-earth concentration. To elaborate,arXiv:2210.14119v1  [cond-mat.mes-hall]  25 Oct 2022this SAF arises from the cancellation of magnetization between the top and bottom FeGd layer. The magnetization in each layer\nis designed to be opposite but equal in magnitude at room temperature. This is obtained by choosing the T Comp of the top layer\nto be 350 K and the T Comp of the top layer to be 250 K. This heterostructure presents several advantages. Compared to SAF with\nferromagnet or multilayer RE/TM films, SAF with RE-TM allows more flexible tuning of each layer. The thickness34, 35and\ncomposition23of each layer can be varied while the net magnetization stays zero, and PMA remains robust. In contrast, SAF\nwith ferromagnet and multilayer RE/TM films are limited in thickness and composition to maintain PMA18, 36. Furthermore,\nthe use of thicker layers diminishes the relative strength of interface exchange on an individual layer. This opens the possibility\nof switching each layer individually. In this study, we explored laser-induced ultrafast switching in SAF with RE-TM by using\na two-temperature model for laser irradiation37, 38. We found deterministic spins switching in this heterostructure, like those\nobserved in single-layer RE-TM films. More importantly, synchronized switching are found within the same sublattice in the\nFeGd bilayer. Furthermore, we stabilized skyrmions in this heterostructure as initial states and found switching remains robust\nwith a laser pulse. These findings pave the way to employ SAF with a bilayer RE-TM for spintronics applications.\nFigure 1. A schematic diagram of a synthetic antiferromagnet used in this study. Two 5 nm thick FeGd combines to form the\nsynthetic antiferromagnet. In this study, layer 1 has T Comp at 350 K, above room temperature, and layer 2 has T Comp at 250 K,\nbelow room temperature.\nResults and Disscussions\nFigure 2. (a) Time evolution of magnetic moment per atom with application of a 100-fs laser pulse with 30 mJ/m2fluence\nevery 100 ps. (b) Time evolution of total magnetic moment of Gd and Fe sublattice in each FeCo layer with application of a\nlaser pulse every 100 ps.\nFigure 2 shows the results of ultrafast switching in SAF with FeGd after laser pulses. Initially, the spins of Fe sublattices\nare pointing down (- z-direction) and the spins of Gd sublattices are pointing up (+ z-direction). The spins are initialized by\nthe application of a 0.01 T out-of-plane magnetic field, and no external fields are applied after initialization and during the\n2/9application of laser pulses. This is a stable configuration in this heterostructure as the exchange couplings between the two\nFeGd layers align the spins within the same sublattice parallel and the spins in different sublattices antiparallel. Also, magnetic\nanisotropy in FeGd holds the magnetic moment in an out-of-plane direction without an external field. After the application\nof a 100-fs laser pulse with 30 mJ/m2fluence, the magnetic moments in both sublattices reverse. As shown in Figure 2 (b),\nspins in Gd sublattices reverse from the positive to the negative direction, and spins in Fe sublattices reverse from the negative\nto the positive direction. Furthermore, Gd spins in both layer 1 and layer 2 reverse simultaneously and the same switching\nis observed in Fe spins of both layers. From Figure 2 (a), the total moment deviates from zero after the excitation by a laser\npulse. This is due to the different exchange couplings and relaxation time of Fe and Gd sublattices. As discussed by previous\npublications14–16, RE and TM sublattice have different relaxation times and lead to a transient state, where spins in RE and TM\nalign in parallel after the first few picoseconds of a laser pulse. In this SAF, within picoseconds the initial spike in magnetic\nmoment corresponds to the transient state, matching the previous study of single-layer RE-TM films. After the initial spike,\na large downward spike is observed. This is the consequence of the high temperature from the laser pulse. As a result, the\nmagnetic moments are not synchronized in one direction and lead to a smaller total moment. Then, as the temperature cools\ndown, the stronger-coupled Fe sublattice is more aligned, which leads to an increase in the total moment. As the temperature\ncools down further, the magnetic moment begins to decrease. This is explained by the more alignment of the Gd sublattice over\nthis period, which points opposite to the Fe atoms, and the spins begin to relax back to one of the ferrimagnetic ground states.\nAfter 100 ps, the total magnetic moment approaches back to zero. From Figure 2 (b), the moments of each sublattice are now\npointing in opposite directions, corresponding to opposite spin directions from the initial configuration. Subsequence laser\npulses, which were applied every 100 ps, show deterministic switching of spins in this heterostructure.\nFigure 3. Switching rate of magnetization in synthetic antiferromagnet with bilayer FeGd as a function of laser fluence.\nSimulations at each fluence is repeated 128 times and the switching rate is the percentage of switching occurred out of 128\nsimulations. Error bars correspond to one standard deviation from avenging. 35 fs (green), 50 fs (blue), 100 fs (red) 150 fs\n(purple) shows similar switching rate.\nTo validate the repeatability of this switching, we repeated the simulations with various laser fluence and laser pulse width.\nFigure 3 shows the switching rate of spins in SAF with FeGd with laser fluence from 25 J/m2to 60 J/ m2and laser pulse widths\nof 35 fs, 50 fs, 100 fs, and 150 fs. The simulations have repeated 128 times for each set of fluence and laser pulse width, and\nthe switching rate is the percentage of switching observations out of 128 simulations. From Figure 3, the switching rate is\nsimilar in all four laser pulse width (35 fs, 50 fs, 100 fs, and 150 fs) for various fluences. On the other hand, varying laser\nfluence has a significant impact on the switching rate. With a laser fluence of 25 J/ m2, the switching rate is near zero. As the\nlaser fluence increases to 30 J/ m2, the switching rate increases to about 70 %. Between laser fluence of 35 to 45 J/ m2, the\nswitching rate is above 90 %. Above 45 J/ m2, switching decreases with increases in laser fluence, reducing to about 60 % at a\n3/9Figure 4. Ultrafast switching of a skyrmion in a synthetic antiferromagnet with bilayer FeGd. (a) Time evolution of magnetic\nmoment after application of a 100-fs laser pulse with 30 mJ/m2fluence. (b) Schematic representation of spins in Fe and Gd\nsublattice before and after switching by laser pulse. Red arrows represent down magnetic moments, blue arrows represents up\nmagnetic moments, and green arrows represents near in-plane magnetic moments. Both layer of FeGd are shown here, where\nthe top half of each schematic correponds to the top FeGd layer and the bottom half corresponds to the bottom FeGd layer.\nlaser fluence of 60 J/ m2. While the mechanisms behind this dependence remain unknown, this phenomenon is likely related\nto the angular momentum exchange between the Fe and Gd sublattices. As discussed in other publications15, 33, 34, angular\nmomentum exchange between the two different sublattices is a crucial component of all-optical switching in RE-TM films.\nFrom intuition, at low laser fluence (< 25 J/ m2), there is not enough energy to initiate the exchange of angular momentum,\nresulting in a zero switching rate. For high laser fluence (> 45 J/ m2), an excess temperature may lead to excessive fluctuations\nin spins and reduces the effectiveness of angular momentum exchange between the two sublattices. While the temperature\ndifferences for different fluences certainly play a role in the switching rate, they also affect the angular momentum exchange\nbetween the two sublattices in the switching process. Further investigations are needed to reveal the underlying reasons, which\nare beyond the scope of this study.\nWe further investigate the potential of using ultrafast switching in other magnetic states in RE-TM ferrimagnets. Figure\n4 shows the ultrafast switching of a 20 nm skyrmion in SAF with FeCo. This skyrmion was initially stabilized through the\ninterfacial Dzyaloshinskii–Moriya interaction39, 40under 0.01 T. Details of this skyrmion calculation were discussed in previous\npublications41. As seen in Figure 4 (b), initially, the spins of Fe and Gd sublattices form a skyrmion. In Fe sublattice, the\nspins in the core of a skyrmion are pointing in the positive direction and the spins outside are pointing in the negative direction.\nThe spins in Gd sublattice align antiparallel to the spins in Fe sublattice. After applying a 100 fs laser pulse with 30 mJ/m2\nfluence, the spins in the heterostructures reverse from the initial configuration. As the spins relax, they relax back to a skyrmion\nconfiguration with spins opposite to the initial configuration, as illustrated in Figure 4 (b). From Figure 4 (a), subsequence\nlaser pulses show the spin reversal process is repeatable. This reversal of spin texture likely arises from the angular momentum\nexchange between the two different sublattices, which leads to maintaining spin texture in a subpicosecond timescale. This result\ndemonstrates another unique feature of all-optical switching in RE-TM ferrimagnet. Furthermore, this bilayer design opens up\nthe possibility to incorporate into a magnetic tunnel junction. One can introduce exchange bias by adding an antiferromagnetic\nlayer on top of the top FeGd layer. By doing so, the exchange bias effect from the antiferromagnetic layer can enhance or\nreduce the barrier of switching in the top FeGd layer, which results in parallel or anti-parallel spins in each sublattice between\nthe top and bottom FeGd layer. Of course, such heterostructure will need to be tested and optimized experimentally.\n4/9Conclusions\nWe have performed atomistic simulations to study all-optical ultrafast switching in a 10-nm thick synthetic antiferromagnet\nwith bilayer amorphous rare-earth transition-metal ferrimagnet. Through this study, we confirmed deterministic spin switching\nin the synthetic antiferromagnet by a femtosecond laser pulse. Furthermore, we demonstrated the reversal of magnetization in a\nskyrmion using a laser pulse. These results indicate promise in the applications of synthetic antiferromagnet with ferrimagnetic\nheterostructures in future energy-efficient high-density spintronic devices.\nMethods\nParameter Value\nFe Magnetic moment ( mFe) 2.22 mB\nGd Magnetic moment ( mGd) 7.60 mB\nFe-Fe Exchange Interaction (J Fe\u0000Fe) 2.83 x 10\u000021J\nGd-Gd Exchange Interaction (J Gd\u0000Gd)1.26 x 10\u000021J\nFe-Gd Exchange Interaction (J Fe\u0000Gd) -1.09 x 10\u000021J\nAnisotropy (K u) 0.30 x 10\u00005J/m3\nDamping ( a) 0.05\nFe Gyromagnetic ratio ( gFe) 1.85 T\u00001s\u00001\nGd Gyromagnetic ratio ( gGd) 1.76 T\u00001s\u00001\nTable 1. Parameters used for modeling magnetization dynamics in GdFe, which were obtained from Ostler el al. and Radu et\nal.14, 42.\nFigure 5. Temporal evolution of electron and lattice temperature after irradiation of a 100-fs laser pulse with 30 mJ/m2\nfluence..\nWe built an atomistic model of Fe and Gd atoms on the FCC lattice with in-plane periodic boundary conditions. The Fe\nand Gd atoms are randomly distributed in the FCC lattice. For a 10 nm thick SAF, this model contains 32 x 32 x 32 sites,\nwith half the sites (32 x 32 x 16) belonging to each layer of 5 nm thick RE-TM in the SAF. A semi-classical two-temperature\nmodel is employed to calculate the temporal evolution of electron and lattice temperature under the application of femtosecond\nlaser irradiation37, 38. Figure 5 shows the temperature profile of a 100 fs laser pulse with 30 mJ/m2fluence. Within 0.1\nps, the electronic temperature reaches a peak of over 1300 K, and the lattice temperature reaches just below 600 K. Both\ntemperatures are above the measured Curie temperature of 540 K42. The atomistic spins are coupled to the electron temperature\nin the two-temperature model37, 38. A stochastic Landau–Lifshitz–Gilbert (LLG) equation is used to model the magnetization\n5/9dynamics43. 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Reports 9, 9964, DOI: 10.1038/s41598-019-46458-4 (2019).\n42.Ostler, T. A. et al. Crystallographically amorphous ferrimagnetic alloys: Comparing a localized atomistic spin model with\nexperiments. Phys. Rev. B 84, 024407, DOI: 10.1103/PhysRevB.84.024407 (2011).\n43.Gilbert, T. A lagrangian formulation of the gyromagnetic equation of the magnetization field. Phys. Rev. 100, 1243 (1955).\nAcknowledgements\nThis work was supported by the DARPA Topological Excitations in Electronics (TEE) program (grant D18AP00009). The\ncontent of the information does not necessarily reflect the position or the policy of the Government, and no official endorsement\nshould be inferred. Approved for public release; distribution is unlimited.\n8/9Author contributions statement\nC.T.M conceived the simulations and analysed the results, W.Z. and S.J.P contributed to discussions. All authors reviewed the\nmanuscript.\nAdditional information\nThe author(s) declare no competing interests.\nData Availability\nThe datasets generated during and/or analyzed during the current study are available from the corresponding author on\nreasonable request.\n9/9"    },    {        "title": "1410.4280v1.Understanding_the_Magnetic_Puzzles_of_Double_Perovskites_A2FeOsO6__A_Ca__Sr_.pdf",        "content": "Understanding  the Magnetic Puzzles of Double Perovskite s \nA2FeOsO 6 (A=Ca, Sr)  \nY . S. Hou, H. J. Xiang, and X. G. Gong  \n \nKey Laboratory of Computational Physical Sciences (Ministry of Education), State \nKey Laboratory of Surface Physics, and Department of Physics, Fudan University , \nShanghai 200433, People’s Republic of  China  \n \n \nAbstract  \nDouble perovskites  Sr2FeOsO 6 and Ca2FeOsO 6 show puzzling magnetic properties,  \nthe former a low -temperature antiferromagnet  while the later a high -temperature \ninsulating ferrimagnet . Here, in order to understand  the underlying  mech anism, we \nhave investigated the frustrated magnetism  of A2FeOsO 6 by employing density \nfunctional  theory and maximally -localized  Wannier functions . We find that lattice \ndistortion enhances the antiferromagnetic nearest -neighboring Fe -O-Os interaction but \nweakens the antiferromagnetic interactions through the Os -O-O-Os and  \nFe-O-Os-O-Fe paths , which  is responsible for the magnetic transition  from the \nlow-temperature antiferromagnetism to the high-temperature ferrimagnetism  with the \ndecrease of the radius of the A2+ ions. We also discuss the 5 d3-3d5 superexchange and \npropose such superexchange is intrinsically antiferromagnetic instead of the expected \nferromagnetic . Our work illustrate that the magnetic frustration can be effectively \nrelieved by lattice distortion, which provides a nother  dimension to  tune the complex \nmagnetism in other 3 d-5d (4d) double perovskites.  \n \n \n \n \nPACS numbers: 75.50.Gg, 75.50.Ee, 71.20. -b, 71.27.+a  \n \n \n I. INTRODUCTION  \nDouble perovskite (DP) oxides \n'\n26A BBO , where A is alkaline -earth or rare-earth \nmetals and B(\n'B ) are transition metals (TMs), have attracted considerable attention \ndue to the numerous interesting physical discoveries, including the room -temperature \n(RT) half -metallicity [1], high -temperature (HT) insulating ferrimagnetism [2, 3], \nmultiferroicity [4], ferromagnetism [5] and so on. Especially , osmium -based DPs \nSr2FeOsO 6 (SFOO), SrCaFeOsO 6 (SCFOO) and Ca 2FeOsO 6 (CFOO) are intensively \ninvestigated recently because of their puzzling magnetic behaviors [3, 6-11]. \n Experimentally, it is showed that SFOO adopts two types of antiferromagnetism, \nnamely , the AF1 and AF2, whose Neel temperatures are \nAF1\nN 140K T  and \nAF2\nN 67K T\n[6], respectively. Theoretically , the mechanism s of the occurrence of the \nAF1 and AF2 are under debate. For the AF1, it was widely  accepted that the \nferrimagnetic (FIM) ab planes are coupled to the neighboring planes by a \nferromagnetic (FM) Fe -O-Os superexchange [3, 8, 9]. However, it was recently  \nsuggested that these FIM ab planes may be coupled by the antiferromagnetic (AFM) \nOs-O-O-Os interactions [10]. For the AF2, Morrow et al. proposed that the long -range \nFe-Fe AFM interaction through the four -bond Fe -O-Os-O-Fe path dominates and \nleads to the AFM -type Fe -Os chains along the c axis [10], while  Kanungo et al . \nshowed that the long -range Os -Os AFM interaction through the four-bond \nOs-O-Fe-O-Os path is responsible [9]. For the magnetic  ordering temperature, there is \nno quantitative understanding why the TN of AF1 is low  up to now.  \nSurprisingly, SCFOO is a ferrimagnet  in comparison  with the antiferromagnet ic \nSFOO and has a higher magnetic ordering temperature \n210KCT  [10] than SFOO, \nalthough the main lattice structure difference between SCFOO and SFOO is that the \nFe-O-Os bond angle along the c axis in SCFOO is sm aller than that in SFOO. More \ninterestingly , CFOO is  an insulating  ferrimagnet  with a n even  higher magnetic \nordering temperature \n320KCT  [3, 10], which could not  be accounted for by the \ngeneralized double exchange mechanism [12]. To the best of our knowledge, there is \nstill a lack of comprehensi ve and unified understanding s on how the low -temperature \n(LT) antiferromagnetism of A2FeOsO 6 transform s to the HT ferrimagnetism with the \ndecrease of the radius of the A2+ ions.  \nIn this paper, to obtain a unified insight into  all these puzzles , we have systematically investigated the frustrated magnetism of the DPs CFOO, SCFOO and \nSFOO by employing density functional theory (DFT) and maximally -localized \nWannier functions (MLWFs). We find that lattice distortion enhances the AFM \nFe-O-Os interaction but weakens the AFM interactions of the Os -O-O-Os and \nFe-O-Os-O-Fe paths. As a result of the serious lattice distortion, therefore, CFOO has  \nthe strong and dominant AFM interaction between the nearest neighboring (NN) Fe3+ \nand Os5+ ions. Consequently , the NN Fe3+ and Os5+ ions are antiparallelly coupled and \nthe experimentally observed ferrimagnetism is established. Simultaneously , its AFM \ninteractions through the Os -O-O-Os and Fe -O-Os-O-Fe paths are weak, so the Os -Os \nand Fe -Fe magnetic frustration is effectively relieved and it has a very high TC. \nBecause SCFOO is less distorted compared to CFOO, its magnetic frustration \nbecomes stronger. Acco rdingly, its TC is lowered. In the tetragonal I4/m structure of \nSFOO, lattice distortion vanishes along the c axis but is very similar to that of CFOO \nin the ab plane. This special lattice distortion pattern results in the in -plane NN Fe3+ \nand Os5+ ions be ing antiparallelly aligned and the FM chains along the c axis. The \nresulting magnetic structure is the strongly frustrated antiferromagnetism AF1 with a \nlow Neel temperature \nNT . Lastly, strong spin -lattice coupling leads to the \ntransformation from AF1 to AF2.  Our work illustrates that the magnetic frustration \ncan be effectively relieved by lattice distortion, which may be responsible for  the \ncomplex magnetism in other 3 d-5d (4d) DPs.  \n \nII. COMOPUTATIONAL DETAILS  \nFirst-principles calculations based on DFT are performed within the generalized \ngradient approximation (GGA) according to the Perdew -Burke -Ernzerhof (PBE) \nparameterization as implemented in Vienna Ab initio  Simulation Package (VASP) [13]. \nThe projector -augment ed wave method [14], an energy cutoff of 500 eV and a \ngamma -centered k -point mesh grid are used. Io n positions are relaxed towards \nequilibrium with the Hellmann -Feynman forces on each ions less than 0.01 eV/Å. \nWe use the simplified (rotationally invariant) coulomb -corrected density functional \n(DFT+U) method according to Dudarev et al.  [15]. \neff\nFeU 4.0 eV , and \neff\nOsU 2.0 eV\n [9] are applied to the Fe 3 d and Os 5 d states, respectively . With these \neffective U, th e calculated band gap of CFOO is about 1.19 eV , very close to the experimentally measured activation energy (\n1.2 eVgapE ) [3]. It has been \ndemonstrated [7] that spin -orbit coupling (SOC) in CFOO is insignificant, so SOC is \nnot taken into account in the present work. Hopping integrals between 3 d/5d orbitals \nare extracted f rom the real -space Hamiltonian matrix elements in the \nnon-spin-polarized MLWFs basis. MLWFs are obtained by employing the \nvasp2wannier90  interface in combination with the wannier90  tool [16]. In order to \nobtain the 3 d/5d-like Wannier functions, we construct MLWFs in a suitable energy \nwindow mainly containing the 3 d/5d antibonding states. All MLWFs are considered to \nbe well converged if the total spread change within  50 successive iterations is smaller \nthan 10−9Å2.   \n \nIII. RESUL TS AND DISCUSSIONS  \nWe first demonstrate the magnetic interaction between the NN Fe3+ and Os5+ ions \nthrough the Fe -O-Os path in CFOO is intrinsically AFM and that lattice distortion can \neffectively relieve the magnetic frustration and increase \ncT . Then we show that the \ncompeting magnetic interactions in the tetragonal I4/m structure of SFOO give rise to \nthe frustrated AF1 antiferromagnetism with a low Neel temperature \nNT . We give \nexplanations to the ferrimagnetism  of SCFOO  with a  lower TC. Last, we propose an \nimportant and general rule on the 3 d5-5d3 superexchange, which will help one to \nunderstand the complex magnetism in other DPs.  \n \nA. Intrinsically AFM interaction between the NN Fe3+ and Os5+ ions and the \neffect of lattice distortion on the ferrimagnetism in CFOO  \nThe lattice  of CFOO is severely distorted. It has the monoclinic structure with the \nspace group of P21/n [3]. Lattice distortion of DPs \n'\n26A BBO , depending on the ionic \nradii of the constituent ions, can be characterized by the tolerance factor  \n' 2 0.5AO\nBO Brrt\nr r r   \n \nwhere\nAr , \nBr, \n'Br  and \nOr  are the ionic radii of the respective ions. The tolerance \nfactor of CFOO (\n0.9639CFOOt ) is smaller than 1. Accordingly, both FeO 6 and OsO 6 \noctahedra are severely tilted and rotated (see inset of FIG.1). The experimental out-of-plane and in -plane Fe -O-Os angles are 151.5o and 154.0o, respectively . After \nthe structure relaxation, these two kinds of angles become 149.7o and 149.1o.  \n  Lattice distortion has been suggested to cause  the ferrimagnetism in CFOO [3]. \nAccording to Goodenough -Kanamori rules [17], the superexchange between the N N \n3 3 2\n2 Fe ( )ggte\n and \n53\n2 Os ( )gt ions is expected to be FM. However, experimental \nmeasurements show that CFOO is FIM [3], indicating that the superexchang e between \nthe NN Fe3+ and Os5+ ions is AFM. Feng et al. [3] proposed this ferrimagnetism was \ndriven by the lattice distortion, since d3-d5 magnetic in teraction switch ed from FM to \nAFM in a range of 125o-150o and the Fe -O-Os angles in CFOO were  just in the \nvicinity of this range. Their proposal is based on such underlying assumptions: (1) the \nAFM interaction due to \n2gt  electrons is  (much) weaker than the FM interaction due \nto the \nge  electrons; (2) the former has a weaker dependence on the Fe -O-Os angle \nthan the latter [18]. However, it is highly possible that the 5 d \n2gt  electrons of the \nOs3+ ions are spatially extended so that the former is stronger than the latter, which \nleads to an AFM interaction betwee n the NN Fe3+ and Os5+ ions. \nIn order to unveil why CFOO is FIM and how lattice distortion affects this \nferrimagnetism, we have systematically explored the effect of lattice distortion on its \nmagnetic interaction. Since the positions of O2- ions are vital to the magnetic \ninteraction, we perform series of calculations using a linear superposition of the \nWyckoff positions of the relaxed structure and the pseudo -cubic one in which the \noxygen atoms are artificially positioned to make Fe -O-Os angles straight. Th e lattice \nconstants and the positions of the Fe3+, Os5+ and Ca2+ ions are fixed. The atomic \npositions can be computed as   \n( ) (1 )relax cubic\nx x x     R R R\n \nwhere \nrelaxR  and \ncubicR  are the position vectors of the relaxed and pseudo -cubic \nstructures, respectively , and \nx  varies between 0 and 1. For example, \n0x\ncorresponds to the relaxed structure and \n1x to the pse udo-cubic one. Thus \nx \ncharacteri zes the lattice distortion induced by O2- ions. The dominant magnetic \ninteractions are divided into three groups shown in the inset of FIG. 1. The first group \nis the superexchange between the NN Fe3+ and Os5+ ions. The second one is the \nsuper -superexchange between the next near -neighboring (NNN) Os5+ ions. The third one is the long -range Fe -Fe interaction through the four -bond Fe -O-Os-O-Fe path. \nTechnically , we adopt the four -states mapping method [19] to evaluate these magnetic \ninteractions. Note that a positive exchange constant J corresponds to the AFM \ninteraction whereas a negat ive one to the FM interaction.  \nWe find the magnetic interaction between the NN Fe3+ and Os5+ ions is intrinsically \nAFM. The calculated magnetic exchange constants of Fe -O-Os paths in the \npseudo -cubic structure are shown in the FIG. 1a. They are all positiv e and thus AFM. \nThe intrinsically AFM interaction of the Fe -O-Os path can be qualitatively understood \nbased on the extended Kugel -Khomskii model [20-22]. According to this model, \nmagnetic interactions can be evaluated based on the hopping integrals and on -site \nenergies, namely ,  \n44\n( ) ( )( )afm afm fm fm\nij ij ij ij H AFM FM\nij ij ij\nafm fm ij ij ij Ht t t t JJ J JU U U J        \n \nwhere U, \nHJ  and \nij  are the on -site Coulomb interaction, the Hund’s coupling and \nthe energy difference between the \nthi  and \nthj  energy levels, respectively . \nafm\nijt  (\nfm\nijt) \nis the hopping integral. The first term in \nijJ  describes the AFM contribution due to \nthe hybridization between two occupied orbitals. The second term describes the FM \ncontribution due to the hybridization between the occupied and empty orbitals. In \norder to elucidate why the magnetic interaction betwe en the NN Fe3+ and Os5+ ions is \nintrinsically AFM, we take the Fe -O-Os path along the c axis in the pseudo -cubic \nCFOO as a typical example. Its detailed hopping integrals and energy levels are given \nin the right panel of FIG. S1 of supplemental material (S M) [23]. Compared with the \nFM interaction between the NN Mn3+ ions in the cubic LaMnO 3 (LMO) [24], two \npivotal factors drive the magnetic interaction between the NN Fe3+ and Os5+ ions in \nthe pseudo -cubic CFOO to be intrinsically AFM. The first one is the very lar ge energy \ndifference \n  (up to 3.0 eV) between the occupied \nge  orbitals of Fe3+ ion and the \nunoccupied one of Os5+ ion. This gives weak FM contribution. The second one is the \nrather large hopping integrals between the occupied \n2gt  orbitals of Fe3+ and Os5+ ions. \nFor instance, the leading hopping integral is 0.27 eV . This gives strong AFM \ncontrib ution. Therefore AFM contribution dominates over the FM one, giving rise to \nthe intrinsically AFM interaction between the NN Fe3+ and Os5+ ions.  \nIn addition, we find that lattice distortion can effectively relieve the magnetic frustration in CFOO, thereby  raising the FIM phase transition temperature \nCT . Shown \nin the FIG. 1a and 1b are the evolutions of the dominant magnetic exchange constants \nas \nx  increases. As lattice distortion is gradually becoming serious, the NN AFM \ninteractions between the NN Fe3+ and Os5+ ions slightly enhance, but the NNN AFM \ninteractions between the NNN Os5+ ions dramatically weaken. Simultaneously , the \nlong-range four -bond Fe-O-Os-O-Fe AFM interactions along the pseudo -cubic [001], \n[010] and [100] axes also become weaker (see Fig. 1c). Since Os5+ ions form a \nface-centered sublattice with geometrically frustrated edge -sharing tetrahedrons, \nantiferromagnetically interacting Os5+ ions are strongly frustrated. On one hand, the \ndominant NN Fe -O-Os AFM interactions require the magnetic moments of Fe3+ ions \nto be parallelly aligned. On the other hand, the AFM magnetic interaction through the \nfour-bond Fe -O-Os-O-Fe paths requires t he magnetic moments of Fe3+ ions to be \nantiparallelly aligned. Obviously, Fe3+ ions are magnetically frustrated. Because \nlattice distortion weakens the NNN Os -O-O-Os and long -range four -bond \nFe-O-Os-O-Fe AFM interactions, it can effective ly relieve the Os -Os and Fe -Fe \nmagnetic frustration. Accompanied with such frustration relieving is the raising of the \nFIM phase transition temperature \nCT . FIG. S2 of SM  shows the evolution of \nCT  \nobtained by Mo nte Carlo (MC) as lattice distortion weakens. It clearly shows that the \nCT\n of the relaxed structure (\n0x ) (about 266K, close to the experimentally \nmeasured one (\n320KCT ) [3]) is higher than that of the less distorted one \n(\n0.25x ). Not e that \nCT  slightly increases along with the weakening of lattice \ndistortion for large \nx . This is because the magnetic ground state of CFOO with \nsmall l attice distortion (see the inse t of FIG. 1c) is no longer FIM but AFM with the \nAF1 order as appearing in the SFOO.  \nFIG. 2a demonstrates the mechanism that lattice dist ortion enhances the NN \nFe-O-Os AFM interaction. For the illustration purpose, we take the Fe -O-Os path \nalong the c axis as an example. FIG. S1 of SM  shows the detailed leading hopping \nintegrals and energy levels in the relaxed and pseudo -cubic structures, respectively . \nThese hopping integrals clearly indicates that lattice distortion tremendously reduces \nthe electron hopping between the occupied \nge  orbitals of Fe3+ ions and the \nunoccupied one of Os5+ ions. Consequently , one can conclude based on the formula of ijJ that lattice distortion extraordinarily reduces the FM contribution to the NN \nsuperexchange. In contrast, lattice distortion has a rather minor effect on the AFM \ncontribution, because it brings into considerable hoppings between the occupied \nge  \norbitals of Fe3+ ions and the occupied \n2gt  orbitals of Os5+ ions, although it reduces \nthe hopping between the occupied \n2gt  orbitals of Fe3+ and Os5+ ions as well. \nTherefore, lattice distortion enhances the NN AFM interaction by both dramatically \nreducing the FM contribution and maintaining  the AFM contribution almost \nunchanged.   \n  We find that the NNN AFM interaction between the NNN Os5+ ions is weakened by \nlattice distortion. This is because such NNN super -superexchange has sensitive \ndependence on the geometry of the Os -O-O-Os path. Shown in the insets of Fig. 2b \nare the geometries of the concerned Os -O-O-Os paths in the relaxed and pseu do-cubic \nstructures. The detailed leading hopping integrals and energy levels between the \ninvestigated Os5+ ions are shown in the FIG. S3 of SM . Note that FM contribution to \nthe NNN super -superexchange is rather weak in the relaxed and pseudo -cubic \nstructu res because of small hopping integrals and large energy differences. So the \nNNN super -superexchange is basically determined by the AFM contribution. \nComparing the two investigated Os -O-O-Os paths (FIG. 2b) clearly indicates that \nlattice distortion increase s the O -O bond length. Such increase reduces the hopping \nbetween the \n2gt  electrons of Os3+ ions, as verified by the reduced hopping integrals \nfrom the pseudo -cubic structure to the relaxed one ( see FIG. S3 of SM ). Therefore \nlattice d istortion blocks the \n2gt  electron hopping through Os -O-O-Os path, thereby \nweakening the NNN AFM interaction.  \n   \nB. Competing magnetic interactions lead to the AF1 antiferromagnetism with \na low TN in SFOO  \nSFOO adopts two different magnetic and lattice structures depending on \ntemperature [6]. With temperature decreasing, its magnetic structure transforms from \nAF1 into AF2 antiferromagnetism and its lattice structure transforms from I4/m into \nI4 with a di merization between the NN Fe3+ and Os5+ ions along c axis. In both AF1 \nand AF2, moments of Fe3+ and Os5+ ions are antiparallelly coupled in the ab plane (FIG. 3a and FIG. 3c). Spins order as \n  along the c axis in AF1 (FIG. 3b) but \n\n in AF2 (FIG. 3d). The experimentally measured out -of-plane and \nin-plane Fe -O-Os angles in the I4/m-AF1 phase are 180.0o and 166.1o [8], respectively . \nThey become 180.0o and 161.2o after structure relaxations.  \n  Our study on the I4/m-AF1 phase (FIG. 4a and FIG. 4b) shows the out -of-plane NN \nAFM interaction (\n2\nFe-OsJ ) is much weaker than the in -plane one (\n1\nFe-OsJ ) and that the \nout-of-plane NNN AFM interaction (\n2\nOs-OsJ ) is stronger than the in -plane one (\n1\nOs-OsJ ), \nwhich are understandable based on our above res ults on CFOO. Because the Fe -O-Os \nangle along the c axis is 180.0o, similar to that in the pseudo -cubic CFOO, the \nout-of-plane NN AFM interaction \n2\nFe-OsJ  is weak. In the tetragonal ab plane, lattice \ndistortion is similar to that in the r elaxed CFOO, so the in -plane NN AFM interaction \n1\nFe-OsJ\n is strong. The weak in -plane NNN AFM interaction \n1\nOs-OsJ is due to the strong \nin-plane lattice distortion blocking the Os5+ ions’ \n2gt  electron hopping, similar to the \nweak NNN AFM interactions in the relaxed CFOO. Besides, the out -of-plane NNN \nAFM interaction \n2\nOs-OsJ is stronger than that of the relaxed CFOO but weaker than \nthat of the pseudo -cubic CFOO. Overall, such  magnitude of the spin interactions is a \nresult of the combination of the absence of the lattice distortion along the c axis and \nstrong lattice distortion in the tetragonal ab plane. Finally, it is expected that the \nlong-range four -bond Fe -O-Os-O-Fe AFM in teraction along the c axis \n(\n3\nFe-Fe 6.2 meV J ) is stronger than the in -plane one (\n2\nFe-Fe 5.1meV J ).  \nHere we demonstrate how the competing magnetic interactions establish the AF1 in \nthe tetragonal I4/m structure of SFOO. First of all, it should be noted that the easy \naxis is the c axis [6], that is, magnetic moments can only point up and down along the \nc axis. Because the in -plane NN AFM interaction \n1\nFe-OsJ  is approximately four times \nof the magnitude of the in -plane NNN AFM interaction \n1\nOs-OsJ  and the in -plane \nlong-range four -bond Fe -O-Os-O-Fe AFM interaction \n2\nFe-FeJ  and their pairwise \nnumbers (Z’s) are all the same (\n4Z ), the optimal configuration is that, in the ab \nplane, magnetic moments of Fe3+ and Os5+ ions are antiparallelly aligned, as observed \nin the experiments [6]. Besides, the out -of-plane NN, NNN and the long -range \nfour-bond magneti c interactions are all AFM. If only the out -of-plane NN AFM interaction is taken into consideration, the FIM Fe3+-Os5+ layers should be \nantiparallelly coupled along the c axis. In this case, the resulting magnetic structure is \nthe FIM (FIG. 4a). If the out -of-plane NNN AFM interaction is taken into \nconsideration, the FIM Fe3+-Os5+ layers should be parallelly coupled along the c axis. \nIn this case, the resulting magnetic structure is the AF1 (FIG. 4b), namely, the \nexperimentally observed one. Finally, if onl y the long -range four -bond Fe -O-Os-O-Fe \nAFM interaction is taken into consideration, it gives rise to the AF2 (FIG. 4c). \nObviously, the out -of-plane NN, NNN and long -range four -bond AFM interactions \ncompete against each other to give rise to the different magnetic ground states. Since \ntheir magnitudes are comparable (see FIG. 3b), their pairwise numbers are the \ndecisive factor in determining the optimal magnetic structure. Their pairwise numbers \nare \n2NNZ , \n8NNNZ  and \n2Fe O Os O FeZ    , respectively . This indicates that the \nout-of-plane NNN AFM interactions overwhelmingly dominate over the out -of-plane \nNN and long -range four -bond AFM interactions. Therefore the optimal magnetic \nconfiguration is the AF1.  \nThe foregoing deduction can be further confirmed as follows. In the FIM, all the \nout-of-plane Fe -Fe and Os -Os pairs are frustrated (see FIG. 4a). In the AF1, all the \nout-of-plane Fe -Os and Fe -Fe pair are frustrated (see FIG. 4b). In the AF2, half of the \nout-of-plane Fe -Os and Os -Os pair are frustrated (see FIG. 4c). In terms of the \nout-of-plane NN, NNN and long -range four -bond AFM interactions , therefore, th e \nformula -unit (f.u.) magnetic energies  of the FIM, AF1 and AF2 are \n2 3 2( ) 2 4 19.8meV / f.u.FIM Fe Os Fe Fe Os OsE J J J J     \n \n2 3 2\n1( ) 2 4 7.4meV / f.u.AF Fe Os Fe Fe Os OsE J J J J     \n \n3\n2( ) 6.2meV / f.u.AF Fe FeE J J \n. \nThis indicates that FIM should have the highest energy , AF2 the media one and AF1 \nthe lowers one . Such  estimation  is in accord with  our DFT calculations: \n34.139meV / f.u.FIME\n > \n231.153meV / f.u.AFE > \n134.156meV / f.u.AFE . \nSo the AF1 is established to relieve the magnetic frustration as much as possible.   \nThe low Neel temperature TN of the AF1 is a result of the strong magnetic \nfrustration. Actually, only the in -plane \n1\nFe-OsJ  and the out -of-plane \n2\nOs-OsJ  AFM \ninteractions are not frustrated in the AF1. However, the in -plane \n1\nOs-OsJ , out-of-plane \n2\nFe-OsJ\n and the long -range four -bond \n2\nFe-FeJ , \n3\nFe-FeJ  AFM interactions violate the AF1 antiferromagnetism and thus induce frustration. Our MC simulations indicate that, if \nonly \n1\nFe-OsJ  and \n2\nOs-OsJ  are taken into consid erations, the TN of the AF1 is very high, \nup to 354K, sharply contradicting with the experimentally observe one (140K). To \ndetermine why the experimentally measured TN is so low, we performed four \nadditional MC simulations: one with the in -plane \n1\nOs-OsJ , one with the out -of-plane \n2\nFe-OsJ\n, one with the long -range four -bond \n2\nFe-FeJ  and \n3\nFe-FeJ , and one with all these \nmagnetic interactions. The resulting specific heat versus temperature plots are \npresented in FIG 4d. As can be seen, \n1\nOs-OsJ , \n2\nFe-OsJ  and the long -range four -bond \n2\nFe-FeJ\n and \n3\nFe-FeJ  can all low er the TN for they are all frustrated. Moreover, the \nout-of-plane Fe -O-Os AFM interactions make the most contribution to the lowering of \nTN of the AF1. If all the dominating magnetic interactions are taken into consideration, \nthe MC simulated TN is 155K, very close  to the experimental one.  \nBy comparing the magnetic exchange constants of I4 structure with that of I4/m \nstructure (see FIG. 3), one can find that the magnetic interactions situation in the \nformer is very similar to that in the latter, with the exception that the rather slight \ndimerization along the c axis in the former prominently enhances the out -of-plane NN \nAFM interactions \n3\nFe-OsJ  (see FIG. 3d), which indicates a very strong spin -lattice \ncoupling. Like in the I4/m structure, the long -range four -bond Fe -O-Os-O-Fe AFM \ninteraction \n3\nFe-FeJ  favors the formation of the AF2. Furthermore, the enhancement of \nthe out -of-plane \n3\nFe-OsJ  favors this formation. Thus we attribute the AF2 \nantiferromagnetism in I4 structure to the strong spin -lattice coupling.  \n \nC. Explanations to the ferrimagnetism  of SCFOO  with a  lower TC  \nComparing SCFOO with S FOO  and CFOO , one can conclude that its mediate  \nlattice distortion causes its ferrimagnetism with a  lower TC. Experiments show that \nSCFOO has a rather similar lattice structure to that of CFOO [10]. However, its \nFe-O-Os bond angles approach a more linear geometry  than that of CFOO , because \nhalf of Ca2+ ions are replaced by larger Sr2+ ions. So it can be inferred that SCFOO \nshould be still ferrimagnetic . Our detailed study shows that SCFOO is indeed FIM \n(see Part IV of SM ), consistent with experiments [10]. Since its Fe -O-Os bond \nbecome more linear, its NN AFM interactions become weaker but its NNN Os-O-O-Os and long -range four -bond Fe -O-Os-O-Fe AFM interactions become \nstronger. This is verified by the calculated magnetic exchange parameters, as listed in \nthe Table II of SM . Consequently , its magnetic frustration gets stronger and its TC \nshould be lowered. Our MC simulated TC of SCFOO is about 100K, lower than the \ncorresponding TC (266K) of CFOO, consistent with the experimental observations [6, \n10].         \n \nD. Discussions on the 3 d5-5d3 superexchange  \nBased on the present work, an important and general rule on the 3 d5-5d3 \nsuperexchange in the DPs can be proposed as follows. It is generally accepted that the \nd5-d3 superexchange changes from FM for \nc  to AFM for \nc  with \noo135 150c\n [17]. However, we demonstrate that the magnetic interaction \nbetween 3 d5 and 5 d3 TMs would be intrinsically AFM (This conclusion is \nindependent of the choice of a reasonable U, see Table III of SM ) and that this AFM \ninteraction would enhance as its angle \n  decreases, as evidenced by the Fe -O-Os \ninteractions in CFOO, SCFOO and SFOO. This intrinsically AFM interaction results \nfrom both the large hopping integrals between the occupied \n2gt  orbitals and the large \nenergy difference between the occupied \nge  orbitals of 3 d TM and the unoccupied \nones of 5 d TM, for the former gives rise to strong AFM contribution and the latter \ngives rise to weak FM contribution to the 3 d5 -5d3 superexchange. As the angle \n  \ndecreases, the electron hopping s between the occupied \nge  orbitals of the 3 d5 TM and \nthe unoccupied ones of the 5 d3 TM will substantially reduce but the electron hopping s \nbetween the occupied orbitals of 3 d5 TM and 5 d3 TM are maintained almost \nunchanged. Thus decreasing the angle \n  means reducing the FM contribution but \nmaintaining the AFM contribution almost unchanged. Consequently , the AFM \ninteraction of the 3 d5-O-5d3 path enhances with \n  decreasing.  \n \nIV. SUMMARY  \nIn conclusion, we have investigated the effect of lattice distortion on the frustrated \nmagnetism of double perovskites Ca 2FeOsO 6, Sr 2FeOsO 6 and Sr 2CrOsO 6. We find \nthat lattice distortion enhances the NN AFM Fe -O-Os interactions but weakens the \nAFM interactions of the Os -O-O-Os and Fe -O-Os-O-Fe paths. As lattice distortion gets more and more severe from Sr 2FeOsO 6 to SrCaFeOsO 6 and then to Ca 2FeOsO 6, \ntherefo re, the NN AFM Fe -O-Os interactions become stronger but the AFM \ninteractions of Os -O-O-Os and Fe -O-Os-O-Fe paths become weaker. Consequently , \nthe magnetic ground state transforms from antiferromagnetism to ferrimagnetism, and \nthe magnetic transition temper ature increases. We propose the 5 d3-3d5 superexchange \nis intrinsically antiferromagnetic instead of the expected ferromagnetic . Our work \nillustrates that the magnetic frustration can be effectively relieved by the lattice \ndistortion in 3 d-5d (4d) double perovskites.  \n  \nACKNOWLEDGMENTS  \nWork was partially supported by NSFC, the Special Funds for Major State Basic \nResearch, Foundation for the Author of National Excellent Doctoral Dissertation of \nChina, The Program for Professor of Special Appointment at Shang hai Institutions of \nHigher Learning, Research Program of Shanghai municipality and the Ministry of \nEducation, and Fok Ying Tung Education Foundation. We thank Dr. Jihui Y ang, Prof. \nXiao Gu, Prof. Zhixin Guo and Dr. Haiyuan Cao for useful discussions.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigures   \nFIG. 1. (Color online) The dependence of the calculated magnetic parameters on \nx  \n(lattice distortion).  (a), (b) and (c) correspond to the NN Fe -O-Os superexchanges, \nNNN Os -O-O-Os super -superexchanges and the long -range four -bond Fe -O-Os-O-Fe \ninteractions, respectively . The inse t between (a) and (b) illustrates the crystal structure \nof Ca 2FeOsO 6 and the dom inant magnetic interaction paths. The NN superexchange \npaths Fe -O-Os and NNN super -superexchange paths Os -O-O-Os are shown by the \nblack and blue solid lines with double arrowheads, respectively. The long -range \nfour-bond exchange paths \n1\nFe-FeJ , \n2\nFe-FeJ  and \n3\nFe-FeJ  (not shown) are along the \npseudo -cubic [001], [100] and [010] axes. Ca2+ ions are omitted from the crystal \nstructure for clarity . The inset in (c) is the dependence on \nx of the energy of the \nFIM (black) and AF1 (red) magnetic structures.  \n \n \nFIG. 2. (Color online) The mechanism that lattice distortion enhances the NN AFM \ninteraction between Fe3+ and Os5+ ions (a) and weakens the NNN AFM interaction \nbetween Os5+ ions (b)  in Ca 2FeOsO 6. Solid (dashed) lines with double arrowheads \nindicate the electron hopping making AFM (FM) contribution to the NN \nsuperexchange or NNN super -superexchange. S and W represent “strong” and “weak”, \nrespectively . In (a), the FM contribution of \n1xa  is weaker  than that of \n0xa . \nHowever, the AFM contribution of \n1xa is stronger  than that of \n0xa . In (b), the \nAFM contribution of \n1xa is weaker than that of \n0xa . Inse ts in (a) and (b) are \nthe local structures of Fe -O-Os and Os -O-O-Os paths, respectively . The relevant  bond \nangles, bond lengths and calculated magnetic exchange constants are given.  \n \n \nFIG.3. (Color online) The dominant magnetic exchange paths and the AF1, AF2 \nmagnetic structures of Sr 2FeOsO 6. All magnetic exchange constants J are in unit of \nmeV . (a) and (b) correspond to the I4/m-AF1 phase. (c) and (d) correspond to the \nI4-AF2 phase. (a ) and (c) are the spin arrangement in the tetragonal ab plane. (b) and \n(d) are the spin ordering  along the c axis. Black arrows represent spins. Given are the \nrelevant  bond distances and angels obtained from DFT calculations.   \n \n \n \n \n \n \nFIG. 4. (Color online) Shown are t he FIM (a), AF1 (b) and AF2(c) magnetic structures. \nIn (a), (b) and (c), the frustrated magnetic ions pairs are connected by the black \ndashed lines with double arrowheads. Fe (Os) sites are represented by the green (gray) \nhorizontal lines. Blue (red) arrows represent up (down) spins. (d) is the specific heat \nof SFOO, calculated as a function of temperature T in terms of the spin exchange \ninteractions.  \n \n \n \n \n \n \nReferences  \n[1] K. I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y . Tokura, Nature 395, \n677 (1998).  \n[2] Y . 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G. Gong, and M. -H. Whangbo, Dalton \nTransactions 42, 823  (2013).  \n[20] K. I. Kugel and D. I. Khomski i, Zh. Eksp. Teor. Fiz. 64, 1429  (1973) ,[Sov . Phys. \nJETP 37, 725 (1973)] . \n[21] K. I. Kugel and D. I. Khomskii, Sov . Phys. Usp. 25, 231 (1982) . \n[22] V . V . Mazurenko, F. Mila, and V . I. Anisimov , Phy . Rev. B 73, 014418 (2006).  \n[23] See supplementary material for the detailed  hopping  parameters , energy levels \nand m agnetic transition temperature s TC of Ca 2FeOsO 6, detailed study  on the ground \nmagnetic  structure of SrCaFeOsO 6, calculate d magnetic exchange parameters of \nSrCaFeOsO 6 and the relaxed structure  of Ca 2FeOsO 6, and the calculated  magnetic \nexchange parameters \n1\nFe OsJ  in the pseudo -cubic phase of Ca 2FeOsO 6 versus \nCoulom b interaction U .  \n[24] R. Kov acik and C. Ederer, Phys . Rev. B 81, 245108 (2010).  \n \n \n "    },    {        "title": "1505.00583v1.Spiral_ferrimagnetic_phases_in_the_two_dimensional_Hubbard_model.pdf",        "content": "Spiral ferrimagnetic phases in the two-dimensional\nHubbard model\nJ. D. Gouveia\u0003, R. G. Dias\nDepartamento de F\u0013 \u0010sica, I3N, Universidade de Aveiro, Campus de Santiago, Portugal\nAbstract\nWe address the possibility of spiral ferrimagnetic phases in the mean-\feld phase\ndiagram of the two-dimensional (2D) Hubbard model. For intermediate values\nof the interaction U(6.U=t.11) and doping n, a spiral ferrimagnetic phase is\nthe most stable phase in the ( n;U) phase diagram. Higher values of Ulead to a\nnon-spiral ferrimagnetic phase. If phase separation is allowed and the chemical\npotential\u0016replaces the doping nas the independent variable, the ( \u0016;U) phase\ndiagram displays, in a considerable region, a spiral (for 6 .U=t.11) and non-\nspiral (for higher values of U) ferrimagnetic phase with \fxed particle density,\nn= 0:5, re\recting the opening of an energy gap in the mean-\feld quasi-particle\nbands.\nKeywords: A. Magnetically ordered materials, C. Crystal structure and\nsymmetry, D. Electron-electron interactions, D. Electronic band structure, D.\nPhase transitions\n1. Introduction\nThe 2D Hubbard model remains the most important open theoretical prob-\nlem in the \feld of the strongly correlated electronic systems, despite all e\u000borts\nfuelled by the advent of the high- Tcsuperconductivity.[1, 2] At half-\flling, the\nspin dynamics of the 2D Hubbard model is described by the Heisenberg anti-\nferromagnetic exchange term.[3] Away from half-\flling, the movement of holes\nthrough the spin background generates additional spin mixing. The competition\nbetween the Heisenberg exchange and the spin con\fguration mixing generated\nby hole hopping in the 2D Hubbard model is still far from understood.[4, 5, 6] In\nparticular, there is no consensus regarding the ground state magnetic phase dia-\ngram of the 2D Hubbard model and di\u000berent authors obtain di\u000berent mean-\feld\n(MF) phase diagrams depending on the magnetic phases allowed.[7] Tradition-\nally, one considered ferromagnetism, antiferromagnetism and paramagnetism\n\u0003Corresponding author. Tel. +351 227 828 940\nEmail address: gouveia@ua.pt (J. D. Gouveia)\nPreprint submitted to Solid State Communications October 30, 2018arXiv:1505.00583v1  [cond-mat.str-el]  4 May 2015B D \nC A Figure 1: The 2D lattice and its four sublattices A, B, C and D. We consider two situations:\n(i)mA=mD=m1andmB=mC=m2; (ii)mA=mC=m1andmB=mD=m2.\nphases.[8, 9, 10, 11, 12] The complexity of the MF phase diagram was increased\nwith the introduction of spiral phases[13], which appear between the \"usual\"\nmagnetic phases in the diagram. This complexity was further increased by the\nconsideration of spatial phase separation.[14, 15, 16]\nIn this paper, we extend the results above mentioned, by introducing the\npossibility of a spiral ferrimagnetic phase, that is, a ferrimagnetic phase such\nthat the orientation of magnetic moments changes along the lattice (see Fig.\n1). More precisely, we study the 2D Hubbard model using the Hartree-Fock\napproximation in a square lattice decomposing the lattice in four square sub-\nlattices (A, B, C and D as in Fig. 1) and allowing di\u000berent amplitudes for\nmagnetizations of the spiral phases in the sublattices. Note that, even under\nthe MF approximation, when four sublattices are considered, it is not possible\nto obtain the analytical form of the spectra of the 2D Hubbard model. Our MF\napproach to the 2D Hubbard model follows that of Dzierzawa and Singh.[17, 18]\n2. Calculations\nIntroducing a di\u000berent creation operator in each sublattice, Ay,By,Cyand\nDy, the tight-binding term of the Hubbard Hamiltonian, is\nHt=P\nx;yAy\nx;yBx;y+Ay\nx;yCx;y\n+By\nx;yDx;y+Cy\nx;yDx;y\n+Ay\nx;yBx;y\u00001+Ay\nx;yCx\u00001;y\n+By\nx;yDx\u00001;y+Cy\nx;yDx;y\u00001+H:c:; (1)\nwhere we set the hopping constant equal to 1.\nWe consider for now only the sublattice A (we add the other sublattice\nterms later on). The interaction term of the Hubbard Hamiltonian is, as usual,\nHU=UP\nrAy\nr\"Ar\"Ay\nr#Ar#. We assume that the magnetic moments align in the\nx-yplane, so thathSzi= 0 and the Hartree term becomesU\n4Phni2, wherehni\n2is the density of electrons on each sublattice (here assumed to be the same on\nall of them).\n0.0 0.2 0.4 0.6 0.8 1.005101520\nU\nnFerro\nParam(q,q)\nAF(0,q)\n(0,π)\n(q,π)\n(a) (n;U) phase diagram\nU\nnFerro\nS\nParam(0,π)\n(q,π)(0,q)(q,q)\nU\nn0.0 0.2 0.4 0.6 0.8 1.005101520\nAFPS (b) (n;U) phase diagram with\nphase separation\nn=1 AF\n5 10 15/Minus4/Minus2024\nFerro\nParam(q,π)\n(0,π)\n(0,q)(q,q)\nU(c) (U;\u0016) phase diagram with\nphase separation\n(d)m(n;U)\n (e)qx(n;U)\n (f)qy(n;U)\n (g)E(n;U)\n (h)\u0016(n;U)\nFigure 2: (a) Mean-\feld phase diagram for the usual 2D Hubbard model: The system displays\nantiferromagnetism (AF), ferromagnetism (F), paramagnetism (P) or spiral phases ( q6= 0;\u0019).\nThe antiferromagnetic state ~ q= (\u0019;\u0019) only occurs for n= 1 (half-\flling). (b) ( n;U) and (c)\n(U;\u0016) mean-\feld phase diagrams for the 2D Hubbard model, allowing for phase separation\n(yellow region). (d) m, (e)qx, (f)qy, (g)EMFand (h)\u0016as functions of the doping nand\nCoulomb interaction U, for the 100\u0002100 2D Hubbard model.\nThe Fock term includes averages like hAy\n\"A#i=hS+\nAi=hSAx+iSAyi, whose\nvalues depend on the magnetic phase. Let us assume the average spin in the\nsublattice A is\nh~S~ rAi=mA\n2[cos(~ q\u0001~ rA);sin(~ q\u0001~ rA);0]: (2)\nThe vector ~ q= (qx;qy) de\fnes the magnetic phase of the system. In ~k-space we\nhave\nhS+\nA~ki=1p\nLX\n~k0hAy\n~k0;\"A~k0\u0000~k;#i=mApLu:c:\n2\u000e~k;\u0000~ q; (3)\nwhereLu:c:is the number of unit cells, which gives\nhAy\n~k;\"A~k+~ q;#i=mA\n2; (4)\nwhile all the other mean values in the summation of Eq. 3 vanish. The Fock\nterm in Fourier space is\n\u0000mU\n2X\n~k\u0010\nAy\n~k+~ q;#A~k;\"+Ay\n~k;\"A~k+~ q;#\u0011\n+ULu:c:\n4m2\nA: (5)\n3Adding the tight-binding, Hartree and Fock terms, the Hamiltonian HMFreads,\nin thefA~k;B~k;C~k;D~k;A~k+~ q;B~k+~ q;C~k+~ q;D~k+~ qgbasis,\n \nHt(~k)Hm\nHy\nmHt(~k+ 2~ q)!\n; (6)\nplus the diagonal term\nULu:c:\n4(m2\nA+m2\nB+m2\nC+m2\nD) +ULhni2\n2: (7)\nHere,Ht(~k) is the tight-binding term (Eq. 1) of the Hamiltonian in ~k-space,\n0\nBB@0 1 + eiky 1 +eikx 0\n1 +e\u0000iky 0 0 1 + eikx\n1 +e\u0000ikx 0 0 1 + eiky\n0 1 + e\u0000ikx1 +e\u0000iky 01\nCCA; (8)\nHmis the diagonal matrix, Hm= diag(\u0001 A;\u0001B;\u0001C;\u0001D) with\n\u0001A=\u0000UmA\n2; \u0001B=\u0000UmB\n2eiqy;\n\u0001C=\u0000UmC\n2eiqx;\u0001D=\u0000UmD\n2eiqx+iqy:(9)\n3. Results and discussion\nBy settingmA=mB=mC=mD=m, we recover the MF magnetic phase\ndiagram of the usual 2D Hubbard model, consistent with the ones obtained by\nseveral authors[17, 13, 15] for zero temperature, as presented in Fig. 2a. In order\nto obtain such a diagram, one minimizes either the MF energy EMFusing the\nelectronic density nas an independent variable, or the thermodynamic potential\n\nMFusing the chemical potential \u0016, with respect to the site magnetization\namplitudemand the order parameter ~ q= (qx;qy). These parameters de\fne the\nmagnetic phase of the system.\nA solution with m= 0 is paramagnetic and is usually ~ q-degenerate, while\nsolutions for m6= 0 are in general unique. In the latter case, the wave vector ~ q\nspeci\fes the type of magnetic ordering. For instance, ~ q= (0;0) for the ferromag-\nnetic phase, ~ q= (\u0019;\u0019) for the antiferromagnetic phase and all other choices for\nspiral phases. In the example shown in Fig. 1, we have qx=\u0019=18 andqy=\u0019=6.\nAdditionally, in the same example, the magnetization amplitudes (denoted by\nthe size of the arrows) are mA=mD=m1andmB=mC=m2<m 1. Com-\nparing, for each pair ( n;U) or (U;\u0016), the data obtained for m(Fig. 2d), qx\n(Fig. 2e) and qy(Fig. 2f), the MF magnetic phase diagram displayed in Fig. 2a\nensues. For some values of \u0016, there is more than one pair ( ~ q;m) which minimizes\nthe thermodynamic potential. In those cases, a \frst-order phase transition in\n40.0 0.2 0.4 0.6 0.8 1.005101520\nU\nn\nAF\n/LParen4m2m1\nm1m2/RParen4/LParen4m2m2\nm1m1/RParen4 /LParen4m2m2\nm1m1/RParen4\n0.0 0.2 0.4 0.6 0.8 1.005101520\nU\nnFerro\nParamAF (q,q)\n(0,π)(q,π)(0,q)Spiral \nFerriFerri Ferri Ferri\nSpiral \nFerri(a) (n;U) phase diagram\n5 10 15/Minus4/Minus2024\nFerroParamn=1 AF\nn=0.5 Ferrin=0.5 Spiral \n          Ferri\nU/LParen4m2m1\nm1m2/RParen4 (b) (U;\u0016) phase diagram with phase\nseparation\n(c)m1;m2(i)\n (d)m1;m2(ii)\nU=9.0\n/Minus6/Minus4/Minus20\n0\n    .5\n   1U=19.0\n/Minus6/Minus4/Minus20 (e)E(n;U)\n (f)\u0016(n;U)\nFigure 3: Top \fgures: (a) Mean-\feld phase diagram for the 2D Hubbard model with sublat-\ntices with independent magnetization amplitudes. In the central (green) region of the phase\ndiagram, con\fguration (i) is the one that minimizes the MF energy, while con\fguration (ii) is\nthe most stable in the red regions. (b) ( U;\u0016) mean-\feld phase diagram with phase separation\noccurring on the borders of the green and red regions (thick solid lines). This phase diagram\ndisplays spiral and non-spiral ferrimagnetic phases (green regions) with \fxed particle density,\nn= 0:5, re\recting the opening of an energy gap in the MF quasi-particle bands. Bottom\n\fgures:m1(n;U) andm2(n;U) for the MF 2D Hubbard model with two sublattices in (c)\ncon\fguration (i) and (d) con\fguration (ii). (e) Ground state MF energy for usual 2D (blue),\ncase (i) (green) and case (ii) (red) for U= 9 andU= 19. (f)\u0016as a function of the doping n\nand Coulomb interaction U. All \fgures represent numerical results obtained for the 100 \u0002100\n2D Hubbard model with two sublattices with di\u000berent magnetization amplitudes.\nthe order parameters occurs. When using nas a basic variable (and posteriorly\ncalculating \u0016=@E=@n\u0019\u0001E=\u0001nusing the data in Fig. 2g), nseems to be\nmultiply de\fned for some values of \u0016, which implies instability (e. g. of the\nspiral phase for U= 15). The use of \u0016as a basic variable solves this ambiguity\nand leads to plateaus in the chemical potential \u0016(n;U) in the regions where\nphase separation (PS) occurs (see Fig. 2h). In each PS region of the diagram,\ntwo spatially separated phases occur: the ones immediately to the left and to\nthe right of the PS region in question (see Fig. 2b). The two phases have dif-\nferent electronic densities, such that the electronic density of the whole system\namounts to n. In Fig. 2c, we show the same phase diagram as in Fig. 2a, but\nusing\u0016as the independent variable. The colors of corresponding regions are\nthe same for easier reading. The thick solid line indicates a discontinuity in n.\n5In this work, the magnetic phase diagram for the Hubbard 2D model com-\nprising four sublattices is obtained by \fnding the magnetization amplitudes\n(mA;mB;mC;mD) and the vector ~ qwhich minimize the energy. We consider\ntwo situations: (i) mA=mD=m1andmB=mC=m2; (ii)mA=mC=m1\nandmB=mD=m2.\nThe ground state magnetization amplitude of the usual 2D Hubbard model\nis proportional to nfor each value of Uin the ferromagnetic phase (see Fig. 2d).\nWhen a spiral ferrimagnetic phase is allowed, it was found that, near zero \flling\n(n= 0) and half-\flling ( n= 1), the ground state magnetization remains the\nsame as in the usual 2D case (see Figs. 3c and 3d). This means that in these\nregions, the ground state magnetization is still constant throughout the whole\nlattice. However, as one moves to intermediate n, one \fnds that m1andm2\nbecome distinct, as shown in Figs. 3c and 3d for cases (i) and (ii) respectively,\nwherem1andm2are displayed as a function of nandU. These \fgures show\ntwo sheets re\recting the separation of the magnetization amplitudes. The colors\ngreen for case (i) and red for (ii) are used on all plots of Fig. 3. For intermediate\n\flling, the system is able to lower its energy by adopting di\u000berent magnetization\namplitudes on sublattices 1 and 2 in both cases (i) and (ii). This is shown in Fig.\n3e forU= 19 andU= 9. Depending on the region of the phase diagram one\nanalyses, con\fguration (i) or (ii) may have the lowest energy, as shown in Fig.\n3a. In this \fgure, we added another layer on top of the usual 2D MF magnetic\nphase diagram, showing which of the two-sublattice con\fgurations considered\nhas the lowest energy in the ferrimagnetic region: green for case (i) and red for\ncase (ii). Furthermore, the energy was minimized with respect to qxandqy,\nwhile using the new magnetization values, but it was found that only very small\nchanges in ~ qoccur, i.e., despite the changes in magnetization amplitudes, the\nmagnetic phases in the diagram remain the same. For this reason, the magnetic\nphases are shown as being the same as those of the usual 2D model.\nThe mean-\feld energy dispersion relation of the usual 2D Hubbard model\ndisplays two bands. Electrons occupy the lowest band until half-\flling ( n=\n1) and then proceed to occupying the higher band. As can be seen in Fig.\n2h, the fermionic density increases with the chemical potential until the phase\nseparation region is reached. In this region, the chemical potential is constant\ndespite any increase in the number of particles, up to half-\flling. At this point,\nany increase in ninduces a jump in the value of \u0016, equal to the energy separation\nbetween the two energy bands (called the energy gap). As the plot in Fig. 2h\nonly goes up to half-\flling, we see \u0016increasing smoothly until it reaches the\nphase separation region, followed by a plateau and a jump at n= 1. In both\ncases studied in this work, with the lattice divided into two sublattices, the\nenergy bands open a gap at quarter \flling ( n= 0:5), as shown in Fig. 3f.\nAnother gap appears at three quarter \flling ( n= 0:75), but only the phases\nwithn\u00141 are shown in Fig. 3b. The plot in Fig. 3b is again the same as\nFig. 3a, but using \u0016as the basic variable. In this diagram, the green region\ncorresponds to n= 0:5, therefore only con\fguration (i) for the spiral and non\nspiral ferrimagnetic phases is present in the phase diagram. The dashed line\nseparates the ferrimagnetic region from the spiral ferrimagnetic one and the\n6thick solid lines denote again discontinuities in n.\n4. Conclusion\nHaving addressed the possibility of a spiral ferrimagnetic phase in the mean-\n\feld phase diagram of the 2D Hubbard model, we conclude that, for interme-\ndiate values of the interaction Uand doping n, the spiral ferrimagnetic phase\nis the most stable phase in the ( n;U) phase diagram. Higher values of Ulead\nto non-spiral ferrimagnetic phases. We emphasize the case of intermediate n\nand higher U, for which the ground state does not appear to be purely ferro-\nmagnetic, contrasting with results by other authors.[15] Additionally, allowing\nphase separation and replacing nby\u0016as the independent variable, the ( \u0016;U)\nphase diagram displays, in a considerable region, spiral (for intermediate values\nofU) and non-spiral (for higher values of U) ferrimagnetic phases with \fxed\nparticle density, n= 0:5, re\recting the opening of an energy gap in the mean-\n\feld quasi-particle bands. We further note that generalizing the ferrimagnetic\nphase to cases where more than two di\u000berent magnetization amplitudes are al-\nlowed should lead to even more stable ferrimagnetic phases in certain regions\nof the phase diagram. Preliminary results with three di\u000berent amplitudes are\nconsistent with this conjecture. All these results provide strong evidence on\nthe stability of the spiral ferrimagnetic phase in the mean \feld magnetic phase\ndiagram of the 2D Hubbard model.\nAcknowledgements\nR. G. Dias acknowledges the \fnancial support from the Portuguese Science\nand Technology Foundation (FCT) through the program PEst-C/CTM/LA0025/2013.\nJ. D. Gouveia acknowledges the \fnancial support from the Portuguese Science\nand Technology Foundation (FCT) through the grant SFRH/BD/73057/2010.\nReferences\n[1] J. G. Bednorz, K. A. M uller, Zeitschrift fur Physik B Condensed Matter\n64 (1986) 189{193. doi: 10.1007/BF01303701 .\n[2] M. A. Kastner, R. J. Birgeneau, G. Shirane, Y. Endoh, Rev. Mod. Phys. 70\n(1998) 897{928. URL: http://link.aps.org/doi/10.1103/RevModPhys.\n70.897 . doi: 10.1103/RevModPhys.70.897 .\n[3] R. G. Dias, J. M. B. Lopes dos Santos, Journal de Physique I 2 (1992)\n1889{1897. doi: 10.1051/jp1:1992252 .\n[4] F. Gebhard, K. Bott, M. Scheidler, P. Thomas, S. W. Koch, Phil. Mag. B\n75 (1997) 13{46.\n[5] M. Ogata, H. Shiba, Physical Review B 41 (1990) 2326{2338.\n7[6] Y. Nagaoka, Solid State Communications 3 (1965) 409{412. doi: 10.1016/\n0038-1098(65)90266-8 .\n[7] M. P. Marder, Condensed Matter Physics, John Wiley and Sons, 2000.\n[8] D. R. Penn, Phys. Rev. 142 (1966) 350{365. URL: http://link.aps.org/\ndoi/10.1103/PhysRev.142.350 . doi: 10.1103/PhysRev.142.350 .\n[9] M. A. J. Dorantes-Davila, J. L. Moran-Lopez, Phys. Rev. B 27 (1983) 575{\n577.\n[10] E. Kaxiras, E. Manousakis, Phys. Rev. B 37 (1988) 656{659.\n[11] S. N. Coppersmith, C. C. Yu, Phys. Rev. B 39 (1989) 11 464 { 11 474.\n[12] A. Richter, G. R opke, F. Goedsche, Physica Status Solidi B Basic Research\n88 (1978) 189{198. doi: 10.1002/pssb.2220880122 .\n[13] S. Sarker, C. Jayaprakash, H. R. Krishnamurthy, W. Wenzel, Phys.\nRev. B 43 (1991) 8775{8778. URL: http://link.aps.org/doi/10.1103/\nPhysRevB.43.8775 . doi: 10.1103/PhysRevB.43.8775 .\n[14] E. Langmann, M. Wallin, Journal of Statistical Physics 127 (2007) 825{840.\n[15] P. A. Igoshev, M. A. Timirgazin, A. A. Katanin, A. K. Arzhnikov, V. Y.\nIrkhin, Phys. Rev. B 81 (2010) 094407.\n[16] W. Schumacher, Physica Status Solidi (b) 119 (1983) 235{238.\n[17] M. Dzierzawa, Z. Phys. B 86 (1992) 49{52.\n[18] A. Singh, Z. Tesanovic, H. H. Kim, Pramana - J. Phys. 38 (1992) 211{217.\n8"    },    {        "title": "1911.11183v1.Identification_of_ferrimagnetic_orbitals_preventing_spinel_degradation_by_charge_ordering_in_Li__x_Mn__2_O__4_.pdf",        "content": "Identi\fcation of ferrimagnetic orbitals preventing spinel degradation by charge\nordering in Li xMn 2O4\nHasnain Ha\fz,1, 2,\u0003Kosuke Suzuki,3Bernardo Barbiellini,4, 1Yuki Orikasa,5Stanislaw\nKaprzyk,6, 1,yNaruki Tsuji,7Kentaro Yamamoto,8Ayumu Terasaka,3Kazushi Hoshi,3\nYoshiharu Uchimoto,8Yoshiharu Sakurai,7Hiroshi Sakurai,3and Arun Bansil1,z\n1Department of Physics, Northeastern University, Boston, MA\n2Current Address: Department of Mechanical Engineering,\nCarnegie Mellon University, Pittsburgh, Pennsylvania, 15213, USA\n3Faculty of Science and Technology, Gunma University, Kiryu, Gunma 376-8515, Japan\n4School of Engineering Science, LUT University, FI-53851 Lappeenranta, Finland\n5Department of Applied Chemistry, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan\n6Faculty of Physics and Applied Computer Science,\nAGH University of Science and Technology, aleja Mickiewicza 30, Krakow 30-059, Poland\n7Japan Synchrotron Radiation Research Institute, SPring-8, Sayo, Hyogo 679-5198, Japan\n8Graduate School of Human and Environmental Studies,\nKyoto University, Sakyo-ku, Kyoto 606-8501, Japan\n(Dated: November 27, 2019)\nSpinel Li xMn 2O4is a key cathode material that is used extensively in commercial Li-ion batter-\nies. A challenge with this material has been that the capacity of the battery fades with cycling,\nan e\u000bect that can be traced to the presence of an anti-ferromagnetic insulator phase in the fully\nlithiated LiMn 2O4(LMO) and the associated charge disproportionation that drives distortions of\nthe MnO 6octahedra. Here, by combining x-ray magnetic Compton scattering experiments with\nparallel \frst-principles computations, we show that the anti-ferromagnetic phase of LMO is sur-\nrounded by a robust ferrimagnetic metallic phase, which becomes stable when even a small amount\nof Li is removed from or added to the charge-ordered LMO. In this surprising ferrimagnetic state,\ncharge-ordering and octahedral distortions are found to be strongly suppressed. We identify the\nnature of the ferrimagnetic orbitals involved through theoretical and experimental analyses of the\nmagnetic Compton scattering spectra.\nI. INTRODUCTION\nSpinel Li xMn2O4is an attractive cathode material\nfor rechargeable batteries1,2because it is less expensive\nand environmentally more friendly than lithiated cobalt\nand nickel oxides. Unfortunately, the lithiated ( x= 1)\ncompound LiMn 2O4(LMO) su\u000bers from the problem of\ncapacity fading due to a structural phase transition3,4.\nAbove the room temperature, LMO assumes the cubic\nstructure of the normal spinel in which lithium ions oc-\ncupy tetrahedral positions, with the manganese ions lo-\ncated only at the octahedral positions. A decrease in\nthe temperature to around 283 K produces the famous\nmetal-insulator (Verwey) transition5, which is associated\nwith the distortion of the MnO 6octahedra. This struc-\ntural transformation limits the application of LMO as a\ncathode material6. The Verwey transition also drives the\nonset of long-range antiferromagnetic (AFM) order below\nthe Curie temperature of 65 K7, which is also supported\nby \frst-principles calculations8. Geometric frustration of\nthe AFM order in the spinel structure9leads to complex\npotential energy landscapes that exhibit multiple mag-\nnetic phase transitions10whose nature remains unclear.\nIn this study, we explore magnetic properties of spinel\nLixMn2O4on a \frst-principles basis by exploiting recent\nadvances in constructing exchange-correlation function-\nals. Speci\fcally, we employ the strongly-constrained-\nand-appropriately-normed (SCAN) functional11, whichhas proven especially accurate for investigating the\nelectronic, geometric and magnetic structures of el-\nemental manganese12, MnO 2polymorphs13, 3dper-\novskite oxides14, oxides superconductors15{17, and lay-\nered lithium intercalated transition metal oxides18. Par-\nallel magnetic Compton scattering experiments on LMO\nsamples are also reported over a wide range of Li con-\ncentrations varying from unlithiated ( x= 0) to over-\nlithiated (x= 1:079) Li xMn2O4. Our analysis reveals\nthat ferrimagnetism competes with the AFM order in\nLMO and leads to ferrimagnetic (FIM) moments even\nfor slight departures from stoichiometry ( x= 1). Liu\net al.8report that the AFM ordering is responsible for\ntriggering changes in the Mn valence and driving Jahn-\nTeller distortions. The FIM phase we have found here,\nhowever, suppresses the octahedral distortions, which are\nresponsible for cathode degradation. We show how the\nmagnetic state associated with this puzzling FIM phase\ncan be visualized through an analysis of our magnetic\nCompton spectra.\nCompton scattering, which refers to inelastic x-ray\nscattering in the deeply inelastic regime, provides a di-\nrect probe of the ground-state momentum density \u001a(p)\nof the many-body electronic system through a measure-\nment of the so-called Compton pro\fle, J(pz), wherepzis\nthe momentum transferred in the scattering process19,20.\nHigh-resolution x-ray Compton scattering studies have\nrevealed that the orbital involved in the lithium inser-arXiv:1911.11183v1  [cond-mat.mtrl-sci]  25 Nov 20192\nFIG. 1. FIM phase in Li xMn 2O4. Magnetic moments for vari-\nous lithium concentrations xobtained via magnetic Compton\nscattering (red squares) and SQUID (yellow squares) mea-\nsurements are compared with the corresponding theoretical\npredictions (green circles). Disappearance of the total mag-\nnetic moment at x= 1 is highlighted by the star symbol.\nThe inset shows how the experimental and theoretical values\nof the average lattice constant vary with xand approximately\nfollow a linear behavior consistent with Vegard's law.\ntion/extraction process in Li xMn2O4is mainly the oxy-\ngen 2porbital21. Although the oxygen 2 porbitals thus\ndominate the redox process, magnetic properties of the\nmaterial are controlled by manganese 3 dorbitals. Here,\nwe use magnetic Compton scattering (MCS) to explore\nhow the manganese 3 dstates evolve with Li intercala-\ntion and how their magnetism drives performance of the\nbattery.\nIn an MCS experiment, one measures the magnetic\nCompton pro\fle (MCP)19,Jmag, which can be expressed\nin terms of a double integral of the spin-dependent elec-\ntron momentum density, \u001amag(p), as\nJmag(pz) =Z Z\n\u001amag(p)dpxdpy: (1)\nHere,\u001amag(p) =\u001a\"(p)\u0000\u001a#(p), where\u001a\"(p) and\u001a#(p)\nare the momentum densities of up-spin (majority) and\ndown-spin (minority) electrons, respectively. The area\nunder the magnetic Compton pro\fle Jmag(pz) yields the\ntotal magnetic spin moment. Therefore, MCS experi-\nments require a strong magnetic \feld and a net total\nmagnetic moment in the sample. In this way, MCS al-\nlows access to magnetic electrons in materials and its\npotential in this regard was recognized quite early in the\n\feld22,23. The technique has proven especially success-\nful in extracting occupation numbers of 3 dMn orbitals\nin bilayer manganites24,25and recent MCS studies have\nrevealed \fne details of the magnetic orbitals in a number\nof materials26{28.\nFIG. 2. Computed spin-dependent partial-density-of-states\n(PDOS) associated with the e gand t 2gorbitals of Mn3+and\nMn4+ions in Li xMn 2O4for Li concentrations (a) x= 1:0\nand (b) x= 0:75. See panel (a) for the meaning of lines of\nvarious colors. Vertical dashed-line marks the Fermi energy\n(EF). Up and down arrows indicate the contributions of spin\nup and down PDOS, respectively.\nII. MATERIALS AND METHODS\nA. Magnetic Compton experiments\nThe MCS experiments were carried out at 10 K on\nbeamline BL08W at SPring-8, Japan29,30. Elliptically\npolarized x-rays emitted from an elliptical multipole wig-\ngler were monochromatized to 183 keV by a bent Si\n620 crystal. Energy spectra of Compton-scattered x-rays\nfrom the sample at a scattering angle of 178\u000ewere mea-\nsured using a ten-segmented Ge solid-state detector with\nexternal magnetic \feld of 25 kOe. The estimated momen-\ntum resolution is 0.50 a.u. full-width-at-half-maximum.\nThe spherically averaged pro\fle Jmag(p) was extracted\nfrom the di\u000berence between two spectra taken under the\nsame experimental conditions with alternating directions\nof magnetization of the sample, aligned by an external\nmagnetic \feld19. The observed spectra were corrected for\nthe energy-dependent scattering cross section, e\u000eciency\nof the detector, and absorption of the sample.\nPolycrystalline samples of Li xMn2O4(x= 0:41, 0:50,\n0:92 and 1:08) were prepared by chemical lithium extrac-\ntion following the method reported previously21. The\ncompositions were determined by inductively coupled\nplasma (ICP) measurements. X-ray powder di\u000braction\nanalyses con\frmed spinel phases and the increase of the\nlattice constant with increasing xover the range x=3\nFIG. 3. Schematics of magnetic con\fgurations of Li xMn 2O4for lithium concentrations x= 1 and x= 0:75. Charge-ordering\nis prominent for the x= 1 phase with an overall AFM con\fguration (a). As soon as the Li ions are removed, see (b), charge-\nordering becomes unstable, and results in partial ordering, spin-\ripping, and reduction of the average local magnetic moment\nper Mn atom. This induces ferrimagnetism at x= 0:75, which is seen also in the PDOS of Fig. 2(b).\n0\u00001 as observed previously21. Total magnetic moments\nwere obtained from SQUID magnetometer (MPMS5-SW,\nQuantum Design, Inc.) measurements.\nB. First-principles calculations\nFirst-principles calculations were performed using\nthe pseudopotential projector augmented-plane-wave\nmethod31as implemented in the Vienna Ab-Initio Simu-\nlation Package (VASP)32,33, with a kinetic energy cuto\u000b\nof 600 eV for the plane-wave basis set. Computations\nwere carried out using both the Generalized Gradient\nApproximation (GGA)34,35and the recently constructed\nstrongly-constrained-and-appropriately-normed (SCAN)\nmeta-GGA11exchange-correlation functional. A 4 \u00024\u00024\n\u0000-centeredk-point mesh was used to sample the Brillouin\nzone of a primitive spinel unit cell (containing eight for-\nmula units with 32 oxygen atoms). Equilibrium positions\nof all atoms were calculated via structural optimization,\nwhere the internal degrees of freedom, along with the\nshape and volume of the unit cell, were allowed to vary\nuntil the residual forces per atom were less than 0.005\neV/\u0017A. We obtained spin momentum densities, \u001amag(p),\nand the spherical Compton pro\fles Jmag(p) of the va-\nlence electrons using Kohn-Sham orbitals following the\nmethod of Makkonen et al.36. This scheme has been re-\ncently used to study the Compton pro\fle of lithium iron\nphosphate (LiFePO 4)37, which is as an exemplar cathodebattery material38,39.\nIII. RESULTS AND DISCUSSIONS\nA. Ferrimagnetic (FIM) phase\nFigure 1 highlights the evolution of the magnetic mo-\nment in Li xMn2O4for Li concentrations varying from\nzero to around 1.1 per formula unit. The experimen-\ntal Compton magnetic moment is seen to follow a linear\nbehavior as shown by the gray dashed line. The AFM\nstate (zero average moment) is realized in the end com-\npounds MnO 2and LiMn 2O4, but the system assumes an\nFIM state with a net non-zero moment at other compo-\nsitions, including the over-discharged regime of x > 1.\nThe moments obtained via SQUID measurements are in\ngood accord with those obtained from MCS experiments.\nSome di\u000berences are to be expected since MCS measures\nonly the spin moment22while the SQUID couples to the\ntotal moment, which includes the orbital component of\nthe magnetic moment. Notably, anomalous behavior of\nthe magnetic moment near the Verwey transition has\nbeen reported previously40. Our theoretical moments\n(green circles) are in reasonable accord with the corre-\nsponding experimental results (red and yellow squares).\nSome excursion from linearity in theoretical moments\ncould be due to the small size of the simulation cell\nused in computations. In fact, Korringa-Kohn-Rostoker-4\ncoherent-potential-approximation (KKR-CPA)41,42cal-\nculations show that the spin moment on Mn atoms in-\ncreases linearly with increasing xif the system is de-\nscribed by a spin-glass-like behavior with randomly ori-\nented Mn moments21. Moreover, the strengthening of\nMn moments with Li insertion is consistent with muon-\nspin-rotation experiments43. However, the random-alloy\nmodel21underlying the KKR-CPA scheme cannot be\nexpected to capture the AFM charge-ordered state at\nx= 1.\nB. Charge, spin and magnetic con\fguration\nThe SCAN functional used in this study successfully lo-\ncalizes Mn 3 delectrons by \flling 3 dz2orbitals, and results\nin the coexistence of Mn3+and Mn4+in LiMn 2O4. Fig-\nure 2, which presents our spin-dependent partial-density-\nof-states (PDOS) in Li xMn2O4, shows that egorbitals of\nsome Mn atoms split upon lithium intercalation. The Mn\n3dz2levels move to lower energies and become \flled, and\neventually open a band gap of about 0.1 eV at x= 1.\nFor 0< x < 1, the system is metallic due to the par-\ntial occupation of the Mn 3 dz2bands. At x= 1, the\nground-state is found to be an AFM charge-ordered state\nwith alternating AFM Mn3+layers and FM Mn4+lay-\ners with spins along the (001) direction as illustrated in\nFig. 3(a). FIM states appear as soon as Li atoms are\nremoved from or added to the unit cell. A stable FIM\ncon\fguration for x= 0:75 is shown in Fig. 3(b). In this\nphase, the spin of the t2gelectrons is not compensated\naccording to the PDOS shown in Fig. 2(b), and there-\nfore, these electrons produce a net magnetic moment.\nInterestingly, we found a similar magnetic PDOS for the\nover-lithiated phase x= 1:125 where the Mn t2gstate is\npartially compensated, and hence, it is mainly responsi-\nble for the magnetic moment as shown in Supplementary\nFig. S1(a). In general, SCAN is able to describe complex\ncon\fgurations of competing stripe and magnetic phases\nin the cuprates17and it also produces a small distortion\nfrom theFd3mcubic symmetry of Li xMn2O4atx= 0:5\n(See Figs. S1(b) and S3(b) in Supplemental Material44),\nwhich is consistent with the recent experimental results\nof Bianchini et al.45In sharp contrast to the preced-\ning SCAN-based results, the GGA functional46fails to\nproduce the insulating AFM state of LiMn 2O4, and the\nassociated charge disproportionation on Mn atoms that\ndrives local Jahn-Teller distortions8of the MnO 6octahe-\ndra. Notably, the theoretical lattice parameters obtained\nvia SCAN are in good agreement with the corresponding\nexperimental values as shown in the inset of Fig. 1.\nC. Magnetic orbitals and Compton pro\fles.\nElectronic states in momentum space preserve their in-\ndividual angular dependencies, which facilitates the de-\ntection of particular states47. Thus, the projection of the\nFIG. 4. Compton pro\fles and visualization of the magnetic\norbitals. (a) Theoretical 2D momentum density map for pro-\njection along (001) for x= 0:75. (b) Experimental (scatter\nplots with error bars) and theoretical (solid lines) spherically-\naveraged magnetic Compton pro\fles. Grey solid-line repre-\nsents atomic d-orbital Compton pro\fle which is solved ana-\nlytically using normalized Slater-type orbitals. All the pro\fles\nare normalized to one.\ntheoretical 3D spin-polarized electron momentum density\n\u001amag(p) along the (001) direction at x= 0:75 given in\nFig. 4(a) determines the nature of the electronic states\nproducing the FIM solution. Clearly, the angular de-\npendence of \u001amag(p) is dominated by the t2gcharacter\nof the 3dMn electrons48. This spin momentum result\nis consistent with our previous observation that the t2g\nelectrons are not compensated according to Fig. 2(b) of\nthe PDOS. Based on these robust arguments, one can\nconclude that the FIM orbitals have mostly t2gcharac-\nter. Figure 4(b) presents the measured and calculated\nspherically averaged MCPs for various lithium concen-5\nFIG. 5. Directional Magnetic Compton pro\fles along the\n(100) and (110) directions. The low-momentum contributions\nmainly come from the (110) directions, which contribute sig-\nni\fcantly to the spherical average in Fig. 4. The SCAN results\nshow good agreement with the KKR-CPA results for the FIM\nphase.\ntrations,x. For all values of x, the shapes of the MCPs\nare quite similar, the main di\u000berence being the area under\nthe pro\fle, which gives the total spin magnetic moment.\nAs we noted already, the spin magnetic moment follows\na linear behavior for all lithium concentrations with the\nexception of the x= 1 compound where the moment van-\nishes. The agreement between theory and experiment in\nFig. 4(b) is considered very good when we keep in mind\nthe delicate nature of the MCP. The shape of the exper-\nimentalJmag(p) is also similar to that of the Compton\npro\fle from a manganese 3 datomic orbital49since the\nmanganese t2gorbitals undergo little hybridization with\nthe 2porbital of oxygen21. Figure 5 shows a comparison\nof the directional Compton pro\fles between the KKR-\nCPA and SCAN calculations. As mentioned above, our\nKKR-CPA calculations describe a spin-glass-like behav-\nior with randomly oriented Mn moments embedded in a\nperfect cubic spinel structure. Therefore, the agreement\nbetween the two methods indicates that SCAN can suc-\ncessfully capture the almost cubic FIM spin-glass phase.\nThus, the Jahn-Teller distortions obtained within SCAN\nare strongly suppressed if the electrons occupy the FIM\norbitals visualized in Fig. 4(a). As shown by our mag-\nnetic Compton scattering experiments, the application\nof an external \feld promoting the FIM phase and the\nassociated orbitals can serve as a basis for preventing\nJahn-Teller distortions.IV. CONCLUSIONS\nWe have shown that a fundamental understanding of\nthe role of magnetic electrons in Li-ion battery materials\ncan be obtained via magnetic Compton scattering ex-\nperiments and \frst-principles calculations. Speci\fcally,\nthe magnetic Compton pro\fles of Li-Mn-O system at\ndi\u000berent lithium concentrations have been analyzed to\nunfold the underlying magnetic transitions in terms of\nthe spin-moment contribution of the Mn 3 dorbitals.\nThe calculated total spin moments are in good agree-\nment with the corresponding experimental values. The\npresent analysis demonstrates that the Mn 3 dmagnetic\nelectrons have mainly t2gsymmetry, and therefore,\nthese electrons prevent Jahn-Teller distortions promoted\nby theegcharacter. Our study explains the intimate\nconnection between the charge, spin and lattice degrees\nof freedom and their role in the bonding of the MnO 6\noctahedra50in the spinel battery materials. Although\nour present analysis is based on spherically-averaged\nexperimental MCPs, it will be interesting to consider\ndirectional MCPs from single-crystal samples to gain\nfurther insight into the nature of the connection between\nthe octahedral bonds and magnetism. Electron transfer\nduring the lithiation and delithiation processes is shown\nto involve octahedral units whose distortion can be\na\u000bected by temperature and magnetic \feld. In this way,\nthe nature of the octahedral bonds and magnetism are\nconnected with the electrochemical performance of the\nbattery materials.\nACKNOWLEDGMENTS\nWe thank Dr. M. Itou for technical support in\nconnection with magnetic Compton scattering experi-\nments. K.S. was supported by a Grant-in-Aid for Young\nScientists (B) (No. 15K17873) from the Ministry of\nEducation, Culture, Sports, Science, and Technology\n(MEXT), Japan, and the work at JASRI was partially\nsupported by the Japan Science and Technology Agency.\nCompton scattering experiments were performed with\nthe approval of JASRI (Proposals Nos. 2014B1335\nand 2015B1171). SQUID measurement was performed\nwith the approval of Gunma University-Industry Col-\nlaboration and Intellectual Property Strategy Center\n(Proposal 2017). The work at Northeastern University\nwas supported by the U.S. Department of Energy,\nO\u000ece of Science, Basic Energy Sciences Grant No. DE-\nFG02-07ER46352, and bene\fted from Northeastern\nUniversitys Advanced Scienti\fc Computation Center\n(ASCC), and the allocation of time at the NERSC\nsupercomputing center through DOE Grant No. DE-\nAC02-05CH11231. S. 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A. Chernova, G. M. Nolis, F. O. Omenya, H. Zhou,\nZ. Li, and M. S. Whittingham, J. Mater. Chem. 21, 9865\n(2011).8\nFIG. S1. Computed spin-dependent partial-density-of-states (PDOS) associated with the e gand t 2gorbitals of Mn3+and\nMn4+ions in Li xMn 2O4for Li concentrations (a) x= 1:125, (b) x= 0:5 and (c) x= 0:0. See panel (a) for meaning of lines\nof various colors. Vertical dashed line marks the Fermi energy (E F). Up and down arrows indicate the contributions of spin\nup and down PDOS respectively. SCAN results yield insulating states for x= 0:0 and 1 :0 (Fig. 2a), and metallic states for\nx= 1:125;0:75 (Fig. 2b), and 0 :5.9\nFIG. S2. Computed projected density-of-states of O-2p states. O Iand O IIatoms indicate the oxygen atoms from the Mn4+\nand Mn3+layers respectively for the AFM con\fguration in x= 1.10\nFIG. S3. Schematics of magnetic con\fgurations of Li xMn 2O4for lithium concentrations, (a) x= 1:125, (b) x= 0:5 and (c)\nx= 0:0. All the structures initially assume a cubic symmetry with space group Fd3m (227) where Li, Mn and O atoms occupy\nthe Wycko\u000b positions: 8 a, 16dand 32 erespectively. For over-lithiated case, the extra Li goes to the octahedral position 16 c\nsince all the tetrahedral positions 8 aare occupied. DFT optimization shows a breaking of the cubic Fd3m symmetry during\nstructural relaxation.11\nFIG. S4. XRD pattern for Li xMn 2O4(x=0.41, 0.50, 0.92 and 1.08)\nLi concentration (x) Lattice constant ( \u0017A)\n0.41 8.0546\n0.5 8.0966\n0.92 8.1616\n1.08 8.2\nTABLE S1. Experimental lattice constant obtained from XRD analysis."    },    {        "title": "2210.11042v1.Strong_variation_of_spin_orbit_torques_with_relative_spin_relaxation_rates_in_ferrimagnets.pdf",        "content": " \n1 \n Strong variation of s pin-orbit torques with relative spin relaxation rates in  ferrimagnet s \nLijun Zhu1,2* and Daniel C. Ralph3,4 \n1. State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese \nAcademy of Sciences, Beijing 100083, China  \n2. College of Materials Science and Opto -Electronic Technology , University of Chinese Academy of Sciences, \nBeijing 100049, China  \n3. Cornell University, Ithaca,  New York 14850, USA  \n4. Kavli Institute at Cornell, Ithaca, New York 14850, USA  \n*ljzhu@semi.ac.cn  \n \nSpin-orbit torque s (SOT s) have been widely understood as an interfacial  transfer of spin that is \nindependent of the bulk  properties  of the magnetic layer. Here, we report that SOTs acting on \nferrimagnetic  Fe xTb 1-x layers  decrease and vanish  upon approaching the magnetic compensation point  \nbecause  the rate of  spin transfer to the magnetization becomes slower than the rate of spin  relaxatio n \ninto the crystal lattice due to  spin-orbit scattering . These results indicate that the relative rates of \ncompeting spin relaxation processes with in magnetic layers  play a critical role in determining the  \nstrength of  SOTs , which provides  a unified understanding for the diverse  and even seemingly puzzling  \nSOT phenomena in ferromagnetic  and compensated  systems. Our work indicates that spin-orbit \nscattering  within the magnet should be minimized for efficient SOT  devices . We also find that the \ninterfacial  spin-mixing conductance of interfaces of ferrimagnet ic alloys  (such as Fe xTb 1-x) is as large as \nthat of 3 d ferromagnet s and insensitive to the degree of magnetic compensation. \nIntroduction  \nEfficient manipulation of magnetic materials is essential for s pintronic devices . While s pin-orbit torques (SOTs)1,2 \nare well established to be an effective tool to manipulate metallic 3 d ferromagnets ( FMs), whether they can \neffectively control antiferromagnetically -ordered systems  has remained elusive  despite the recent blooming of \ninterest in ferrimagnets (FIMs) and antiferromagnets (AFs)3-9. Experimentally, for reasons unclear, the SOTs \nexerted on nearly compensated FIMs5-7,10 are often measured to be considerably weaker than those on 3 d FMs for \na given spin -current generator ( by up to >20 times, see below). More strikingly, it remains under debate whether  \nuniform, perfectly compensated FIMs (Ms = 0 emu/cm3) can be switched  at all by SOTs 11-13.  \nMicroscopically , SOTs have been widely assumed  as an interfacial  transfer of spin (i.e., spin dephasing \nlength λdp ≈ 0 nm for transverse spin current) that is independent of the bulk properties of the magnetic layer , such \nas in drift -diffusion analyses14-16. Under this assumption , spin current entering the magnet  from an adjacent spin -\ngenerating layer  is absorbed fully by the magnetization via dephasing to generate SOTs , and  the dampinglike SOT \nefficiency per current density ( 𝜉𝐷𝐿𝑗) will depend only on the spin Hall ratio ( θSH) of the spin-generating layer  and \nthe spin transparency ( Tint) of the interface which determines what fraction of the spin current enters the magnet \n17,18, i.e., \n𝜉𝐷𝐿𝑗= TintθSH.                                                  (1)  \nThis picture is a reasonabl e approximat ion for  sufficiently thick  metallic FMs that have  a short λdp (≤1 nm) due to \nstrong exchange coupling 19-21 and a long spin diffusion length  associated with spin relaxation due to spin -orbit \nscattering22,23. However, in antiferromagnetically  ordered systems λdp can be quite long, as predicted more than a  \n2 \n decade ago24-27, which, as we discuss below, question s the widely accepted model s of “interfacial torques ”, at \nleast, in FIMs and AFMs. So far , any role s of the bulk properties of the magnetic layer , e.g., the competing spin \nrelaxation rates , in the determination of  𝜉𝐷𝐿𝑗 have been overlooked in SOT analyses . \nHere , we report measurements of  SOTs acting on ferrimagnetic FexTb1-x layers  with strong spin -orbit coupling \n(SOC)8 by tuning the FexTb1-x composition  and temperature ( T). We find that , in contrast to the prediction  of Eq. \n(1), 𝜉𝐷𝐿𝑗varies strongly with the degree of magnetic compensation  for a  given  Tint, due to  changes in the fraction \nof spin current that relaxes directly to the lattice via SOC instead of being absorbed by the magnetization to apply \nSOTs.  These results uncover the critica l role of spin relaxation rates of the magnetic layer and provide a unified \nunderstanding for the diverse SOT phenomena in ferromagnetic and antiferromagnetically ordered systems . \n \nSample details  \nFor this work, we sputter -deposited  Pt0.75Ti0.25 (5.6 nm )/Fe xTb1-x (8 nm ) bilayers with different Fe volumetric \nconcentration s (x = 0.3-1). The Pt0.75Ti0.25 layer , a dirty -limit Pt alloy  with strong intrinsic spin Hall effect17, \nsourc es spin current  that exert s SOT on the FIM FexTb1-x (the spin diffusion length  is expected to be ≤ 8 nm at \ntemperatures in this study 22). Each sample was deposited by co -sputtering on an oxidized Si substrate with a 1 \nnm Ta seed layer, and protected  by a 2 nm MgO and a 1.5 nm Ta layer that was oxidized upon exposure to \natmosphere. For electrical measurements, t he samples were patterned into 5× 60 µ m2 Hall bars by \nphotolithography and ion milling  with a water -cooled stage. After processing , the magnetization hysteresis of the  \nFexTb1-x measured from the anomalous Hall voltage  (VAH) in patterned Hall bars  shows fairly close  coercivity  \n(perpendicular depinning field)  and squareness as the  magnetization of  unpatterned regions of the films  measured \nby a superconducting quantum interference device  (see Figs. 1(a) and 1(b), more details  about the magnetization \nmeasurement s can be found in Sec. 1 in the Supplementary Materials ). As shown in Figs. 1(a) -1(d), the Fe xTb1-x \nhas strong bulk perpendicular m agnetic anisotropy (PMA) for 0.3  ≤ x ≤ 0.62 and well -defined in -plane magnetic \nanisotropy for 0.75 ≤ x ≤ 1. All the PMA samples have large anisotropy  fields (1 4.4-72.2 kOe, as estimated from \nthe fits in Fig. S3 ) and square h ysteresis loops for  both the out -of-plane magnetization and anomalous Hall voltage . \n \n \nFig. 1. (a) Magnetization vs out -of-plane field ( Hz) and (b) Anomalous Hall voltage ( VAH) vs Hz for Pt 0.75Ti0.25 (5.6 \nnm)/Fe 0.59Tb0.41 (8 nm ), indicating strong perpendicular magnetic anisotropy and  a high coercivity of ≈1 kOe. VAH \nvs Hz for Pt 0.75Ti0.25 (5.6 nm)/Fe xTb1-x (8 nm ) with (c) perpendicular ( x = 0.3, 0.43, and 0.61) and (d) in -plane \nmagnetic anisotropy ( x = 0.75, 0.85, and 1).  \n \n \n3 \n Strong Variation of Spin-Orbit Torque s with Composition and temperature  \n \nFig. 2. (a) Ms and (b) 𝜉DL𝑗 for Pt 0.75Ti0.25/Fe xTb1-x with different Fe concentration ( x) at 300 K. (c) Ms and (d) 𝜉DL𝑗 \nfor Pt 0.75Ti0.25/Fe 0.59Tb0.41 at different temperatures. ( e) Frequency dependence of ferromagnetic resonance \nlinewidth ( ∆H) of the FeCoB layer in FeCoB (5.2 nm )/Ti (1 nm), FeCoB (5.2 nm )/Ti (1 nm)/Fe 0.5Tb0.5 (8 nm ), \nand FeCoB (5.2 nm )/Ti (1 nm)/Fe 0.61Tb0.39 (8 nm ) samples . The solid lines represent linear fits, the slopes of which \nyield the damping. In (a) -(e) some error bars  are smaller than the data points. (f) 𝐺eff ↑↓ of the FeCoB/Ti/Fe xTb1-x \ninterfaces measured from spin pumping into the Fe xTb1-x. The blue circles are for the composition series (300 K) \nand the red dots for the temperature series  (x = 0.59) . The blue dashed  line represents 𝐺eff ↑↓ = 0.31× 1015 Ω-1 m-2 \npreviously reported for typical Pt/3d FM interfaces [41].  \n \nWe performed harmonic Hall voltage response (HHVR) measurements37,38 by carefully separating out any \nthermoelectric effects (see details in Sec. 1 in the Supplementary Materials ). We calculate the SOT efficiency \nusing 𝜉DL𝑗\n = (2e/ℏ)Ms tFeTb HDL/jc, 18 where e is elementary charge, ℏ reduced Plank’s constant, tFeTb the Fe xTb1-x \nthickness, Ms the saturation  magnetization of the Fe xTb1-x (see Sec. 2 in in the Supplementary Materials ), and jc \nthe current density in the Pt 0.75Ti0.25. HDL is the current -driven damping -like SOT field. The “planar Hall correction” \n32,33 ,38-40 is negligible for the PMA FexTb1-x samples ( VPh/VAH<0.1, see Fig. S4  in in the Supplementary Materials ).  \nIn Figs. 2(a) and 2(b) we show the measured values of Ms and 𝜉DL𝑗 at 300 K for the Pt 0.75Ti0.25/Fe xTb1-x bilayers \nwith different Fe xTb1-x composition s (we refer  to this as the composition series) . Ms decreases monotonically by a \nfactor of 33, from 1560 emu/cm3 for x = 1 (pure Fe, 3 d FM) to 47 emu/cm3 for x = 0.5 (nearly full  compensation ), \nand then increase s slowly  as x further decrease s. More details about  the composition dependent magnetic properties  \nare shown in Sec. 2 in in the Supplementary Materials . As x decreases  in the Fe -dominated  regime  (x ≥ 0.5) , 𝜉DL𝑗 \ndecreases  by a factor of 7 at 300 K, first slowly from 0.38 ±  0.02 for x =1 to 0.27 ±  0.01 for x = 0.61  and then more \n \n4 \n rapidly to 0.054 ±  0.002 for x = 0.5 . 𝜉DL𝑗 increases slowly with decreasing x in the Tb -dominated regime ( x< 0.5) . \nThe fieldlike  SOT from the same HHVR measurements is smaller than 𝜉𝐷𝐿𝑗 for each x and also varies with x (Fig. \nS8). We also measured Ms and 𝜉DL𝑗 of Pt 0.75Ti0.25/Fe 0.59Tb0.41 as a function of temperature (we refer  to this  as the \ntemperature series) . Upon cooling from 350 K to 25 K, Ms and 𝜉𝐷𝐿𝑗 for the Pt0.75Ti0.25/Fe 0.59Tb0.41 sample are tuned \nby > 2 times ( Fig. 2(c)) and by > 7.5 times ( Fig. 2(d)), respectively. The dramatic tuning of 𝜉DL𝑗 by the Fe xTb1-x \ncomposition and temperature is a striking observation because it suggests a strong dependence of SOTs on some \nbulk properties of the Fe xTb1-x, in contrast to 𝜉DL𝑗 for heavy metal ( HM)/3d FM samples which is insensitive to the \ntype of the FM 41 and temper ature42. \n \nRobustness o f the  Spin Hall ratio and the effective spin -mixing conductance   \nThese strong variations cannot be explained by changes in either θSH or Tint (Eq. (1) ). First, θSH is a property \nof the Pt 0.75Ti0.25 layer, not the Fe xTb1-x layer. The Pt 0.75Ti0.25 layer is made identically for all of the samples,  and \nhas a sufficiently large resistivity ( ρxx = 135 µΩ cm) such that its properties can hardly be affected significantly by \neither a neighboring layer or temperature . We have verified that 𝜉DL𝑗 for a ferromagnetic Pt 0.75Ti0.25 (5.6 nm )/FePt \n(8 nm ) bilayer  only ha s very weak temperature dependence  (Fig. S1 4 in the Supplementary Materials ), in good \nconsistence with previous reports on other HM/3 d FM samples42. We have also measured negligible SOT signal \nfrom the  8 nm  FexTb1-x layers in control samples with out a Pt 0.75Ti0.25 layer  (Sec. 7 in the Supplementary Materials ), \nindicating that ch anges in our signals are not due  to SOT arising from the  FexTb1-x bulk. Note that  a bulk torque of \na magnetic layer is strongly thickness dependent43,44 and vanishes at small thickness es of a few nm 43. \nAs for the possibility of changes in Tint, if we employ a drift-diffusion analysis 14-16, the effect on  Tint of spin \nbackflow at  the Pt 0.75Ti0.25/Fe xTb1-x interface should be  proportional to the effective spin -mixing conductance ( 𝐺eff ↑↓) \nof the interface, i.e. Tint ≈ 2𝐺eff ↑↓/GPtTi,45 with GPtTi = 1/λsρxx ≈ 1.3×105 Ω-1 m-1 18,46 being the spin conductance of the \nPt0.75Ti0.25. To quantify 𝐺eff ↑↓ , we measur e the change in the damping  (α) of a precessing Fe60Co20B20 (= FeCoB) \nlayer  due to the absorption of the FeCoB -emitted spin current at the Fe xTb1-x interfaces (Fig. 2(e) and Fig. S 9). The \nsamples  used here had the structure  FeCoB  (5.2 nm )/Ti (1 nm ) and FeCoB  (5.2 nm )/Ti (1 nm )/Fe xTb1-x (8 nm ). \nEach of these samples is sputter -deposited on  a 1 nm Ta seed layer and protected by  capping with  MgO (2 nm )/Ta \n(1 nm ). The value of α for the FeCoB layers  is determined from the linear dependence of the ferromagnetic \nresonance linewidth ( ∆H, half width at half maximum) on the frequency ( f) using the relation ΔH = ΔH 0 + 2παf/γ, \nwhere ΔH 0 is the inhomogeneous broadening of the linewidth and γ the gyromagnetic ratio. The damping \nenhancement  of the FeCoB layer induced by spin pumping into  the 8 nm Fe xTb1-x layers, ∆ α = αFeCoB/Ti/FeTb - αFeCoB/Ti , \ncan be  related to 𝐺eff ↑↓  as 47-49 \n∆𝛼 = 𝛾ℏ2𝐺eff↑↓/2𝑒2𝑀FeCoB 𝑡FeCoB                  (2) \nwhere tFeCoB = 5.2 nm and  MFeCoB = 1255 emu/cm3 is the saturation magnetization of the FeCoB layer as measured \nby SQUID . The value of  α = 0.0053 for  the bare  FeCoB /Ti sample with no Fe xTb1-x coincides  closely  with the \nintrinsic damping of FeCoB (≈ 0.00641), indicating that the damping in this system does not contain any significant \ncontributions from interfacial two -magnon scattering or spin memory loss . As shown in Fig. 2(f) , 𝐺eff ↑↓ of the \nFeCoB/Ti/Fe xTb1-x interfaces is insensitive to temperature and the Fe xTb1-x composition within experimental \nuncertainty , and has a value  as high as that of typical 3 d FM/Pt interfaces (≈ 0.31× 1015 Ω-1 m-2) 41. This indicates  \n5 \n that compensated Fe xTb1-x alloys act as spin sinks that are  just as good as 3 d FMs and Pt , and that there is no \nenhance ment in the amount of spin reflection and backflow due to magnetic compensation . In principle, changes \nin Tint for SOT measurements could also arise from spin memory loss induced by interfacial SOC38,50,51, but this \nshould be a minor effect for Tint of our un -annealed Pt 0.75Ti0.25/Fe xTb1-x just as is the case of un -annealed Pt/Co 52. \nAs noted above, we also do not observe any enhancement in damping due to spin memory loss in the spin -pumping \nmeasurements.   \n \nVariation of SOT with t he relative spin relaxation  rates  \nSince we ha ve ruled out any significant change in θSH or Tint as contributing to the large variations we measure \nin 𝜉𝐷𝐿𝑗 as a function of composition and temperature, these large variations must be due to physics that is not \ncaptured in the simple Eq. (1) . We suggest that spin relaxation induced by SOC in the bulk of the Fe xTb1-x layer is \nthe most likely physics that is neglected in Eq. (1) . As schematically shown in Fig. 3(a) , a spin current in general \ncan be relaxed in a magnetic layer through two competing m echanisms: exchange -interaction -induced angular \nmomentum transfer from spin current to magnetization (with a relaxation rate τM-1) and bulk spin -orbit -scattering -\ninduced loss of spin angular momentum to the lattice (with a relaxation rate τso-1). Theory 24-27 and experiments \n53,54 have suggested that, in fully or partially compensated magnetic systems, the rate of spin angular momentum \ntransfer via exchange interaction can be greatly decreased because of cancellations between exchange fields of \nantiferromag netically aligned magnetic sub -lattices, resulting in long λdp and low τM-1. This makes it possible for \nspin currents in ferrimagnets to relax partially or even primarily via spin -orbit scattering to the lattice, instead of \napplying a spin -transfer torque to the magnetization.  \nWe can consider how the SOT should scale as a function of the ratio τM-1/τso-1. Quantitative measurements of \nthese rates (e.g., from the dependence on layer thicknesses of spin valve or spin -pumping experiments) are quite \nchallenging because the bulk properties of Fe xTb1-x 55 and other ferrimagnetic alloys53,54 vary sensitively with the \nlayer thicknesses 53-55(e.g., the magnetic compensation, the bulk PMA, the orientation of the magnetic easy axis, \nand resistivity all chang e with thickness). Nonetheless, it is reasonable to expect τM-1 ∝ Ms for such ferrimagnetic \nalloys considering the cancelling effects of the exchange fields from the antiferromagnetically -aligned magnetic \nsub-lattices. For the spin -orbit scattering rate, t he Elliot -Yafet mechanism 56-58 predicts τso-1 ∝ ζsoτe-1, where ζso is \nthe bulk SOC strength and τe-1 is the momentum scattering rate. One can thus expect τM-1/τso-1 = kM s/ζsoτe-1, with k \nbeing a constant. Since only the spin current relaxed through exchange interaction with magnetization contributes \nto SOTs, we propose that  \n𝜉DL 𝑗 ≈ 𝜉DL,0 𝑗 τM-1/(τM-1+ τ so-1) = 𝜉DL,0 𝑗(1+(kM s/ζsoτe-1)-1)-1, (3) \nwhere 𝜉DL,0 𝑗 is the value of 𝜉DL 𝑗 in the limit τM-1/τso-1 >> 1.  \nFigures 3(b) -3(d) show the estimated values of τe-1, ζso, and ζsoτe-1 for both our composition series ( x = 0.3 -1, \nT = 300 K, black symbols) and our temperature series ( x = 0.59, T = 25-350 K, red symbols). Here, τe-1 is estimated \nfrom the resistivity of the Fe xTb1-x following ρFeTb = m*/ne2τe, where m*≈ 9.1×10-31 kg is an effective mass of \ncarriers and n is the carrier density measured from the ordinary Hall coefficient ( ROH = 1/ne for a single -band \nmodel [59], see Sec. 6  in the Supplementary Materials ). τe-1 of the Fe xTb1-x varies by a factor of 2 by composition \nand a factor of 6 by temperature, suggesti ng a significant tuning of the Fermi surface properties. The average SOC \nstrength of the Fe xTb1-x is estimated as ζso ≈ xζso,Fe + (1-x)ζso,Tb using the theoretical values of ζso,Fe = 0.069 eV for  \n6 \n Fe and ζso,Tb = 0.283 eV for Tb60. We estimate that ζsoτe-1 decreases by a factor of > 16 as x varies between 0.3 to 1 \nand by a factor of > 7 as temperature increases from 25 K to 350 K ( Fig. 3(d) ). In Fig. 3(e)  we plot 𝜉𝐷𝐿𝑗 as a function \nof Ms/ζsoτe-1 for both the composition series and the temperature series. As Ms/ζsoτe-1, or equivalently τM-1/τso-1, \ndecreases, we find that 𝜉𝐷𝐿𝑗 decreases first slowly and then rapidly towards a vanishing value61. The variation of \n𝜉𝐷𝐿𝑗 with Ms/ζsoτe-1 can be fit very well by Eq. (3)  with 𝜉DL,0 𝑗 = 0.395 ±  0.022 and k = (1.66 ±  0.23)× 1011 s-1 emu-1 \ncm3 eV.  \n \nFig. 3. (a) Schematic of the spin relaxation processes that can influence the SOT, highlighting the competition \nbetween exchange interaction (with relaxation rate τ M-1 ∝ Ms) and spin -orbit scattering (τ so-1 ∝ ζsoτe-1). Only the \nspin current relaxed by exchange interaction contributes to SOTs. (b) Momentum scattering ti me (τe), (c) Estimated \nSOC strength ( ζso), (d) ζsoτe-1, and (e) 𝜉DL𝑗 of Pt 0.75Ti0.25/Fe xTb1-x vs Ms/ζsoτe-1 for the composition series ( x = 0.3-1, \nT = 300 K, black circles) and for the temperature series ( x = 0.59, T = 25-300 K, red circles). The solid curve in \n(e) represents the fit of the data to Eq. (3) . \n \nSince the strong variation of 𝜉𝐷𝐿𝑗with relative spin relaxation rates we propose here is unlikely to be specific \njust to the HM/ FexTb1-x system  (as indicated by the general fact that the SOT  provided by a given spin -current \ngenerator is significantly weaker on FIMs than on FMs, see Table 1 ), we generalize Eq. (1)  as \n𝜉𝐷𝐿𝑗= TintθSH τM-1/(τM-1+ τ so-1).      (4) \n \n7 \n Only in FMs and FIMs, which have relatively low resistivity, sma ll SOC, and high -magnetization so that τM-1 /τso-\n1 >> 1, should the simple form of Eq. (1)  apply. In perfectly compensated FIMs (Ms = 0 emu/cm3) in which τM-1 \ngoes to zero, we expect 𝜉𝐷𝐿𝑗 to go to zero as well. In FIMs or “AF” domains that are  only partially  uncompensated  \n(Ms > 0 emu/cm3)62-65, τM-1 can be comparable to τso-1 so that 𝜉𝐷𝐿𝑗 is reduced, but not zero.  \nWe note that our conclusions are contrary to some previous experiments  which  reported  𝜉DL𝑗 to remain  constant  \n66-68 or even  diverge69 near the magnetic compensation  point  of HM/CoTb or HM/ CoFeGd bilayers.  While  it might \nbe possible that  τM-1 /τso-1 is different in Co Tb and CoFeGd compared to Fe xTb1-x (e.g. Gd has zero atomic orbital \nangular moment um8 and thus considerably weak er SOC  than Tb ), we also question these previous conclusions for \na variety of  technical  experimental  reasons. In three  of the previous experiments, the PMA of the FIM layer was \nweak  and showed gradual magnetization hysteresis68, non -parabolic first -harmonic signal in HHVR measurements \n66,68,69, and/or non -linear second -harmonic  signal in  HHVR measurements66,68 as a function of a small in -plane \napplied magnetic field. This indicates magnetization behavior outside of the simple macrospin model assumed in \nthe HHVR analysis. References 68,69 also applied  “planar Hall correction ” to their HHVR results; this ofte n \ncauses erroneous estimates of 𝜉𝐷𝐿𝑗 (see Tab. 1 for a few examples and refs. 32,33,38, 39,40 for m ore discussions ). \nReference [ 66,67] reported substantial changes of sample properties before and after device patterning, resulting \nin large uncertainties in the estimation of Ms and 𝜉𝐷𝐿𝑗for their Hall -bar samples. Reference  68 studied CoTb layers \nwith thicknesses (1.7 -2.6 nm) much t hinner than the likely spin dephasing length (≈ 10 nm53) so that the escape \nrate from the film was likely faster than either τM-1 or τso-1.   \n \nTable 1. The out -of-plane HHVR results of the PMA  HM/FM samples  without and with the “ planar Hall correction ” \nvs the in -plane HHVR results on in -plane magnetized samples  with the similar HM resistivities  and the same FM \nlayer . Applying a large “planar Hall correction ” gives unrealistic numbe rs for the fieldlike and/or dampinglike \ntorque efficiencies and  alters the sign of the dampinglike torque of the Pd 4/Co 0.64  and the sign of  both \ndampinglike and fieldlike torque of the W 2.5/CoFeB 1. The PMA results for 𝜉DL𝑗 are in good agreement  with in -\nplane HHVR results  only if the “correction”  is not applied . \n \nPMA samples  VPH/VAH PMA sample  \nNo “ correction ” PMA sample  \nwith “correction ” In-plane sample  Refe rence \n𝜉DL𝑗 𝜉FL𝑗 𝜉DL𝑗 𝜉FL𝑗 𝜉DL𝑗 𝜉FL𝑗  \nW 2.2/CoFeB  1 0.486 -0.132 -0.064  -3.52 -3.25 - - [33] \nW 2.5/CoFeB 1  0.54 -0.15 -0.005 0.93 1.00 - - [33] \nPt 4/Co 0.75  0.31 0.21 -0.049  0.29 0.13 0.19 -0.046  [38] \nPd 4/Co 0.64  0.56 0.07 -0.050  -0.1 -0.16 0.06 -0.0002  [32] \nAu 0.25Pt0.75 4/Co 0.8 0.33 0.30 -0.12 0.39 -0.14 0.32 -0.020  [38] \n \nScientific i mplications  \nTaking into account the relative rates of spin -orbit -induced relaxation to the lattice versus spin transfer to the \nmagnetization can resolve outstanding puzzles in previous experiments, and has other important implications. \nFirst, the condition τM-1 ≪ τso-1 can explain why the measured 𝜉𝐷𝐿𝑗values for SOT acting on nearly -compensated \nFIMs are often several to over 20 times smaller than for corresponding measurements using 3 d FMs ( see Tab. 2 \nfor a few representative examples  with spin current sources that have similar resistivities, thicknesses, and thus  \n8 \n similar values of θSH and Tint). We have also verified from the variation of measured 𝜉𝐷𝐿𝑗values of a large number \nof samples that τM-1/(τM-1+ τ so-1) = 0.58 for Co 0.65Tb0.35 layers such that Pt -X/Co 0.65Tb0.35 is only 58% of that of  Pt-\nX/Co for given Pt -X (Pt -X being Pt -based alloys and multilayers, the detail of which will be published elsewhere). \nMoreover, t he diminishment of SOTs in fully -compensated systems  explains the absence of current -induced \nswitching of some HM/AF 11-13, while the sizable SOTs in nearly but not fully compensated systems explains  the \noccurrence of switching of uncompensated “antiferromagnetic” domains by in -plane current62-65. Our result s \nsuggest that in general SOTs will be reduced in any experiments which use magnetic free layers for which τso-1 is \nnot much less than τM-1. Spin -orbit scattering within the magnetic layer should therefore be minimized and the \naverage exchange coupling max imized for efficient SOT devices. Finally, it will be essential to modify spin \ntransport models to include spin decoherence by spin -orbit scatting on an equal footing with dephasing by the \nexchange interaction.  \nTable 2. Comparison of 𝜉DLj for FIMs and 3d FMs in contact with spin current sources that have  similar \nresistivit ies, thickness es, and thus similar  values of  θSH and Tint. \nspin current \nsource  𝜉DLj \nFIM 3d FM ratio \nTa -0.03 (CoTb)5 -0.12 (FeCoB)10 4 \nW -0.04 (CoTb)70 -0.44 (FeCoB)71 11 \nPt 0.017 ( CoTb)5 0.15 (Co,FeCoB)10 8.8 \nPt/NiO  0.09 (CoTb) 6 0.6 (FeCoB)72 6.7 \nPt0.75Ti0.25 0.05 (Fe 0.5Tb0.5) 0.38 (Fe)  7.6 \nBi2Se3 0.13 ( GdFe Co)7 3.5 (NiFe)73 27 \nConclusion s \nIn summary, we have shown that the strength of SOTs depends critically on the ratio of rate of spin -orbit -\ninduced spin relaxation within a magnetic layer relative to the rate of exchange -induced spin transfer to the \nmagnetization. We find experimentally that SOT effic iencies decrease strongly upon approaching the magnetic \ncompensation point in ferrimagnetic FexTb1-x due to a decrease in the rate of exchange -induced spin transfer on \naccount of partial cancellation between the oppositely -directed exchange interactions fr om the magnetic sub -\nlattices. Near the compensation point, spin -orbit -induced spin relaxation dominates over spin transfer to the \nmagnetization so that the measured SOT goes to zero. These results suggest the breakdown of the “ interfacial \ntorques ” concept in FIMs and AFs. We find no indication of any dependence of the spin transparency of FexTb1-x \ninterfaces on the degree of compensation. This work  provides not only a unified understanding of the very different \nefficiencies of SOTs that have been reported i n the literature for FMs, FIMs, and AFs , but also insight about how \nthe different sources of spin relaxation should be optimized in the design of FIMs and AFs for spintronic \ntechnologies 8,9.  \n9 \n Acknowledgement  \nWe thank Dahai Wei, Xin Lin, Qianbiao  Liu for help with sample deposition. We also thank Tianxiang Nan, \nYanan Chai, Wei Han, Liangliang Guo for help with ferromagnetic resonance measurements. This work was \nsupported in part by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB44000000), \nin part by the Office of Naval Research (N00014 -19-1-2143), in part by the NSF MRSEC program (DMR -1719875) \nthrough the Cornell Center for Materials Research, and in part by the NSF ( NNCI -2025233 ) through the Cornell \nNanofabrication Facility/National Nanotechnology Coordinated Infrastructure .  \nData availability  \nThe data that support this study are available from the corresponding authors upon reasonable request.  \nConflict of Interest  \nThe authors declare no conflict of interest.  \nReferences  \n[1] I. M. Miron , K. Garello, G. Gaudin, P. -J. Zermatten, M. \nV. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl , \nP. Gambardella , Perpendicular switching of a single \nferromagnetic layer induced by in -plane current injection, \nNature 476, 189  (2011) . \n[2] L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph , R. 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Ansermet1\n1Ecole Polytechnique F ´ed´eralede Lausanne, ICMP, Station 3, CH-1015 Lausanne-EPFL, Switz erland\n2Max Planck Institut f ¨urPhysik Komplexer Systeme, N¨ othnitzer Str. 38, 01187 Dresd en, Germany\n3CRANN and School of Physics, Trinity College, Dublin 2, Irel and\n4Ecole Polytechnique F ´ed´eralede Lausanne, ITP-CTMC, CH-1015 Lausanne, Switzerland\nWe present a thorough77Se NMR study of a single crystal of the magnetoelectric compo und Cu 2OSeO3.\nThe temperature dependence of the local electronic moments extracted from the NMR data is fully consistent\nwitha magnetic phase transitionfrom the high-T paramagnet ic phase toalow-T ferrimagnetic statewith3/4of\nthe Cu2+ions aligned parallel and 1/4 aligned antiparallel to the ap plied field of 14.09 T. The transition to this\n3up-1down magnetic state is not accompanied by any splittin g of the NMR lines or any abrupt modification in\ntheirbroadening, hence thereisnoobservable reductionof thecrystallinesymmetryfromitshigh-Tcubic P213\nspace group. These results are in agreement with high resolu tion x-ray diffraction and magnetization data on\npowder samples reported previously by Bos et al.[Phys. Rev. B, 78, 094416 (2008)]. We also develop a mean\nfield theory description of the problem based on a microscopi c spin Hamiltonian with one antiferromagnetic\n(Jafm≃68K) andone ferromagnetic ( Jfm≃ −50K) nearest-neighbor exchange interaction.\nPACS numbers: 76.60.Jx, 75.50.Gg,75.25.-j, 77.84.Bw\nI. INTRODUCTION\nMultiferroic and magnetoelectricmaterials are currently at\nthe center of intense research activity.1–4In magnetoelectric\ncompoundstheapplicationofanelectricfieldcaninducea fi-\nnite magnetization and similarly an electric polarization can\nbe induced by an applied magnetic field.5,6Combining elec-\ntronic and magnetic properties is an exciting playground fo r\nfundamentalresearch which may also lead to new multifunc-\ntional materials with potential technological applicatio ns.3,7,8\nSince both time-reversal and spatial-inversion symmetrie s\nmust be broken in ferroic materials,9magnetoelectric (ME)\neffects are allowed only in 58 out of the 122 magnetic point\ngroups.9,10AlongwiththeweaknessoftheMEeffect,thisre-\nsults in a limited numberof compoundsdisplayingME prop-\nerties, such as Cr 2O311, Gd2CuO412, and BaMnF 413. Even\nthoughMEmaterialshavebeenstudiedextensivelyinthepas t\ndecades the recent discovery of the multiferroic compounds\nTbMnO 314and TbMn 2O515, where ferroelectricity (FE) is\ndrivendirectlybythespinorder,hasledtoarevivalofinte rest\nin ME systems. The magneticstate in these compoundsis ei-\ntheraspiral(asinTbMnO 3,14Ni3V2O8,16andMnWO 417,18),\nor a collinear configuration (as in Ca 3(CoMn) 2O6,19and\nFeTe2O5Br20).\nConsiderable interest has also been drawn to the investi-\ngation of ME effects appearing in non-polar systems such as\nSeCuO 3,21TeCuO 3,21and Cu 2OSeO322. The latter, as re-\nported recently by Bos et al.,22, undergoes a ferrimagnetic\nphase transition at Tc≃60K which is accompanied by a\nsignificant magnetocapacitance signal and an anomaly of the\ndielectric constant. However, high resolution powder x-ra y\ndiffraction (XRD) data show that the lattice remains metri-\ncally cubic down to 10 K, and this excludes a ME coupling\nmechanism that involves a spontaneous lattice strain. This\nresult is further supported by recent infrared,23and Raman24\nstudies. In this respect, Cu 2OSeO3appears to be a unique\nexample of a metrically cubic material that allows for piezo -\nelectric as well as linear magnetoelectric and piezomagnet ic\nFIG.1: (Coloronline)CrystalstructureofCu 2OSeO3. Copperatoms\n(Cu1in blue and Cu 2in green) form a network of distorted corner\nsharingtetrahedra. Onlytheionsinsidetheunitcellaresh own. Oxy-\ngen atoms are shown in yellow while Se Iand Se IIions are shown in\nmagenta andredrespectively.\ncoupling.\nHere we present an extensive77Se Nuclear Magnetic Res-\nonance(NMR)studywhichhighlightsthelocalmagneticand\nstructuralpropertiesof Cu 2OSeO3. Our NMR measurements\nareperformedinasinglecrystalofCu 2OSeO3asafunctionof\ntemperature and by varying the direction of the applied mag-\nneticfieldwithrespecttothecrystallineaxes. Adetaileda nal-\nysis of the77Se spectra based on space groupsymmetry con-\nsiderationsleadstothefollowingmainconclusions: Thete m-\nperaturedependenceofthelocalelectronicmomentsextrac ted2\nfromthe NMR data is fully consistent with a phase transition\nbetweenthehigh-Tparamagneticphaseandalow-Tferrimag-\nnetic configurationwith 3/4of theCu2+moments(typeCu 2)\naligned parallel and 1/4 (type Cu 1) aligned antiparallel to the\napplied field. This 3up-1down state has 1/2 of the saturated\nmagnetization value and is the one proposed by Bos et al22.\nThe transition is not accompanied by any observable change\nin the broadening of the NMR lines or any clear splitting,\nshowing that there is no observable symmetry reduction in\nthecrystallinestructurefromthehigh-temperature P213space\ngroup,inagreementwiththereportedpowderXRDdata.22In\naddition, we provide a microscopic description of the prob-\nlem based on a spin Hamiltonian with two nearest-neighbor\nexchangecouplings: oneantiferromagnetic( Jafm≃68K)be-\ntween Cu 1and Cu 2ions, and one ferromagnetic( Jfm≃ −50\nK) between Cu 2ions. The mean field theory predictions of\nthismodelareingoodqualitativeagreementwiththebehavi or\nofthelocalmomentsextractedfromNMRandmagnetization\ndata.\nThe article is organized as follows. In Sec. II we sum-\nmarize the synthesis process and give some technical detail s\nof our experimental procedure. In Sec. IIIA we present our\nNMR spectra and give a first discussion of some central find-\nings. The theoretical framework for the explanation of the\nNMR spectra is given in Sec. IIIB based on symmetry con-\nsiderations. The comparison to the NMR data is then given\nin Sec. IIIC, where we also extract the relevant transferred\nhyperfinefield parametersas well as the T-dependenceof the\nlocal moments of both types of Cu2+ions. A comparison to\nthemeanfieldtheorypredictionsisalsomadehere. OurNMR\nresultsforthenuclearspin-lattice 1/T1andspin-spin 1/T2re-\nlaxation rates are presented in Sec. IV. We conclude in Sec.\nV witha briefdiscussionofourresults.\nII. SYNTHESISANDEXPERIMENTALDETAILS\nSingle crystals of Cu 2OSeO3were grown by the standard\nchemical vapourphase method. Mixturesof highpurity CuO\n(Alfa-Aesar,99.995%)andSeO 2(Alfa-Aesar,99.999%)pow-\nder in molar ratio 2:1 were sealed in quartz tubes with elec-\ntronic grade HCl as the transport gas for the crystal growth.\nThe ampoules were then placed horizontally into a tubu-\nlar two-zone furnaces and heated very slowly by 50◦C/h to\n600◦C. The optimum temperatures at the source and deposi-\ntion zones for the growth of single crystals have been 610◦C\nand 500◦C, respectively. After six weeks, many dark green,\nalmost black Cu 2OSeO3crystals with a maximum size of\n8x6x3mmwereobtained.\nThe orientation of the crystal axes with respect to the\ncrystal faces were determined by Laue X-ray back-scatterin g\nmeasurements and by single crystal X-ray diffraction. The\ndiffraction data are in agreement with previously publishe d\ndata.22,25,26The crystal structure of Cu 2OSeO3which be-\nlongs to the cubic space group P213 is presented in Fig. 1.\nThestructureisathree-dimensionalarrayofdistortedcor ner-\nsharing copper tetrahedra. The unit cell consists of 16 Cu2+\nionswhichbelongtotwocrystallographicallydifferentgr oups/s49/s49/s52/s46/s48 /s49/s49/s52/s46/s50 /s49/s49/s52/s46/s52 /s49/s49/s52/s46/s54 /s49/s49/s52/s46/s56 /s49/s49/s53/s46/s48\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s77/s72/s122/s41/s80/s52/s80/s52/s80/s51\n/s80/s51/s80/s50\n/s80/s50/s80/s50 /s80/s49\n/s80/s49\n/s91/s49/s48/s48/s93/s91/s49/s49/s49/s93\n/s91/s49/s49/s48/s93/s80/s49\nFIG. 2:77Se NMR line shape measurements at 220 K in an exter-\nnal field of 14.09 T applied parallel to the [111], [110] and [1 00]\ncrystallographic directions.\nTABLE I: The experimentally observed paramagnetic shifts ∆νexp\n(in kHz) at T= 220K measured with the external magnetic field\nH0along the [111], [110] and[100] crystallographic directio ns.\n∆νexp\n[111]∆νexp\n[110]∆νexp\n[100]\nP1 -121.8±0.1 -142.9 ±0.1 -96.6 ±0.01\nP2 -49.4±0.2 -53.9 ±0.1 137.5 ±0.04\nP3 89.9±0.1 60.9 ±0.1\nP4 277.1±0.2 206.8 ±0.1\ndenoted here by Cu 1and Cu 2, in 4αand 12bsites respec-\ntively. Theoxygenatomsformtwodifferenttypesofdistort ed\nCuO5polyhedra,i.e., trigonalbipyramidalpolyhedrafor Cu 1\nsitesandsquarepyramidalpolyhedraforCu 2sites. TheCuO 5\npolyhedra are connected by sharing edges and corners. Sim-\nilarly, the unit cell contains two crystallographicallyin equiv-\nalent groupsof Se4+ions, Se Iand Se II, each one with multi-\nplicity4. ThetwotypesofSeO 3(lonepair)trigonalpyramids\nsharecornerswiththeCuO 5polyhedra.\nPulsed NMR experiments were carried out on77Se nuclei\nwith nuclear spin I= 1/2, gyromagnetic ratio γN/2π=\n8.1179MHz/T and natural abundance 7.58 %. The NMR ex-\nperiments were performed on a 125 mg single crystal at an\nOxford14.09T magnetequippedwith a variabletemperature\ninsert utilizing a home built spectrometer. The echo signal\nwasproducedbyastandardHahnechopulsesequencewitha\ntypicalπ/2pulse length of 6 µs. The separation between the\ntwo echo-generating pulses was 20 −60µsdepending on the\nexperimental conditions. The77Se NMR spectra were mea-\nsured by Fourier transform (FT) of the half spin-echo signal\nwheneverthewholelinecouldbeirradiatedwithoneradiofr e-\nquencypulse. For the broadlines, the spectrumwas obtained\nbyplottingtheareaoftheechoasafunctionoftheirradiati on3\nfrequency(frequencysweep). Thenuclearspin-lattice T1and\nspin-spin T2relaxation times were measured using the stan-\ndard spin echo pulse sequence combined with the saturation\nrecoverymethodfor T1measurements. Fortheanalysisofthe\nNMRdatawehavemeasuredthemagnetizationofa28.86mg\nsingle crystal of Cu 2OSeO3at 14 T using a QuantumDesign\nPPMS (Physical Properties Measurement System) apparatus\nlocatedintheTrinityCollegein Dublin,Ireland.\nIII.77SENUCLEARMAGNETICRESONANCESPECTRA\nA. General results\nThe nuclear77Se spins provide a powerful local probe of\nthe behavior of the electronic Cu2+moments through the\ntransferred hyperfine and the magnetic dipolar mechanisms.\nHere we have performed detailed77Se NMR lineshape mea-\nsurements at 14.09 T with the magnetic field applied parallel\nto thecrystallographicdirections[111],[110]and[100]. Fig-\nure 2 shows the77Se NMR spectrum at T= 220K. There\nappear four distinct resonance lines when the field is applie d\nalong[111]and[110],whiletwolinesareobservedalongthe\n[100] direction. The specific values of the shifts of the line s\n∆ν=ν−νLfromthebareLarmorfrequency νLareprovided\nin Table I. The integrated areas under these lines are found\nwiththerelativeratios3:1:3:1for[111],2:2:2:2for[110 ],and\n4:4for[100]. Sincetheareaunderagivenlineisproportion al\ntothenumberofthenucleithat resonateinthecorrespondin g\nfrequency window, we conclude that there are four magneti-\ncally inequivalentgroups of77Se sites in the [111] and [110]\ndirections and only two groups for [100]. It is clear that the\nlatter correspond to the two crystallographically inequiv alent\ngroups of selenium sites Se Iand Se IIwhich have the same\nmultiplicity (four per unit cell). On the other hand, both Se I\nand Se IIgroups split into two magnetically inequivalent sub-\ngroupswhenthefieldisalong[111]and[110]withmultiplic-\nities 3:1 and 2:2 respectively. Below in Sec. IIIB, we shall\nbe able to identify these specific subgroups of Se sites (and\neven retrieve the values of the correspondingelements of th e\ntransferred hyperfine tensor) by taking into account the loc al\nsymmetryaroundeachSesiteundertheconditionsthat(1)we\nareinthe3up-1downstateand(2)thatthecrystallinestruc ture\nbelongsto P213 space group. The second conditionis an im-\nportantissuesinceanysymmetryreductionofthecrystalfr om\nP213 to either of the two possible crystallographicsubgroups\nR3 orP212121,27will result in partially or fully splitting of\ntheabovemultiplelines.\nLet us now examine what happens as we cool down the\nsystem. We have studied the temperature dependence of the\nNMR spectra in the range 20-290 K with the magnetic field\napplied along [111] and [110]. Some representative NMR\nspectra are shown in Fig. 3. We first note that the lines P1\nand P2 get gradually closer to each other below 100 K and\nmergeintoasinglepeakatlowertemperatures. Wealsofinda\ngradualincreasein theline broadeningsorFull Widthat Hal f\nMaximum (FWHM). However, as we show in Fig. 4, all the\nNMRlinebroadenings(shownwithsymbolsinFig. 4)follow/s49/s49/s50/s46/s56 /s49/s49/s51/s46/s54 /s49/s49/s52/s46/s52 /s49/s49/s53/s46/s50 /s49/s49/s50/s46/s56 /s49/s49/s51/s46/s54 /s49/s49/s52/s46/s52 /s49/s49/s53/s46/s50/s80/s50 /s80/s51\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s77/s72/s122/s41/s80/s49\n/s80/s50/s80/s51/s80/s49\n/s80/s52\n/s72\n/s48/s47/s47/s91/s49/s49/s49/s93/s80/s52\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s77/s72/s122/s41/s50/s57/s48/s32/s75/s50/s51/s48/s32/s75/s49/s52/s48/s32/s75/s49/s48/s48/s32/s75/s54/s48/s32/s75/s53/s48/s32/s75\n/s72\n/s48/s47/s47/s91/s49/s49/s48/s93\nFIG. 3:77Se NMR line shape measurements for selected temper-\natures measured at external field of 14.09 T applied parallel to the\n[111] and [110] crystallographic directions.\nquite closely the corresponding gradual increase of the mag -\nnetization as measured by PPMS (solid line). This being the\ntypicalbehaviorofinhomogeneousbroadening,togetherwi th\nthefactthatwefindnoobservablesplittingofanyofthemul-\ntiple lines, we are lead to conclude that there is no clear sig n\nof any symmetry reduction of the crystalline structure as th e\nsystementerstheferrimagneticstate.\nAnotherimportantissue is thebehaviorofthelineshiftsas\nwe cooldownthe system,since thisis governedbythe corre-\nspondingorderingbehavior of the electronic Cu2+moments.\nFigure5showstheT-dependenceoftherelativeshifts ∆ν/νL\nmeasured at H 0= 14.09T for fields applied along the direc-\ntions [111] and [110]. From these data, as we show below in\nSec. IIIB,we shall beabletoextracttheT-dependenceofthe\nlocalmomentsofbothCu 1andCu 2ionsandthusconfirmthe\nferrimagneticnatureofthelow-Tphaseofthiscompound.\nB. Theoreticalanalysis\nInordertounderstandandanalyzetheaboveNMRdatawe\nfirst note that the most relevant terms in the nuclear spin-1/ 24\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54/s48/s46/s50/s48/s48/s46/s50/s52/s32/s70/s87/s72/s77/s32/s40/s77/s72/s122/s41\n/s32/s32\n/s80/s49/s32/s91/s49/s49/s49/s93/s32\n/s80/s50/s32/s91/s49/s49/s49/s93\n/s80/s51/s32/s91/s49/s49/s49/s93\n/s80/s52/s32/s91/s49/s49/s49/s93\n/s80/s49/s32/s91/s49/s49/s48/s93\n/s80/s50/s32/s91/s49/s49/s48/s93\n/s80/s51/s32/s91/s49/s49/s48/s93\n/s80/s52/s32/s91/s49/s49/s48/s93\n/s32/s57/s46/s52\n/s109/s111/s108/s45/s67/s117\n/s84/s32/s40/s75/s41\nFIG. 4: (Color online) Temperature dependence of the77Se NMR\nfull width at half maximum (FWHM) for each of the four spectra l\nlinesmeasuredat14.09Talongthe[111] and[110] crystallo graphic\ndirections. The FWHM for lines P1 and P3 along [111] are multi -\nplied by a factor 0.9 and 0.7 respectively in order to match th e other\ncurves. The temperature dependence of the magnetic suscept ibility\nper copper mol (multiplied by a factor of 9.4) as measured wit h the\nmagnetometer at 14T isshown witha solidline.\nHamiltonian for the k-th Se site are (i) the Zeeman coupling\nwith the external field H0, (ii) the transferred-hyperfine in-\nteractionswith the relevantneighboringCu spinsand(iii) the\ndipolarinteractionswithall theelectronicCuspins,name ly\nHk=−γN/planckover2pi1Ik·/parenleftig\n(1+Kc\nk)·H0+Htr\nk+Hdip\nk/parenrightig\n,(1)\nwhereKc\nkdenotes the chemical shift tensor, while Htr\nk\nandHdip\nkare the local effective fields resulting from the\ntransferred-hyperfineandthedipolarinteractionsrespec tively.\nTransferred-hyperfine contribution .— Let us discuss the\ntransferred-hyperfinecouplingfirst. Thetransferred-hyp erfine\ninteraction is local, i.e. it involves a77Se nuclear spin and\nits six nearest neighbor Cu ions, and is mediated through the\nCu-O-Sebonding. Thelocal transferred-hyperfinefield expe -\nriencedbythe k-thnuclearspincanbewrittenas\nHtr\nk=q/summationdisplay\njAj\nk·Mj, (2)\nwhereq≡1/(γN/planckover2pi1gµB),Mj=−gµB/angbracketleftSj/angbracketrightdenote the local\nmoments of the six neighbor Cu ions, µBis Bohr’s magne-\nton, andgis the spectroscopic factor with g≃2.11.26To go\nbeyond this general description we need to exploit the local\nsymmetrypropertiesoftheelectronicenvironmentofeachS e\nsite. As we mentionedin the introduction,each unit cell con -\ntainsfourcrystallographicallyequivalentsitesofSe Iionsand\nsimilarlyfourequivalentsitesofSe IIions. Wedenotetheseby\nSen\nIand Sen\nIIrespectively, where n= 1,...,4. We start our\nanalysiswiththefirsttypeofseleniumions. Ascanbeseenin\nFig.6,eachSen\nIionsitsonahighsymmetrycrystalsite. Each\noxygenofthe trigonalpyramidSeO 3is connectedtotwo dif-\nferent types of copper ions, denoted as Cu 1and Cu 2. The/s45/s49/s54/s45/s49/s50/s45/s56/s45/s52/s48/s52/s56\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s49/s54/s45/s49/s50/s45/s56/s45/s52/s48/s52/s32/s32 /s32/s47\n/s76 /s32 /s32/s47\n/s76 \n/s40/s98/s41/s32/s72\n/s48/s47/s47/s32/s91/s49/s49/s48/s93/s40/s97/s41/s32/s72\n/s48/s47/s47/s32/s91/s49/s49/s49/s93\n/s32/s80/s49\n/s32/s80/s50\n/s32/s80/s51\n/s32/s80/s52\n/s32/s32/s32\n/s84/s32/s40/s75/s41/s32/s80/s49\n/s32/s80/s50\n/s32/s80/s51\n/s32/s80/s52\nFIG.5: (Coloronline) Temperaturedependence oftherelati veNMR\nshifts∆ν/νLat H0= 14.09T and for each of the four spectral\nlines observed along the (a) [111] and (b) [110] crystallogr aphic di-\nrections.\nthree Se I-O-Cu1-Cu2bonding groups are equivalent and are\nmapped to one another by a rotation of 120◦around a 3-fold\nsymmetryaxis. This C3axis passes throughthe selenium site\nand is vertical to the plane of the three Cu 2ions. This plane\nis parallel and slightly above the plane formed by the three\nCu1ions, as well as to the plane formed by the three oxygen\natoms (see Fig. 6). Thus the transferred hyperfineinteracti on\nat any given Sen\nIsite can be described by two hyperfine ten-\nsors,ACu1\nSen\nIandACu2\nSen\nI. The former (latter) represents the sum\nof the three hyperfine tensors between the n-th SeIand the\nthree Cu 1(Cu2) ions. Now, one of the two 3-fold rotations\nreads(x,y,z)/mapsto→(z,x,y), and this transforms the elements\nofanysecondranktensoras\n\nxx xy xz\nyx yy yz\nzx zy zz\n/mapsto→\nzz zx zy\nxz xx xy\nyz yx yy\n. (3)\nSince this is a symmetryoperationwe must have xx=yy=\nzzandxy=xz=yz. These conditions describe an axially\nsymmetric tensor with the third principal eigenvector alon g\nthe [111] direction. Thus the hyperfine tensors ACu1\nSen\n1and5\nFIG. 6: (Color online) The local environment of each Se Inuclear\nsite. There is a 3-fold symmetry axis through the Se site whic h is\nvertical to the planes of the Cu 1and Cu 2ions. Note that the plane\nof the Cu 2ions lies slightly above the plane of the Cu 1ions. Shown\ndistances are inunits of ˚A.\nACu2\nSen\nIare also axially symmetric andtheir principalaxes sys-\ntem can be assigned easily. One is the 3-fold axis ( e3) while\nfor the remaining two we may take any pair of mutually per-\npendicular axes (denoted by e1ande2) on the plane formed\nby the three Cu 2(or Cu 1) neighboring ions. Thus we may\nwrite\nqACu1\nSen\nI=\nαtr0 0\n0αtr0\n0 0βtr\n,qACu2\nSen\nI=\nα′\ntr0 0\n0α′\ntr0\n0 0β′\ntr\n,(4)\nwhereweemphasizethateachtensoriswritteninitsownprin -\ncipalaxesframe. Inwhatfollowstheabovediagonalelement s\naretreatedasfittingparameters.\nWe now turn to the local symmetry of the Sen\nIIions. These\narebondedthroughoxygensitestosixCu 2ions(Fig.7)which\nbelongto two parallel planes. As in the case of the Sen\nIsites,\na 3-fold rotation axis is again present and this passes throu gh\nthe selenium site of the trigonalpyramidSeO 3andis vertical\nto the set of planes defined by the Cu 2ions. The hyperfine\ninteractions at each Sen\n2site can thus be described with one\nhyperfine tensor, ACu2\nSen\nII, which stands for the sum of the hy-\nperfine tensors of all six neighboring Cu 2ions. Due to the\n3-fold axis this tensor is axially symmetric and, as explain ed\nfor the Se Icase, the principalaxes are the 3-foldaxis and the\ntwo mutuallyperpendicularaxes(theirplaneis parallelto the\ntwo Cu planes). Written in its own principal axis system, the\nhyperfinetensor ACu2\nSen\nIIcanthusbewrittenas\nqACu2\nSen\nII=\nδ′\ntr0 0\n0δ′\ntr0\n0 0ϑ′\ntr\n. (5)\nSummarizing, the hyperfine tensors for each one of the four\nSen\nIsites(respectivelySen\nIIsites)oftheunitcellhavetheform\nof Eqs. 4 (resp. 5) as long as they are written in their own\nlocal principal axis coordinate frame. In Table II we provid e\nFIG.7: (Coloronline)Thelocalenvironment ofSe IItypeions. Asin\nthe case of Se Iions, there is a 3-fold symmetry axis through the Se\nsite which is vertical to the planes of Cu 2ions. Shown distances are\ninunits of ˚A.\nthe coordinates of the unit vector e3in the (x,y,z) frame for\neach of the eight selenium sites in the unit cell. The coordi-\nnate frames of the hyperfine tensors for the four Sen\nIand the\nfourSen\nIIsitesarerelatedwithproperrotationoftheirprincipal\naxes[e1,e2,e3].\nDipolar contribution .— In order to provide a more ac-\ncurate quantitative description of the measured NMR spec-\ntra we must also consider the dipolar coupling. In contrast\nto the transferred-hyperfineinteractions, the dipolar cou pling\nis long-ranged and since we are dealing with a ferrimag-\nnet we must also consider the effect of the demagnetization\nfield. To this end we follow the Lorentz method28which\nconsists of splitting the summation over the dipolar contri bu-\ntions into two parts. In the first we perform a discrete sum-\nmation over all individual dipolar contributions from all C u\nions enclosed in a sphere of radius Rmuch larger than the\nlattice spacing and much smaller than the size of the sam-\nple. The summation outside this Lorentz sphere gives rise\nto the demagnetization field Hdemwhich can be evaluated\nusing a continuous integration. This gives the well known\nexpression28Hdem= 4π(1\n3−Nd)·/tildewiderM, where the demag-\nnetizing tensor Ndis specific to the shape of the sample and\n/tildewiderM= (4M1+12M2)/a3= 4M/a3is the total moment di-\nvidedbythevolumeofthesample. Thedemagnetizingtensor\nisnotknownbutweshallbeabletoadjustitsothatweobtain\na reasonable T-dependence of the extracted moments at high\ntemperatures(cf. below).\nFollowing the above, the total dipolar field at the k-th Se\nnuclearsite canbewrittenas\nHdip\nk=q/summationdisplay\nrjk≤RDj\nk·Mj+Hdem, (6)\nwhereasusuallythecartesian components( α,β=x,y,z)of\nthedipolartensorsaregivenby\nq/parenleftig\nDj\nk/parenrightig\nαβ=−δαβ\nr3\njk+3rα\njkrβ\njk\nr5\njk, (7)\nwhererjkstands for the displacement vector from the k-th\nnuclearsitetothe j-thelectronicspin. Summingoverthetwo\ndifferenttypesofCu2+ionsin Eq. (6) separatelywe get\nHdip\nk=q/parenleftig\nDCu1\nk·M1+DCu2\nk·M2/parenrightig\n+Hdem.(8)6\nTABLE II: The second column gives the unit vector e3in the fixed\n(x,y,z)frameforeachoftheeightSesitesperunitcell(firs tcolumn).\nThe following columns give the theory predictions for the su m of\nthe dipolar (excluding the demagnetization field) and the tr ansferred\nhyperfine contributions to the relative NMR shifts, when the field is\nalong the [111], [110], and [100] crystallographic directi ons. The\ndimensionless parameters /tildewideα,/tildewideβ,/tildewideδ,/tildewideϑaredefined inEqs. (10)-(13).\nSe e3 [111] [110] [100]\nSe1\nI(1,1,1)/√\n3 /tildewideβ (/tildewideα+2/tildewideβ)/3 (2/tildewideα+/tildewideβ)/3\nSe2\nI(1,1,-1)/√\n3 (8/tildewideα+/tildewideβ)/9 (/tildewideα+2/tildewideβ)/3 (2/tildewideα+/tildewideβ)/3\nSe3\nI(-1,1,1)/√\n3 (8/tildewideα+/tildewideβ)/9 /tildewideα (2/tildewideα+/tildewideβ)/3\nSe4\nI(1,-1,1)/√\n3 (8/tildewideα+/tildewideβ)/9 /tildewideα (2/tildewideα+/tildewideβ)/3\nSe1\nII(1,1,1)/√\n3 /tildewideϑ (/tildewideδ+2/tildewideϑ)/3 (2/tildewideδ+/tildewideϑ)/3\nSe2\nII(1,1,-1)/√\n3 (8/tildewideδ+/tildewideϑ)/9 (/tildewideδ+2/tildewideϑ)/3 (2/tildewideδ+/tildewideϑ)/3\nSe3\nII(-1,1,1)/√\n3 (8/tildewideδ+/tildewideϑ)/9 /tildewideδ (2/tildewideδ+/tildewideϑ)/3\nSe4\nII(1,-1,1)/√\n3 (8/tildewideδ+/tildewideϑ)/9 /tildewideδ (2/tildewideδ+/tildewideϑ)/3\nUsing the known values22,25,26of the positions of the Cu2+\nmoments one may evaluate the summations over the Lorentz\nsphere with very good accuracy (we find that the sums con-\nvergealreadyfor R∼5a). Now, the symmetryof the crystal\nnecessitates that the resulting dipolar tensors have the sa me\nprincipalaxesas thecorrespondingtransferred-hyperfine ten-\nsors. Indeed written in the corresponding local coordinate\nframewe find\nqDCu1\nSen\nI=\nαdip0 0\n0αdip0\n0 0βdip\n,qDCu2\nSen\nI=\nα′\ndip0 0\n0α′\ndip0\n0 0β′\ndip\n,\nqDCu1\nSen\nII=\nδdip0 0\n0δdip0\n0 0ϑdip\n,qDCu2\nSen\nII=\nδ′\ndip0 0\n0δ′\ndip0\n0 0ϑ′\ndip\n,(9)\nwhere, in units of 1022cm−3:αdip= 1.68,βdip=−3.37,\nα′\ndip= 1.18,β′\ndip=−2.36,δdip=−0.58,ϑdip= 1.17,δ′\ndip=\n−1.3,andϑ′\ndip= 2.61.\nPredictions for the relative shifts .— Including both the\ntransferred-hyperfine and the dipolar contributions it is\nstraightforward to deduce the relative shifts for each Se si te\nand for each direction of the applied field considered in our\nexperiments. The resulting expressions – without the contr i-\nbution for the demagnetization field and the chemical shift –\nare providedin Table II in termsof the dimensionlessparam-\neters\n/tildewideα= (αtr+αdip)χ1+(α′\ntr+α′\ndip)χ2(10)\n/tildewideβ= (βtr+βdip)χ1+(β′\ntr+β′\ndip)χ2(11)\n/tildewideδ=δdipχ1+(δ′\ntr+δ′\ndip)χ2 (12)\n/tildewideϑ=ϑdipχ1+(ϑ′\ntr+ϑ′\ndip)χ2, (13)\nwhereχ1=M1/H0andχ2=M2/H0denote the local sus-\nceptibilitiesperCu 1andCu 2respectively.TABLEIII:Extractedestimatesofthecomponents ofthetran sferred\nhyperfine tensors ACu1\nSen\nI,ACu2\nSen\nIandACu2\nSen\nIIdefined in Eqs. (4) and (5)\n(here in units of 1022cm−3) for two possible values of the constant\nr1definedinthe text.\nr1αtrβtrα′\ntrβ′\ntrδ′\ntr θ′\ntr\n−2−2.3 12.6−5.0 26.8−14.6−14.0\n−1.2−1.7−5.4−1.2−3.8−16.8−16.2\nC. Comparison withexperiment\nLetusnowcomparethetheoreticalpredictionsgiveninTa-\nble II to our experimental data. We find exact agreement for\nthe number of distinct lines as well as for their relative in-\ntensities. When the magnetic field is applied along [111] our\nmodel predicts that the group of Se Isites give one spectral\nlineforSe1\nIandaseparate,threetimesmoreintenselinefrom\nSe2\nI, Se3\nIand Se4\nI. A similar result holds for the group of Se\nnuclear spins of the second type. In the [110] direction we\nexpect four resonance lines with relative ratio 2:2:2:2, wh ile\nin the [100] direction we expect two lines with relative rati o\n4:4 (see Table II). These predictionsare in perfectagreeme nt\nwithourexperimentalresultsshowninFigs.2 and3.\nFor consistency reasons we would like next to contrast the\nT-dependenceof /tildewideα,/tildewideβ,/tildewideδ, and/tildewideϑobtained from the NMR data\nalong the [111] direction with that obtained from the NMR\ndata along the [110] direction. The results (taken after cor -\nrectingthedataforthechemicalshiftandthedemagnetizat ion\nfield, cf. below) are shown in Fig. 8 and are almost identi-\ncal. This providesa much stronger confirmation of the inter-\nnal consistency of the above theory. In addition, Fig. 8 tell s\nus that we may use either the [111]or the [110] data in order\nto extract, in conjunction with Eqs. (10)-(13), the transfe rred\nhyperfinefield parametersas well as the T-dependenceof the\nlocalmomentsM 1andM2. Inwhatfollowswehavetakenthe\ndataalong[111].\nAssignment of the NMR lines .— The next step is to iden-\ntify the specific subgroupof Se sites associated to each give n\nNMR line. To this end, we first note that the local fields of\nSen\nIand Sen\nIIgiven in Table II map to one another when in-\nterchanging (/tildewideα,/tildewideβ)/mapsto→(/tildewideδ,/tildewideϑ). This preventsa straightforward\nassignmentoftheNMRlinestospecificsubgroupofselenium\nsites. Forexample,we cannotinferifit isP1 andP2 orrather\nP1andP4(cf. Fig.3)thatcomefromthesametypeofSesites.\nWeovercomethisdrawbackbycomparingthefitsindifferent\ndirections of the applied field. Indeed, by fitting the experi -\nmental data of ∆νexp\n[111](see Table I) with the theoretical ex-\npressionsgiveninTableIIwecanreproducetheexperimenta l\ndataofboth ∆νexp\n[110]and∆νexp\n[100]onlyundertheconditionthat\nP1 and P2 come from one group of selenium sites while P3\nand P4 come from the other. This result is further supported\nby the nuclearspin-lattice and spin-spinrelaxationtime m ea-\nsurements presented in the following section. However it is\nstill not possible at this point to tell whether the pair P1-P 2\ncomes from type-I or type-II Se sites. We have extracted the\ntemperaturedependenceofthelocalmomentsusingbothpos-7\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s48/s46/s48/s49/s50/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s52/s48/s46/s48/s48/s48/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s48/s46/s48/s49/s50/s32\n/s84/s32/s40/s75/s41/s126\n/s126\n/s126/s91/s49/s49/s48/s93\n/s32/s91/s49/s49/s48/s93\n/s32/s91/s49/s49/s48/s93\n/s32/s91/s49/s49/s48/s93/s126/s126\n/s126\n/s126/s91/s49/s49/s49/s93\n/s32/s91/s49/s49/s49/s93\n/s32/s91/s49/s49/s49/s93\n/s32/s91/s49/s49/s49/s93/s126\nFIG.8: (Coloronline) A comparison of the values of the param eters\ngiveninEqs. (10)-(13) extractedfromthe NMRline shiftsal ongthe\n[111] directionwiththose extractedfrom the [110] directi on.\nsibilities and we have found that the choice which gives the\nmost physicallyreasonablebehavioris the one whichassign s\nthepairP1-P2totype-IISe sites.\nHyperfine parameters and local susceptibilities .— We are\nnow ready to extract the local moments from the NMR data.\nWefirstwritethemeasuredsusceptibilityperCuandthemea-\nsuredrelativeshifts K≡∆ν/νLas\nχtot(T) =χspin(T)+χdia+χvv (14)\nK(T) =Kspin(T)+Kc. (15)\nHereχspin= (χ1+ 3χ2)/4is the total spin susceptibility\nper Cu, and χdia,χvvare the T-independent diamagnetic and\nvan Vleck contributions respectively. Similarly, Kspin(T)in-\ncludes the dipolar and the transferredhyperfine contributi ons\nwhich were given above in Table II, while Kcstands for the\nT-independentchemicalshift.\nFigure 9(a) shows the so-called Clogston-Jaccarino plot29\nof the four measured shifts Kin both [111] and [110] direc-\ntionsversusthemeasuredsusceptibilitypermolCuinthehi gh\ntemperature regime. The linear behavior shows that both χ1\nandχ2are proportional to χspinin this regime. We can ex-\nploit this linear behavior in order to extract the values of Kc\nfor each line if we make an estimate of χdia+χvv. For the\nlatter we take the upper bound of χdia+χvv≃1×10−4\ncm3/mol-Cu which is obtained from the comparison of the\nmeasured magnetization with the mean field theory predic-\ntion (cf. below). We extract the following estimates respec -\ntively for the lines P1, P2, P3 and P4: Kc≃5.60×10−4,\n8.06×10−4,14.2×10−4,15.7×10−4along [111], while\nKc≃4.68×10−4,6.73×10−4,13.8×10−4,15.1×10−4\nalong[110].\nAfter subtractingthe abovevaluesof χdia+χvvandKchem\nfrom the bare data, we adopt the following procedure. We\nfirst extract an estimate for θ′\ntrby using the data for /tildewideθat the\nlowest available temperature ( T= 20K) and the relation/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s45/s50/s45/s49/s48/s49/s50/s51/s45/s50/s45/s49/s48/s49/s50/s51/s52\n/s32\n/s32/s49/s48/s51\n/s47\n/s76 \n/s50/s57/s48/s32/s75/s49/s48/s51\n/s47\n/s76 \n/s32 /s32/s40/s49/s48/s45/s51\n/s99/s109/s51\n/s47/s109/s111/s108/s45/s67/s117/s41/s49/s55/s48/s32/s75\n/s32/s40/s97/s41/s32/s72\n/s48/s47/s47/s32/s91/s49/s49/s49/s93\n/s40/s98/s41/s32/s72\n/s48/s47/s47/s32/s91/s49/s49/s48/s93\n/s80/s49/s80/s50/s80/s51/s80/s52/s80/s49/s80/s50/s80/s51/s80/s52\nFIG. 9: (Color online) The fractional shift ∆νPi/νLfor each one of\nthe four NMR peaks as measured along (a) [111] and (b) [110], v s.\nthe susceptibility χmeasured with the magnetometer with Tbeing\nan implicit parameter. The dotted lines indicates the point s where\nT= 170K,and290K.\nM1(20K) =r1M(20K), wherer1is a constant which mea-\nsures the effect of quantum fluctuations at low temperatures .\nUsingEq. (13),therelationM =(M1+3M2)/4,andthePPMS\ndataforM,wemayextractthewholetemperaturedependence\nofM1andM2. Inturn,Eqs. (10)-(12)canprovidetheremain-\ningtransferredhyperfineparametersinastraightforwardw ay.\nFor comparison, we have used two values of r1: The first,\nr1=−2, corresponds to the case without quantum fluctua-\ntion effects, while the second value, r1=−1.2, corresponds\nto the reduction in the spin length found by Neutron diffrac-\ntiondata.22Asforthedemagnetizationfactor,it turnsoutthat\nforNd/lessorsimilar0.6theextractedM 1(T)isnotpositiveatroomtem-\nperatures but saturates to a negative value. However such a\nbehaviorwouldbequiteunphysical: Althoughthereisalarg e\nantiferromagneticexchangefieldoneachCu 1site sinceithas\n6neighboringCu 2sites, onestill expectsthatM 1shouldulti-\nmatelychangesignatsometemperature T∗whichisabove Tc\nbut certainlywell belowroomtemperature. On this issue, th e\nmean-fieldtheorydescribedbelowpredictsthat T∗≃152K.\nSo we adjust Nd≃0.7, which in conjunction with the value8\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s77/s101/s97/s110/s32/s70/s105/s101/s108/s100/s32\n/s32/s32 /s32/s77\n/s116/s111/s116/s97/s108\n/s32/s32 /s32/s77\n/s49/s32\n/s32/s32 /s32/s32/s77\n/s50/s32\n/s32/s32/s77/s32/s40\n/s66/s47/s67/s117/s41/s32\n/s84/s32/s40/s75/s41/s77\n/s116/s111/s116/s97/s108/s58/s32/s80/s80/s77/s83\n/s77\n/s49/s32/s58/s32/s78/s77/s82/s32/s40/s78\n/s100/s61/s48/s46/s49/s44/s32/s114\n/s49/s32/s61/s32/s45/s50/s41\n/s77\n/s50/s32/s58/s32/s78/s77/s82/s32/s40/s78\n/s100/s61/s48/s46/s49/s44/s32/s114\n/s49/s32/s61/s32/s45/s50/s41\n/s77\n/s49/s32/s58/s32/s78/s77/s82/s32/s40/s78\n/s100/s61/s48/s46/s55/s44/s32/s114\n/s49/s32/s61/s32/s45/s50/s41\n/s77\n/s50/s32/s58/s32/s78/s77/s82/s32/s40/s78\n/s100/s61/s48/s46/s55/s44/s32/s114\n/s49/s32/s61/s32/s45/s50/s41\n/s77\n/s49/s32/s58/s32/s78/s77/s82/s32/s40/s78\n/s100/s61/s48/s46/s55/s44/s32/s114\n/s49/s32/s61/s32/s45/s49/s46/s50/s41\n/s77\n/s50/s32/s58/s32/s78/s77/s82/s32/s40/s78\n/s100/s61/s48/s46/s55/s44/s32/s114\n/s49/s32/s61/s32/s45/s49/s46/s50/s41\n/s32\nFIG.10: (Coloronline)Comparisonbetweenthetotalmagnet ization\nmeasured by PPMS (corrected for the van Vleck and diamagneti c\ncontributions)andthelocalmomentsextractedfromNMR(wi thtwo\ndifferentvalues oftheratio r1,cf. text)withthecorrespondingmean\nfield theory predictions. We also demonstrate how the value o f the\ndemagnetizingfactor NdaffectsthesignofM 1athightemperatures.\nofr1can provide a physically more reasonable behavior at\nhightemperatures.\nTheresultingestimatesofthehyperfineparametersarepro-\nvidedinTableIII,whiletheextractedT-dependenceofthel o-\ncal moments is shown in Fig. 10. The major result contained\nin Fig. 10 is that the NMR data are fully consistent with the\n3up-1downferrimagneticstate,namelythattheCu 2moments\narealignedparallelandtheCu 1momentsarealignedantipar-\nallel to the applied field. Hence NMR on single crystals pro-\nvides a more direct local probe of this state, mapping out the\nwholeT-dependenceofthelocalmoments.\nComparison to microscopic model :— It is worthwhile to\ncontrast the above NMR findings with a microscopic spin\nmodelfor Cu 2OSeO3. Followingthe Kanamori-Goodenough\nrules (see also discussion in Ref. 22) we introduce two ex-\nchange parameters: one ferromagnetic ( Jfm) between Cu 2-\nCu2ions and one antiferromagnetic ( Jafm) between Cu 1-Cu2\nions. Looking at the structure one finds that each Cu 1neigh-\nbors six Cu 2ions while each Cu 2neighbors four Cu 2ions\nand two Cu 1ions. The self-consistent equationsof the corre-\nspondingmeanfieldtheoryaregivenby\nm1=tanh/parenleftbiggb−3Jafmm2\n2T/parenrightbigg\n(16)\nm2=tanh/parenleftbiggb+2|Jfm|m1−Jafmm2\n2T/parenrightbigg\n(17)\nwherem1,2≡M1,2/(gµB/2),b≡gµBH0/kB, and the ex-\nchangeconstantsareinunitsofKelvin. Themeanfieldtransi -\ntiontemperatureis TMF\nc=1\n2/parenleftig\n|Jfm|+/radicalbig\nJ2\nfm+3J2\nafm/parenrightig\n, while\nthe predictions for the local and the total spin susceptibil ityperCuat T > T caregivenbytheexpressions\nχ1=(gµB)2\n8kB2T−2|Jfm|−3Jafm\n(T−Tc)(T+Tc−|Jfm|)(18)\nχ2=(gµB)2\n8kB2T−Jafm\n(T−Tc)(T+Tc−|Jfm|)(19)\nχspin=(gµB)2\n8kB2T−1/2|Jfm|−3/2Jafm\n(T−Tc)(T+Tc−|Jfm|).(20)\nUsing the exact numerical solution of Eqs. (16) and (17) we\nhaveobtainedaquiteaccuratefit(cf. Fig. 10,solidblackli ne)\nof the measured magnetization data in the range 110-300 K,\nusingg≃2.11,26Jafm≃68KandJfm≃ −50K.Thisfitputs\nan upper bound on the diamagnetic and van Vleck contribu-\ntionstothesusceptibility χdia+χvv≃1×10−4cm3/mol-Cu\nwhich is of the right order of magnitude. Figure 10 shows\nalso the solution for the temperature dependence of the loca l\nmagnetizations M 1and M 2(solid red and blue lines) which\nare to be contrasted with the behavior extracted from NMR.\nThe agreement is quite satisfactory. Of particular interes t is\nthe behavior of M 1at high temperatures. As we discussed\nabove, the Cu 1moments remain antiparallel to the field even\naboveTcdue to the large negative exchange field that is ex-\nerted from the 6 neighboring Cu 2moments. The mean field\ntheory prediction for the temperature T∗at which M 1even-\ntually changes sign is given by T∗=3\n2Jafm+|Jfm| ≃152\nK.\nSothemeanfieldtheoryprovidesasemi-quantitativeagree-\nment with the measured magnetization data and captures the\nessential local physics of the problem, being in agreement\nwith the picture obtained from NMR. On the other hand, the\nmean-fieldtheorydoesnotprovideagooddescriptionattem-\nperatures close to Tcsince it neglects long-range correlation\neffects. In addition it can overestimate the value of Tc, and\ndoes not capture quantum fluctuation effects which give rise\nto an overall reduction in the length of the local moments at\nlowenoughtemperatures.\nIV. NUCLEARSPIN-LATTICEAND SPIN-SPIN\nRELAXATIONRATES\nThe77Sespin-latticeandspin-spinrelaxationtimes, T1and\nT2respectively,weremeasuredforthefourdifferentpeaksof\nthe spectrum when the magnetic field is parallel to the [111]\naxis. We first note that the lines P1 and P2 have very similar\nT1andT2valuesoverthewholerangeofT’sinvestigatedhere\nand the same holds true for the lines P3 and P4. This is yet\nanother confirmation that the lines P1-P2 belong to one type\nofSesites andthe linesP3-P4belongtothesecondtype.\nTherecoveryofthelongitudinalmagnetizationfollows,fo r\neach peak, a single exponentialbehavior in time, as expecte d\nfor aI= 1/2nuclear spin. The values of spin-lattice relax-\nation rate 1/T1derived from the fit of the recovery data are\nshownin Fig. 11(a). At T >170K,1/T1variesweakly with\ntemperature for all lines. We may obtain a theoretical fit of\n1/T1byconsideringtheRamanscatteringofmagnonsoffthe\n77Se nuclei which is expected to be the dominant process (at9\n/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48\n/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54/s49/s48/s45/s54/s49/s48/s45/s52/s49/s48/s45/s50/s49/s48/s48\n/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48 /s50/s52/s48 /s50/s56/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s32\n/s40/s97/s41/s84/s32/s45/s49 /s49\n/s32/s40/s109/s115/s32/s45/s49\n/s41\n/s32/s32\n/s49/s47/s84\n/s49/s32/s40/s109/s115/s45/s49\n/s41\n/s49/s47/s84/s32/s40/s75/s45/s49\n/s41/s32\n/s40/s98/s41/s32\n/s84/s32/s40/s75/s41/s83/s105/s110/s103/s108/s101/s32/s69/s120/s112/s111/s110/s101/s110/s116/s105/s97/s108/s32\n/s32/s32/s32/s32/s32/s32/s68/s101/s99/s97/s121/s32/s80/s49\n/s32/s80/s50\n/s32/s80/s51\n/s32/s80/s52/s84/s32/s45/s49 /s50\n/s32/s40/s109/s115/s32/s45/s49\n/s41\n/s32/s32/s32/s32/s32/s32/s32 /s78/s111/s110\n/s69/s120/s112/s111/s110/s101/s110/s116/s105/s97/s108/s32\n/s32/s32/s32/s32/s68/s101/s99/s97/s121\nFIG. 11: (Color online) Temperature dependence of the77Se NMR\n(a) spin-lattice T−1\n1and (b) spin-spin T−1\n2relaxation rates with the\nmagnetic field of 14.09 T applied parallel to [111]. The inset shows\n1/T1vs. 1/T for the line P1 and the corresponding fit (solid line)\nwiththe contributionfrom Ramanscatteringof magnons.\nlowenoughtemperatures)thatconservesboththetotalener gy\nand the total angularmomentumof the nuclear+electronspin\nsystem. Tothisend,weshallmakethereasonableassumption\nthat the relevant lowest spin-wave excitation branch aroun d\ntheminimum ∆(whichissetbytheexternalfield)hasatypi-\ncalparabolicdispersionoftheform ǫk= ∆+λ(ak)2,where\naisthelatticeconstantand λisanenergyscaleoftheorderof\nthe actual exchange couplings in the system which gives the\ncurvature around the minimum of the lowest magnon band.\nFollowingsimilarargumentswith Refs. [30,31]oneobtains\n1\nT1=z(A/z)2sin2θ\n25π3/planckover2pi1λ/parenleftbiggkBT\nλ/parenrightbigg2/integraldisplay∞\n∆/kBTdx\nex−1(21)\nwherezisthenumberofCu2+ionsclosesttothenuclearspin,\nA/zstandsforthehyperfinecouplingwithasingleelectronic\nspin, and θis the angle between the quantization axes of the\nelectronic and the nuclear spin. Our optimal fit based on a\nnumericalevaluationoftheintegralofEq.(21)with ∆/kB=\ngµBH(H= 14T) is shown in the inset of Fig. 11(a) (solidline)andgives A2sin2θ/λ3≃1.6×107erg−1. TakingA/q∼\n2×1023cm−3for the line P1, gives λ/kB∼50K which is\noftherightorderofmagnitude.\nFigure 11(b) shows the T-dependence of the spin-spin re-\nlaxation rate 1/T2. These results were obtainedby fitting the\ndecay of the spin echo signal M(2τ), after aπ/2−τ−π\npulse sequence, with the proper functional form. We have\nfound that the decay of the transverse nuclear magnetizatio n\nfollows a single exponential decay in the range 80-290 K\nwhile it deviates from the single exponential law below 60\nK. The irreversible decay M(2τ)of the77Se echo signal has\nthe following contributions. First, the77Se nuclear dipole-\ndipole interaction which can be estimated from the calcula-\ntionofthevanVlecksecondmoment32MSe-Se\n2takingintoac-\ncount the natural abundance of77Se.33We have found that /radicalbig\nMSe-Se\n2≈85s−1with a slight variation among the dif-\nferent Se sites, which is much smaller than the experimen-\ntal values reported in Fig. 11. The second contribution to th e\nspin echo decay is the Redfield term,34which is of the form\nexp(−2τ/T1). From the 1/T1values reported in Fig. 11(a)\nit is obvious that the contribution of the Redfield term in the\nspinechodecayisalsoverysmall here.\nThethirdcontributiontothespinechodecayisduetofluc-\ntuating dipolar fields between unlike nuclear spins with the\nmost dominant being the one among Se and Cu nuclei. De-\npending on the time scale (or correlation time τc) of these\nfluctuating dipolar fields we have the following two limit-\ning regimes: (i) The fast motion regime, characterized by\nγN/angbracketleftδHz/angbracketrightτc≪1, where the spin echo decay is a single ex-\nponential, exp (−2τ/T2), with1/T2=γ2\nN/angbracketleftδH2\nz/angbracketrightτc,34and\nγ2\nN/angbracketleftδH2\nz/angbracketrightisthesecondmomentoftheselenium-copperdipo-\nlar interaction for which a straightforward lattice sum giv es\nMSe-Cu\n2≃7.7×106s−2. The fast motion approximation is\nin our case applicable for T/greaterorsimilar80K. (ii) The quasi-static\nregime, characterized by γN/angbracketleftδHz/angbracketrightτc≫1, where the slow\nfluctuations of the selenium-copper dipolar interaction gi ve\nrise to a non-exponentialspin echodecay. In the nearlystat ic\nregime the decay goes as exp [−(2τ/T2)3], where1/T2=\n(γ2\nN/angbracketleftδH2\nz/angbracketright/12τc)1/3.35Identifying τcwith thespin-latticere-\nlaxation time of Cu nuclei, which at 30 K is measured to be\nabout 1ms,36we find1/T2≃862s−1in agreementwith our\nlow-T data. As we noted above the decay of the transverse\nmagnetizationis not exponentialat low T’s. A goodfit of the\ndatacanbeobtainedbybothasquareexponentialdecay(half\nGaussian) and by a third power exponential. The discrimina-\ntion between them is preventedby the weakness of the NMR\nsignalat longtimes.\nV. SUMMARY\nWe have presented extensive77Se NMR measurements in\nsinglecrystalsofthemagnetoelectricferrimagnetCu 2OSeO3.\nTheanalysisofthedatahasprovidedanumberofcentralfind-\nings both for the crystalline as well as the magnetic phase of\nthis compound. First, the T-dependence of the two types of\nCu2+moments, extracted from the NMR data, is fully con-\nsistent with a phase transition from the high-T paramagneti c10\nphase to a low-T ferrimagnet whereby 3/4 of the Cu2+mo-\nments are aligned parallel and 1/4 antiparallel to the appli ed\nfield. Belowthetransitiontemperaturewedonotobserveany\nclearchangein the broadeningofthe NMRlinesoranysplit-\nting of the NMR lines, which shows that there is no mea-\nsurable symmetry reduction in the crystalline structure fr om\nits high-T space group P213. These results are in agreement\nwithpreviousdatafrommagnetization,high-resolutionx- ray,\nandNeutrondiffractionmeasurementsonpowdersamplesre-\nported by Bos et al,22but also from infrared,23and Raman24\nstudiesin singlecrystalsofCu 2OSeO3.\nWehavealsodevelopedamicroscopicspinmodelwithtwo\nnearest-neighbor exchange interactions: One antiferroma g-\nnetic (Jafm≃68K) between Cu 1and Cu 2ions which forces\nthem to be antiparallel to each other, and a second ferromag-\nnetic interaction between nearest-neighbor Cu 2ions which\nalignsthen in the same direction,givingrise to the abovefe r-\nrimagnetic state. We have shown that a mean field solution\nof this model providesa very gooddescriptionof the physics\nof the problem and is in excellent agreement with measured\nmagnetizationdatainawidetemperaturerangeand,moreim-portantly, it is also consistent with the local picture extr acted\nfrom NMR. A first-principles study of this system may pro-\nvidea moreaccurateandrefinedmicroscopicspinmodel.37\nMore generally, our NMR study provides a strong local\nconfirmation that the ferrimagnetic ordering in Cu 2OSeO3\ndoesnot proceedvia a spontaneouslattice distortionand th us\nthis material providesa unique exampleof a metrically cubi c\ncrystalthatallowsforpiezoelectricaswellaslinearmagn eto-\nelectricandpiezomagneticcoupling.\nVI. ACKNOWLEDGMENTS\nWe would like to thank K. Schenk, O. Janson, A. Tsirlin,\nL. Hozoi, and M. Abid for useful discussions. We also ac-\nknowledge experimental assistance from K. Schenk, M. Za-\nyed, P. Thomas, and S. Granville. One of the authors (H. B.)\nacknowledges financial support from the Swiss NSF and by\ntheNCCR MaNEP.\n∗Electronic address: mariaeleni.belesi@epfl.ch\n1W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 442, 759\n(2006).\n2N.A. Spaldin,M. 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Soc.Jpn. 55, 1233 (1986).\n36The copper NMR signal (not shown here) could be observed in\nCu2OSeO3onlybelow 30K.\n37O.Janson, and A.Tsirlin(private communication)."    },    {        "title": "1802.03874v1.Canted_ferrimagnetism_and_giant_coercivity_in_the_non_stoichiometric_double_perovskite_La2Ni1_19Os0_81O6.pdf",        "content": " 1 Canted  ferrimagnetism and giant coercivity in the non -stoichiometric \ndouble perovskite La2Ni1.19Os0.81O6  \n    \nHai L. 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 . 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Mater. 22, 5712  (2010) . \n "    },    {        "title": "1903.04293v1.High_Field_Anomalies_of_Equilibrium_and_Ultrafast_Magnetism_in_Rare_Earth_Transition_Metal_Ferrimagnets.pdf",        "content": "1 \n High Field Anomalies of Equilibrium and Ultrafast Magnetism  in Rare -Earth -Transition \nMetal  Ferrimagnet s \nA. Pogrebna , 1 K. Prabhakara ,1 M. Davydova,2 J. Becker ,1,3 A. Tsukamoto,4 Th. Rasing ,1A. Kirilyuk1 \nA. K. Zvezdin,2 P. C. M. Christianen,3 and A.V. Kimel1 \n1 Radboud University, Institute of Molecules and Molecules,  Nijmegen, The Netherlands  \n2Prokhorov General Physics Institute of the Russian Acad emy of Sciences, Moscow, Russia  \n3High Field Magnet Laboratory  (HFML –EMFL) , Radboud University, Nijmegen, The Netherlands  \n4College of Science and Technology, Nihon University, Chiba, Japan  \nABSTRACT : Magneto -optical spectroscopy in fields up to 30 Tesla reveals anomalies in the equilibrium and ultrafast mag-\nnetic properties  of the ferrimagnetic rare -earth -transition metal alloy TbFeCo. In particular, in the vicinity of the magneti-\nzation compensation temperature , each of the magnetizations of the antiferromagnetically coupled Tb and Fe Co sublattices \nshow triple hysteresis loops. Cont rary to  state-of-the-art theory , which  explains such loops  by sample inhomogeneit ies, \nhere we  show that the y are an intrinsic property  of the rare -earth ferrimagnet s. Assuming  that the rare-earth ions are  par-\namagnet ic and have a non -zero orbital momentum in the ground state  and, therefore,  a large magnetic anisotropy , we are \nable to reproduce the experimentally observed behavior in equilibrium. The same theory is also able to describe  the exper-\nimentally observed critical slowdown of the spin dynamics in the vicinity of the magnetization compensation tempera ture, \nemphasizing the role played by the orbital momentum in static  and ultrafast magnetism of ferrimagnets.  \nIntroduction  \nThe need for ever faster and energy -efficient data storage and \ninformation processing is a strong motivation to search for un-\nconventional ways to control magnetism by means other than \nmagnetic field s [1–4]. Several successful realizations of mag-\nnetiz ation control  with  the help of an electric current  [5,6], \nelectric field  [2] and light  [3] have been demonstrated . This \nhas become heavily debated topic in  modern magnetism and \nrevealed  a lack of understanding of the mechanisms  that are  \nresponsible for these phenomena  [3,7]. It is clear, however, \nthat in all these cases the spin -orbit and the exchange spin -\nspin interactions play a decisive role. For instance, spin -orbit -\ntorques [8–12], multiferro icity [13,14] , opto - and photomag-\nnetic   [3] phenomena cannot be understood wi thout taking \ninto account the spin-orbit interaction as well as orbital mo-\nmenta of the ground and excited states. The exchange inter-\naction , in turn,  can be efficiently harnessed for spin manipu-\nlation  in multi -sublattice spin systems or multilayers  [15–20]. \nFerrimagnetic rare -earth  intermetalli cs, and rare -earth -transi-\ntion metal (RE- TM) alloys  in particular,  are among the most \nstudied  systems  in fundamental and applied magnetism. For \nexample,  unique functionalities of GdFeCo, GdFe, and GdCo \nalloys have been demonstrated in spintronics  [21,22]  and ul-\ntrafast magnetism  [23–25]. The interplay between  the ex-\nchange and the spin -orbit interaction in rare-earth ferrimag-\nnets facilitate electric field, current and optical control of \nspins . In particular, due to  the antiferromagne tic coupling be-\ntween  the non-equivalent magnetic sublattices in GdFeCo , it \nis possible to reverse its magnetization solely with a single \nfemtosecond laser pulse  [26]. Anomalous hysteresis loop s and \ncritical slow ing down of laser -induced spin dynamics in high \nmagnetic field s were reported for GdFeCo   [27], but a theory \nof such  behavior has not been developed until now.  Interest-\ningly , as the Gd ion in the alloy is in an S-ground state , its s ub-\nstitution with other rare -earth ions having non -zero orbital \nmomentum in the ground  state  must enhance the spin -orbit \ninteraction and thus open up fundamentally new opportuni-\nties in the field  of spintronics and ultrafast magnetism. TbFeCo is an example of such a material, that, due to large \ncoercive fields  above 10 T , is well suitable for high density \nmagnetic recording   [28,2 9]. Although several attempts of \nmodeling the laser -induced spin dynamics in TbFeCo have \nbeen performed  [30–32], not only spin dynamics, but even the \nstatic spin structure of unperturbed TbFeCo are poorly under-\nstood.  Moreover, experimental studies of high -coercive -field \nmaterials are seriously hampered by the need for even higher \nmagnetic fields and thus require unique experimental instal-\nlations.  \nHere we report the experimental observation of anomalous \nhysteresis loops of the magnetization s of the antiferromag-\nnetically coupled FeCo and Tb sublattices of ferrimagnetic \nTbFeCo in high magnetic fields.  The loops appear to be  very \nsensitive to temperature near the magnetization compensa-\ntion point, where the magnetizations  of the  two sublattices  \ncancel  each other . We show that  the observed  behavior  can be \nexplained in the framework of an f-d ferrimagnet by taking \ninto account  the orbital  momentum and, as a result , the large \nmagnetic anisotropy of the rare -earth ions. In order to bring \nthe developed theory into an ultimate test , we experimentally \ninvestigate the magnetization dynamics in TbFeCo trigg ered \nby a femtosecond laser pulse in high magnetic fields and com-\npare the outcome  of the experiment with the mode ling. It is \nsurprising that the theory developed  to explain  equilibrium \nmagnetic properties i s also able to predict the experimentally \nobserved  critical slow ing down of the spin dynamics that was \nobserved in the experiment .  \nThe paper is organized as follows . Section I describes static \nmagneto -optical measurements in which anomalous hystere-\nsis loops were observed. Next, in section II we introduce the \ntheoretical model aimed to describe  the ground state of a fer-\nrimagnet with large rare -earth anisotropy. We plot magnetic \nfield -temperature ( H-T) phase diagram and expla in the equi-\nlibrium magnetism of TbFeCo  using the developed theory . In \nsection III we present the experimental results on laser -in-\nduced ultrafast dynamics of TbFeCo in high magnetic field , 2 \n and theoretically explain the observed anomalies using the de-\nrived p hase diagram. We conclude the manuscript with a \nsummary, which emphasizes the simplicity,  and at the same \ntime predictive power of the proposed theor y. Finally, we \nhighlight experimentally observed features whose explana-\ntion, being beyond the capabilities of the presented model, is \nthe next challenge in the physics of rare -earth alloys.  \n \nI. Magneto -optical spectroscopy in high \nmagnetic fields  \nThe s tudied material was an amorphous  rare-earth -transi-\ntion metal alloy  with stoichiometric composition  \nTb22Fe68.2Co9.8. Without applied magnetic field,  the antifer-\nromagnetic exchange coupling between the rare -earth and \nthe transition metal magnetic sublattices favor a collinear  \nantiparallel alignment. The magnetizations of the transi-\ntion met al and the rare -earth  sublat tices have  different \ntemperature dependences so that  at the magnetization \ncompensation temperature  TM the net magnetization is \nzero, if no magnetic field is applied.  In the studied sample \nTM=322  K. Below this temperature, T < T M, the net magnet-\nization  is dominated by the RE sublattice. Above the com-\npensation point , the situation is  the opposite , and the TM \nsublattice dominates  the net magnetization . The strong in-\nter-sublattice 3d-4f exchange interaction between  the Tb \nand FeCo magnetic moments defines t he Curie tempera-\nture, TC, which is around 700 K  [33]. The studied sample is \na thin film  structure  with composition  glass /SiN(5 nm)/RE -\nTM(20 nm)/SiN(60 nm). The TbFeCo magnetic layer has an \neasy-axis type of magnetic anis otropy, oriented perpendic-\nularly to the sample plane.   \n \nFig. 1  (color online)  Schematic of the experiment. The sam-\nple was insert ed into 37 T dissipative magnet . Dashed lines \n– pump and probe beams.  \nThe e xperiments were performed at the High Field Magnet \nLaboratory (HFML)  in Nijmegen . Magnetic field s up to 30 \nT were applied along the normal to the sample  (see Fig. 1) . \nTo benefit from elemental specificity of the magneto -opti-\ncal phenomena in TbFeCo in the visible spectral \nrange  [34], we performed the  measurements of the polar \nmagneto -optical Kerr effect (MOKE)  with light of two pho-\nton energies . In particular, for photon energy =1.55 eV \nthe effect is expected to probe the normal component of the magnetization of the FeCo sublattice, while light with \n=2.41  eV is mainly sensitive to the normal magnetization \ncomponent of Tb.  \nFigure 2 shows the MOKE as a function of the applied mag-\nnetic field at different temperatures near the magnetiza-\ntion compensation temperature.  Above TM = 322 K t he ob-\nserved behavior is anomalous  as the loops  have a triple \nshape and are very sensitive to the sample temperature . At \ntempe ratures  below TM = 322 K, the anomalies are far less \npronounced, but sti ll visible ( see arrows in  Fig. 2 ) \n \nFig. 2  (color online)  Static MOKE data of t he TbFeCo sam-\nple measured at 2.41 eV (a) and 1.55 eV  (b) photon energies  \nat different temperatures from 267.0 K to 342. 5 K. A para-\nmagnetic background was subtracted from the measure-\nments. The magnetization compensation temperature TM is \naround 3 22 K. Black arrows correspond to the  hysteresis \nedges around  the first-order field -induced phase transitions  \ndiscussed in section II , while a second -order phase transition \nis shown with a blue arrow.  \nSimilar hysteresis loops were observed in rare -earth ferri-\nmagnetic alloys and intermetallics earlier and explained by \ninhomogeneities with the strongest ever reported ex-\nchange  bias fields  [35,36] . In particular, an application of \nthat theory to our case would imply that the exchange bias \nbetween the inhomogeneities reaches 10 T. However, it is \nalso known that hysteresis loop dependencies of the mag-\nnetization on temperature or field is a signature of a first \norder phase transition. Gradual changes of the magnetiza-\ntion upon a change of temperature or field are generally \nexplained as second -order phase transition s. Despite in-\ntense interest in rare-earth - transition metal alloys, their \nequilibrium prop erties in high magnetic field s and H-T \nphase diagram s near the compensation temperature, in \nparticular, are still unexplored. H-T phase diagram of  uni-\naxial ferrimagnets has been studied theoretically ear-\nlier [37], under the assumption that only the TM-sublattice \nis anisotropic. In systems with different symmetry  [38,39]  \nit was found tha t the behavior of the phase diagram is \ngreatly affected by magnetic anisotropies of both sublat-\ntices.  \n3 \n In order to reveal the origin of the observed anomalous \nhysteresis loops, we develop ed a theory of magnetism of \nTbFeCo in thermodynamic equilibrium  and calculate d the  \nH-T phase diagrams of this compound .  \nMagnetic fields at which magnetization changes were sub-\ntracted from the static MOKE data (see Fig. 2) in a wide \nrange of temperatures  are shown in phase diagram in Fig. \n3(a). The blue line data poin ts correspond  to a second or-\nder phase transition, where the magnetization changes \ngradually. Black diamonds and associated black d ashed \nlines correspond to the edges of the experimental hystere-\nsis, where the magnetization changes abruptly.  \nII. H-T phase diagram  and anomalous hyste-\nresis loops  \nSimilarly to Refs  [37,40] , the calculations are  based on  \nanalysis  of the free energy for a two -sublattice  f-d (RE-TM) \nferrimagnet  assuming that rare-earth ion is paramagnetic . \nWe take advantage of the fact that t he single -domain \nmodel work s especially well in the vicinity of the compen-\nsation point. This is because the domain size diverges at \nthe compensation point   [41] as the magnetostatic energy \ndrops to zero  when  the magnetization \nfd M M M   \nvanishes at T M. The free energy has the form:  \n0- ( , ) ( , )effH\nd f a d f W h T d W  M H  M h M M\n, (1) \nwhere \neff dH H M  is the effective magnetic field act-\ning upon the rare -earth ion, \n22\n22f d\na d f\ndfW K KMM Mn Mn\n is the uniaxial anisotropy \nenergy, with\n,dfKK  being d- and f-sublattice anisotropy \nconstants , respectively , and \nn  is an easy axis unit vector. \nThe magnetization function for rare -earth ions was taken \nin the following form:  \n35\n1 2 3 ( , ) ( ) ( ) ( )f eff eff eff effM H T T H T H T H     \n, (2) \nand it is directed  along the effective field.  Whereas for Gd \nthe f-sublattice magnetization is accurately described by \nthe Brillouin  function, it doesn’t hold  well for other rare -\nearth ions with non -zero orbital momentum in the ground \nstate  [42], which is further compl icated  by the amorphous \nnature of the alloy.  Therefore, w e fit the sus ceptibilities \nnumerically, s tarting from the functions obtained by an \nexpansion of the Langevin  function \n()eff\nTb eff\nBHLkT  in series \nand numerically adapt ing them  and the effective  magnetic  \nparameters to reproduce the experimentally observed \nfeatures.  We present the calculations performed  for the \nfollowing parameters: \nex dHM\n  80 T, TM = 322 K, TC = \n700 K  [33]. Magnetic moments of the Tb and FeCo \nsublattices used in the calculations are  \n10eff\nTb B   and \n1.8eff\nFeCo B\n, respectively  [31]. As it was shown earlier  [38], the rare -earth anisotropy is one of the defining \nfactors for the character of the phase diagram near the \ncompensation point . In o ur model we assume  that \n63 10fK\n erg/cc and \ndK  = 0 (we assume the d-sublattice \nanisotropy to be much smaller than that of the rare-earth \nsublattice, as Fe ions are in the S-ground state).  \n \nFig. 3 (color online)  (a) Experimental and (b)  theoretical \nmagnetic field - temperature phase diagram  for TbFeCo . The \nlines in the experimental phase diagram are guides for the \neyes.  The black points (dashed black curves)  correspond to \nthe points  stability loss, the red curve  (TMP) is the first -order  \ntransition line , and the blue points (blue curve BR) corre-\nspond to the second -order phase transition.  (c) Schematics \nof the three phases AF 1, AF 2 and NC that are present in the \ndiagram . The states of the magnetization of both sublattices \nare shown with arrows, where the red arrow correspond s to \nthe FeCo  magneti zation , and the green arrow  correspond s \nto the Tb magnetization . \nUsing expression (1) for the thermodynamic potential we \nnumerically calculate the magnetic (H-T) phase diagram. \nThe ground states of the system are found by minimization \nof the thermodynamic potential w ith regard to the order \nparameter \nd , which denotes the polar angle of the FeCo \nmagnetization in the coordinate system with  the z-axis \naligned along  the external magnetic field. At the local \nminima one finds \n0\nd  and \n2\n20\nd . The lines  of \nstability loss, where \n2\n20\nd , are found for each phase.  At \n4 \n the lines of the first -order phase transition two phases, \ncorresponding to solutions \n(1)\nd and \n(2)\nd , correspond to the \nlocal minima of the thermodynamic potential  and the \ncondition  \n(1) (2)\ndd   is fulfilled.   \nThe phase diagram in Fig. 3(a,b)  shows  three phases: two \nantiferromagnetic collinear phases AF 1 and AF 2 (rare -earth \nmagnetization is along the magnetic field below the com-\npensation temperature TM in phase AF 1, and opposite to it \nabove TM in phase AF 2) and one canted phase NC. The blue \nline BR represents the second -order phase transition be-\ntween phases AF 1 and NC and defines the spin -flop field de-\nnoted as HBR below the compensation point. Above the \ncompensation temperature the s pin-flop occurs discontin-\nuously,  via a first-order phase transition ( see line RP in Fig. \n3 (a) ). The red line  (TMP) corresponds to a first-order phase \ntransition  between the collinear phases AF 1 and AF 2 along \nsegment  TMR as well as between the phases AF 2 and NC \nalong the rest of the line , i.e. along segment RP. Each of the \ndashed curves corresponds to the stability loss of one of the \nphases. Following the markup in Fig. 3 (a), lines AA’, QQ’ \nand RB’ (we denote the corresponding fields HAA’, HQQ’ and \nHRB’, respectively ) are the lines of stability loss of phases \nAF 2, NC and AF 1, respectively. One might notice that the \nfirst-order phase transition TMRP goes to the outside of the \narea shown in the phase diagram. The point P is a tricritical \npoint at which the order of the phase transition between \ncollinear and noncollinear phases changes from first to sec-\nond. A number of  unusual phenomena are expected to oc-\ncur in ferrimagnets near this point  [43,44] , being an inter-\nesting subject for future studies.  \nThe features of the magnetic phase diagram can be ob-\nserved experimentally by mea suring the dependence of the \nmagnetization on external magnetic field . In particular, de-\nducing magnetic fields corresponding to jumps in the ex-\nperimentally measured hysteresis loops, we define the \nfields and the temperatures at which collinear and non-col-\nlinear phases loose stability. In this way we plotted  the ex-\nperimental phase diagram shown in Fig. 3(a). The structure \nof the theoretical phase diagram (see Fig. 3(b)) explains \nwell the behavior of the experimentally observed phase \ntransitions.  To demonstrate the agreement of the theoret-\nical predictions with the experimental results , we also cal-\nculate the dependence  of all possible equilibrium values of \nthe order parameter as a function of  external magnetic \nfield \n()()i\ndH  at fixed temperature . The obtained \nmagnetization plots explain  the experimental results \nshown in Fig. 2.  Above the compensation temperature, as \nillustrated in Fig. 4  (a), the theory reproduces the triple \nhysteresis loops observed  in the experiment. The large cen-\ntral loop at lower fields encompasses the first -order phase \ntransition between the two collinear phases. Two loops \nthat appear at higher positive and negative fields are due \nto the two first -order phase transitions between collinear \nand non -collinear phases (from AF 2 to NC and vice versa , \nrespectively ). From Fig. 3  (a) one can see that the first -or-\nder transition, from which the additional loop at positive \nfield originate s, occurs at the line RP. The hysteresis \naround these first -order phase tra nsitions is defined by the position of the stability loss lines in the ( H-T) phase dia-\ngram and corresponds to the dashed lines in Fig. 3 (a).  The \nloops disappear if the phase transition to the non -collinear \nphase NC becomes of second -order. At 290 K, i.e. below the \ncompensation point, we see that the second -order phase \ntransition occurs below the coercive field (see Fig. 4(b)).  \n \nFig. 4 (color online) Calculated Tb magnetization curves  (a) \nbelow and (b) above compensation temperature , at T = 290 \nK and T = 32 4 K, respectively .  \nTherefore, our relatively simple theory is able to qualita-\ntively explain the observed anomalous hysteresis loops \nwithout involving inhomogeneities and huge exchange \nbias fields.  The single -domain picture holds well  near the \ncompensation point, where the domain size in the magnet \ndiverges. The triple hystere sis loops can be explained as a \nseries of first -order phase transitions.  The quantitative dif-\nference between the theory and the experiment can be im-\nproved by taking into account such  features of realistic \namorphous alloys as random single ion anisotropy , result-\ning in sperimagnetism  [45]. \nTo test our theory further, it is interesting to check if the \ntheory can also explain anomalies in ultrafast magnetism \nof rare -earth -transition metal ferrimagnets . Some of these \nanomalies were seen in earlier experimental studies of ul-\ntrafast laser -induced spins dynamics in  GdFeCo  in the vi-\ncinity of the spin-flop transition  [29], but n either theory \nnor simulations of the dicovered high field dynamics have \nbeen reported up to now.   \nIII. Ultrafast m agnetism and critical slow-\ndown in high magnetic fields   \nTo investigate the dependence of ultrafast spin dynamics \non bias temperature and high magnetic field , we per-\nformed time -resolved measurements of the polar mag-\nneto -optic al Kerr effect (tr -MOKE) using a pum p-probe \ntechnique , with 50 fs optical pulses generated by a 1-kHz \nTi:Al2O3 regenerative amplifier seeded with a Ti:Al2O3 os-\ncillator . The central photon energy of the pulses could be \ntuned with the help of an Optical Parametric Amplifier. Re-\nlying on the conventionally accepted approximation that \nthe main effect of optical pump pulse on a metallic magnet \nis ultrafast heating and r elying on conclusions  of earlier \n5 \n studies [33] , we assumed that the laser -induced sp in dy-\nnamics is independent of the photon energy of the pump . \nTuning the photon energy of probe one can be sensitive to \nFeCo and Tb sublattices  [34], respectively. Therefore, we \nperformed two types of pump -probe experiments. In order \nto monitor the laser -induced spin dynamics of  the Tb-sub-\nlattice, the pump and probe photon energies were chosen \nat 1.55 eV  and 2.48 eV , respectively . In order to monitor the \ndynamics of the FeCo -sublattice, the pump and probe pho-\nton energies were altered. The pump beam was focus ed on \nthe sample into a spot around 90 µm in diameter and the \ndiameter of the  probe  beam was smaller - around  30 µm. \nThe fl uence of the pump p ulses was 0.15 mJ/cm2, while the \nprobe fluence was kept around 1.5 µJ/cm2. All time -re-\nsolved measurements were performed at magnetic fields \noutside the hysteresis loops.  \n \nFig. 5 (color online)  Transient magneto -optical Kerr effect \nmeasured from TbFeCo  at different magnetic fields.  Traces \nmeasured with probe photon energy at 1.55 eV  (FeCo sublat-\ntice)  are shown by open  orange squares . Experiments  per-\nformed at photon energy 2.48  eV (Tb sublattice)  are shown \nby filled green circles.  In the left panel shown traces meas-\nured below the compensation point, T = 160 K (1.55 eV) and \nT = 220 K (2.48 eV). In the right panel shown traces meas-\nured above the compensation point , T = 350 K. The lines are \ncorresponding fits with functions , discussed in the text.   \nThe results of tr -MOKE measurements on TbFeCo at vari-\nous magnetic fields and at diff erent probe photon energies  \nbelow and above the compensation point are shown in left \nand right panels of Fig. 5, respectively.  To analyze the magnetization dynamics, we will distin-\nguish two time -domains: (i) sub -10 ps longitudinal dynam-\nics i.e. demagnetiz ation of the magnetic sublattices and (ii) \nsub-100 ps transversal dynamics of the magnetic sublat-\ntices, i.e. tilt of the magnetization. The data shown in Fig. \n5 were fitted with a double -exponential function  \n0 0 1exp( / ) exp( / )rise A t A t   \n, where \n0  and \nrise  are \nthe characteristic times of the ultrafast longitudinal and \ntransversal dynamics, respectively. A0 and A1 are the ampli-\ntudes. Assuming that the ultrafast demagnetization of both \nTb and FeCo sublattices is completed within a few  ps [33], \nin the fit we set \n0  = 1.5 ps , while  A0, A1,\nrise were taken as \nfitting parameters (see Supplementary II for details ).  \nIn the collinear phase (Fig. 5 (e)) the dynamics is very fast \nand associated with the longitudinal demagnetization (i), \nwhich is in good agreement with previous re-\nports  [27,33,46,47] . \nFigure 6 show s the field dependencies of the rise time \nrise\n, as deduced from the fit, below and above the compensa-\ntion temperature in the noncollinear phase . It is clearly \nseen that the dynamics slows down close to the spin -flop \ntransition . It is remarkable that below the compensation \ntemperature , decreasing the external magnetic field from \nHAA’+5 T to HAA’, which  is close to the spin -flop field HBR at \nthat temperature , leads to a 400% increase of the rise time. \nA similar decrease of the field from HAA’+5 T to HAA’ above \nthe compensation temperature result s in a rather moder-\nate increase of \nrise by 25%.  Finally, we find that  above the \ncompensation temperature the magnetization of Tb and Fe \nhave clearly different dynamics with a slower response of \nthe Tb spins.  \n \n \nFig. 6 (color online)  (a, b) The rise time of the tr-MOKE sig-\nnal (see Fig. 5)  below and above the compensation point, re-\nspectively . Orange circle s (open)  and curves show charac-\nteristic times  constants  which correspond to  the FeCo mag-\nnetic sublattice, while green circle s (filled)  and curves corre-\nspond to the  Tb magnetic sublattice.  Grey dashed lines cor-\nrespond to the hysteresis edge around the magnetic field -in-\nduced  first order  phase transitions.  \nTo assign the observed features of the presence  and ab-\nsence of the critical slowing down to the peculiarities of the \nphase diagram, we  simu late the ultrafast laser -induced \nmagnetization dynamics . We start with the corresponding \n6 \n magnetic structure in thermodynamic equilibrium and as-\nsume that a femtosecond laser pulse demagnetizes both \nsublattice s by 10%  (see Suppl ementary  II). The subsequent \ntransversal magnetization dynamics was modeled with the \nhelp of the Landau -Lifshitz -Gilbert ( LLG) equation . We \nshow that the observed dynamics in the canted phase can \nbe explained in the framework of coherent magnetization \nprecession of thermalized sublatt ices brought out of equi-\nlibrium by ultrafast demagnetization. After the demagnet-\nization, the spins of the sublattices will relax towards the \nequilibrium orientation via a heavily damped precession.   \nIn the framework of the LLG equation , one can derive the \nout-of-equilibrium position of the magnetization right af-\nter the demagnetization   [48]. We divide the magnetiza-\ntion dynamics into three time -domains, as earlier:  after the \ninitial longitudinal demagnetization  (i), the coherent rota-\ntion of spins further away from the initial equilibrium ori-\nentation  occurs (ii). We f ind that  a change in magnetiza-\ntion of any of the sublattice s of the order of one percent is  \nalready  enough to trigger the magnetization dynamics \nsimilar to that observed in the experiment.  Using the \nframework described above w e derived  analytical expres-\nsions for the rise time, corresponding  to the strongly \ndamped dynamical regime as observed in the experiment:  \n \n \n 2 2\n2 2arctanh / 2 / / 2\n/2ex r ex\npr\nrise\nex r  \n\n \n  , (3) \nwhere  \n is the effective Gilbert damping constant for the \nferrimagnet, the exchange frequency \nex ex H , and the \nresonance frequency  is defined as  \n2 2 2\n002\ncos cos 2fd\nrKK mHH           \n, where \ndf m M M\n and \n  is the gyromagnetic ration of \nelectron . The angle \n0  is the angle between the external \nmagnetic field and the antiferromagnetic vector  \ndfL M M\n and \nis the component of the magnetic \nsusceptibility perpendicular to the external magnetic field. \nWe assume the effective Gilbert  damping  constant, which \nis a function of the composition and temperature  [49], to \nbe equal to 0.2 . An example of the calculated dynamics can \nbe found in Fig. S4 (see Supplementary).  \nThe phase diagram predicts th at below the compensation \npoint the transition to the angular phase upon an increase \nof the external magnetic field occurs via a second -order \nphase transition  (Fig. 3) . At the phase transition  the fre-\nquency of the ferromagnetic resonance softens ,\n0r , \nand the dynamics of the order parameter  slows critically \n(diverges to infinity) down as predicted by  equation (3) \nand calculations shown in Fig. 7 (a). Similar slowing down \nis seen experimentally ( Fig. 6 (a)).  \nFigure 7 (b) summarize s the calculated field dependenc y of \nthe response time above the compensation temperature, \nwhere the phase transition between collinear and angular phases is of first  order. The slowing down at the phase tran-\nsitions is not critical , as it would be expected at a first-order \ntransition from  the general theory of phase transi-\ntions  [50]. Therefore , the experimental results reported in \nFig. 7  (b) agree well with the theoretically predicted behav-\nior based on the magnetic phase diagram. Note that exper-\nimental verification of the theoretically predicted features \nof first -order phase transitions is often obscured by such \nfactors as sample inhomogeneities .  \n \nFig. 7 (color online)  Dependence of the calculated rise time  \non the external magnetic field . The calculations were per-\nformed  below  (a) and above (b)  the co mpensation tempera-\nture at T 1 = 290 K and T2 = 324 K , respectively . The charac-\nteristic  fields  at the given temperatures are  HBR (T1) = 7 T  and \nHQQ’ (T2) = 8.7 T . \nFinally, our model could not reproduce the dramatic dif-\nference between the time scales for the  sublattices  of Tb \nand FeCo observed experimentally  (Fig. 7 (b)) . We must \nnote that a difference in dynamics of the two sublattices at \nthe timescales of the order of 60 ps has been reported ear-\nlier  [34], but despite several efforts of computation al stud-\nies [19,51] , the origin of such a behavior  is still not under-\nstood . Different excitation times for TM and RE magnetic \nsublattices were previously observed in time -resolved X -\nray studies  [15]. The mechanism was explained by a larger \nmagnetic moment per rare -earth ion in comparison to the \nmagnetic moment of transition metal ion s. However, re-\nported experiments were done in the collinear phase where \nthe rise time is on a 1 ps timescale, and the difference was \nobserved only on timescales where the electron -phonon \nsystem is still not thermalized.  Distinct spin dy namics on \na time -scale 10 -100 ps must have a different origin. We sug-\ngest that a possible explanation of such a behavior can be \nsperimagneti sm reported for  realistic TbFeCo alloys   [45]. \nAs a matter of fact, m odeling spin dynamics of sperimag-\nnets is one of the challenges in modern computational \nmagnetism.  \nConclusions  \nWe performed experimental and theoretical studies of \nanomalous hysteresis loops of the magnetizations of the \nantiferromagnetically coupled FeCo and Tb sublattices \nof ferrimagnetic TbFeCo in high magnetic fields. Unlike \nprevious theories which ex plained such loops by ex-\nchange bias between the surface and bulk layers within \none fi lm, here we showed that such a loop can  be an in-\ntrinsic feature  a f-d ferrimagnet. By taking into account \n7 \n the orbital momentum that results in a large magnetic \nanisotropy of the rare -earth ions, we computationally \nexplored and defined the phase diagram of TbFeCo in \nH-T coordinates.  In order to bring the developed theory \ninto an ultimate test, we experimentally investigate d \nthe magnetization dynamics in TbFeCo triggered by a \nfemtosecond laser pulse and compare d the outcome of \nthe experiment s with the modeling. It is surprising that \nthe theory developed to explain equilibrium magnetic \nproperties is also able to predict the experimentally ob-\nserved dynamics , including critical slow ing down of the \norder parameter in the vicinity of the magnetic c ompen-\nsation temperature.  Finally , we note that above the \ncompensation temperature , we experimentally ob-\nserved clearly different dynamics of the magnetization \nof Tb and Fe sublattices. These  features call  for further \ntheoretical studies  that would  take into account such  \nfeatures of realistic amorphous alloys as random single \nion anisotro py and sperimagnetism . \n \nAcknowledgments  \nThe work was funded by the Nederlandse Organisatie voor \nWetenschappelijk Onderzoek (NWO), the European Re-\nsearch Council ERC Grant agreement No. 339813 (Ex-\nchange) , HFML -RU/FOM member of the European Mag-\nnetic Field Laboratory (EMFL) , the European Union Hori-\nzon 2020 and innovation program under the FET -Open \ngran t agreement no.713481 (SPICE) . \nReferenc es \n[1] H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, \nT. Dietl, Y. Ohno, and K. Ohtani, Nature 408, 944 \n(2000).  \n[2] F. Matsukura, Y. Tokura, and H. Ohno, Nat. \nNanotechnol. 10, 209 (2015).  \n[3] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. \nPhys. 82, 2731 (2010).  \n[4] A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y. \nLyanda -Geller, and L. P. Rokhinson, Nat. 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In contrast, the proximity to a ferromagnetic/ferrimagnetic insulator may improve \nthe device quality, thus promises a better way to open the gap while minimizing the side-effects. Here, we grow thin single-crystal Sb\n1.9Bi0.1Te3 micro flakes on insulating ferrimagnet BaFe12O19 by using \nthe van der Waals epitaxy technique. The micro flakes show a negative magnetoresistance in weak perpendicular fields below 50 K, which can be quenched by increasing temperature. The signature \nimplies the weak localization effect as its origin, which is absent in intrinsic topological insulators, unless a surface state gap is opened. The surface state gap is estimated to be 10 meV by using the \ntheory of the gap-induced weak localization effect. These results indicate that the magnetic proximity effect may open the gap for the topological surface attached to BaM insulating ferrimagnet. This \nheterostructure may pave the way for the realization of new physical effects as well as the potential \napplications of spintronics devices.\nA gap opened for the surface states by breaking time reversal symmetry in topological insulators is anticipated to \nhost many novel physics\n1–8. Experimentally, the gap may be realized either by magnetic doping9–15, or by magnetic \nproximity to a ferromagnetic insulator16–18. One of the signatures of the gap openings is the weak localization \neffect19. The effect can give rise to positive low-field magnetoconductivity at low temperatures19–22. In contrast, \nfor gapless surface states, a π  Berry phase always leads to weak anti-localization and an associated negative mag-\nnetoconductivity23–26. However, in actual samples, the magnetic doping inevitably introduces magnetic scattering \ncenters, defects, as well as magnetic clusters, which lead to mixed surface and bulk phases in magnetotrans -\nport27,28. As a result, it is hard to distinguish magnetically-doped topological insulators from diluted magnetic \nsemiconductors20, in the latter the weak localization-like magnetoconductivity is also anticipated and not attrib-\nuted to the gap of the surface states. Compared to the magnetic doping, the magnetic proximity effect may also induce a gap for the surface states of topological insulator. A higher Curie temperature magnetic order can be achieved in a heterostructure of topological insulator and ferromagnetic insulator if the Curie temperature of the ferromagnetic insulator is high enough\n29. Moreover, the topological insulator-ferromagnetic insulator het-\nerostructure is expected to suppress external magnetic impurities and magnetic clusters; therefore, it may be a better experimental candidate to induce the gap for the topological surface states. A number of heterostructures have been studied\n29–34 with different ferromagntic insulator substrates, such as EuS30,31, yttrium iron garnet29,33, \nGdN32 and BaFe12O19 (BaM)34. In the experiments, only a suppressed weak antilocalization effect with a negative \n1High Magnetic Field Laboratory, the Chinese Academy of Sciences, Hefei 230031, the People’s Republic of China; \nUniversity of Science and Technology of China, Hefei 230026, The People’s Republic of China. 2Collaborative \nInnovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, The People’s Republic of China. \n3Department of Physics, South University of Science and Technology of China, Shenzhen, China. 4Hefei Science \nCenter, Chinese Academy of Sciences, Hefei 230031, Anhui, China. *These authors contributed equally to this work. \nCorrespondence and requests for materials should be addressed to H.-Z.L. (email: luhz@sustc.edu.cn) or M.T. (email: \ntianml@hmfl.ac.cn)Received: 21 September 2015\naccepted: 09 December 2015\nPublished: 19 February 2016OPENwww.nature.com/scientificreports/2\nScientific  RepoRts  | 6:21334 | DOI: 10.1038/srep21334magnetoconductivity was achieved. The suppressed weak antilocalization cannot be unambiguously attributed \nto gap opening because random magnetic scattering can also induce the suppression of the weak antilocalization effect\n19,26. A negative magnetoresistance in low fields has been demonstrated in a Bi2Se3-EuS heterostructure with \na Bi2Se3 layer thinner than 4 nm31. However, it is not sufficient to conclude that the magnetic proximity has indeed \nopened the gap since the finite-size effect can also open gaps in thin films35 and leads to the weak localization \neffect22. Very recently, a low-field positive magnetoconductivity was observed in a Bi2Se3-BaM heterostructure in \nparallel magnetic fields34, but the perpendicular magnetoconductivity remains negative. Domain walls may be the \npossible origins of the positive parallel-field magnetoconductivity in very weak parallel fields. Most heterostruc-tures have been fabricated by the molecular beam epitaxy (MBE) method, and the size of these heterostructures is much larger than the magnetic domains of the ferrimagnetic insulator. Thus massive Dirac electrons would be expected in magnetic domain areas, but remain massless at the domain walls\n29. The domain walls may suppress \nconductivity28; then a positive magnetoconductivity arises as the magnetic field removes the domain walls.\nIn this work, we fabricated BaFe12O19-Sb1.9Bi0.1Te3 heterostructures by using the van der Waals epitaxial tech-\nnique. The size of the topological insulator flakes can be controlled to be comparable with the magnetic domains of BaM. In perpendicular magnetic fields, a positive magnetoconductivity possibly associated with the weak localization effect is observed, indicating that the magnetic proximity has opened a gap for the surface state of topological insulator. The parallel-field magnetoconductivity shows a negative magnetoconductivity near zero \nfield as the in-plane magnetization of BaM is not able to open the gap for the surface states. Using the magneto-\nconductivity formula for the competition between weak antilocalization and weak localization effects, we fitted the magnetoconductivity curves in perpendicular fields and found that the surface gap ∆ induced by the mag -\nnetic proximity is about 10  meV . Our results demonstrate that the magnetic proximity can break time-reversal \nsymmetry and open a sizable gap for the surface states of topological insulator. This topological insulator-BaM heterostructure thus may pave the way for further experimental research on novel physics and potential applica-tions of spintronics devices.\nResults\nHeterostructures of topological insulator and BaFe12O19. Hexagonal BaFe12O19 is a well-known fer -\nrimagnetic insulator with a uniaxial anisotropy along the c crystallographic axis. The magnetic domain structure of a single-crystal BaM has been verified with positive and reversed magnetic domains along the c axis under different directions of magnetization. The size of the magnetic domains is about 5 μ m. The domains exhibit laby-\nrinth, stripe, honeycomb-type patterns as the magnetization field tilts from the c axis to the a-b plane\n36,37. We pre-\npared the BaM single crystals by the floating zone method. We chose large and flat single crystals with a natural cleavage plane (0001) as the ferrimagnetic insulator substrate. Figure1(a) shows the magnetization of the BaM substrate measured by the Magnetic Property Measurement System, where M is the magnetic moment, and M\nS is \nthe saturation magnetic moment. The saturation magnetization field HS of BaM along the in-plane (perpendicu-\nlar to the c axis) and out-of-plane directions are 1.5 T and 0.5 T, respectively. There is no obvious change of HS in \nFigure 1. Device characteristics. (a) The magnetic moments of the single crystal ferromagnetic insulator \nBaFe12O19 (BaM). The out-of-plane and in-plane magnetic moments are indicated by “|| c” and “⊥  c ”,  \nrespectively. The magnetic moments in the two directions do not change as the temperature increases from 2 K \nto 50 K. Inset: the XRD pattern of the single crystal BaM. Only (00l) peaks related to the hexagonal phase can \nbe observed. (b) The R-T curves of the BaM substrate only and the heterostructure of topological insulator and BaM, respectively. Inset: The scanning electron microscope image of the Sb\n1.9Bi0.1Te3-BaFe12O19 heterostructure, \nwith the current (I +  and I-) and voltage (V +  and V-) probes. The white points are redundant tellurium \nparticles generated during cooling. The warping edges show the large lattice mismatch between Sb1.9Bi0.1Te3 and \nBaFe12O19. The scale bar is 10 μm.www.nature.com/scientificreports/3\nScientific  RepoRts  | 6:21334 | DOI: 10.1038/srep21334both directions when the temperature increases from 2  K up to 50  K. Due to the free moving of domain walls in \nresponse to the variations of the external fields, no evident hysteresis is observed at low fields38.\nThe van der Waals epitaxy is a facile method to grow high-quality nanostructures on a clean surface of sub-\nstrate irrespective of large lattice mismatch39,40. By this method, we successfully fabricated Sb1.9Bi0.1Te3-BaM het-\nerostructures in a 1-inch horizontal tube furnace via the catalyst-free vapor-solid (v-s) growth technique similar \nto that in ref. 41. We choose the stoichiometric Sb1.9Bi0.1Te3, because it can effectively lift the position of the Dirac \npoint out of the bulk valence band while tuning the Fermi level inside the bulk gap through charge compensa-tion\n42. The inset of Fig.1(b) presents the scanning electron microscope (SEM) image of the Sb1.9Bi0.1Te3 nanoplate \non the BaM substrate. The warping edges of the nanoplate indicate the large lattice mismatch between Sb1.9Bi0.1Te3 \nand BaM. The white points on the Sb1.9Bi0.1Te3 nanoplate are redundant tellurium generated during the cooling. \nFigure1(b) shows the R-T curves of both the heterostructure and BaM substrate. BaM is a ferrimagnetic insula-tor with high room temperature resistance. After the growth of the topological insulator on BaM, the resistance of BaM was reduced to 200 Ω  at 300 K, revealing that the BaM becomes conductive after annealing for 1 hour \nnear 300 °C. When the temperature decreased to 125 K, the resistance of BaM increased sharply. In contrast, the \nresistance of the heterostructure increased slowly with a decrease of temperature. For instance, at T =  100 K, the \nresistance of the BaM substrate reached above 2 ×  10\n6 Ω , which is 100 times larger than the resistance of the het-\nerostructure. This high resistance indicates that when T <  100 K, the current mainly flows through the topological \ninsulator film. Here, our temperature range of interest is below 55 K, at which the BaM substrate becomes a full \ninsulator.\nMagnetoconductivity and weak localization. We measured the magnetoconductivity of our samples \nby the standard four-probe transport measurement. Figure2(a) shows the magnetoconductivity of the Sb\n1.9Bi0.1Te3-BaM heterostructure at different temperatures in perpendicular magnetic fields. We observed a pos -\nitive magnetoconductivity (i.e., negative magnetoresistance) in the perpendicular field up to 7 T at low tempera-\ntures. The positive magnetoconductivity weakened as the temperature increased and finally the magnetoconductivity displayed mixed behavior from positive in low fields to negative in high fields when the temperature increased up to 50 K. In the perpendicular field, the classical magnetoconductivity arising from the \nLorentz force always gives a negative magnetoconductivity (positive magnetoresistance). In the strong disorder \nregime, the transport can exhibit a negative magnetoresistance\n43, but it requires that the conductivity σ/eh2. \nWhile at 2 K, the conductivity σ in our heterostructures ranges between σ /< </ ehe h 1022, not in the strong \ndisorder regime. The positive magnetoconductivity here is most likely the consequence of the weak localization \neffect of the gapped surface states and, therefore, has a quantum interference origin19. As the temperature \nincreased to T =  55 K, the magnetoconductivity became entirely negative. There is no obvious change in the mag-\nnetization of our BaM substrates near 50 K, so the observed sign change of the magnetoconductivity near 50 K is \nnot due to the gap closing with vanishing magnetization, but possibly because the temperature quenched the quantum interference and then the classical magnetoconductivity became overwhelming. In order to conduct a \ncomparative study, we also grew Sb\n1.9Bi0.1Te3 on a SiO2 substrate. As shown in Fig.3(a), the perpendicular-field \nmagnetoconductivity of Sb1.9Bi0.1Te3 on the SiO2 substrate exhibited a sharp cusp near zero field. The cusp is a \ntypical signature of the weak antilocalization effect for the topological surface states due to their nature as mass-\nless Dirac fermions.\nMagnetization induced surface state gap.  Functionally, the magnetoconductivity of the weak antilo-\ncalization and weak localization effects can always be fitted by the Hikami-Larkin-Nagaoka (HLN) formula44,\nFigure 2. Magnetoconductivity of heterostructures. (a) In perpendicular magnetic fields. (b) In parallel \nmagnetic fields. The solid curves within 1 T in (a) are the fitting curves by using Eq. (2). The data curves at \ndifferent temperatures are offset for clarity.www.nature.com/scientificreports/4\nScientific  RepoRts  | 6:21334 | DOI: 10.1038/srep21334πψ ∆σ() =− α\n\n+\n−\n\n\n,\n() φφe\nh el elB1\n2 4ln\n412\n22\nwhere ψ is the digamma function. The prefactor α  and the phase coherence length φl serve as two fitting parame-\nters. However, the information about the gap opening cannot be given by the HLN formula. To quantitatively \nstudy the gap opening in the surface states, we employed the magnetoconductivity formula derived for the weak antilocalization-localization crossover of the massive Dirac fermions\n19,\n∑α\nπψ ∆σ() =\n\n+\n−\n\n\n,\n() φφ =,e\nhl\nll\nlB1\n2ln\n2 iiB B\n0122\ni22\ni2\nwhere the magnetic length =/le B4B2, /= /+ /φφll l 1 11ii22 2, and α0, α1, l02, l12 are explicit functions (see \nMethods) of Δ  /2EF, with Δ   the gap of surface states and EF the Fermi level measured from the Dirac point. In the \nweak antilocalization limit, α=00 , α=− .051 , while in the weak localization limit, α=.050 , α=01 . Equation \n(2) is valid when the mean free path is much shorter than the magnetic length ( )l leB22. We used formula (2) to \nfit the magnetoconductivity in perpendicular fields. The solid lines in Fig.2(a) are the fitting curves by using \nequation (2). Figure4(a) shows the fitting result of Δ  /2EF as a function of temperature. At T =  2 K, Δ /2EF is about \n0.329. With increasing temperature, Δ  /2EF decreases. At T =  50 K, Δ /2EF =  0.304, changed by less than 10%. \nThese values of Δ  /2EF are in the regime between the unitary and orthogonal symmetry classes, where a sup-\npressed weak localization effect can give rise to a positive magnetoconductivity19. In the earlier experiments45,46, \nthe higher Fermi energies in topological insulators Bi2Se3 and Bi2Te3 may greatly reduce Δ  /2EF, leading to only \nweak antilocalization-like magnetoconductivity according to the theory19.\nIn order to evaluate the surface gap, we used 2D carrier density nD2 derived from the Hall measurement  \nto estimate the Fermi level. By measuring the Hall resistance of two similar devices on the SiO2 and BaM  \nsubstrates, respectively, as shown in Fig.3(b), we obtain that =. ×−−n ecm 6261 0D213 12 on SiO2 and \n=. ×−−n ecm 5381 0D213 12 on the BaM substrate. The values of the carrier density means that the Fermi energy is \nat the bottom of the bulk conduction band, which is about 0.1–0.2 eV from the Dirac point of the surface states, \naccording to the ARPES data in ref.42. We estimate that the surface state gap induced by the magnetic proximity is of the order of\n∆= /≈ .( ) E23 10meV3F\nAs shown in Fig.4(b), with increasing temperature, the fitted phase coherence length is reduced by half, from \n28 nm at 2 K to 14 nm at 50 K, showing a much stronger temperature dependence compared to that of Δ  /2EF. \nTherefore, the fitting results indicate that the reduction of the phase coherence length is the main reason why the positive magnetoconductivity of weak localization is suppressed in Fig.2.\nFigure 3. Transport properties of the control sample. (a) The comparison of the magnetoconductivity in \nperpendicular fields between the control sample (topological insulator on SiO2) and the topological insulator on \nBaM at 2 K. The black curve is the experimental data while the red curve is the fitting. The fitting yields \nφl =  135 nm. (b) The Hall resistance of the topological insulator grown on the SiO2 and BaM substrates, \nrespectively.www.nature.com/scientificreports/5\nScientific  RepoRts  | 6:21334 | DOI: 10.1038/srep21334Discussion\nWe also show the magnetoconductivity data in parallel fields in Fig.2(b). The positive magnetoconductivity \ncan still be recognized in the high-field range, but a deflection appears near 1.5  T and the magnetoconductivity \nbecomes negative near the zero field, as shown in Fig.2(b). The negative magnetoconductivity near the zero field indicates that there is no gap opening for the surface states in parallel magnetic fields. This absence of gap is because while applying the external magnetic field in the in-plane direction, the spin of BaM substrate will be \nrotated to the in-plane direction, and correspondingly, the surface state of topological insulator remains gapless. \nAs the temperature goes up to 30 K, the deflection of the magnetoconductivity curves is weakened and disap-\npears at 50 K. This deflection behavior has also been observed in magnetically doped topological insulators\n47 \nand undoped ultrathin topological insulator films22,48,49. In theory, if the magnetoconductivity curves exhibit a \ndeflection, the magnetoconductivity are contributed by multiple channels and the bulk states might contribute to the transport besides the surface states\n50,51. In topological insulator ultrathin films, this can be well explained \nby the 2D modified Dirac model35, while in our heterostructure, the thickness of the topological insulator nano-\nflakes is about 20 nm, which is much thicker than the ultrathin limit (<  5 nm). Besides, in the parallel fields, the \nobserved deflection in magnetoconductivity happens at 1.5 T, at which the weak-field limit (usually within 0.5 T) \nfor the weak antilocalization and weak localization effects has been violated. The observed magnetoconductivity in parallel fields cannot be explained by the competition between weak antilocalization and weak localization. \nThe mechanism for the deflection in the parallel-field magnetoconductivity is not clear so far. One possible expla-\nnation is similar to that in ref.28, where domain walls suppress the conductivity while the magnetic field may remove the domain walls gradually, leading to the positive magnetoconductivity at high fields. The size of our heterostructure is comparable to the single magnetic domain in BaM, but the observed deflection cannot be explained by this mechanism. Another possible explanation is related to the canting of magnetization as discussed in refs.29,30. When the parallel field is larger than 1.5 T, the magnetization of BaM saturates in the in-plane direc-\ntion. A canting of magnetization in the out-of-plane direction might form at the interface of our heterostructure due to strong spin-orbit coupling as well as a large anisotropy of the Lande g factor for the interface electrons\n29,30. \nThe surface state gap might also be opened by the canting of magnetization in the high-field range. Also, we can see in Fig.2, the ∆σ   induced by the magnetization canting is much smaller than that in the perpendicular field \ndirection.\nThe magnetic proximity effect usually is short-range, especially for a material with weak spin-orbit interaction. \nIf the ferrimagnetic substrate cannot affect the top surface, there should be no apparent change in the resist-ance. However, in our Fig.1(b), the resistance increases by one order, as the SiO\n2 substrate is replaced with the \nferrimagnet BaFe12O19. Considering the resistance of BaFe12O19 is several orders higher, the resistance increase \nis not due to BaFe12O19 alone, but most likely from the topological insulator thin film under the influence of a \nlong-range magnetic proximity effect. One of the possible reasons for the long-range proximity effect is strong spin-orbit coupling. It has been shown that the magnetic proximity effect can induce an obvious negative mag-netoresistance in a Platinum film attached to a ferrimagnetic insulator YIG, up to a film thickness of 14 nm\n52. \nThe topological insulator Sb1.9Bi0.1Te3 also has strong spin-orbit coupling, so the magnetic proximity between \nSb1.9Bi0.1Te3 and BaFe12O19 might be also long-range. In the device for the Hall-bar meassurement in Fig.3(b), the \nthickness of the topological insulator is 30 nm. In the Hall-bar device, we did not observe the negative magnetore-\nsistance, probably because of poor interface contact or thicker topological insulator layer.\nFigure 4. Fitting results of the magnetoconductivity in perpendicular magnetic fields. (a) The fitted Δ /2EF \nas a function of temperature, where Δ  is the gap of the surface states and EF is the Fermi level. Inset: a schematic \nillustration of the band structure of the topological insulator on BaM. The bulk conduction band, bulk valence band, and surface states are indicated by BCB, BVB, and SS, respectively. (b) The fitted phase coherence length, which is suppressed with increasing temperature. The relative change of Δ /2E\nF with temperature is much \nsmaller than that of the phase coherence length. The fitting is performed by using Eq. (2).www.nature.com/scientificreports/6\nScientific  RepoRts  | 6:21334 | DOI: 10.1038/srep21334Methods\nSynthesis of heterostructure.  The mixed powder of Sb2Te3 and Bi2Te3 with molar ratio 19:1 was placed in \nthe hot center of the tube furnace. Hexagonal BaM single crystals with a clean cleavage plane (0001) were put at \na location about 9–12 cm away from the hot center. The tube is cleaned by ultrapure Ar gas for five times prior to \nthe growth, and then the Ar gas flow is fixed at 50 s.c.c.m. In order to avoid the escape of tellurium, the hot centre \nwas heated to 475° C within 5 min and kept at this temperature for one hour, followed by natural cooling. During \nthe whole process, we maintained the pressure in the tube at 2.8 Torr. We obtained high-quality van der Waals \nepitaxy heterostructures of topological insulator and ferromagnetic insulator only when the substrate was placed 11.5 cm away from the hot center.\nDevice fabrication and characterization. To make standard four-probe devices on the topological \ninsulator-BaM heterostructure for transport measurement, we transferred the heterostructure into a FEI SEM/FIB dual beam system for the deposition of Pt electrodes. The four Pt-stripe electrodes have a nominal width of 0.8 μ m and a thickness of 200 nm deposited from a precursor gas of (methylcyclopentadienyl) trimethyl platinum \n(CH\n3)3CH3C5H4Pt. To avoid Pt contamination and Ga ion irradiation, we designed the distance between two \nadjacent electrodes to be larger than 2 μ m. During the whole fabrication process, we only used the electron beam \nto guide the Pt deposition.\nMagnetoconductivity fitting formula.  We use equation (2) to fit the magnetoconductivity in perpendic-\nular magnetic fields. The parameters in equation (2) are defined as19\nα\nα=( +− )( −) ,\n=( +) (− ),\n=− (+ )(+− ),\n=( +) (− )( +− ),−\n−\n−\n−ll aa ba bb ab\nll ab ab ab\nab ab ab ab\nab ab ab ab[]\n[]\n[]\n[2 ]e\ne022 44 42 24 22 21\n122 44 22 22 221\n144 44 44 221\n044 22 244 22 21\nwhere le is the mean free path, θ =( /) a cos2 , θ =( /) bs in 2, and θ=∆/E cos2F. We assume that the effects of \nrandom magnetic scattering is to reduce the phase coherence length and have been included in the definition of \nφl.\nReferences\n1. Chang, C. Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. 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This work was supported by the Natural Science Foundation of \nChina (Grant No. 11174294, 11374302, 11574127, U1432251 and U1332209), and the program of Users with Excellence, the Hefei Science Center of CAS and the CAS/SAFEA international partnership program for creative research teams of China.\nAuthor Contributions\nG.Z., N.W ., H.Z.L. and M.T. conceived and designed the experiment. G.Z. and N.W ., synthesized the samples and fabicated the devices. W .N., H.D., J.Y ., W .W . and Z.Y . carried out the transport measurement. H.Z.L., G.Z., M.T. and Y . Z. provided the analysis and the theoretical interpretation. G.Z., H.Z.L. and M.T., wrote the paper, with assistance from all the co-authors.\nAdditional Information\nCompeting financial interests: The authors declare no competing financial interests.\nHow to cite this article: Zheng, G. et al. Weak localization effect in topological insulator micro flakes grown on \ninsulating ferrimagnet BaFe 12 O 19 . Sci. Rep. 6, 21334; doi: 10.1038/srep21334 (2016).\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/"    },    {        "title": "2004.11406v2.Temperature_dependent_Magnetic_Transitions_in_CoCrPt_Ru_CoCrPt_Synthetic_Ferrimagnets.pdf",        "content": " 1 Title:  Temperature -dependent Magnetic Transitions in CoCrPt -Ru-CoCrPt Synthetic Ferrimagnet s \nAuthors:  Bradlee Beauchamp  and Ernesto E. Marinero  \nSchool of Materials Engineering,  Neil Armstrong Hall of Engineering, Purdue University, 701 W Stadium Ave, West \nLafayette, IN 47907  \nCorresponding Author:  eemarinero@purdue.edu  \nDOI:  https://doi.org/10.1016/j.jmmm.2020.167586  \n© 2020.  This manuscript version is made available under the CC -BY-NC-ND 4.0 license \nhttp://creativecommons.org/licenses/by -nc-nd/4.0/  \nAbstract:  \n The magnetic orientations and switching fields of a CoCrPt -Ru-CoCrPt synthetic ferrimagnet with \nperpendicular magnetic  anisotropy have been studied in the temperature  range from 2 K to 300 K. It was \nfound that two sets of magnetic transitions occur in the CoCrPt -Ru-CoCrPt ferrimagnet across this \ntemperature range . The first set exhibits three magnetic transitions in the 50  K – 370 K range, whereas  the \nsecond involves only two transitions  in the 2 K and 50 K range. The observed magnetic  hysteresis curves of \nthe synthetic ferrimagnet are explained  using the energy diagram technique  framework  pioneered  by Koplak \net al.  [1] which accurately describes the competition between interlayer exchange  coupling  energy, Zeeman \nenergy, and anisotropy energy in the system. In this work we expand the framework to include synthetic \nferrimagnets ( SFMs ) comprising higher perpendicular magnetic anisotropy materials and large (4X) interlayer \nexchange coupling energies  which are promising for the development of ultrafast (ps) magnetic switching free \nlayers in MTJ structures .  Furthermore, we apply the analysis  to predict SFM magnetic hysteresis curve s in a \ntemperature regime that includes  temperature  extrema that  a synthet ic ferrimagnet would be expected to \nreliably operate at, were it to be utilized as a free layer in a memory  or sensor  spintronic device.  \nKeywords:  \nSynthetic ferrimagnet,  Magnetic films , CoCrPt , Interlayer e xchange coupling, Perpendicular magnetic \nrecording, Magnetic switching , spintronics, magnetic tunnel junctions  \nIntroduction:  \nSynthetic ferrimagnet ( SFM ) trilayers consist of two  antiparallel  ferromagnetic  (FM) films separated by \na thin non-ferromagnetic  metallic interlayer. For the case of identical FM layers, if the films are dissimilar in \nthickness, the SFM structure will exhibit a net magnetic moment  (uncompensated ferrimagnet) . The  interlayer  \nexchange coupling energy  (IECE)  of the SFM  varies with the interlayer thickness in an oscillatory fashion  [2] \nand it has been attributed to various physical processes that include  dipolar magnetostatic interaction s and  2 Ruderman -Kittel -Kasuya -Yosida (RKKY) coup ling. First observed  by Gr ünberg et al.  [3], films exhibiting  \nantiferromagnetic coupling  were utilized  shortly thereafter  in magnetic sen sor devices based on the  giant \nmagnetoresistance (GMR) observed in Fe/Cr antiferromagnet s tructures  [4][5]. More recently, SFMs  and \nantiferromagnets have been utilized in magnetic tunnel junctions (MTJs) to provide exchange bias to the \nreference layer (typically a single, magnetically hard ferromagnet) such that its magnetization is “fixed” [6,7] . \nSFM trilayers or antiferromagnets have been considered to replace the recording or reference layer of the MTJ \n[8,9] . When utilized in an MTJ device the strength  of coupling can determine if the SF M is acting as a reference \nor free layer . The coupling strength is derived from  measuring the magnetic field required for ov ercoming the \nIECE which renders the magnetization orientation of the individual layers to be parallel.  It is noted that free \nlayer  SFM structures incorporated into MTJ devices  have demonstrated lower critical switching current s than \nsingle FM free layers  with out negatively  affecting  thermal stability  [10]. Additionally , we have  proposed that \nSFM free layers can exhibit ultrafast switching speeds down to the picosecond time regime [11]. \nMost  MTJ devices utilize  CoFeB as the FM electrode due to high tunneling magnetoresistance \nmeasured when used with MgO tunneling barrier s [12]. However, the  maximum  thickness of CoFeB exhibiting \nperpendicular magnetic anisotropy  (PMA ) is limited to around 1 .5 nm [13]. The magneti zation of CoFeB is also \nrelatively high, which increases the charge current needed for spin -transfer torque switching. CoCrPt is a \nmaterial of interest for MTJ applications  due to its low magnetization and its large anis otropy  [14], resulting in  \nlower switching current s, improve d thermal stability , and the use of thicker FM layers with concomitant \nprocess control improvements. In addition, the SFM configuration circumvents the materials -restrictive low \nmagnetic damping requirement for selection of the FM thin film for MTJ devices [11]. \n It is essential to tailor the  IECE and switching properties of the S FM structure  for use in memory \ndevices . However, the magnetic properties of the SFM are temperature dependent ,  and memory devices \ncould  be expected to operate  under extreme conditions  within the range of 2 00 K to 370  K. In this paper  the \nIECE of CoCrPt -Ru-CoCrPt trilayer structures has been  investigated from  2 K to 300  K. It has been observed  by \nKoplak et al . [1] that with decreasing temperature , the hysteresis loops of S FMs vary dramatically , and these \nauthors  developed  a formalism  to describe  the observed changes in the hysteresis loop as a function of \ntemperature in a CoFeB -Ta-CoFeB synthetic antiferromagnet . They employ an  energy balance approach that \nincludes the Zeeman energy, the IECE, and energy barriers for switching arising from the effective magnetic \nanisotropy energy.  It was found that t he two main  parameters  controlling the switching behavior with \ndecreasing temperature is the ratio of the magnetic moments of the two constituent ferromagnetic layers  as \nwell as the energy barrier  for switching  of each film, which is temperature dependent . In this paper th e energy \ndiagram technique introduced by Koplak et al. is used to describe the  magnetic  transitions measured  in a  3 CoCrPt -Ru-CoCrPt SFM as a function of  temperature.  This is compared  with their results on the CoFeB -Ta-\nCoFeB S FM structure.  Predictions are also  made for the magnetic transitions of the CoCrPt -Ru-CoCrPt SFM at \n200 K to 370  K to exemplify  the practical use of the energy diagram technique  for assessing the robustness of \na potential sensor or memory device employing a SFM read layer.  \n \nMaterials and Methods : \nAll films were deposited  without substrate heating or bias in a magnetron sputter system with  a base \npressure < 10-7 Torr. The films were grown on oxidized silicon (100) substrates, with dimensions ~ 5mmx5mm. \nThese small coupons were obtained by manual dicing of larger (2” diameter) Si wafers. Thus, the actual \ndimensions of each sample exhibited small variations. Area measurem ents for each coupon were performed \nwith optical microscopy with a reference scale. The error in surface area determination is  ~8%. The thin film \nstructure consisted of the following: Ta(5  nm)/Ru(10  nm)/CoCrPt( 1.7 nm)/Ru( X nm)/CoCrPt( 1.3 nm)/Ru(5  nm). \nThe CoCrPt sputtering target has a nominal composition of Co 70Cr18Pt12. The Ta/Ru seed layer was used to \npromote CoCrPt growth  with its basal (002) plane parallel to the thin film plane (c -axis out of plane) . Magnetic \nhysteresis loops were collected using a Quantum Design MPMS -3 superconducting quantum interference \ndevice (SQUID) magnetometer with 10-8 emu sensitivity.  All samples were mounted in the magnetometer \nusing a plastic straw attachment. Measurements at 370  K could not be performed using this method d ue to \nerrors on account of warping of the straw mount at this temperature.  All magnetic hysteresis loops presented \nin this work were performed  with the substrate aligned perpendicular to the direction of the applied magnetic \nfield.  \nResults and Discussion:  \n \nFigure 1 . Left: Schematic representation of  the film stack  cross -sectio n: red arrows indicat e the direction of \n                                                     \n                   \n               \n                             \n                            \n           \n   4 magnetization of the constituent FM layers at remanence . Right:  Hysteresis curves of the S FM structures with \ntwo different Ru interlayer thicknesses  measured at 300  K. \n As shown in Fig. 1, the IECE of the CoCrPt -Ru-CoCrPt S FM can be tailored by varying the Ru interlayer \nthickness. The IECE per unit area of a S FM with dissimilar FM layers was esti mated  by Koplak et al.  using  the \nexpression:  JEX = -HBm2/S. Here H B is the bias field which  indicates the center of the outer loop . In Fig. 2a-b, H B \nis labeled and measured by finding the center of the outer loops  at which point the SFM becomes saturated. \nAt 2 K (Fig. 2c), there is no outer loop unlike in Fig. 2a -b. In this case a minor loop must be taken to locate the \nHB field, shown by the red curve in Fig. 2c . Peng et al. [15] performed similar minor loop measur ement to \ndetermine  the exchange coupling field in CoCrPt (18 nm)/Co(1 nm)/Ru(0.9 nm)/CoPtCr(8 nm) A FM structures \nas the magnetic measurements of these samples exhibited no outer loops present in the major loop of the \nhysteresis curve. A comparable measureme nt was later performed by Bandiera et al. [9], where a minor loop \nwas measured to deduce the exchange coupling field in a SFM structure comprising Co/Pt multilayers , Co \ninterlayers, Ru  and CoFeB. Both studies demonstrate that the exchange coupling field ca n be obtained by   the \nreversal of the softer magnetic layer in the SFM. Peng et al. measured this minor loop in the first quadrant of \nthe hysteresis loop, while Bandiera et al. measure d the minor loop in the third quadrant . \nIn the J EX  equation, m2 is the magnetic moment from the thinner magnetic layer  (m1 being the moment from \nthe thicker layer) , and S is the surface area of the film. The measured surface areas, S, are determined to be \n0.21 ± 0.01  cm2 and 0.09 ± 0.01 cm2 for the hyst eresis loops shown in Fig. 1 for the SFMs with 0.5 nm and 0.8 \nnm Ru spacer layers, respectively. The corresponding JEX values are calculated -0.07 erg/cm2 and -0.03 erg/cm2 \nfor Ru=0.5  nm and 0.8  nm respectively.  We note that the magnitude of IECE stronlgy depends not only on the \nRu interlayer thickness, but also on the magnetic and structural properties of the Ru/CoCrPt interface. In \nparticular, Peng et al. [15] in their study of CoCrPt(18 nm)/Ru(0.9 nm)/CoCrPt(8 nm) AFM system, \ndemonstrated that the additi on of a Co(1  nm) interlayer between Ru and CoPtCr, incremented  J EX from 0.13 \nerg/cm2 to 0.8 erg/cm2. Generally, higher IECE is desirable for memory applications , including novel devices \nsuch as a double MTJ containing two S FM reference layers and a S FM free layer discussed in [11] which is \npredicted to switch in ps time scales.    5  \nFigure 2 . Hysteresis curves of CoCrPt(1.7)/Ru(0.8)/CoCrPt 1.3) S FM at a) 300 K, b) 50 K, and c) 2 K. The \nswitching behavior from 50-300 K includes three magnetic transitions while two transitions  are observed  at 2 \nK. In Figs. 2 a and 2b, the center of the o uter loops is indicated by H B. In Fig. 2c, the HB indicates the center of \nthe minor loop (red curve) associated with the switching of the thinner magnetic layer.  \n An SFM has four possible magnetic configurations ( ↑↑, ↑↓, ↓↑, ↓↓), where the left and right \narrows indicate the bottom  (thicker)  and top  (thinner)  FM layers, respectively  as shown in Fig. 2.  The number \nof transitions found in the hysteresis loop at a given temperature depend on the IECE, the Zeeman energy, and \nthe energy barrier for magnetic reversal. From the literature, it is also evident  that the  magnetic  field \nsweeping rate  influences the magnetic switching behavior  [16]. However, in this work  each  data  point of the \nhysteresis curve  is collected once the applied field has stabilized.  Also, each hysteresis curve is collected once \nthe sample temperature has fully stabilized . \n Koplak et al.  found for the CoFeB -Ta-CoFeB S FM three types of  hysteresis loop s over the temperature \nrange studied. In the 180 K-300 K range,  three magnetic transitions were  observed  for Type I hysteresis  (↑↑-\n↑↓, ↑↓-↓↑, ↓↑-↓↓). Two transitions are present  between  120 K-170 K for Type II hysteresis  (↑↑-↑↓, \n↑↓-↓↓) and 2  K-110 K (↑↑-↓↑, ↓↑-↓↓) for Type III hysteresis . These transitions  are observed  when \nthe applied magnetic  field is swept from positive to negative . The reverse transitions are encountered  when  \nthe field is swept from negative to positive. In the case of t he CoCrPt -Ru-CoCrPt system , we observe  three \ntransitions ( ↑↑-↑↓, ↑↓-↓↑, ↓↑-↓↓) in the  50 K to 300 K  range (Fig. 2a -b). Whereas t he number of \nmagnetic  transitions  reduces to two  when the sample is cooled down to 2 K (Fig. 2c) , (↑↑-↑↓, ↑↓-↓↓) \nsimilar to the hysteresis loop measured  in the  120 K-170 K range in the CoFeB -Ta-CoFeB system. The third set \nof magnetic transitions (↑↑-↓↑, ↓↑-↓↓) that are reported  for the CoFeB -Ta-CoFeB S FM (Type III \nhysteresis) are not observe d for the CoCrPt -Ru-CoCrPt S FM.  To understand the  differences in magnetic \ntransitions between the SFM system here reported (CoCrPt -Ru-CoCrPt) and that studi ed by Koplak et al.  \n(CoFeB -Ta-CoFeB) , we provide a comparative analysis  based on the magnetic properties of the constituent \nlayers and on the higher IECE  (4X at 300  K) provided by the Ru in our study.  An important contribution  of this \n       6 study is the extension of  the energy balance formulation  developed by Koplak et al.  to SFM structures \ncomprising higher perpendicular magnetic anisotropy ferromagnets that are critical for thermally stable single \nnm scale magnetic structures. Table 1 summarizes the types of hysteresis curves observed in both the CoCrPt -\nRu-CoCrPt and the CoFeB -Ta-CoFeB SFMs as a function of temperature . \n \nIn Table 1, the  hysteresis types and the associated magnetic transitions observed in the CoCrPt -Ru-CoCrPt SFM \nare shown in the second column  and they  are compared to the  CoFeB -Ta-CoFeB SFM results reported  by \nKoplak et al . [1] in the third column . The indicated magnetic transitions occur when the magnetic field is swept \nfrom positive saturation to negative saturation. The bold arrows represent  the magnetic moment,  m1, of the \nthicker magnetic layer. The temperature range where each type of hysteresis curve is observed are provided  \nunder the heading of the different SFM structures.  Three magnetic transitions are reported by Koplak et al. for \nthe CoFeB -Ta-CoFeB, while two magnetic transitions are observed for the CoCrPt -Ru-CoCrPt SFM.  \nHere we analyze the magnetic transitions exhibited by the SFM with a 0.8 nm Ru spacer (Fig. 1) using \nthe energy diagram technique. This technique relies on a simple e nergy balance (Eq. 1) which contains the \nIECE (EEX), Zeeman energy (E Z), and the potential barriers E eff1 and E eff2 (corresponding to the 1.7 nm and 1.3 \nnm CoCrPt, respectively).  \nEq. 1)  𝑬𝑻𝒐𝒕𝒂𝒍 =𝑬𝑬𝑿+𝑬𝒁+𝑬𝒆𝒇𝒇𝟏 +𝑬𝒆𝒇𝒇𝟐 \nThe IECE, EEX, is proportional to the surface area of the sample and can be estimated from  |𝐸𝐸𝑋|=𝐻𝐵∙\n𝑚2. Here HB represents the bias field, which is measured by locating the center of the minor loop of the softer \nmagnet  as previously described . In the case of the f irst type of switching shown in Fig s. 3a and 3 b, there are \nthree loops: the center field of the outer loops is HB and indicates the strength of IECE. The potential  energy  \nbarrier  separates the two perpendicular orientations of magnetization . Notably the hysteresis curves are \nmeasured along the easy -axis, therefore,  Eeff1 and E eff2 are not equal to the anisotropy energy determin ed \nfrom  the hard axis hysteresis. In a hysteresis loop  with three transitions  (Figs. 3a and 3 b), E eff1 can be \n                            \n                           \n                 \n   \n                 \n     ,     ,                      \n                  \n     ,                   \n                   \n     ,              7 estimated from  the coercive field of the outer loops as 𝐻𝐶−𝑜𝑢𝑡𝑒𝑟 =𝐸𝑒𝑓𝑓 1\n2∙𝑚1. Then E eff2 can be calculated from  the \ncoercive field of the inner loop expressed by 𝐻𝐶−𝑖𝑛𝑛𝑒𝑟 =𝐸𝑒𝑓𝑓 1+𝐸𝑒𝑓𝑓 2\n2(𝑚1−𝑚2). These estimat es arise  from the equations \nderived by  Koplak et al.  describing  the possible magnetic transition s. The Zeeman energy, E Z, is proportional to \nthe applied magnetic field and can be expressed as 𝐸𝑍=−(𝑚1+𝑚2)∙𝐻.  \nThe second type of hysteresis curve shown in Fig. 3c has no outer loops, which are needed to estimate \nEEX. However, one can still calculate H B and thus E EX by measuring the minor loop as s hown in Fig. 2c. This is \nobtained by switching the softer, thinner m 2 magnetic layer after the SFM has been saturated [9,15] . This \nmethod of measuring the bias field advances the application of  the energy balance framework to SFM \nstructures where there are no outer loops in the hysteresis curve . HB, labeled in Fig. 2c, was determined to be \n-1798 Oe. The minor  loop  in Fig. 2c is  measured by saturating the SFM to the ↓↓ orientation , sweeping the \nmagnetic field to just beyond the ↓↓-↓↑ magnetic transition, and then saturating the SFM back to the ↓↓ \norientation.  This indicates an IECE of -0.11 erg/cm2 for the S FM at 2 K. After E EX is obtained, E eff1 and E eff2 can \nbe calculated using equations 𝐻↑↑−↑↓=2|𝐸𝐸𝑋|−𝐸𝑒𝑓𝑓 2\n2𝑚2 and 𝐻↑↓−↓↓=−2|𝐸𝐸𝑋|+𝐸𝑒𝑓𝑓 1\n2𝑚1, respectively.  The resulting \nenergy diagram (Fig. 3c) is consistent with the transition fields of the minor loop and the satu rated hysteresis \nloop.  Eeff1 and E eff2 are plotted at each temperature in Fig. 5a. We note that , Eeff1 is larger than E eff2 until the \ntemperature is lowered to 2 K , most likely due to the changing m 1/m 2 ratio as temperature is decreased.  It is \nnoted that the error associated with the E eff parameters is around ± 1x10-5 erg so the observed difference in \ntheir values is significant. A similar behavior was observed by Koplak et al . which is observed at 100 K  and is \nattributed  to the changing ratio of m 1 to m 2 as temperature is decreased.  \n \n \nFigure 3 . Energy diagrams of the CoCrPt(1.7)/Ru(0.8)/CoCrPt(1.3) S FM at a) 300 K, b) 50 K, and  c) 2 K. The solid \nlines indicate the total energy , excluding the energy barriers, while the dashed lines include the temperature -\ndependent energy barrier  term, E eff. The hysteresis curves are shown in each pane  with corresponding \n      \n      \n        \n        \n   8 magnetic moments on the secondary axis.  The red hysteresis loop in c) displa ys the minor loop measured  to \ndetermine  HB. Dashed arrows indicate  the energies associated with the minor loop and the corresponding \ntransitions.  \nThe energy diagrams shown in Fig. 3 describe the magnetic transitions occurring for each temperature. \nSolid li nes indicate the energy of the S FM system with zero E eff, i.e. when there are  no energy barriers to \novercome. The solid lines are thus , the addition of the Zeeman energy and the IECE, with a y -intercept equal \nto the IECE. The dashed lines represent the total energy of the system after the E eff1 and E eff2 are included, as \ndescribed by  Eq. 1. As the magnetic field is swept, the magnetic orientation present  is the one with the lowest \nenergy. Shown in Fig. 3a, the blue solid line represents the  ↑↑ orientation as the field is lowered from +2 T. If \nthe energy barrier for reversal of each layer is zero, the ↑↑-↑↓ will occur at the intersection of  the blue  \nsolid line and the orange solid line representing the total energy of the ↑↓ orientation. At 300  K, the Eeff \nenerg ies for m 1 and m 2 are negligibly low such that the magnetic transition s occur approximately at the solid \nline intersections.  \nThe potential barriers become larger  as the temperature is decreased to 2 K (Fig. 5a). Since the E eff \nterms are not field -dependent, they shift the dashed lines up along the y -axis.  For a magnetic transition to \noccur , the magnetic field must be changed such that the energy barrier between the solid and dashed line is \ncrossed. As seen in Fig.  3b, the ↑↑-↑↓ transition no longer occurs at the intersection of the solid blue and \norange lines, but at the point where the potential barrier of another orientation energy is crossed. As the \npotential barriers increase with lower temperature, certain tra nsitions are prohibited from occurring due to \nthe existence of lower energy states from other magnetic orientations. At 2  K (Fig. 3c ), the ↑↓-↓↑ transition \ndoes not occur as it does  at 300  K and 50  K (Fig. 3a -b) since the potential barrier of the ↓↓ state is lower in \nenergy than the ↓↑ state.   \nThe third set of magnetic transitions  (↑↑-↓↑, ↓↑-↓↓), or Type III hysteresis, occurs when the \ncondition 𝐸𝑒𝑓𝑓 1<𝐸𝑒𝑓𝑓 2∙𝑚1\n𝑚2−2|𝐸𝐸𝑋|∙𝑚1−𝑚2\n𝑚2 is satisfied [1]. The CoCrPt -Ru-CoCrPt SFM studied does not \nmeet this requirement and does not show this set of transitions even down to 2  K. Compared to the CoFeB -Ta-\nCoFeB  SFM reported by  Koplak et al . [1], which has an IECE at 300  K of EEX/S = -0.01 erg/cm2, the CoCrPt -Ru-\nCoCrPt SFM has an IECE at 300  K of E EX/S = -0.04 erg/cm2. The CoFeB -Ta-CoFeB SFM and the CoCrPt -Ru-CoCrPt \nSFM have m 1/m 2 ratio s of 1.38 and 1.79, respectively . At 300  K the effective anisotropy energy barriers for \nboth systems are: E eff1/S = 4.0x10-3 erg/cm2 and E eff2/S = 2.5x10-3 erg/cm2 for CoFeB -Ta-CoFeB, E eff1/S = 1.7 x10-3 \nerg/cm2 and E eff2/S = 0.73x10-3 erg/cm2 for the CoCrPt -Ru-CoCrPt. The energy barriers for the CoCrPt -Ru-\nCoCrPt are lower for E eff1 and E eff2 by a factor of 2.3 and 3.4, respectively. This disparity in E eff causes the right  9 side of the inequa lity to be lower, which explains why the third set of magnetic transitions are absent  in the \nCoCrPt -Ru-CoCrPt SFM.   \nFigure 4 illustrates the dependence of the left and right side s of the inequality 𝐸𝑒𝑓𝑓 1<𝐸𝑒𝑓𝑓 2∙𝑚1\n𝑚2−\n2|𝐸𝐸𝑋|∙𝑚1−𝑚2\n𝑚2 on the IECE for both the CoCrPt and CoFeB SFM systems. The plot shows the energies \ncalculated at 100  K, since CoFeB -Ta-CoFeB shows the third set of magnetic transitions at this temperature. At a \ngiven IECE, the third set of magnetic transitions should be observed  whe n the right side of the equation is \nlarger than E eff1. When plotted as a function of E EX, the right side of the inequality is a line whose  slope is \ndetermined  by the m 1/m 2 ratio and the intercept by  the product of  Eeff2 and m 1/m 2. The dependence of the \nright side of the inequality on m 1/m 2 is also illustrated in Fig. 4a -b. As m 1/m 2 approaches 1, the slope  and the \ny-axis intercept  of the right side of the inequality  is lower ed. Since it has been shown that fast spin transfer \ntorque switching can be achiev ed with a low m 1/m 2 ratio [11], this analysis is important in understanding the \ntype of hysteresis curve s that will be present in the SFM when tailoring the ratio of magnetic moments.  It is \nevident from Fig. 4a that the magnetic switching behavior of the CoCrPt -Ru-CoCrPt SFM will not exhibit the \nthird type of magnetic switching for either stronger or weaker IECE as the intercept of the right side of the \ninequality is lower than E eff1. \nAs mentioned earlier, the SFM can be used as a replacement for a single FM layer in a memory device. \nSuch devices are expected to operate successfully over  a wide range of temperatu res. Therefore, it is \nimportant to predict the behavior of the SFM at any temperature. The energy diagram technique can be used \nto predict the transition fields of the SFM if the temperature dependence of E eff, EEX, and the magnetization, \nm, are  known. Fig ure 5a-c show s the temperature dependence of E eff, EEX, and m, respectively.  Eeff, Eeff1, and \nEeff2 are  proportional to 𝑚𝑛(𝑛+1)\n2 at lower temperatures (<150  K), while at higher temperatures the potential \nenergy barriers are proportional to 𝑚𝑛 similar to the temperature dependence of the magnetic anisotropy \nobserved for  other materials  [17,18] . Here, m is the magnetic moment and n is the exponent  of the magnetic \nanisotropy function (n=2 is typical for uniaxial anisotropy). Fig. 5a shows the fit for E eff2 based on the 𝑚𝑛(𝑛+1)\n2 \nproportionality. Good agreement is observed  with the fit until around 150  K, above which there are significant \ndifference s between the meas ured E eff and the fit.  At higher temperatures,  Eeff2 is proportional to  𝑚𝑛 as \npredicted in [17]. Eeff1 shows a similar temperature dependence as Eeff2  and the same  exponent for the  \nmagnetic anisotropy function  fits the data , however  the data point at 50  K significantly deviates from the \ntrend. The parameters E EX and m change linearly  in this temperature range . The m values are later \nextrapolated to 370  K using this linear fit. This is appropriate since this temperature is well below the Curie  10 temperature reported for Co 70Cr18Pt12 (~ 673 K [19]), around which the temperature dependence of m would \nsignificantly depart from the linear trend . It is noted in Fig. 5c that the m 1/m 2 ratio across the temperature \nrange studied is larger than the value of 1.3 expected from the nominal layer thickness of 1.7  nm and 1.3  nm. \nAs an example , at 2  K and 300  K the ratio is 1.38 and 1.7, respectively. The discrepancy may result from the \nmanner in which the magnetic moments are determined for the results presented in Fig. 5c, namely, they are \nestimated from the magnetization measurements of the SFM hysteresis loops at saturation and remanence. It \nis feasible that at remanence, contributions from the thin film multi -domain magnetic state may influence the \nmagnitude of the measured magnetization. In addition, whereas the film thicknesses and moment s of the \nconstituent layers were determined in  samples  grown on Ta(5  nm)/Ru(10  nm)  underlayers, in  the SFM, m2  is \ngrown on thin Ru , therefore its crystalline quality may differ from m 1.  This could also result in difference s in \nmagnetic properties of the layers as a function of temperature.  \nFitting the trends seen in Fig. 5 allows one to predict the behavior at 2 00 K and 370  K, the temperature \nextrema that a spintronic sensor or memory device could potentially expected to ope rate reliably.   The energy \ndiagrams for the CoCrPt(1.7)/Ru(0.8)/CoCrPt(1.3) SFM at these two temperatures are shown in Fig. 6. The \nenergy diagram in Fig. 6a is interpolated from the parameters depicted in Fig. 5, while the energy diagram in \nFig. 5b is extr apolated from fitted parameters in Fig 5.   These energy diagrams are constructed using the fitted \nparameters from Fig. 5. Both energy diagrams in Fig. 6a and 6b show magnetic transitions corresponding to \nthe type I regime [1]. The transition field for ↑↑-↑↓ (where m 1 reversal occurs) is predicted to be  1400 Oe \nand 950 Oe at 2 00 K and 370 K, respectively.  In Fig. 6a the hysteresis diagram measured at 200  K is shown and \nagrees with the predicted transitions from the energy diagram.  The hysteresis curve at 370  K was not be \nacquired due to errors introduced by deformation of the straw  sample mounting arrangement employed for \nall other measurements as discussed in the materials and methods section . \nThe CoCrPt -Ru-CoCrPt SFM system provides significant advantages over CoFeB -Ta-CoFeB as free layers \nin MTJ structures. These derive from the lower saturation magnetization and its superior magnetic anisotropy \n[14]. The anisotropy in CoCrPt is largely determined by its magnetocrystalline anisotropy as opposed to \ninterface anisotropy for the case of CoFeB. Therefore, Co CrPt  films exhibit PMA for film thicknesses up to 15 \nnm. Thus, the m 1/m 2 ratio in the films can be controlled more precisely, allowing for more tunability of the \nSFM properties. This is of particular interest for the implementation of ps magnetic switching employing SFM \nstructur es as proposed by Camsari et al. [11]. Said SFMs require larger IECE than that provided by Ta layers in \norder to achieve ps magnetic switching. Therefore, it is important to extend the energy diagram technique  11 developed by Koplak et al . to this material st ructure to understand its behavior at temperature regimes of \ninterest for practical utilization of spintronic memory devices exploiting these SFM s. \n \nFigure 4 . The left and right sides of the ine quality  𝐸𝑒𝑓𝑓 1<𝐸𝑒𝑓𝑓 2∙𝑚1\n𝑚2−2|𝐸𝐸𝑋|∙𝑚1−𝑚2\n𝑚2 plotted as a function of \nEEX for  the a) CoCrPt -Ru-CoCrPt and  b) CoFeB -Ta-CoFeB SFM.  The black solid and dashed lines labeled “Right – \nm1/m 2” represents the right side of the inequality at different magnetic ratios. The left side of the inequality is \nrepre sented by the red curve and is labeled E eff1. The black squares shown in a) represent the observed IECE of \nthe CoCrPt -Ru-CoCrPt SFM at 100  K. \n \nFigure 5. a) Eeff1, Eeff2, and E eff (Total) plotted versus temperature.  b) |E EX| plotted versus temperature.  c) The \nmagnetic moments, m 1 and m 2 of the 1.7  nm and 1.3  nm thick CoCrPt layers, respectively, plotted versus \ntemperature.  \n    \n       12  \nFigure 6. The energy diagram for the CoCrPt(1.7)/Ru(0.8)/CoCrPt(1.3) SFM at 2 00 K and 370  K. The energy \ndiagram in a) is interpolated from the parameters de rived in Fig. 5, while the energy diagram in b) is \nextrapolated from fitted parameters in Fig 5.  The t ransit ion fields are indicated  by the vertical dashed lines.  \nThe hysteresis curve measured at 200  K is shown in a).  \nConclusions:  \nMTJ devices incorporating SFM structures as free layers are expected operate reliably over a wide \ntemperature range that under extrem e conditions could range from 200 K to 370 K. Therefore, it is important \nto determine the temperature dependence of their magnetic transitions to engineer key material properties \nsuch as IECE and the energy barriers associated with magnetization reversal. In this paper the IECE of \nCoCrPt/Ru/CoCrPt SFMs has been investigated from 2 K to 300 K. Building on previous work by Koplak et al. \n[1], this work further elucidates the temperature -dependence of the potential barrier for magnetization \nreversal. The magnit ude of  H B was derived from minor loop measurements, as the hysteresis curves exhibited \nno outer loops, a procedure reported also in refs. [9,15]. H B is needed to estimate IECE and details of the \nenergy diagram technique first described by Koplak et al. [1]. In this  stduy, we observe two types of hysteresis \ncurves in CoCrPt/Ru/CoCrPt SFMs: one above 50 K with three subloops and the other at 2 K with two magnetic \ntransitions. Type III hysteresis, seen in the CoFeB -Ta-CoFeB SFM system in Koplak et al., is not observed in the \nCoCrPt/Ru/CoCrPt SFM, due to the large E eff1 which prevents the Eeff1<Eeff2∙m1\nm2−2|EEX|∙m1−m2\nm2  inequality \nfrom being satisfied. E eff1 corresponds to the energy barrier between magnetization directions present in the \nCoCrPt film.  \nThe utilization of  the energy diagram technique to understand magnetic transitions in CoCrPt /Ru/CoCrPt SFMs \nenabled us to predict  the magnetic orientations present in th is SFM system at any given temperature. Future \nwork will include studies in SFM structures comprising diff erent ferromagnetic materials , alternative  exchange \n        \n      \n        \n       13 coupling interlayers such as Rh to increment the magnitude of  IECE , as well as the incorporation of nanoscale \ninterlayer  magnetic materials between the FM constituent layers and the non -magnetic spacer t o enhance \nIECE, as demonstrated by Peng et al [15] will also be investigated . In addition, we plan to extend the \ntemperature range investigated in this study and to utilize synchrotron structural and magnetic \ncharacterization probes to better characterize their structural and magnetic properties.  \n Funding:  \nThis research was funded in part by Dr. Marinero’s startup funds from Purdue University School of Materials \nEngineering and in part by the Office of Naval Research ( Award# N00014 -18-1-2481) . \nReferences:  \n[1] O. Koplak, A. Talantsev, Y. Lu, A. Hamadeh, P. Pirro, T. Hauet, R. Morgunov, S. Mangin, Magnetization \nswitching diagram of a perpendicular synthetic ferrimagnet CoFeB/Ta/CoFeB bilayer, J. Magn. Magn. \nMater. 433 (2017) 91 –97. https://doi.org/10.1016/j.jmmm.2017.02.047.  \n[2] S.S.P. Parkin, N. More, K.P. Roche, Oscillations in Exchange Coupling and Magnetoresistance in Metallic \nSuperlattices Structures: Cu/Ru, Co/Cr and Fe/Cr, Phys.  Rev. Lett. 64 (1990) 2304 –2308. \nhttps://doi.org/10.1103/PhysRevLett.64.2304.  \n[3] P. 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Kaiser, A. Panchula, P.M. Rice, B. Hughes, M. Samant, S.H. Yang, Giant tunnelling \nmagnetoresistance at room temperature with MgO (100) tunnel barriers, Nat. Mater. 3 (2004) 862 –867.  14 https://d oi.org/10.1038/nmat1256.  \n[8] D. Watanabe, Et Al., Interlayer exchange coupling in perpendicularly magnetized synthetic ferrimagnet \nstructure using CoCrPt and CoFeB, J. Phys. Conf. Ser. 200 (2010) 72104. https://doi.org/10.1088/1742 -\n6596/200/7/072104.  \n[9] S. Bandiera, R.C. Sousa, Y. Dahmane, C. Ducruet, C. Portemont, V. Baltz, S. Auffret, I.L. Prejbeanu, B. \nDieny, Comparison of synthetic antiferromagnets and hard ferromagnets as reference layer in magnetic \ntunnel junctions with perpendicular magnetic anisotr opy, IEEE Magn. Lett. 1 (2010). \nhttps://doi.org/10.1109/LMAG.2010.2052238.  \n[10] J. Hayakawa, S. Ikeda, K. Miura, M. Yamanouchi, Y.M. Lee, R. Sasaki, M. Ichimura, K. Ito, T. Kawahara, R. \nTakemura, T. Meguro, F. Matsukura, H. Takahashi, H. Matsuoka, H. Ohno,  Current -induced \nmagnetization switching in MgO barrier magnetic tunnel junctions with CoFeB -based synthetic \nferrimagnetic free layers, in: IEEE Trans. Magn., 2008: pp. 1962 –1967. \nhttps://doi.org/10.1109/TMAG.2008.924545.  \n[11] K.Y. Camsari, A.Z. Pervaiz, R . Faria, E.E. Marinero, S. Datta, Ultrafast Spin -Transfer -Torque Switching of \nSynthetic Ferrimagnets, IEEE Magn. Lett. 7 (2016) 1 –5. https://doi.org/10.1109/LMAG.2016.2610942.  \n[12] S. Ikeda, J. Hayakawa, Y. Ashizawa, Y.M. Lee, K. Miura, H. Hasegawa, M. Tsu noda, F. Matsukura, H. \nOhno, Tunnel magnetoresistance of 604% at 300 K by suppression of Ta diffusion in CoFeBMgOCoFeB \npseudo -spin-valves annealed at high temperature, Appl. Phys. Lett. 93 (2008). \nhttps://doi.org/10.1063/1.2976435.  \n[13] S. Ikeda, K. Miura,  H. Yamamoto, K. Mizunuma, H.D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, \nH. Ohno, A perpendicular -anisotropy CoFeB – MgO magnetic tunnel junction, Nat. Mater. 9 (2010) 721 –\n724. https://doi.org/10.1038/nmat2804.  \n[14] D. Watanabe, S. Mizukami, M. O ogane, H. Naganuma, Y. Ando, T. Miyazaki, Fabrication of MgO -based \nmagnetic tunnel junctions with CoCrPt perpendicularly magnetized electrodes, J. Appl. Phys. 105 (2009) \n126–129. https://doi.org/10.1063/1.3062816.  \n[15] Y. Peng, J.G. Zhu, D.E. Laughlin, Int erfacial Co nanolayers for enhancing interlayer exchange coupling in \nantiferromagnetic interlayer exchange coupling media, J. Appl. Phys. 91 (2002) 7676 –7678. \nhttps://doi.org/10.1063/1.1452262.  \n[16] R.B. Morgunov, E.I. Kunitsyna, A.D. Talantsev, O. V. Kopl ak, T. Fache, Y. Lu, S. Mangin, Influence of the  15 magnetic field sweeping rate on magnetic transitions in synthetic ferrimagnets with perpendicular \nanisotropy, Appl. Phys. Lett. 114 (2019). https://doi.org/10.1063/1.5096951.  \n[17] J.B. Staunton, L. Szunyogh,  A. Buruzs, B.L. Gyorffy, S. Ostanin, L. Udvardi, Temperature dependence of \nmagnetic anisotropy: An ab initio approach, Phys. Rev. B - Condens. Matter Mater. Phys. 74 (2006) \n144411. https://doi.org/10.1103/PhysRevB.74.144411.  \n[18] B.D. Cullity, C.D. Graham , Introduction to Magnetic Materials, John Wiley & Sons, Inc., Hoboken, NJ, \nUSA, 2008. https://doi.org/10.1002/9780470386323.  \n[19] M.F. Doerner, T. Yogi, D.S. Parker, S. Lambert, B. Hermsmeier, O.C. Allegranza, T. Nguyen, Composition \nEffects in High Densit y CoPtCr Media, IEEE Trans. Magn. 29 (1993) 3667 –3669. \nhttps://doi.org/10.1109/20.281263.  \n "    },    {        "title": "0808.3922v1.Inhomogeneous_ferrimagnetic_like_behavior_in_Gd2_3Ca1_3MnO3_single_crystals.pdf",        "content": "Inhomogeneous ferrimagnetic-like behavior in Gd 2/3Ca 1/3MnO 3\nsingle crystals\nN.Haberkorn\nComisión Nacional de Energía Atómica, Centro Atómico Bariloche, S. C. de Bariloche, 8400 R. N., \nArgentina. and\nInstituto Balseiro, Universidad Nacional de Cuyo and Comisión Nacional de Energía Atómica, S. C. de \nBariloche, 8400 R. N., Argentina\nS. Larrégola\nFacultad de Química, Bioquímica y Farmacia, Universidad Nacional de San Luis, Chacabuco y Pedernera, \nSan Luis 5700, Argentina.\nD. Franco\nFacultad de Ciencias Químicas, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina\nG. Nieva\nComisión Nacional de Energía Atómica, Centro Atómico Bariloche, S. C. de Bariloche, 8400 R. N., \nArgentina. and\nInstituto Balseiro, Universidad Nacional de Cuyo and Comisión Nacional de Energía Atómica, S. C. de \nBariloche, 8400 R. N., Argentina\nWe present a study of the magnetic properties of Gd 2/3Ca1/3MnO 3single crystals at low temperatures. We \nshow that this material behave as an inhomogeneous ferrimagnet. In addition to small saturation \nmagnetization at 5 K, we have found history dependent effects in the magnetization and the presence of \nexchange bias. These features are compatible with microscopic phase separation in the clean Gd 2/3Ca1/3MnO 3\nsystem studied.PACS numbers: 75.50.Gg; 75.30.Kz\nnhaberk@cab.cnea.gov.ar\n  TE / FAX: 542944445171 / 2944445299* ManuscriptTransition metal oxides with perovskite structure have generated great interest due to the rich variety \nof their electrical and magnetic properties. These include, for example, high temperature superconductivity \nand colossal magnetorresistance (CMR).1, 2 In hole-doped perovskites R1í[AxMnO 3 (RA xMO R: Rare earths, \nand A: Ca, Sr) showing CMR, the electrical and magnetic properties are known to be very sensitive to the \nlattice parameters, the Mn3+ / Mn4+ ratio and the oxygen content.1 By modification of these parameters it is \npossible to obtain complex phase diagrams that include ferromagnetism (FM), antiferromagnetism (AF), \nweak FM, spin canting and, in some cases, spatial inhomogeneity related to multiphase coexistence.1,3The\nphenomenon of multiphase coexistence or phase separation corresponds to the simultaneous existence of \nmutually penetrating submicrometer sub-phases with slightly different electronic density giving rise to\ndifferent magnetic behavior. This electronic phase separation is associated with spatial inhomogeneity which \nin turn is related to local crystalline distortion. The mismatch between the size of different ions could be \nexpressed by the values of the tolerance factor,\n) (2;\nO MnO AR\nr rr rt\n\u000e\u000e²¢ , and size disorder at the R,A-site, \n¦ ²¢\u00102 2 2\ni ii r rxV , where ²¢ARr, corresponds to the average size of the R,A-site cations, and, x i and r i are \nthe fractional occupancies and ionic radii of the i cations.4, 5 In GdCa xMO for x § 1/3, we can estimate t and \nV2, 0.891 and 0.0025 A2, respectively, a result quite similar to YCa 1/3MO (t §0.884,V2§0.0014 A2).\nWhereas YCa 1/3MO shows short ferromagnetic order and a spin glass-like (SGL) behavior at low \ntemperatures (T < 30 K),6 in GdCa 1/3MO a ferrimagnetic behavior was reported, associated with the \nantiferromagnetic (AF) order of the Gd and Mn sublattices at low temperatures.7, 8 The presence of a SGL \nbehavior in YCa 1/3MO could be related to the large local lattice distortion and associated with phase \ncoexistence,3,6 similar to that found in PrCa 1/3MO.9 The GdCa 1/3MO compound presents ferrimagnetic \nbehavior with Curie temperatures of around 50 and 80 K, for the Gd and Mn sublattices, and a compensation \ntemperature (T comp) of § 15 K.7,8 At T comp, the rare-earth and transition metal sublattice magnetizations at zero \nfield exactly cancel.\nIn this work, we analyze the magnetic behavior at low temperatures of GdCa 1/3MO single crystals.\nThe samples were grown by the floating zone technique from isostatically pressed and pre-sintered rods of the \nsame nominal composition. The phase purity of the single crystals was probed by x-ray diffraction, and the composition was checked by energy dispersive spectroscopy (EDS). Magnetic properties were measured in a commercial SQUID magnetometer. Curie temperature (T c) was estimated from the inflection point of the \nfield-cooled (FC) magnetization ( M) versus T curves at low applied magnetic fields. In the FC procedure the \nsamples were cooled from 150 K, under an applied field between 0 and 5 T. Cooling (FCC) and warming \n(FCW) M versus Tmeasurements were performed. The results that will be presented correspond to the same \ncrystal, being representative of all measured crystals.\nPowder x-ray diffraction patterns obtained by grinding several GdCa 1/3MO single crystals show \nsingle phase orthorhombic Pbnm(n°62) structure. Two crystalline directions were identified in the single \ncrystal used in the magnetic measurements. Figure 1 shows a schematic picture of the crystalline axis and its \nrespective x-ray diffraction pattern. The (020) and (200) orthorhombic reflections are equivalent to the family \nof (110) reflection expected for the pseudo orthorhombic or pseudo cubic (p-cub) lattice formed by the \ncations. The lattice parameters are b/2§\u00030.393 nm, a/2§ 0.381 nm. Taking these into consideration, a \nface rotated 90° from those is equivalent to the (100) axis (see figure 1).\nFigure 2a shows M versus T curves for field H = 7.5 kOe applied along different crystal orientations\nof GdCa 1/3MO single crystal. The results show an inflection of the magnetization at approximately 80 K \nassociated with the ferromagnetic order of the Mn cations. Below 50 K the magnetization decreases due to the \ncompensation originated by the magnetic order of the Gd sublattice (Gd-Mn interaction). Depending on the \napplied magnetic field the magnetization goes to negative values (H < 2.5 kOe, not shown) or begins to \nincrease at T comp§ 16 K.7, 8 Figures 2b and 2c show the hysteretic M versus T behavior around Tcomp for two \ndifferent applied magnetic fields in the (100) p-cub axis. The differences in cooling and warming measurements \ncould be associated with a change in the domain size. This fact is also manifested in the coercive field. \nHysteresis loops at the same temperature range present different coercive fields when the temperature is\nreached by cooling or warming (not shown). Hysteresis in magnetization was previously reported in YCa\n1/3MO,6 and it was associated with a spin glass like (SGL) behavior. In this case, as in our experiments a \ndynamic phase coexistence could be present since long local distortions are present in both materials.3 As we \nwill show later, a possible phase separation is also supported by two different facts: the low Ms at low \ntemperatures, and the presence of exchange bias ( EB) near Tcomp. Different curves in figure 2a make evident \nthe crystalline anisotropy effect. This anisotropy is also manifested in the hysteresis loops presented in figure \n3. At 60 K it is easier to magnetize the Mn along the (020) direction than along (100) p-cub axis (see figure 3a). While at the lowest measured temperatures, where the Gd influences the magnetization, the easier axis \ncorresponds to the (100) p-cub axis(see figures 3b and 3c). Although more studies are necessary to clarify this \npoint, the anisotropy difference of the Mn and Gd sublattices could produce canting between them.\nFigure 4 shows the saturation magnetization ( Ms) obtained from hysteresis magnetic loops like those \nshown in figure 3. In the Ms estimated a paramagnetic signal was subtracted. The possible phase coexistence \nis supported by the low Ms value at 5 K § 80 emu / cm3. This value is approximately half of the expected \nvalue considering ferrimagnetic order. The saturation magnetization per mole of GdCa 1/3MO expected from \nthe Gd3+ s = 7/2, l = \u0013\u0003LV\u0003ȝ\u0003 \u0003\u000b\u0015\u0012\u0016\u0003[\u0003\u0015\u0003[\u0003\u001a\u0012\u0015\f ȝ%  \u0003\u0017\u0011\u0019\u001a\u0003ȝ%\u000f\u0003ZKLOH\u0003KLJK -VSLQ\u0003PDQJDQHVH\u0003JLYHV\u0003VSLQ\u0003RQO\\\u000f\u0003ȝ\u0003 \u0003\nJVȝ%\u000f\u0003 J\u0003  \u0003 \u0015\u0003 VR\u0003 ȝ\u0003  \u0003 \u0015\u0003 ȝ%>\u0015\u0012\u0016[\u0015\u0003 \u000bIRU Mn3+)+ 1/3x3/2 (from Mn4+\f@ \u0003 \u0016\u0011\u0019\u001a\u0003 ȝ%\u00117 Considering these \nmagnetizations, we expect a Ms§\u0003\u0014\u0011\u0013\u0013\u0003ȝ%\u0003§  160 emu/cm3.7 Magnetic hysteresis loops also show a high \nparamagnetic like signal at different temperatures (see figure 3). Although at low temperatures it could be \nassociated with sublattice rotation due to canting,7 it could also be associated with phase coexistence. It is \nimportant to remark that the contribution of paramagnetic Gd moment alone can not explain the high\nparamagnetic signal, because in this case a high Ms should be expected from the non compensated\nferromagnetic Mn moments. Figure 5 shows the temperature dependence of the coercive field, \nHc =| Hc1íHc2 | /2, where Hc1 and Hc2are the fields for zero magnetization at both branches of the hysteresis \nloops for field excursions up to 1 and 3 T. The Hctemperature dependence shows a non monotonic decrease\nwhen the temperature is raised, resulting quite different to the continuous and smooth decrease measured by \nO. Peña et al.8 We observe a drop of Hc around Tcomp, which is a consequence of the superposition of two \nsignals: a ferrimagnetic one, responsible for the loops, and a paramagnetic one.10 As we discuss previously, \nthe paramagnetic like behavior could be a consequence of weakly coupled sublattices,7 or phase coexistence. \nThe coercive field also makes evident the crystalline anisotropy (see figure 5). We observe that the differences in H\nc for the (100) p-cub and (020) axis are more important at temperatures lower than 30 K, in this \ntemperature range the Gd sublattice magnetization starts to play a more important role. However, in our case, \nconsidering a possible phase coexistence, the shape anisotropy of the small domains could be affecting the Hc\nvalues.11Another feature in the hysteresis loops (see inset figure 3b) is the shift with respect to zero field of \nthe magnetization, i. e. exchange bias (EB). The presence of EB, associated to the presence of FM / AF \nmagnetic interfaces is usually found in artificially designed materials like FM / AF multilayers or more recently, in manganite with phase coexistence.12, 13 Materials with AFM / ferrimagnetic and FM / \nferrimagnetic interfaces also could show EB.14 The magnitude of the effect depends on a number of \nparameters including AF and FM domain size, interfacial roughness, AF and FM anisotropy, etc.\nFigure 6 shows the T dependence of the EB field ( Heb =| Hc1+Hc2 | /2), obtained from hysteresis loops \nat different FC field, H = 1 and 3 T, in two crystalline directions. The exchange bias fields show a sharp increase and change sign around T\ncomp. This behavior is similar to that found in ferrimagnetic / FM \nmultilayer.10, 15 Since we are measuring a single crystalline sample we expect a homogeneous ferrimagnet \ndue to the absence of interfaces. However the presence of EB should be associated in our case with a very \nsmall domain size distribution with different Tcomp. This is supported by the suppression of the Heb when the \nmagnetic fields excursions in the loops are increased. For example, loops in a range -1 T < H < 1 T show Heb \nseveral times higher than loops in a range -3 T < H < 3 T, and the EB effect disappear for loops in a range -5\nT < H < 5 T. The change of sign in Heb near of Tcomp is a consequence of domain rotation when the Gd\nsublattices dominates the magnetization.\nIn summary, we studied the magnetic properties of GdCa1/3MO single crystals. This cleans system, \nhighly locally distorted, shows characteristics typically found in inhomogeneous ferrimagnets. Several \nfeatures are compatible with phase coexistence in the single crystals: the Ms at 5 K is approximately half of \nthe expected value considering a ferrimagnetic order; hysteric behavior on cooling and warming M versus T\ncurves; and the presence of exchange bias near Tcomp.\nAcknowledgments\nThis work was partially supported by CONICET PIP5251 and ANPCYT PICT PICT00-03- 08937. N. H. and \nG. N. are member of CONICET.\n.\n1Lev P. Gor’kov, and Vladimir Z. Kresinc. Physics Reports 400, 149 (2004).\n2Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen. Rev. Mod. Phys. 77, 721 (2005).\n3 Weida Wu, Casey Israel, Namjung Hur, Soonyong Park, Sang-Wook Cheong and Alex De Lozanne. Nature \nMaterials, 5, 881 (2006).4 J. P. Zhou, J. T. McDevitt, J. S. Zhou, H. Q. Yin, and J. B. Goodenough, Y. Gim and Q. X. Jia. Appl. Phys. \nLett., 75, 1146 (1999).\n5 A. Maignan, C. Martin, G. Van Tendeloo, M. Hervieu, and B. Raveau. Phys. Rev . B, 60, 15214 (1999).\n6 R. Mathieu, P. Nordblad, D. N. H. Nam, N. X. Phuc, and N. V. Khiem. Phys. Rev. B, 63, 174405 (2001).\n7 G. J. Snyder, C. H. Booth, F. Bridges, R. Hiskes, S. DiCarolis, M. R. Beasley and T. H. Geballe. Phys. Rev. \nB,55, 6453 (1997).\n8 Octavio Peña, Mona Bahout, Karim Ghanimi, Pedro Duran, Dionisio Gutierrez, and Carlos Moure. J. Mat \nChem. 12, 2480 (2002).\n9 P. G. Radaelli et al.Phys. Rev. B 63, 172419 (2001); V.N. Smolyaninova et al. Phys. Rev. B 65, 104419 \n(2002); D. Saurel et al. Phys. Rev. B 73, 094438 (2006).\n10 D. J. Webb, A. F. Marshall, Z. Sun, T. H. Geballe, and Robert M. White. IEE Trans. Magn. 24, 588 (1988).\n11 J. Tao, D. Niebieskikwiat, M. B. Salamon, and J. M. Zuo. Phys. Rev. Lett. 94, 147206 (2005).\n12 T. Qian, G. Li, T. Zhang, T. F. Zhou, X. Q. Xiang, X. W. Kang, and X. G. Lia. Appl. Phys. Lett. 90, 12503 \n(2007).\n13 D. Niebieskikwiat and M. B. Salamon. Phys. Rev B 72, 174422 (2005).\n14 J. Nogués and I. K. Schuller. J. Magn. Magn. Mater. 192, 203 (1999).\n15 David J. Webb, F. Marshall, Arnold M. Toxen, T. H. Geballe, Robert M. White. IEE Trans. Magn. 24, 2013 \n(1988).Figure 1. X-ray diffraction patterns for different crystalline axis in the studied GdCa 1/3MO single crystal.\nArrows indicate the equivalent crystalline orientations in the pseudo cubic structure given by the cations\nsublattices.\nFigure 2. (a) Magnetization ( M) versus Temperature ( T) at 7500 Oe for different crystalline axis in a\nGdCa 1/3MO single crystal. Open Circle: (020); Close square: (200); and, Open triangle: (100) p-cub. (b) and (c) \nMagnetization ( M) versus Temperature ( T) in the (100) p-cub axis at 2500 and 5000 Oe, respectively. Close \ncircle: cooling; Open circle: warming.\nFigure. 3: Magnetization ( M) versus magnetic field ( H) at different temperatures: (a) 60 K; (b) 30 K; and (c) 5 \nK. The inset in (b) shows the presence of exchange bias at 18 K for a magnetic loop in a magnetic field range \n-1 T < H < 1 T.\nFigure 4. Saturation magnetization ( MS) versus Temperature ( T) obtained from magnetic hysteresis loops in a \nGdCa 1/3MO single crystal. Dashed line is guide by to the eye.\nFigure 5. Coercive field ( Hc) vs Temperature ( T) for different crystalline axis in a GdCa 1/3MO single crystal.\nFigure 6. Exchange bias field ( Heb) vs Temperature ( T) for different crystalline axis in a GdCa 1/3MO single\ncrystal.30 40 50 60 70(400)\n69.8 °I [arb. units]\n2T\u0003>q@(200)\n33.3 °(020)\n20 30 40 50 60 70(040)67.2 °I [arb. units]\n2T\u0003>q@(020)32.1 °(200)(100) p-cubFigure0 20 40 60 80 100 1200255075100M[emu / cm3]\nT [K](a)\n10 20 3004080\n10 20 30(b)M[emu/cm3]\nT [K](c)\nT [K]Figure0 5 10 15 200255075100M[emu/cm3]\nH [kOe] (020)\n (100) p-cub(a)\n(b)\n-30 -20 -10 0 10 20 30-120-80-4004080120\n-30 -20 -10 0 10 20 30-180-120-60060120180M[emu/cm3]\nH [kOe] (020)\n (100)p-cub(c)-0.3 0.0 0.3-202M[emu/cm3]\nH [kOe] (020)\n (100)p-orth\nM [emu/cm 3]\nH [kOe]Figure0 1 02 03 04 0020406080MS[emu/cm3]\nT[ K ]Figure10 20 30 40 50 6001000200030004000Hc[Oe]\nT[ K ](020) H = 3T\n(020) H = 1 T\n(100)p-cubH=3 T\n(100)p-cubH=1 TFigure5 1 01 52 02 53 0-900-600-3000300600900Heb[Oe]\nT[ K ](020) H = 3T\n(020) H = 1T\n(100)p-cubH= 1 TFigure"    },    {        "title": "0712.2269v1.Ordered_Phases_of_the_Anisotropic_Kagome_Lattice_Antiferromagnet_in_a_Field.pdf",        "content": "Ordered Phases of the Anisotropic Kagome Lattice Antiferromagnet in a Field\nE.M. Stoudenmire and Leon Balents\nDepartment of Physics, University of California, Santa Barbara, CA 93106-9530\n(Dated: April 25, 2022)\nThe antiferromagnetic Heisenberg model on an anisotropic kagome lattice may be a good minimal\nmodel for real magnetic systems as well as a limit from which the isotropic case can be better\nunderstood. We therefore study the nearest-neighbor Heisenberg antiferromagnet on an anisotropic\nkagome lattice in a magnetic \feld. Such a system should be well described by weakly interacting\nspin chains, and we motivate a general form for the interaction by symmetry considerations and by\nperturbatively projecting out the inter-chain spins. In the spin 1 =2 case, we \fnd that the system\nexhibits a quantum phase transition from a ferrimagnetic ordered state to an XY ordered state as the\n\feld is increased. Finally, we discuss the appearance of magnetization plateaux in the ferrimagnetic\nphase.\nI. INTRODUCTION\nFrustrated antiferromagnets are of considerable inter-\nest because frustrating interactions lead to strong \ruc-\ntuations both of classical (thermal) and quantum me-\nchanical origin. Unfortunately, even the simplest model\nHamiltonians for such materials are often not easily\nanalyzable by conventional theoretical techniques. An\noutstanding example is the nearest-neighbor quantum\nHeisenberg antiferromagnet on the kagome lattice. This\nstructure is frustrated to a particularly high degree, and\nthe extensive classical ground state degeneracy may be\nidenti\fed as a mechanism for strongly enhanced \ructua-\ntions. This strongly limits the usefulness of the standard\nsemi-classical spin wave technique. The behavior of the\nspinS= 1=2 case in zero \feld is particularly puzzling,\nwith di\u000berent numerical approaches yielding contradic-\ntory and/or puzzling results1,2,3,4,5,6,7. Analytical stud-\nies have not been any more illuminating as they necces-\nsarily involve numerous approximations that often lower\nthe symmetry of the problem substantially8,9,10,11,12,13.\nHowever, it should be noted that recent experimental\nwork on ZnCu 3(OH) 6Cl2seems to favor a gapless, mag-\nnetically disordered ground state14,15.\nOne window into the behavior of frustrated antifer-\nromagnets is their magnetization process , i.e. the curve\nof magnetization versus applied \feld. Variation of the\nmagnetization over the full range from zero to satura-\ntion opens another dimension of phase space in which\nto explore the phase diagram of these systems, and per-\nhaps \fnd physical and analytical insights. A qualitative\nfeature to be understood is the occurrence of plateaux\nwith quantized magnetization. The structure of these\nplateaux reveal some aspects of the correlations of the\nsystem. For the low-spin kagome lattice, there are sug-\ngestions of a plateau with M= 1=3Ms(Msis the satura-\ntion magnetization).6,16,17,18,19,20This is a very natural\nstructure for the kagome lattice, and can be understood\nas a state with two parallel and one anti-parallel spins per\ntriangle. One experiment, however, suggests a plateau\nwithM= 1=2Ms, which is not an obvious state21.\nIn this work, we consider an antiferromagnet on the\nanisotropic kagome lattice illustrated in \fg. 1, which may\nJJ`FIG. 1: The anisotropic kagome lattice with antiferromag-\nnetic coupling Jbetween nearest-neighbor chain spins ( S\nspins) and coupling J0betweenSspins and inter-chain spins\n(Ispins). We make the approximation that the Ispins are\nfully polarized by the external magnetic \feld.\nbe viewed as a set of spin chains (whose spins we call\nSspins) coupled together through intermediate spins ( I\nspins). The anisotropy could stem from an inhomogene-\nity in the spin interactions, or from an actual spatial\nanisotropy. A well known example of such a system is\nvolborthite, in which spin 1 =2 moments residing on cop-\nper atoms form a kagome network that is anisotropic due\nto di\u000bering superexchange bond angles.22Recently, there\nhas been considerable progress in developing techniques\nto analyze such systems of quantum spin chains weakly\ncoupled together by frustrating interactions. Such meth-\nods have the advantage that they can apply directly to\nlow spin and full Heisenberg ( SU(2)) symmetry, and fully\ninclude the e\u000bects of quantum \ructuations. Here we de-\ntermine the physics of the magnetization process in such\na limit of the kagome lattice.\nWe assume that the spins interact via a nearest neigh-\nbor Heisenberg interaction, and we take the magnetic\n\feld to be applied in the + zdirection. Taking the con-\nvention that lowercase indices only sum over sites on the\nspin chains ( S-spin sites) while uppercase indices only\nsum over intermediate I-spin sites, we decompose thearXiv:0712.2269v1  [cond-mat.str-el]  14 Dec 20072\nHamiltonian as\nH=H0+H0\nH0=JX\n<ij>~Si\u0001~Sj+J0X\n<Ai>Iz\nASz\ni\u0000hX\niSz\ni\u0000hX\nAIz\nA\nH0=J0\n2X\n<Ai>S+\niI\u0000\nA+S\u0000\niI+\nA:\nThis decomposition is convenient in the limit we focus\nupon, namely J0\u001cJandJ0\u001ch. Note that we need\nnot assume h=J is either large or small, and so can ex-\nplore (almost) the full magnetization process. The region\naroundh= 0 is excluded however by J0\u001ch. We leave\nthis more challenging limit for future work. We have cho-\nsen to incorporate the J0term coupling the z-components\nof chain spins to inter-chain spins into H0, which is con-\nvenient since that term commutes with the large \feld ( h)\nterm. This however means that the expansions in powers\nofH0and inJ0do not strictly coincide, since the latter\narise from two sources.\nBecauseJ0is much smaller than the applied \feld, the\nsystem may be described by a low-energy e\u000bective the-\nory involving only states in which the Ispins are fully\npolarized. In the limit J0=J!0, one expects that this\nlow-energy e\u000bective theory will be a set of independent\nLuttinger liquids describing each chain (this will always\nbe the case for S= 1=2 and will also hold for S= 1\nas long as the external \feld is larger than the zero-\feld\ngap to magnetic excitations). For very small J0, we con-\ntinue to assume that the Luttinger liquid description is\naccurate, but there must also be weak inter-chain inter-\nactions. While the inter-chain interactions may be calcu-\nlated, in principle, by symmetry considerations alone, we\n\fnd it convenient to use an approach combining pertur-\nbation theory and symmetry arguments that allows the\ncouplings to be estimated by hand.\nFor the case of a spin 1 system in a large enough \feld,\nwe \fnd that only one relevant interaction term emerges in\nan RG analysis of the general inter-chain coupling. Fur-\nther analysis predicts that the resulting long-wavelength\ntheory favors an XY antiferromagnetically ordered phase\nat zero temperature.\nFor the case of a spin 1 =2 system, however, an RG\nanalysis of the inter-chain coupling reveals two relevant\ninteraction terms. For large applied \feld, one interaction\ndominates the other in the long wavelength limit, leading\nto the same XY antiferromagnetically ordered phase as\nin the spin 1 case. However, for small applied \feld, the\nother relevant interaction dominates, leading to a fer-\nrimagnetically ordered phase. Finally, we reconcile the\ndiscrete translational symmetry of the microscopic lattice\nwith our continuum theory and show that magnetization\nplateaux may appear in the ferrimagnetic phase close to\nthe transition.II. PERTURBATIVE CALCULATION OF\nEFFECTIVE HAMILTONIAN\nSince we consider the limit J0=J\u001c1, the low energy\nstates are those in which the Ispins are completely polar-\nized by the external \feld, and so we work with an e\u000bec-\ntive Hamiltonian projected into the space of such states.\nWe approximately calculate this e\u000bective Hamiltonian as\nfollows.\nA. Elimination of inter-chain spins\nLetPbe the projection operator onto the subspace of\nall states with the Ispins fully polarized, and let Q= 1\u0000\nP. Furthermore, for a general state j iletj \"i=Pj i\nandj Qi=Qj iso thatj i=j \"i+j Qi.\nActing on the eigenvalue equation Hj i=Ej iwith\nPand then with Qgives the equations\nH0j \"i+PH0j Qi=Ej \"i (1)\nH0j \"i+H0j Qi+QH0j Qi=Ej Qi: (2)\nLettingR= (E\u0000H0)\u00001, (2) may be written as\nj Qi=RH0j \"i+RQH0j Qi\nand then iterated to give\nj Qi=1\n1\u0000RQH0RH0j \"i: (3)\nFinally, upon replacing j Qiin (1) by the expression\nin (3), we obtain an equation for j \"ialone:\n\u0014\nH0+PH0 1\n1\u0000RQH0RH0\u0015\nj \"idef=Heffj \"i=Ej \"i:\n(4)\nNote, however, that Eq.(4) is not really a linear eigen-\nvalue equation, since Heffitself depends on Ethrough\nR.\nB. Second Order Approximation\nWe would like to use the assumption that J0\u001cJ\nandJ0\u001chto simplify the expression for Heffand \fnd\nan approximate eigenvalue equation for j \"i. First, we\nmake the generalization\nH!H\u000b=H0+\u000bH0\nin order to more conveniently keep track of powers of H0.\nHenceforth, take j ito be the ground state of H\u000b. The\nground state energy admits an expansion\nE=E0+E1+E2+:::3\nwhereEn/\u000bnandE0is the ground state energy of H0.\nHeffmay then be expanded as\nHeff=H0+H2+O(\u000b3);\nwith\nH2=PH0R0H0; (5)\nandR0= (E0\u0000H0)\u00001.\nThe second term in the expansion of Heffmay be sim-\npli\fed by use of the fact that we are restricting it to act\nonly on states in the image of P. One \fnds\nH2j \"i\n=(\u000bJ0)2S\n2X\n<Aik>S\u0000\nk1\n(E0\u0000H0)\u0000h+J0P\njnnASz\njS+\nij \"i:\nWithin the Psubspace, we may drop the ket and write\nH2=X\nAHA\n2; (6)\nwith\nHA\n2=(\u000bJ0)2S\n2X\n<ik;A>S\u0000\nk1\n(E0\u0000H0)\u0000h+J0P\njnnASz\njS+\ni;\n(7)\nwherei;kare summed over the neighbors of A. Note\nthat, because of the J0dependence of the denominator\nin Eq.(7),HA\n2contains non-trivial terms at all orders of\nJ0greater than or equal to O[(J0)2]. This is convenient\nbecause it allows us to obtain some O[(J0)4] contributions\nby an only O(\u000b2) calculation.\nIII. CONTINUUM LIMIT AND SYMMETRY\nCONSIDERATIONS\nIt is well-known that the low energy physics of an\nisolated antiferromagnetic Heisenberg chain in a uni-\nform applied \feld is described by Luttinger liquid theory,\nwhenever the \feld range is such that the system remains\ngapless. In the limit J0\u001cJ;h, which we consider, all\nin\ruences of the inter-chain spins upon the spin chains\nindeed occur at low energies, and so can be considered in\nthis framework.\nLuttinger liquid theory consists of replacing the Hamil-\ntonian of each spin chain by that of a free boson \feld (and\ncorrections that can be analyzed perturbatively):\nH0!X\nnH(LL)\nn; (8)\nwhere\nH(LL)\nn=v\n2\u0019X\nnZ\nx\u00141\ng(@x'n)2+g(@x\u0012n)2\u0015\n(9)\n(a)(b)(c)FIG. 2: The generating symmetries of the space group cmm\nof the anisotropic kagome lattice: (a) translations, (b) ro-\ntations by \u0019about the hexagon centers, and (c) re\rections\nthrough a vertical line passing through the hexagon centers.\nis the Hamiltonian of a Luttinger liquid on chain n. Here\nthe \\spin velocity\" vand the Luttinger parameter gare\nknown functions of h=JandS. The variables \u0012nand'n\nare dual boson \felds living on the nthchain and satisfying\n['n(x);\u0012m(y)] =\u0000i\u0019\u000enm'(x\u0000y).\nThe physical interpretation of the phase \felds is un-\nderstood from their relation to the microscopic spin op-\nerators:\nS\u0006\nj;n\u0018(\u00001)je\u0006i\u0012n(xj)[b0+b1cos(2'n(xj) +Qxj)] (10)\nSz\nj;n\u0018mch+1\n\u0019@x'n(xj) +a1cos(2'n(xj) +Qxj):(11)\nThe coe\u000ecients a1,b0andb1depend on the chain mag-\nnetization per site mchand, forS= 1=2, the wavevector\nQis given by Q= 2\u0019(mch+ 1=2).\nFrom the above considerations, we expect that H2(and\nhigher order corrections) can also be expressed within the\ncontinuum (conformal) \feld theory. Formally, one may\nexpand any local Hamiltonian in terms of the scaling op-\nerators of the decoupled \fxed point theory. Speci\fcally,\nthe density can be written as\nHA\n2=X\n\u001b\u0015\u001bO\u001b\nnA;nA+1(xA); (12)\nwhereO\u001b\nn;n+1(x) is a local symmetry allowed operator\ninvolving degrees of freedom from the chains n;n+ 1\nimmediately above and below the site A, at horizontal\npositionx, with scaling dimension \u0001 \u001b. This implies that\nthe two point functions of these operators obey\nhO\u001b\nn;n+1(x)O\u001b\nn;n+1(x0)i=C\u001b\njx\u0000x0j2\u0001\u001b; (13)\nwith constants C\u001bthat are dependent upon the conven-\ntion for normalizing the \felds. The expectation value is\ntaken in the continuum conformal \feld theory (CFT) de-\nscribing the decoupled Heisenberg chains in a \feld. The\ntwo point function of two di\u000berent operators\u001b6=\u001b0van-\nishes if the operators have di\u000berent symmetry (or more\nformally are descended from di\u000berent primaries in the\nCFT). The sum over \u001bmay be thought of as being in4\norder of increasing scaling dimension. Terms with large\nscaling dimension are strongly irrelevant, and can there-\nfore be neglected.\nThe \frst few terms in this expansion are strongly con-\nstrained by symmetry. Considering the full microscopic\nHamiltonian, the symmetries consist of the lattice space\ngroup ( cmm - see \fg. 2) and the U(1) rotational symme-\ntry of the spins about the zaxis. One \fnds\nHA\n2=\u0015?cos[\u0012n\u0000\u0012n+1] +\u0015zcos[2'n\u00002'n+1] (14)\n+\u00150\nz@x'n@x'n+1\n+\u00150\n?@x\u0012n@x\u0012n+1cos[\u0012n\u0000\u0012n+1] +::::\nHere we have suppressed the Asubscript on nandxon\nthe right hand side to keep the formula compact.\nThe second order Hamiltonian, HA\n2, is actually fur-\nther constrained by an additional symmetry which one\nobtains only at this order. Speci\fcally, for each term\ninHA\n2associated with an inter-chain site A, the chains\nabove and below this site can be independently re\rected\nabout a vertical axis through site A. This symmetry in\nfact requires that \u0015?= 0 +O[(J0)4].\nA. General prescription for coe\u000ecients\nIn this subsection, we demonstrate that the above ef-\nfective continuum Hamiltonian is uniquely determined\nfrom the microscopic model, by showing how the coe\u000e-\ncients (\u0015\u001b) may be obtained in principle from correlation\nfunctions of decoupled Heisenberg chains. In the follow-\ning subsection, we will make some additional simpli\fca-\ntions in order to provide explicit expressions.\nIn general, we can proceed by demanding equality be-\ntween Eq.(7) and Eq.(14) (Eq.(12)). The coe\u000ecients canthen be extracted by taking expectation values of these\nquantities with microscopic operators whose continuum\noperator content is known, i.e.\n^Q\u001b\nj;n;n+1=c\u001bO\u001b\nn;n+1(xj) +\u0001\u0001\u0001; (15)\nwhere ^Q\u001b\nj;n;n+1is a microscopic expression composed of\nlattice spin operators in the vicinity of site j(horizontal\npositionx0) on chains n;n+ 1, and the omitted terms\nindicated by the ellipses contain only operators of larger\nscaling dimension than O\u001b\nn;n+1. For our problem, the\nneeded ^Q\u001boperators and coe\u000ecients are\n1\n2(S+\nj;nS\u0000\nj;n+1+S\u0000\nj;nS+\nj;n+1) =b2\n0cos[\u0012n\u0000\u0012n+1] +:::\nSz\nj;nSz\nj;n+1=a2\n1\n2cos[2'n\u00002'n+1]+1\n\u00192@x'n@x'n+1+:::\n1\n2(S+\nj;n+S+\nj+1;n)(S\u0000\nj;n+1+S\u0000\nj+1;n+1) + c.c. =\n=b2\n0@x\u0012n@x\u0012n+1cos[\u0012n\u0000\u0012n+1] +:::; (16)\nwherea1;b0are the coe\u000ecients in Eqs. (10, 11).\nKnowing these, one has\nh^Q\u001b\nj;nA;nA+1HA\n2i=c\u001b\u0015\u001bhO\u001b\nnA;nA+1(xj)O\u001b\nnA;nA+1(xA)i\n=c\u001b\u0015\u001bC\u001b\njxj\u0000xAj2\u0001\u001b+\u0001\u0001\u0001: (17)\nThe neglected terms decay faster with jxj\u0000xAj, and\ncan therefore be distinguished from the dominant term\nabove. Thus by calculating the left hand side in Eq.(17),\nand extracting its long-distance behavior, one obtains the\ncoe\u000ecient\u0015\u001b, sincec\u001b;C\u001bare presumed known.\nFrom this formulation, we can already deduce the order of the various \u0015\u001bcoe\u000ecients. Consider \frst \u00150\n?. We require:\n*\"\n(S+\nj;nA+S+\nj+1;nA)(S\u0000\nj;nA+1+S\u0000\nj+1;nA+1)\n2+ c.c.#\nHA\n2+\n=b2\n0C0\n?\u00150\n?\njxj\u0000xAj2\u00010\n?: (18)\nThe only obvious constraint to obtain a non-zero result for this correlation function is that the sites k;iin Eq.(7)\nde\fningHA\n2reside on neighboring chains nA;nA+ 1. There are 4 such combinations included in the sum in Eq.(7).\nTaking the limit J0!0 in the resolvent denominator, we obtain\n(\u000bJ0)2S\u001c\"\n(S+\nj;nA+S+\nj+1;nA)(S\u0000\nj;nA+1+S\u0000\nj+1;nA+1)\n2#\u0014\n(S\u0000\njA;nA+S\u0000\njA+1;nA)1\nE0\u0000H0\u0000h(S+\njA;nA+1+S+\njA+1;nA+1)\u0015\u001d\n=b2\n0C0\n?\u00150\n?\njxj\u0000xAj2\u00010\n?: (19)\nThere is no reason for the correlation function on the left hand side to vanish. To estimate it, we note that the\nresolvent denominator is negative for all eigenvalues and bounded below in magnitude by jE0\u0000H0\u0000hj>h. Hence\nwe estimate\n\u00150\n?.(\u000bJ0)2\nh; (20)5\nup toO(1) coe\u000ecients that can be smooth functions of h=J.\nConsider now the remaining two non-zero coe\u000ecients, \u0015z;\u00150\nz. Both may be obtained from the expectation value,\nwhich we denote G:\nG=\nSz\nj;nASz\nj;nA+1HA\n2\u000b\n=G2+G3+O[(J0)4]: (21)\nSince the number of particles in chains nAandnA+1 must be separately conserved, the sites i;kin the sum in Eq.(7)\nmust be on the same chain to obtain a non-zero result. Hence we may write\nG\u0018(\u000bJ0)2*\nSz\nj;nASz\nj;nA+1\u0002\nS\u0000\njA;nA+S\u0000\njA+1;nA\u0003 1\n(E0\u0000H0)\u0000h+J0P\njnnASz\nj\u0002\nS+\njA;nA+S+\njA+1;nA\u0003+\n: (22)\nFirst consider G2, the second order in J0contribution obtained by taking the J0!0 limit in the above resolvent. If\none takes this limit, then the only remaining operator on chain nA+ 1 which is not translationally invariant is the\nsingleSz\nj;nA+1. Applying a translation on chain nA+ 1, then, one sees that the Sz\nj;nA+1can be replaced by the same\noperator at any other site of the chain, i.e. Sz\nj0;nA+1, with arbitrary j0. Hence, it can be replaced by average over\nall sites, i.e. the uniform magnetization, which is a good quantum number and non-\ructuation. Upon making this\nsubstitution, one \fnds\nG2\u0018(\u000bJ0)2mch\u001c\nSz\nj;nA\u0002\nS\u0000\njA;nA+S\u0000\njA+1;nA\u0003 1\n(E0\u0000H0)\u0000h\u0002\nS+\njA;nA+S+\njA+1;nA\u0003\u001d\n(23)\n.\u0000(\u000bJ0)2\nhmch\nSz\nj;nA\u0002\nS\u0000\njA;nA+S\u0000\njA+1;nA\u0003\u0002\nS+\njA;nA+S+\njA+1;nA\u0003\u000b\n;\nwhere the .is to be understood as indicating an inequality of magnitude (not sign), using the same bound on the\ndenominator as before. The \fnal line is straightforwardly analyzed from known results for Heisenberg chains. Because\nthere is only a single \feld ( Sz\nj;nA) at position xj, the long distance decay of G2contains two pieces: an oscillatory\nterm decaying with the exponent \u0001 zand a non-oscillatory term decaying with the exponent \u00010\nz. The oscillatory term\nindicates a vanishing contribution in the continuum limit. The non-oscillating term has halfthe expected exponent,\n2\u00010\nzto correspond to a contribution to \u00150\nz. It instead represents the generation of a term proportional to @z'n, which\ngenerates a small smooth renormalization of the magnetization curve but is otherwise redundant. Thus neither\u0015z\nnor\u00150\nzare generated at O[(J0)2].\nIn order to calculate the leading contribution to \u0015zand\u00150\nzthen, one must calculate G3by expanding the resolvent\nin Eq.(22) to \frst order in J0, giving\nG3\u0018 \u0000J0(\u000bJ0)2*\nSz\nj;nASz\nj;nA+1\u0002\nS\u0000\njA;nA+S\u0000\njA+1;nA\u0003 1\n(E0\u0000H0)\u0000h\u0014X\nknnASz\nk\u00151\n(E0\u0000H0)\u0000h\u0002\nS+\njA;nA+S+\njA+1;nA\u0003+\n.\u0000J0(\u000bJ0)2\nh2*\nSz\nj;nASz\nj;nA+1\u0002\nS\u0000\njA;nA+S\u0000\njA+1;nA\u0003\u0014X\nknnASz\nk\u0015\u0002\nS+\njA;nA+S+\njA+1;nA\u0003+\n\u0018a2\n1\n2\u0015zCz\njxj\u0000xAj2\u0001z+1\n\u00192\u00150\nzC0\nz\njxj\u0000xAj2\u00010z(24)\nand so we \fnd that \u0015zand\u00150\nzare generally of order ( J0)3.\nB. Explicit Calculation for Spin 1=2System\nThe above results provide a general formulation to\ncalculate the coe\u000ecients in the e\u000bective Hamiltonian,\nEq.(14), from correlation functions of a single Heisen-\nberg chain in a \feld. Because of the complicated form\nofHA\n2, however, even these one-dimensional correlation\nfunctions are not simple to obtain analytically, or ex-\ntract from known results. In this subsection, therefore,\nwe make some additional simplifying assumptions which\nallow an explicit calculation of the e\u000bective Hamiltonian\nfor the spin 1 =2 system. This calculation con\frms thegeneral structure obtained above and allows for some\nsemi-quantitative estimates.\nConsider again the the second order (in \u000b) contribution\nto e\u000bective Hamiltonian density\nHA\n2=\n\u0012\u000bJ0\n2\u00132X\n<ik;A>S\u0000\nk1\n(E0\u0000H0)\u0000h+J0P\njnnASz\njS+\ni:\nWhen the operator above acts within the low energy sub-\nspace, one would like to say that both the term propor-\ntional toJ0and the (E0\u0000H0) term in the denominator\nare small, but it is not clear that ( E0\u0000H0) is small6\nunless it acts directly on the low energy states. So, we\nexchange the S+\nioperator and the resolvent using spin\n1=2 anticommutation relations to obtain\nHA\n2=\n\u0012\u000bJ0\n2\u00132X\n<ik;A>S\u0000\nkh\nSx\niRAi;y+iSy\niRAi;xi\n:\nwhere\nRAi;\u001b=2\n4(E0\u0000H0)\u0000h+J0X\njnnASz\nj\n\u00002hSz\ni+ 2JX\njnni(Sz\niSz\nj+S\u001b\niS\u001b\nj)3\n5\u00001\n:\nHowever, to be able to expand the RAi;\u001boperators,\nwe must make two rather unphysical assumptions. First,\nwe takeJ\u001chand second, we only let the magnetic\n\feld couple to Ispins. This second assumption may not\nbe too bad, however, since our eventual replacement of\nthe chains of Sspins with Luttinger liquids takes the\ne\u000bect of the external \feld into account via the Luttinger\nparameter.\nAfter making the approximations just discussed and\nexpanding, we \fnally obtain the following leading contri-\nbutions:\nHA\n2\u0019 \u0000(\u000bJ0)2\n4hX\n<ik;A>S\u0000\nkS+\ni\n\u0000(\u000bJ0)2J0\n4h2X\n<ijk ;A>S\u0000\nkS+\niSz\nj\n+(\u000bJ0)2J\n4h2X\n<ik;A>X\njnniS\u0000\nkS+\niSz\nj\u0000S\u0000\nkSz\niS+\nj:\n(25)\nIn what follows, we set \u000bback to 1.\nTo connect this result to the continuum limit of the\ninterchain coupling Eq.(14), we now proceed to bosonize\nthe spin operators through the identi\fcations in Eq.(10)\nand Eq.(11). Keeping only contributions that involve\nproducts of operators from both chains, we \fnd from the\n\frst term\nX\n<ik;A>S\u0000\nkS+\ni=\n(S\u0000\njA;nA+S\u0000\njA+1;nA)(S+\njA;nA+1+S+\njA+1;nA+1)\n+ [nA$nA+ 1]\n= 2b2\n0@x\u0012n@x\u0012n+1cos[\u0012n\u0000\u0012n+1] +:::\nwhich con\frms that \u00150\n?is of order J02, as we already\nargued by symmetry.\nThe second term in Eq.(25) must be treated with more\ncare, however. Now, we only need to keep operators that\nconserve particle number on each chain, since we havealready found the leading contribution to \u00150\n?and since\n\u0015?=O[(J0)4] by symmetry. After also dropping opera-\ntors that do not have contributions from both chains, we\n\fndX\n<ikj ;A>S\u0000\nkS+\niSz\nj= (S\u0000\njA;nA+S\u0000\njA+1;nA)(S+\njA;nA+S+\njA+1;nA)\n\u0002(Sz\njA;nA+1+Sz\njA+1;nA+1) + [nA$nA+ 1]\n= (Sz\njA;nA+1+Sz\njA+1;nA+1) (1\u0000Sz\njA;nA\u0000Sz\njA+1;nA\n+fS\u0000\njA;nAS+\njA+1;nA+S+\njA;nAS\u0000\njA+1;nAg)+[nA$nA+1]:\nIt is the last operator in curly braces (the dimerization\noperator) that must be bosonized carefully. Since it in-\nvolves products of operators at very short distances, all\nsubleading bosonic operators omitted in Eq.(10) must be\nresummed, leading to the following continuum limit23\n(S\u0000\nj;nS+\nj+1;n+S+\nj;nS\u0000\nj+1;n)\u0018c0+c0\n0\n\u0019@x'n\u0000c1cos[2'n+Qx]:\n(26)\nAfter using the above identity and collecting all relevant\nterms, we \fnd that\nX\n<ikj ;A>S\u0000\nkS+\niSz\nj=2\n\u00192(c0\n0\u00004)@x'nA@x'nA+1 (27)\n\u0000(2a2\n1+a1c1)(1 + cosQ) cos[2'nA\u00002'nA+1] +:::\ndemonstrating that \u0015zand\u00150\nzare both of order J03. The\nlast term in Eq.(25) does not generate any relevant in-\nteractions in the bosonic theory.\nTo summarize, then, we have explicitly calculated the\ncouplings in the e\u000bective Hamiltonian Eq.(14) for the\nspin 1=2 system up to third order in J0and the re-\nsults con\frm the predictions of our symmetry arguments\n(which apply to systems with any spin). The explicit\ncouplings are\n\u0015?= 0\n\u00150\n?=\u0000J02b2\n0\n2h\n\u0015z=J03\n2h2(2a2\n1+a1c1)(1 + cosQ)\n\u00150\nz=J03\n\u00192h2(4\u0000c0\n0): (28)\nIV. ANALYSIS OF THE CONTINUUM MODEL\nWe have arrived, then, at a low energy e\u000bective the-\nory describing the Heisenberg antiferromagnet on the\nanisotropic kagome lattice that takes the general form\nHeff=\nX\nnZ\nx\u0014v\n2\u00191\ng(@x'n)2+vg\n2\u0019(@x\u0012n)2+\u00150\nz@x'n@x'n+1\n+\u0015?cos[\u0012n\u0000\u0012n+1] +\u0015zcos[2'n\u00002'n+1]\n+\u00150\n?@x\u0012n@x\u0012n+1cos[\u0012n\u0000\u0012n+1] +:::\u0015\n: (29)7\nA. RG Analysis\nTo better understand the behavior of the above model\nat large length scales, we employ the momentum space\nRG procedure, lowering the cuto\u000b from \u0003 !\u0003\u0000d\u0003\nand de\fning dl=d\u0003=\u0003. Including the marginal term\n\u00150\nz@x'n@x'n+1in the \fxed-point action and expanding\nall quantities to \frst order in \u00150\nzleads to the following\ntree-level \row equations for the other couplings (note\nthat\u00150\nzmay be regarded as a constant independent of\nthe scale of the e\u000bective theory):\nd\u00150\n?\ndl=\u0012\n\u00001\n2g+\u0019\u00150\nz\n4v\u0013\n\u00150\n?\nd\u0015?\ndl=\u0012\n2\u00001\n2g+\u0019\u00150\nz\n4v\u0013\n\u0015?+\u0019\u00032\u00150\nz\n16v\u00150\n?\nd\u0015z\ndl=\u0012\n2\u00002g\u0000\u0019g2\u00150\nz\nv\u0013\n\u0015z: (30)\nThese \row equations imply that for \u00150\nz6= 0 and\n\u00150\n?6= 0, a \fnite value of \u0015?will be generated in the\nlong-wavelength theory. Also, one can see that the \u0012-\ninteraction (with coupling \u0015?) is relevant for g&1\n4\u0000\u0019\u00150\nz\n32v\nwhile the'-interaction (with coupling \u0015z) is relevant for\ng.1\u0000\u0019\u00150\nz\n2v.\nNow for the case of spin 1 chain, F\u0013 ath has determined\nnumerically that gis always greater than one (see F\u0013 ath24,\n\fg. 5). In particular, gis one at a chain magnetization\nper site ofmch= 0, reaches a maximum value of g\u00191:46\natmch\u00190:36 and approaches one again in the limit of\nfull polarization. So, for a spin 1 system described by\n(29), the\u0012-interaction is always relevant, while the '-\ninteraction is always irrelevant.\nOn the other hand, for a spin 1 =2 chain, the exact\nBethe ansatz solution shows that gincreases nearly lin-\nearly from 1 =2 to 1 as the chain magnetization per site\nincreases from mch= 0 tomch= 1=2 (see Hikihara and\nFurusaki25, \fg. 2). So for a spin 1 =2 system described by\n(29), the'-interaction and the \u0012-interaction are nearly\nalways relevant, and we must \fnd a further criterion to\ndetermine which interaction dominates the behavior of\nthe system. So, we solve the \row equations and deter-\nmine which coupling constant becomes \u00181 \frst as one\ncontinues to integrate out high momentum modes.\nB. Competing Interactions in the Spin 1/2 System\nWe take the initial conditions given by Eq.(28). Then,\nde\fning\n\f0\n?=\u0012\n\u00001\n2g+\u0019\u00150\nz\n4v\u0013\n;\n\f?=\u0012\n2\u00001\n2g+\u0019\u00150\nz\n4v\u0013\n\fz=\u0012\n2\u00002g\u0000\u0019\u00150\nzg2\nv\u0013the solutions to the \row equations for \u0015?and\u0015zare\n\u0015?(l) =J05\nh3\u00032b2\n0\n64\u0019v(4\u0000c0\n0)\u0010\ne\f?l\u0000e\f0\n?l\u0011\n\u0019J05\nh3\u00032b2\n0\n64\u0019v(4\u0000c0\n0)e\f?l\n\u0015z(l) =J03\n2h3(a1c1+ 2a2\n1)(1 + cosQ)e\fzl:\nWe can ignore the term \u0018e\f0\n?lin\u0015?(l) since it rapidly\ngoes to zero unless g&1=\u00150\nz. Identifying elasL=a, i.e.\nthe ratio of the renormalized cuto\u000b length scale to the\nbare lattice scale, we \fnd that\n\u0015?(l)\u0018J05\u0012L\na\u0013\f?\n\u0015z(l)\u0018J03\u0012L\na\u0013\fz\nIf we then de\fne L?andLzas the length scales at\nwhich the\u0012and'couplings, respectively, become \u00181,\nwe \fnd that\nLz\nL?\u0018(J0)5=\f?\u00003=\fz\nThe exponent is positive for 0 <g<gcand negative for\ngc<g.1, wheregc\u00190:636\u00000:635\u00150\nz\nv+:::This means\nthat forg < gc,Lz\u001cL?and the'interaction is the\ndominant one. For g>gc, the\u0012interaction is dominant.\nC. Ordered Phases in the Gaussian Approximation\n1. Ferrimagnetic Phase\nFirst, consider a system described by the Hamiltonian\n(29) in the limit where g<gc, that is a spin 1 =2 system\nwith magnetization per site between m= 0 andm\u00191=5\n(on a scale where full polarization is m= 1=2). In this\nlimit, the dominant relevant coupling is the '-interaction\nwith coupling \u0015z, so we drop all other couplings not in the\n\fxed-point action. The \u0012\felds may then be integrated\nout to yield an action entirely in terms of the '\felds:\nS\u0019X\nnZ\n\u001cZ\nxv\n2\u0019g\u0014\n(@x'n)2+1\nv2(@\u001c'n)2\u0015\n+\u00150\nz@x'n@x'n+1+\u0015zcos[2'n\u00002'n+1]: (31)\nSince the interaction term in the above theory is rele-\nvant, the long-wavelength modes of the '\felds will \ruc-\ntuate very little from chain to chain if the coupling \u0015zis\nnegative, while they will di\u000ber by \u0019=2 from chain to chain\nif the coupling is positive. If \u0015zis positive, then, rede\fne\nthe'\felds by'n(x)!'n(x) +\u0019\n2n. The coupling term\nfor these new \felds will now be negative.\nAfter the appropriate rede\fnition, \ructuations of the\n'\felds from chain to chain will be suppressed at zero8\nFIG. 3: An example of a ferrimagnetically ordered ground\nstate of the spin 1/2 system with magnetization m.1=5 and\n\u0015z<0.\ntemperature, and so it is reasonable to expand the cosine\nterm in the action to second order about zero. Upon\nperforming a fourier transform in the chain index n, the\nresulting gaussian action is\nS\u0019v\n2\u0019gZ\u0019\n\u0000\u0019dk\n2\u0019Z\nx;\u001c\u0014\n(1\u00002\u0019g\nv\u00150\nzcosk)j@x'kj2+1\nv2j@\u001c'kj2\n+m2\n'(k)v2j'kj2\u0003\n(32)\nwhere the e\u000bective mass m'is de\fned as\nm2\n'(k) =8\u0019g\nv3(1\u0000cosk)\u0015z: (33)\nA system described by the action (32) has excitations\nwith a dispersion relation\n!2\n'(~ p;k) = (1\u00002\u0019g\nv\u00150\nzcosk)p2v2+m2\n'(k)v4(34)\nwhich vanishes as ~ p; k!0, implying the existence of\ngapless excitations. Indeed, the action (32) is invariant\nunder the transformation\n'k=0(x)!'k=0(x) + \u0016'( \u0016'constant)\nbecause of the invariance of the system under translations\nalong the chain or x-direction. The gapless excitations\nare the Goldstone modes associated with the breaking of\nthis symmetry.\nRestoring the original de\fnition of the \felds 'n(x) as\nappropriate, we see that for small enough g, the'n\felds\nwill be ordered as h'n(x)i= \u0016'+\u0019\n2nif\u0015zis positive and\nh'n(x)i= \u0016'if\u0015zis negative. Then, from Eq.(11) we see\nthat the spins will be ordered ferrimagnetically:\nhSz\nn(x)i\u00188\n<\n:mch+\u000em(\u00001)ncos(2 \u0016'+Qx) if\u0015z>0\nmch+\u000emcos(2 \u0016'+Qx) if\u0015z<0:\n(35)\n(see \fg. 3 for an illustration). The amplitude of\nthe ferrimagnetic spin density wave oscillation, \u000em\u0018\nSo, in this regime we expect that the    fields order, which, in terms of the spins corresponds to XY antiferromagnetic ordering:                         .ϕ/angbracketleftS+j/angbracketright∼(−1)jei¯ϕFIG. 4: The XY antiferromagnetically ordered ground state\nthat occurs for both the spin 1/2 and spin 1 systems in a large\nenough \feld ( \u0015?<0 case shown).\na1(J0=J)3g=(2\u00002g)is small for small J0, due to the fact\nthatSz\nn(x) is a continuum operator with all \ructuations\non length scales below Lzintegrated out. Since g<1 in\nthe spin 1=2 model, this factor will tend to suppress the\noscillations in the magnetization.\n2. XY Antiferromagnetic Phase\nNow, consider a system described by the Hamiltonian\n(29) but in the limit where g > gc, namely a spin 1 =2\nsystem with magnetization per site m&1=5 or a spin 1\nsystem in a su\u000eciently large \feld (larger than both the\nHaldane gap and J0). In this limit, the dominant relevant\ncoupling is the \u0012-interaction with coupling \u0015?, so we drop\nall other couplings not in the \fxed-point action.\nWe proceed much as before, integrating out the '\felds\nand rede\fning the \u0012\felds if\u0015?is positive. In this\ncase, however, the appropriate rede\fnition is \u0012n(x)!\n\u0012n(x) +\u0019n. Once the \felds are de\fned such that the co-\nsine interaction makes them slowly varying at zero tem-\nperature, we expand the cosine to obtain a gaussian the-\nory once again, but with an e\u000bective mass\nm2\n\u0012(k) =2\u0019\ngv3(1\u0000cosk)\u0015? (36)\nand a dispersion\n!2\n\u0012(p;k) = (1\u00002\u0019g\nv\u00150\nzcosk) [p2v2+m2\n\u0012(k)v4]: (37)\nThe gapless modes in this limit are the Goldstone\nmodes associated with the breaking of the shift symmetry\n\u0012k=0(x)!\u0012k=0(x) +\u0016\u0012(\u0016\u0012constant)\nassociated with the global rotational symmetry of all\nspins about the magnetic \feld direction.\nRestoring the original \felds, we expect that, for large\nenoughg,h\u0012n(x)i=\u0016\u0012+\u0019nif\u0015?>0 orh\u0012n(x)i=\u0016\u0012if\n\u0015?<0. From the identi\fcation (10), we therefore expect9\nan XY antiferromagnetic ordering of the spins:\nhS+\nj;ni\u0018\u001a\n(\u00001)j+nmsei\u0016\u0012if\u0015?>0\n(\u00001)jmsei\u0016\u0012if\u0015?<0:; (38)\nwherems\u0018b0(J0=J)5=(8g\u00002)is the staggered magneti-\nzation, which is reduced by \ructuations below the scale\nL?. See \fg. 4 for an illustration.\nD. Lattice E\u000bects in the Spin 1=2Ferrimagnetic\nPhase\n1. Commensurate Ferrimagnetic Order\nFor a spin 1 =2 system with g < gc, the ordering of\nthe zero mode of the '\feld corresponds to the break-\ning of the translational symmetry of the system in the\nxdirection (along the chains). However, this symme-\ntry is not actually continuous in the microscopic theory,\nbut reduces to the discrete translational symmetry of the\nlattice.\nIn order to reconcile the continuum and microscopic\ntheories then, we observe that the Hamiltonian (29) may\nalso contain terms of the form cos[2 p'n(x)] for certain\nvalues ofpthat must be determined by symmetry.\nUnder translation by one lattice spacing along the\nchain direction, the '\felds transform as\n'n(x)!'n(x\u00001)\u0000Q=2: (39)\nSo, for a term of the form cos[2 p'n(x)] to respect the\ntranslational symmetry of the system, it must be that\npQ= 2\u0019k, wherekis an integer. Moreover, the action\nshould be invariant under 'n(x)!'n(x) +\u0019, sokmust\nbe chosen to make pan integer.\nRecalling that Q= 2\u0019(mch+ 1=2), both requirements\nimply that only the set fpjgare allowed, where\npj=kj\nmch+1\n2(40)\nandkjis thejth positive integer (in increasing order)\nsuch thatpjis an integer. We may identify\n\u0016\u0015=1\nmch+1\n2=2\u0019\nQ(41)\nas the wavelength of the (continuum) ferrimagnetic wave\n(35). But, for a general magnetization, the wavelength\n\u0016\u0015will not be an integer multiple of the lattice spacing.\nInstead, the ferrimagnetic wave above, when restricted to\nvalues ofxthat coincide with the lattice, will appear to\nbe periodic with an e\u000bective wavelength \u0015eff=k0\u0016\u0015=p0.\nSo then, given that for a \fxed magnetization the cor-\nresponding term cos[2 p0'n] will appear in the Hamilto-\nnian, it will bring the discrete translational symmetry of\nthe lattice to bear on the ordered phase by restricting the\nvalues ofh'n(x)i= \u0016'to coincide with its minima. For\nhm\nFerri-MagnetXY AntiferromagnetSaturated1/21/6J`1/5FIG. 5: Schematic graph of the magnetization curve and\nphases of the spin-1/2 system. Note the hypothetical plateau\njust below the crossover point. The shaded region indicates\nthe parameter space where our model does not apply.\na negative coe\u000ecient of cos[2 p0'n], this will force the\nferrimagnetic oscillations to be symmetric about a par-\nticular lattice site. On the other hand, if the coe\u000ecient\nis positive the oscillations will be symmetric about the\npoint on the lattice between two particular neighboring\nsites.\n2. Magnetization Plateaux\nIn addition to pinning the ferrimagnetic order to the\nlattice, the symmetry-allowed cos[2 p'n(x)] terms in the\nHamiltonian may lead to the appearance of plateaux in\nthe magnetization curve of the spin 1/2 system.\nFor general (but small) values of the external \feld, the\nground state of the spin 1 =2 system will be, in bosonic\nterms, a state in which the 'n(x) \felds on each chain \ruc-\ntuate about some constant average value as we discussed\nin detail above (we take the coupling \u0015zto be negative in\nthis section for simplicity - the other case follows straight-\nforwardly). In such states, the chain magnetization per\nsite smoothly oscillates as a function of xabout its av-\nerage (to zeroth order in J0) valuemch. Note that mch\nis simply the magnetization of a decoupled Heisenberg\nchain and as such, is a monotonially increasing function\nof the external magnetic \feld.25\nNow consider some particular magnetization mP\n(which corresponds to a chain magnetization mP\nch=\n3mP=2\u00001=4) and \frst assume that the external \feld\nhas been tuned to a value hPsuch thatmch(hP) =mP\nch.\nIn such an external \feld, the most important symmetry\nallowed term of the type cos[2 p'n(x)] that may be added\nto the Hamiltonian is the one with the smallest value of10\npgiven by Eq.(40), namely\np0=k0\nmP\nch+1\n2; (42)\nwherek0is de\fned as the smallest positive integer such\nthatp0is an integer.\nTo illustrate, then, how a plateau may arise at this\nmagnetization, we consider the Hamiltonian\nHeff=X\nnZ\nxv\n2\u0019g\u0014\n(@x'n)2+1\nv2(@\u001c'n)2\u0015\n+\u00150\nz@x'n@x'n+1\n\u0000j\u0015z0jcos[2('n\u0000'n+1)]\u0000\u00110cos[2p0'n+\u000ex] +:::\n(43)\nand no longer think of the external \feld as \fxed, but\nas varying in some small range about hP. Since the co-\ne\u000ecient\u000eis zero for h=hPby the de\fnition of p0,\nsymmetry considerations show that it must be given in\ngeneral by\n\u000e(h) = 2\u0019p0[mch(h)\u0000mP\nch]: (44)\nNow, if we integrate out the short wavelength modes\nof the system in an iterative RG procedure (but with-\noutrescaling the high momentum cuto\u000b), the e\u000bective\ncoupling\u0015zwill vary as\n\u0015z[L]\u0018\u0015z0(L)\u00002g(45)\nuntil we reach a length scale Lzde\fned such that\n\u0015z[Lz]\u00181=(Lz)2(i.e. this is the scale such that \u0015zwould\nbeO(1) if we were rescaling at every RG step). There-\nfore, we \fnd that\nLz\u0018(\u0015z0)\u00001=(2\u00002g): (46)\nAt this scale, the chains will be strongly coupled and the\n'\felds will vary slowly from chain to chain.\nThen, since the '\felds vary slowly from chain to\nchain, it is reasonable to take a continuum limit 'n(x)!\n'(x;y)=p\nd(wheredis the chain spacing). Within\nsuch an e\u000bective (2 + 1)-dimensional theory, the long-\nwavelength \ructuations of the \feld '(x;y) are bounded\nand so the term in the action \u00110cos[2p0'n], which may\nhave been an irrelevant perturbation to a decoupled chain\nis now a relevant perturbation, regardless of the value of\np0.\nNow, since the scaling dimension of the cos[2 p0'] term\nisp2\n0g, the e\u000bective coupling \u0011becomes\n\u0011(Lz)\u0018\u00110\u0001(Lz)\u0000p2\n0g\u0018\u00110\u0001(\u0015z0)p2\n0g\n2\u00002g (47)as the momentum cuto\u000b is lowered to 1 =Lzunder the\nRG procedure. Since \u0015z0is small (generally \u0018J03) and\ng<1 for the spin 1 =2 system in its ferrimagnetic phase,\nthis renormalized value of \u0011will decrease rapidly as a\nfunction of p0.\nFinally, then, we determine if a plateau develops\naround the magnetization mPby considering the com-\npetition between the kinetic energy terms in Heffand the\nterm with coupling \u0011. The latter term is minimized if\nh'n(x)i=\u0000\u000e(h)\n2p0x(that is, ifhSz\nn(x)i=mP\nchregardless\nof the external \feld - a plateau). However, such a \feld\ncon\fguration costs some kinetic energy and the resulting\ntotal change in the linear energy density is\n\u0001En(x)\u0018v\n2\u0019g\u0012\u000e(h)\n2p0\u00132\n\u0000\u0011=v[\u000e(h)]2\n8\u0019gp2\n0\u0000\u0011 : (48)\nTherefore, we see that this arrangment remains favorable\n(i.e. the plateau exists) as long as\nj\u000e(h)j<p\u00112p0r\n2\u0019g\nv(49)\n=) jmch(h)\u0000mP\nchj.(J0)3p2\n0g=4(1\u0000g):\nA signi\fcant plateau will only appear, then, for a rel-\natively small p0, which is itself determined by mP\nch\nthrough (42).\nWe therefore expect the widest plateau, if one appears\nat all, to appear just below the crossover point from the\nferrimagnetic to the XY antiferromagnetic phase in the\nspin 1=2 system (see \fg. 5). Furthermore, we may esti-\nmate the dependence of the plateau width on J0from our\nearlier estimate that gc\u00180:636 and from the fact that\nthe smallest p0allowed by Eq.(42) near the crossover is\nabout 9. Using these values in Eq.(50), the power of J0is\n\u0018100, so any plateau near the crossover will be narrow\nindeed.\nAcknowledgments\nThis work was supported by the Packard Founda-\ntion and the National Science Foundation through grant\nDMR04-57440.\n1J. Chalker and J. Eastmond, Phys. Rev. B 46, 14201\n(1992).2P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and11\nP. Sindzingre, Phys. Rev. B 56, 2521 (1997).\n3W. et al., Eur. Phys. J. B 2, 501 (1998).\n4P. Nikolic and T. Senthil, Phys. Rev. B 68, 214415 (2003).\n5G. Misguich and P. Sindzingre, J. Phys. Condens. Matter\n19, 145202 (2007).\n6D. C. Cabra, M. D. Grynberg, P. C. W. Holdsworth, A. Ho-\nnecker, P. Pujol, J. Richter, D. Schmalfuss, and J. Schu-\nlenburg, Phys. Rev. B 71, 144420 (2005).\n7R. Singh and D. 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Goennenwein4, 7, 8, 9,†\n1Institut f¨ ur Physik, Johannes Gutenberg-Universit¨ at Mainz, 55099 Mainz, Germany\n2Graduate School of Excellence Materials Science in Mainz, 55128 Mainz, Germany\n3Quantum Condensed Matter Division,\nOak Ridge National Laboratory, Oak Ridge, 37830 TN, United States\n4Walther-Meißner-Institut, Bayerische Akademie\nder Wissenschaften, 85748 Garching, Germany\n5Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE),\nKing Abdullah University of Science and Technology (KAUST), Thuwal, 23955, Saudi Arabia\n6now at: Erich Schmid Institute of Materials Science,\nAustrian Academy of Sciences, A-8700 Leoben, Austria\n7Physik-Department, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany\n8Nanosystems Initiative Munich (NIM),\nSchellingstraße 4, 80799 M¨ unchen, Germany\n9Institut f¨ ur Festk¨ orperphysik, Technische Universit¨ at Dresden, 01062 Dresden, Germany\n1arXiv:1703.03218v2  [cond-mat.mtrl-sci]  10 Mar 2017Abstract\nWe investigate the generation of magnonic thermal spin currents and their mode selective spin\ntransport across interfaces in insulating, compensated ferrimagnet/normal metal bilayer systems.\nThe spin Seebeck effect signal exhibits a non-monotonic temperature dependence with two sign\nchanges of the detected voltage signals. Using different ferrimagnetic garnets, we demonstrate\nthe universality of the observed complex temperature dependence of the spin Seebeck effect. To\nunderstand its origin, we systematically vary the interface between the ferrimagnetic garnet and\nthe metallic layer, and by using different metal layers we establish that interface effects play a\ndominating role. They do not only modify the magnitude of the spin Seebeck effect signal but\nin particular also alter its temperature dependence. By varying the temperature, we can select\nthe dominating magnon mode and we analyze our results to reveal the mode selective interface\ntransmission probabilities for different magnon modes and interfaces. The comparison of selected\nsystems reveals semi-quantitative details of the interfacial coupling depending on the materials\ninvolved, supported by the obtained field dependence of the signal.\n2In recent years, the generation and flow of spin currents and thus the transport of spin\ninformation has been studied intensively as it is an essential prerequisite for the realization\nof spintronic devices. In spintronics not only the electron charge but also its spin state is\nused for information transmission and manipulation1,2, which promises reduced power con-\nsumption compared to conventional charge-based electronics that entails ohmic losses. The\nviability of this concept has recently been corroborated for instance by the demonstration\nof multi-terminal logic majority gates based on the superposition of spin currents3–6.\nIn general, spin currents can be generated in magnetic materials in different ways as, for\ninstance, ferromagnetic resonance (FMR) based spin pumping7–9, electrical injection by the\nspin Hall effect (SHE)10,11or the excitation of thermal spin currents via the spin Seebeck\neffect (SSE)12–14. The induced spin currents are then detected in an adjacent normal non-\nmagnetic metal (NM) layer by the inverse spin Hall effect (ISHE)15. SSE measurements on\nY3Fe5O12/Pt (YIG/Pt) bilayers with different YIG thicknesses revealed a strong thickness\ndependence of the effect, confirming that in ferromagnetic insulators (FMI) thermally gen-\nerated spin currents are carried by magnonic excitations of the spin system14. Beyond the\nkey influence of bulk properties for the spin current generation and transport in insulat-\ning ferrimagnets, indications were found that the SSE signal also depends on the YIG/NM\ninterface16–18. Guo et al.18have demonstrated that in different YIG/NM hybrids both the\nSSE amplitude and its temperature dependence are influenced by the NM material. This\nsuggests a more complex behavior of the SSE signal strength and sign with a possible expla-\nnation being based on different frequency and momentum dependencies of spin transmission\nprobabilities across different interfaces to different NMs. While different materials are shown\nto yield different SSE characteristics, it is not clear to what extent material properties or the\nmorphology of the interface dominate spin transmission properties. Previous findings have\nnot allowed for a quantitative understanding, as the magnetic structure of YIG prohibits\na distinction and separation between the different magnon modes that make up the total\nspin current. Furthermore, given indications by spin pumping measurements that the signal\namplitude measured only weakly depends on the excitation frequency19, it is perceived as\ncommon wisdom that the spin transmissivity across interfaces is magnon mode independent.\nThe possible mechanism of different transmission amplitudes across the interface is therefore\ndifficult to resolve as there is no means to distinguish between the modes excited.\nRecent SSE experiments performed using the compensated ferrimagnet Gd 3Fe5O12\n3(GdIG), which comprises a more complex magnetic sublattice configuration, have revealed\na more complicated temperature dependence of the SSE signals20. When measured as a\nfunction of temperature, two sign changes of the signal amplitude appear. The sign change\noccurring near the so-called magnetic compensation point, where the magnetizations of\nthe sublattices cancel each other out, is explained by the reorientation of the magnetic\nsublattices20. A second sign change at lower temperatures, however, is the result of a com-\npetition between two bulk magnon modes with opposite precession yielding spin currents of\nopposite spin polarity. The first mode is the fundamental ferrimagnetic magnon mode ex-\nhibiting a gapless dispersion with a narrow bandwidth, while the second mode is gapped by\nthe highly temperature dependent exchange coupling between the Gd and the Fe sublattice.\nThis demonstrates that the SSE is a probe of magnon properties of the respective material.\nWhile established for GdIG, it is unclear if this complex behavior is a universal phenomenon\nthat occurs for other ferrimagnetic materials as well. Given this particular temperature\ndependence, the SSE experiments in different compensated ferrimagnets provide the unique\nopportunity to infer the importance of bulk vs interface effects for the spin transport across\nthe interface. The evolution of the SSE signal at temperatures around the low temperature\nSSE sign change allows one to select the dominating magnon mode and thus to probe the\nmode-selective spin transmission across the interface to the NM, which has not been possible\nso far. Furthermore, intentional modifications of the interface as well as using different NM\nlayers for the spin current detection yield access to the magnon selective interface-related\nspin transmissivity properties.\nIn this Letter we compare the SSE response of different rare-earth iron garnets in order\nto extract the mode- and interface-dependent spin transmission. We start by checking\nwhether the complex SSE behavior found in Gd based iron garnets indeed is a universal\nphenomenon for rare-earth iron garnets with varying thickness. To isolate the interface effect,\nwe systematically vary the interfaces of the FMI/NM bilayers by either changing the NM\ndetection layer or using different surface treatments of the FMI before the NM deposition.\nFinally, we combine this with the high magnetic field influence on the temperature dependent\nspin transmission efficiency and identify the spin current contribution of distinct magnon\nmodes.\nFor our investigation of the temperature dependent excitation of thermal magnons we\nhave selected ferrimagnetic garnet thin films of the stoichiometry Gd 3Fe5O12, Dy 3Fe5O12and\n4(Gd 2Y)(Fe 4In)O 12(GdYInIG). We chose GdYInIG to study the effect of combining mag-\nnetic (Gd) and non-magnetic (YIn) rare earth ions while Dy 3Fe5O12is used to investigate\nthe effect of other magnetic rare-earth ions (Dy) compared to the previously used Gd (in\ncontrast to Gd3+ions, Dy3+ions feature a finite orbital moment ( L= 5)). All thin (20 nm\nto 200 nm) films are grown epitaxially on either Gd 3Ga5O12(GGG) or Y 3Al5O12(YAG)\nsubstrates by pulsed laser deposition in an oxygen atmosphere. Afterwards, the Pt or Pd\ndetection layer is deposited by magnetron-sputtering (ex-situ) or electron beam evaporation\n(in-situ) on top of the garnet thin film. Similarly to our previous work18, high resolution\nx-ray diffraction as well as transmission electron microscopy have been used in order to check\nfor the quality of the deposited films and the interfaces between them. X-ray reflectivity and\natomic force microscopy reveal smooth surfaces of the grown garnet films (see supplemental\ninformation21). The respective high resolution scanning TEM (HR-STEM) images and elec-\ntron energy loss spectroscopy (EELS) elemental maps clearly reveal the crystalline structure\nof the grown garnet films as well as well-defined interfaces between epitaxial ferrimagnetic\ngarnet thin film and the normal metal layer. Additionally, superconducting quantum in-\nterference device (SQUID) magnetometry measurements were carried out for each sample\nto determine the magnetization compensation point. For garnet films grown on YAG sub-\nstrates, the temperature dependence of the magnetization M has been measured by sweeping\nthe system temperature while applying an external magnetic field of µ0H = 1 T. For thin\nfilms on GGG substrates, however, the large paramagnetic moment of GGG prohibits this\nprocedure. M thus has been determined via magnetic field sweeps at constant temperatures.\nSSE measurements are performed in different setup geometries and measurement meth-\nods. Samples produced at the WMI are patterned into a Hall bar mesa structure, as shown in\nFig. 1 a), which allows for simultaneous heating and signal detection20,22. To detect the SSE\nsignal, either measurements as a function of the magnetic field magnitude at a fixed in-plane\nfield orientation (hysteresis measurements) or in-plane angular-dependent measurements at\na fixed magnetic field strength are performed. The latter enable to quantify the SSE signal\nby the characteristic sinusoidal angular dependence, while in the hysteresis measurements\nthe signal is extracted from the difference between positive and negative magnetic field, as\nshown in Fig. 1 b) and d) for different temperatures. A second type of SSE setup has\nbeen used for samples fabricated and measured in Mainz. An additional heater is placed\non top of the measurement-stripe structure electrically isolated from the detection layer20,23,\n5as shown in Fig. 1 c). For this setup, hysteresis measurements14and temperature sweep\nmeasurements23are performed. In the latter the SSE signal is obtained by the difference of\ntemperature dependent measurements at a fixed positive and negative magnetic field. All\nmeasurement methods yield a quantitatively identical SSE signal and show the character-\nistic linear dependence of the signal strength on the applied heating power. Simultaneous\nmeasurements of the detection (metal) layer and bottom layer resistance ( RtandRb) are\nused to determine the temperature gradient applied to the sample stack and the actual tem-\nperature of the sample surface. This temperature is used in the following as the reference\ntemperature for the sample. Note that we show here the measured spin Seebeck voltage\nand not the scaled data in µV K−1. This is to obtain robust information as the flattening of\ntheRvs.Tdependence at very low temperatures complicates and partially prohibits the\naccurate determination of the temperature gradient. Since we have measured all samples\nthat have similar thermal properties in nominally identical conditions, a robust comparison\nof the data is possible. In the following measurement data are thus plotted in units of µV.\nAll SSE measurement methods consistently yield two sign changes of the SSE voltage\namplitude as a function of temperature for the garnets studied here (Fig. 2). At 300 K,\nthe signal has the same sign as the SSE voltage commonly observed in YIG/Pt24,25. Upon\ndecreasing the temperature, the signal amplitude decreases, resulting in a first SSE voltage\nsign change. At this first sign change the SSE-hysteresis flips orientation as shown in Fig.\n1 d). The temperature of the first sign change T sign,1 matches with the temperature of the\nmagnetic compensation point T compwithin the error bar for all investigated garnets, as can\nbe seen in Fig. 2 a)-c).\nWe further investigate the GdIG thickness dependence of this sign change of the SSE\nvoltage. We find that the temperature for magnetic compensation, T comp, changes with\nthe thickness of our GdIG films grown epitaxially on GGG as shown in Fig. 2 d)-f). For\nincreasing film thickness we observe a shift of the compensation point temperature towards\nthe reported bulk value of 286 K26. The shift of the compensation point T comp to higher\ntemperatures for increasing the GdIG film thickness is reflected in the shift of the first sign\nchange of the SSE signal as T sign,1 coincides with T compfor all investigated samples. As the\ntemperature is decreased further, a second sign change of the SSE voltage is observed in\nall investigated samples at T sign,2. Around T sign,2 a continuous increase of the net magnetic\nmoment is observed, indicating that no qualitative change of the sublattice magnetizations\n6takes place in this temperature range. This is further strengthened by recent observations\nmade by Ganzhorn et al.27, where the spin Hall magnetoresistance (SMR) has been inves-\ntigated in GdYInIG/Pt bilayers as a function of temperature. The SMR reveals a distinct\ndependence on the orientation of the magnetic sublattices with the result that in the vicinity\nof T comp, where a canted spin state appears, the SMR changes sign. In the region of T sign,2,\nhowever, the SMR does not exhibit unexpected behavior. Generally, the temperature of\nthe second sign change T sign,2 increases slightly as the temperature of the first sign change\nTsign,1 increases, however, there is no direct proportionality of T sign,1 and T sign,2 (cf. Fig.\n2 d)-f)). Two possible explanations were put forward to explain the low temperature sign\nchange in GdIG20. The first explanation is based on a competition of the two dominating\nbulk magnon modes with temperature and the second one on different interfacial magnon\nmode transmission probabilities at the GdIG/Pt interface. To check the relevance of these\ntwo different models, we investigate the influence of different detection layer materials on the\nSSE signal. For this purpose we cleave a GdIG/GGG (100) sample into two parts, which\nallows us to ensure identical GdIG thin film properties. The two equal GdIG thin films\nare then covered by either Pt or Pd layers. The HR-STEM analysis indicates a very good\nepitaxial quality of the GdIG film as well as a high surface flatness of both metallic films.\nIn addition to that, EELS elemental maps reveal similar sub-nanometer interfaces between\nPt or Pd and GdIG. Fig. 3 a) and b) show the SSE signal as a function of temperature\nfor both samples. While the first sign change occurs at the same temperature T sign,1, as\nexpected for one and the same GdIG thin film if the bulk magnetic properties are decisive,\nthe temperature of the second low temperature sign change at T sign,2 depends strongly on\nthe NM layer. Compared to the Pt capped sample, in the sample with Pd as detection layer\nTsign,2 is shifted to lower temperatures and the overall signal amplitude is reduced.\nTo probe the influence of the interface quality on the SSE signal, a second GdIG/GGG\nsample is cleaved into two parts, whose surfaces are treated differently before the deposition\nof nominally identical Pt detection layers. One sample is only treated by acetone and\nisopropyl alcohol cleaning prior the deposition (marked as “untreated”), while the other\nsample is etched by an O-ion bombardment in-situ prior to the Pt deposition (marked as\n“etched”). As shown in Figs. 3 c) and d), again, T sign,1 remains unaltered while T sign,2\nchanges. Compared to the untreated sample, T sign,2 of the etched sample is shifted to a\nhigher temperature accompanied by an overall reduced signal amplitude. These results\n7demonstrate the strong influence of the interface quality on the temperature of the second\nsign change.\nAs shown by Kikkawa and coworkers28, the magnetic field strength has a significant influ-\nence on the SSE amplitude. To further investigate this matter in compensated garnet/NM\nheterostructures, we probe the dependence of the magnetic field on the SSE signal. Fig. 4\nshows the SSE voltage as a function of temperature for different magnetic field strengths in\nGdIG/Pt and DyIG/Pt heterostructures. T sign,1 slightly shifts towards lower temperatures\nas the magnetic field strength is increased. While for GdIG the temperature shift of T sign,2\nwith magnetic field strength is similar to the temperature change of change T sign,1, a more\npronounced temperature shift of the T sign,2 can be observed for DyIG. The sign change even\nvanishes at low magnetic field strengths, indicating a more complex SSE in DyIG as com-\npared to GdIG. Above the compensation point, both GdIG and DyIG reveal an increase of\nthe SSE signal with increasing external field.\nWe now analyze the experimental findings presented above, considering that the SSE volt-\nage signal can be decomposed into a contribution arising from (i) the temperature dependent\ndominating magnon modes within the ferrimagnet20and (ii) the interfacial spin transmission\nacross the garnet/metal interface. In particular, the comprehensive investigation with dif-\nferent magnetic garnet materials, different metal layers and different interface preparation\nprocedures enables us to determine how the observed interface and magnetic field effects\nreveal the influence of the interfacial boundary between ferrimagnet and metal. As shown\nabove, the temperature of the first sign change T sign,1 coincides with the temperature of the\nmagnetic compensation point determined by SQUID magnetometry measurements within\nthe experimental error for all investigated garnets, independent of the metal detection layer\nor interface quality. Especially the interface quality and metal detection layer insensitivity\nof T sign,1 confirm the interpretation of the origin of this sign change put forward in Ref.20,\nbased on the theoretical proposal of Ohnuma et al.29. All investigated garnets consist of at\nleast three coupled magnetic sublattices. Two of these sublattices (a and d lattice site) are\noccupied by Fe ions, which are antiferromagnetically coupled. This sublattice configuration\nis also present in the classical ferrimagnetic insulator YIG, which shows no magnetic com-\npensation point. By adding a further magnetic material on the c lattice site (e.g., Gd in\ncase of GdIG), a third magnetic sublattice is introduced. For Gd or Dy on the c lattice site,\nthis c sublattice couples ferromagnetically to the Fe on the a lattice site20,30. Furthermore,\n8this third sublattice possesses a strong temperature dependent magnetic moment, leading to\na temperature below the N´ eel point (the magnetic ordering temperature) at which the net\nmagnetic moment of the material is zero. The temperature T compof this so-called magnetic\ncompensation point depends on the magnetic ions on the c sublattice as well as on their\ntemperature dependence26. In a finite magnetic field the magnetic moments invert their ori-\nentation when the temperature is swept across T comp. As a result, the sign of the SSE voltage\nis inverted if measured above and below the compensation point, as the polarization of the\nemitted magnonic spin current follows the orientation of the magnetic lattices20. Any influ-\nence of the metal detection layer or the quality of the interface on T sign,1 would be evidence\nthat interface - and not bulk magnetic properties – are relevant for T sign,1. The fact that we\nobserve T comp≈Tsign,1 irrespective of the thicknesses of the garnet layer, its composition,\nand its interface properties is a strong evidence that the sign change of the SSE voltage at\nTsign,1 indeed simply reflects the change in the ‘bulk’ sublattice moment orientations viz.\nbulk magnon properties. It must be noted that besides the magnetic compensation point,\ncompensated ferrimagnets can also exhibit an angular momentum compensation point. The\ntheoretical model by Ohnuma et al.29however does not predict any change of the thermal\nspin current due to the latter. Indeed, we do not observe any features in our experimental\ndata which would point to a modification of the thermal spin currents around the angular\nmomentum compensation point. While in GdIG the angular momentum of Gd is quenched\nsuch that magnetic compensation and angular momentum compensation temperature coin-\ncide, there would be a distinct difference between the two in DyIG.\nA more complex behavior must be invoked to understand the second SSE voltage sign\nchange at lower temperatures (T sign,2), as we discuss next. As mentioned above, no changes\nof the net magnetization are observed in the vicinity of T sign,2 that could explain the observed\nSSE behaviour. However, the shift of T sign,2 with both the material of the detection layer\n(Fig. 3 a) and b)) and the garnet surface treatment (Fig. 3 c) and d)) demonstrates the\ninfluence of the interface, which was previously neglected. Expanding the original theory\ninvoked by Ohnuma et al.29, Tsign,2 is explained by a temperature dependent competition\nbetween two bulk magnonic modes in the garnet. While at temperatures below T sign,2\nthe spin current is mainly given by the low-energy mode (acoustic like mode), the second\nmode (optical like mode) becomes dominating above T sign,2. Since the two modes are of\nopposite precession, their contributions to the spin current are of opposite spin polarity\n9which results in the sign change of the SSE signal at T sign,2. A more detailed description\nof this process is provided by Gepr¨ ags et al.20. The efficiency of the spin transfer from the\ngarnet into the metal detection layer, which is described by the element specific spin mixing\nconductance G ↑ ↓31, depends on the detection layer material. Due to a generally lower spin\nmixing conductance as well as spin Hall angle of Pd as compared to Pt32, the overall signal\nis reduced as seen in Fig. 3 c)-d). Nevertheless, the observed shift of the second sign change\nto lower temperatures as compared to Pt indicates that magnons of the high energy mode\ncan be transferred more efficiently across the interface into Pd. This is either attributable\nto intrinsic properties of the metal detection layer or the chemical surface termination of the\ngarnet film. While in case of GdIG/Pt a Gd/Fe rich interface is observed, GdIG/Pd exhibits\na Gd/O rich interface (see supplemental information21). One thus might speculate that\nthe decreased Fe concentration at the GdIG/Pd interface results in an increased interface\ndamping of the low energy mode. Besides changing the metal detection layer the interface\ntransparency is manipulated via O-ion bombardment, yielding an universal reduction of the\nsignal strength. This points to a reduction of G ↑ ↓for both modes into the detection layer.\nHowever, as compared to the unetched interface the second sign change is found at higher\ntemperatures, suggesting that the high energy mode is more affected by the reduced interface\ntransparency.\nFinally we corroborate the magnon mode sensitivity of the second sign change by ana-\nlyzing the magnetic field dependence shown in Fig. 4. For YIG it has been shown that high\nmagnetic fields cause a reduction of the magnonic propagation length, which is relevant for\nthe whole temperature range33. An energy gap (Zeeman effect) is opened by the magnetic\nfield, which suppresses the excitation of magnons with energies lower than the gap28,33,34.\nThe field gap is especially important at low temperatures, as in YIG the field induced energy\ngap is in the order of a few Kelvin34. It therefore causes a shift of the whole dispersion re-\nlation to higher energies, thus mainly introducing a constant offset with respect to the zero\nfield state33. The resulting shift of the dispersion relation towards higher energies moves\nTsign,2 to higher temperatures, as it now requires more thermal energy to sufficiently occupy\nthe high energy mode. The different sensitivities of T sign,2 on the magnetic field for GdIG\nand DyIG might originate from the different electronic states of the Gd3+and Dy3+ions.\nIn contrast to Gd3+, Dy3+has a finite orbital moment ( L= 5), which may lead to differ-\nences in the magnetic field dependences of magnon modes due to the different multiplett\n10states occupied. The comparison to field dependent magnetization data (see supplemental\ninformation21) shows that the increase of VSSEat high magnetic fields above the compen-\nsation point cannot be explained by an increase of the magnetization. Instead, one has to\nconsider that in contrast to YIG, where the thermal signal is governed by the fundamental\nferrimagnetic magnon mode, in compensated iron garnets the spin current reflects the field\ndependence of a complex magnon mode structure. The quantification of the latter as well\nas the magnon mode dependence on different electronic states of the magnetic ions require\nfurther theoretical input, which is beyond the scope of this work.\nIn conclusion, we have shown that the ferrimagnetic nature and magnon mode structure\nof compensated garnets can be accessed by temperature dependent SSE measurements. Our\nstudy of different compensated iron garnets, detection metal layer materials and interface\nqualities allows us to clearly show the universal correlation between the magnetic compen-\nsation point and the sign change of the SSE voltage at high temperatures, T sign,1. We find\nthat the second sign change at the lower temperature T sign,2 is strongly influenced by the\ndetection layer material as well as the interface nature, which can be rationalized in terms\nof a modified coupling strength of the two magnon modes. We study these individually by\nvarying the temperature to identify the mode selective magnon transmissivity. The mag-\nnetic field dependence of T sign,2 is consistent with our developed magnonic mode selective\npicture of the SSE. T sign,2 is shifted towards higher temperatures for higher magnetic fields.\nOur results reveal that the SSE provides an elegant solution to gain insight into the magnon\nmode structure of a ferrimagnet and, on the other hand, show for the first time evidence for\nan interface sensitive magnon mode selective coupling towards normal metal layers.\nI. ACKNOWLEDGMENTS\nThe authors would like to express their gratitude to Prof. Gerrit E.W. Bauer and Dr. Joe\nBarker for valuable discussions and the Institute for Materials Research at Tohoku University\nfor the hospitality during a visiting professor stay (MK). Furthermore, they would like to\nthank Prof. Kathrin D¨ orr for the help in sample preparation. This work was supported by\nDeutsche Forschungsgemeinschaft (DFG) SPP 1538 “Spin Caloric Transport,” the Graduate\nSchool of Excellence Materials Science in Mainz (MAINZ), and the EU projects (IFOX,\n11NMP3-LA-2012246102, INSPIN FP7-ICT-2013-X 612759).\n∗Klaeui@uni-mainz.de\n†Sebastian.Goennenwein@tu-dresden.de\n1S. 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).\n2I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Reviews of Modern Physics 76, 323 (2004).\n3M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Applied Physics\nLetters 87, 153501 (2005).\n4S. Klingler, P. Pirro, T. Br¨ acher, B. Leven, B. 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Du, W. Zhang, and H. F. Ding,\nJournal of Applied Physics 115, 17C504 (2014).\n33U. Ritzmann, D. Hinzke, A. Kehlberger, E.-J. Guo, M. Kl¨ aui, and U. Nowak, Physical Review\nB92, 174411 (2015).\n34H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Physical Review B 92, 054436\n(2015).\n35S. Geller, G. P. Espinosa, and P. B. Crandall, J. Appl. Crystallogr. 2, 86 (1969).\n36G. P. Espinosa, J. Chem. Phys. 37, 2344 (1962).\n14−10 0 10−4−2024VSSE(µV)\n−10 0 10\n−10 0 10\n0 180−2024VSSE(µV)\n0 180\n0 180 360\nYAG (111) / GdIG (26 nm)YAG (111) / GdIG (26 nm) a) b)\nc) d)25 K 166 K 297 K\n20 K 158 K 272 KFigure 1. Schematics of the setups used to measure the spin Seebeck effect. At WMI, a) the\nout-of-plane temperature gradient is generated by driving a large current I dthrough a Hall bar\nmesa structure while the thermo-voltage is recorded as b) a function of the in-plane magnetic\nfield orientation as well as d) a function of the magnetic field strength at a fixed magnetic field\norientation. In Mainz c) the out-of-plane temperature gradient is generated by an external heater\nattached on top of the sample. The SSE amplitude is obtained by sweeping a perpendicular\nmagnetic field.\n150100 200 300\nT(K)−4−202VSSE(µV)\n101102103\n0100 200 300\nT(K)−4−2024VSSE(µV)\n200400600\n0100 200 300\nT(K)−10−505VSSE(µV)\n100101102103YAG (111) / GdIG (26 nm) YAG (111) / DyIG (30 nm) YAG (111) / GdYInIG (61 nm)\na) b) c)\nGGG (001) / GdIG (41 nm) GGG (001) / GdIG (100 nm) GGG (001) / GdIG (155 nm) \nd) e) f)Figure 2. In each panel, the upper plot shows the magnetic moment of the garnet film as a\nfunction of temperature determined by SQUID magnetometry measurements. The SSE voltage as\na function of temperature is shown in the bottom plot for the same sample. The upper three figures\na), b), c) show the data of REIG/Pt heterostructures with REIG = GdIG, DyIG and GdYInIG\ngrown on YAG. The lower three figures d), e) and f) contain the data for GdIG/Pt heterostructures\ngrown on GGG, featuring different GdIG layer thicknesses. The dashed lines mark the temperature\nof the magnetic compensation point T comp determined from SQUID magnetometry data and the\ntemperatures at which the SSE voltage changes sign. For all measurements a magnetic field of µ0H\n= 1 T is applied.\n16−1.0−0.50.00.51.0VSSE(µV)\n0 100 200 300\nT(K)−0.50.00.5VSSE(µV)GGG (001) / GdIG (100 nm)\nPt\nPd179 K\n179 K72 K\n58 K\n0 100 200 300\nT(K)−0.50.00.5VSSE(µV)\n−1.0−0.50.00.51.0VSSE(µV)GGG (001) / GdIG (41 nm)\nuntreated\netched153 K\n153 K44 K\n71 Ka) c)\nb) d)Figure 3. SSE voltage as function of temperature measured in GdIG/NM heterostructures grown\non GGG (001). (a), (b) V SSEof GdIG/NM heterostructures using Pt as well as Pd as detection\nlayer. The same GdIG thin film sample was used for both heterostructures. (c), (d) V SSEof two\nGdIG/Pt heterostructures. The Pt thin films were deposited on both an untreated and etched\nsurface of the same GdIG thin film sample. The dashed lines mark the temperatures at which the\nSSE voltage changes sign.\n0 100 200 300\nT(K)−6−4−2024VSSE(µV)\n0 100 200 300\nT(K)−6−4−2024VSSE(µV)YAG (111) / DyIG (30 nm) YAG (111) / GdIG (26 nm)\n1T\n15T1T\n15T\nFigure 4. SSE voltage as function of temperature for (a) GdIG/Pt and (b) DyIG/Pt heterostruc-\ntures for different magnetic field strengths.\n17Magnon mode selective spin transport in compensated\nferrimagnets\nSupplemental Information\nS2. SAMPLE CHARACTERIZATION\nS2.1. X-ray diffraction & Reciprocal space mapping\nThe crystalline quality of the investigated rare earth iron garnet films has been exam-\nined via X-ray diffractometry (XRD) and reciprocal space mapping (RSM). In Fig. S1,\n2Θ−ωscans of the garnet films grown at the WMI are depicted, each showing the YAG\n(444) reflection as well as the respective REIG (444) reflection ( REIG = GdIG, DyIG, and\nGdYInIG). The presence of the latter indicates crystalline growth of the films, reinforced\nby the appearance of Laue fringes for GdIG and DyIG signifying high crystalline quality.\nFor GdIG and DyIG the (444) reflection appears at a scattering angle which is below the\ntheoretical value obtained for the lattice constants aGdIG = 12.471˚A,aDyIG = 12.405˚A35,36\nand the Bragg equation\n2dhklsin (Θ) =λ\nwithdhkl=a√\nh2+k2+l2,\nassuming a cubic geometry. The shift to lower scattering angles signifies an enhance-\nment of the out-of-plane lattice constant, which in turn can be explained by an in-plane\ncompressive strain considering the lattice constant aYAG= 12.008˚A of the substrate35. To\nthe knowledge of the authors there are no reported values of afor GdYInIG, but regarding\nits close chemical relation to GdIG a similar lattice constant and thus a strained state is\nexpected.\nXRD data of the GdIG films investigated at JGU Mainz are shown in Fig. S2. Since\nin this case the GdIG films of varying thickness are grown on GGG susbtrates with (001)\norientation, the 2Θ −ωscans cover the GGG (004) and GdIG (004) reflection. Again,\nthe presence of the reflection peaks and the clear appearance of Laue fringes for GdIG with\n18( ( (\n49 50 51 52 53 54100101102103104105106Intensity (a.u.)\n49 50 51 52 53 54100101102103104105106\n49 50 51 52 53 54100101102103104105106\n2Θ◦) 2Θ ) 2Θ )YAG (111) / GdIG (26 nm) YAG (111) / DyIG (30 nm) YAG (111) / GdYInIG (61 nm)\n◦ ◦\nYAG (444)YAG (444)YAG (444)GdIG (444)\nDyIG (444)\nGdYInIG (444)(a) (b) (c)\nBulk value Bulk valueFigure S1. X-Ray diffraction data of the rare earth iron garnets Gd 3Fe5O12(GdIG), Dy 3Fe5O12\n(DyIG), and (Gd 2Y1)(Fe 4In1)O12(GdYInIG). The films are grown epitaxially via pulsed laser\ndeposition (PLD) in an oxygen atmosphere on Y 3Al5O12(YAG) substrates with (111) surface\norientation. The vertical dashed lines indicate the (444) reflection calculated from the respective\nbulk lattice constant.\nthickness d= 41 nm and 100 nm) indicates high crystalline quality of the films. Furthermore,\nas in the case of YAG, the lattice constant of GGG aGGG= 12.375˚A is smaller than the bulk\nvalue ofafor GdIG, leading to in-plane compressive stress and a shift of the scattering angle\nof GdIG to lower values. The coherent growth of high quality GdIG films up to 155 nm is\nfurther emphasized by the reciprocal space maps around the GGG (228) reflection presented\nin Fig. S3.\n1926 27 28 29101102103104105106107\n26 27 28 29101102103104105106107\n26 27 28 29101102103104105106107Intensity (a.u.)\n2Θ(◦) 2Θ( ) 2Θ( ) ◦ ◦GGG (001) / GdIG (41 nm) GGG (001) / GdIG (100 nm) GGG (001) / GdIG (155 nm)GGG (004)\nGGG (004)GdIG (004)\nGdIG (004)\nGdIG (004)\nGGG (004)Bulk value Bulk value Bulk valueFigure S2. X-Ray diffraction data of Gd 3Fe5O12(GdIG) thin films of various thickness ( d=\n41 nm, 100 nm and 155 nm). The films are grown epitaxially via pulsed laser deposition (PLD)\nin an oxygen atmosphere on Gd 3Ga5O12(GGG) susbtrates with (001) surface orientation. The\nvertical dashed lines indicate the GdIG (004) reflection calculated from the respective bulk lattice\nconstant.\n0.225\n0.230\n0.235\n0.63\n0.64\n0.65\nGdIG (41 nm)\nRSM 228\nl (rlu)\nh (rlu)\nGGG\n(a)GGG/GdIG(41 nm)\n0.230\n0.235\n0.63\n0.64\n0.65\nl (rlu)\nh (rlu)\nGdIG (155 nm)\nRSM 228\nGGG (b)GGG/GdIG(155 nm)\nFigure S3. Reciprocal space maps (RSM) around the GGG (228) reflection for GGG/GdIG(41 nm)\nand GGG/GdIG(155 nm).\nS2.2. X-ray reflectometry & Atomic force microscopy\nTo obtain information about the thickness and surface roughness of the grown garnet\nfilms X-ray reflectometry (XRR) as well as atomic force microscopy (AFM) measurements\nare performed. From the XRR data (Fig. S4 and S5) the film thickness as well as the root\nmean square roughness σrmsof the sample surface are extracted by fitting a simulation curve\n200.05 0.10 0.15 0.20 0.25\nQz(Å−1)10−410−310−210−1100Intensity (a.u.)Data\nSim\n0.05 0.10 0.15 0.20 0.25\nQz(Å−1)10−410−310−210−1100\nData\nSim\n0.05 0.10 0.15 0.20\nQz(Å−1)10−410−310−210−1100\nData\nSimYAG (111) / GdIG (26 nm) YAG (111) / DyIG (30 nm) YAG (111) / GdYInIG (61 nm)\n(a) (b) (c)Figure S4. X-Ray reflectometry data of rare earth iron garnets Gd 3Fe5O12(GdIG), Dy 3Fe5O12\n(DyIG), and (Gd 2Y1)(Fe 4In1)O12(GdYInIG) investigated at the WMI.\n0.05 0.10 0.15 0.20 0.25\nQz(Å−1)101102103104105106107108Intensity (a.u.)Data\nSim\n0.05 0.10 0.15 0.20 0.25\nQz(Å−1)101102103104105106107108\nData\nSim\n0.05 0.10 0.15 0.20 0.25\nQz(Å−1)101102103104105106107\nData\nSimGGG (001) / GdIG (41 nm) GGG (001) / GdIG (100 nm) GGG (001) / GdIG (155 nm)\nFigure S5. X-Ray reflectometry data of Gd 3Fe5O12(GdIG) thin films of various thickness ( d=\n41 nm, 100 nm and 155 nm) investigated at JGU Mainz.\nto the raw data. In case of large thickness ()GGG/GdIG(155 nm, Fig. S5 (c)) no Kiessig\nfringes can be observed so that the film thickness is estimated by the growth time.\nThe AFM measurements (Fig. S6) allow to extract σrmsdirectly from the topographical\ninformation yielded by the data. As seen in Fig. S6 (a)-(c), the sample surfaces are partially\nor significantly contanimated by dirt particles resulting from sample handling, leading to an\nincrease of σrmsas compared to the values obtained by XRR.\nTheσrmsvalues obtained by the different methods for the investigated films are summa-\nrized in Tab. S1.\n212.89 nm\n0.171.502.00\n2 µm\n1.59 nm\n0.200.801.001.20\n2 µm\n3.07 nm\n0.411.502.002.50\n2 µm(a) GdIG\n(b) DyIG\n(c) GdYInIGFigure S6. Atomic force microscopy data of rare earth iron garnets (a) Gd 3Fe5O12(GdIG), (b)\nDy3Fe5O12(DyIG), and (c) (Gd 2Y1)(Fe 4In1)O12(GdYInIG).\nTable S1. Thickness and roughness data for the investigated garnet films obtained by XRR and\nAFM.\nFilm d(nm)σrms,XRR (nm)σrms,AFM (nm)\nYAG (111) / GdIG 26 0.60 0.54\nYAG (111) / DyIG 30 0.40 0.80\nYAG (111) / GdYInIG 61 0.35 0.43\nGGG (001) / GdIG 41 1.37 -\nGGG (001) / GdIG 100 1.71 -\nGGG (001) / GdIG 155 2.27 -\n22S2.3. Transmission electron microscopy characterization\nA direct insight into the crystalline structure of part of the investigated samples is ob-\ntained via high resolution scanning transmission microscopy (HR-STEM). For HR-STEM\nthe samples have been prepared using the standard focused ion beam (FIB) protocol. Re-\nspective HR-STEM images and electron energy loss spectroscopy (EELS) spectra have been\ntaken using an aberration-corrected scanning transmission electron microscope (FEI Probe\nCorrected Titan3TM80-300 S/TEM) operating at 200 kV and equipped with a Gatan Quan-\ntum ERS spectrometer.\nFig. S7 (a),(b) show two examples of high-angle annular dark field (HAADF) cross\nsections of the GGG(001)/GdIG/Pt and GGG(001)/GdIG/Pd samples, whose spin Seebeck\ndata are presented in Fig. 3(a),(b) in the main text. The images already indicate the\nepitaxial growth of the garnet films as well as reveal the overall flatness of the metal films.\nIn Fig. S7 (c) we show an image example of the interface between substrate and grown\ngarnet film of the GGG(001)/GdIG/Pd, combining the information obtained by annular dark\nfield (ADF) and electron energy loss spectroscopy (EELS) imaging. The clear crystalline\nstructure of substrate and grown film are pictured with atomic resolution, revealing the high\nepitaxial quality of the latter and a well defined interface. The elemental maps given by the\nEELS data allow us to assign or rather identify elements at the lattice sites, which show\nlittle interdiffusion of Gd and Ga atoms between GGG and GdIG.\nIn Fig. S8 we show an example of a detailed HR-STEM images of the GdIG/Pt interface,\nincluding ADF information as well as information about the distribution of single elements\nvia EELS. On the one hand, the images reveal that the garnet film retains its crystalline\nstructure up to at least 100 nm thickness. On the other hand, a sub-nanometer, Gd/Fe\nrich interface between Pt and GdIG is observed via the EELS elemental map. In case of\nGdIG/Pd (not shown here) a Gd/O rich interface is observed.\n23GGGGdIGPt\nGGGGdIGPd(a)\n(b)\n(c)\nFigure S7. (a),(b) Annular dark field (ADF) images of (a) GGG(001)/GdIG/Pt and (b)\nGGG(001)/GdIG/Pd samples investigated at JGU Mainz. (c) ADF (left) and electron energy\nloss spectroscopy (EELS, right) image of the GGG/GdIG interface of the sample shown in (b).\nThe images are taken alongside the [110] crystal orientation using a scanning transmission electron\nmicroscope. For comparison, the expected crystal structure is shown schematically showing good\nagreement.\n0 1 2 3 4 5 6\nPosition (nm)0.00.20.40.60.81.0Nomralized counts (a.u.)\nPt\nGd\nFe\nOGGG (001) / GdIG (100 nm) / Pt (5 nm)\n(a) (b)\nPt\nGdIGPosition\nFigure S8. (a) High resolution scanning electron microscopy ADF image of the GdIG/Pt interface\nof the GGG(001)/GdIG/Pt sample investigated at JGU Mainz. Respective spin Seebeck data are\nshown in Fig. 3 in the main text. (b) Distribution of single elements across the interface obtained\nby an electron energy loss spectroscopy elemental map. The direction of the scan is indicated by\nthe arrow in (a).\n24S3. SQUID MAGNETOMETRY OF GDIG AND DYIG\nIn Fig. S9, SQUID magnetometry data obtained for GGG(001)/GdIG(100 nm) and\nYAG(111)/DyIG(30 nm) are shown, depicting the sample magnetization as a function of\nexternal magnetic field at different temperatures. While measuring the respective data, in\naddition to the magnetic moment of the grown iron garnets, the paramagnetic (diamag-\nnetic) moment of GGG (YAG) is recorded as well. To remove this contribution a linear fit is\napplied to the raw magnetiation data in the field range where the sample is assumed to be\nfully magnetized. Subsequently the raw data are rectified by means of the obtained linear\nslope. In case of YAG/DyIG the diamagnetic background is determined once for 2 K and\nthereafter, assuming a constant diamagnetic contribution of YAG, all temperature datasets\nare corrected using the determined parameters. For GGG/GdIG the paramagnetic back-\nground of GGG is calculated for each temperature. Since the latter is large as compared to\nthe GdIG magnetization, this procedure may lead to an error in the slope of M(H) in the\nsaturated state.\n20 K\n40 K\n60 K\n80 K\n100 K\n120 K\n140 K\n160 K\n180 K\n200 K\n220 K\n240 K\n260 K\n280 K\n300 K2 K\n5 K\n10 K\n20 K\n40 K\n60 K\n100 K\n140 K\n180 K\n190 K\n195 K\n200 K\n205 K\n215 K\n225 K\n250 K\n300 K −100 −50 0 50 100\nµ0H(mT)−400−2000200400M(kA/m)\n−8−6−4−20 2 8\nµ0H(T)−800−600−400−2000200400600800M(kA/m)\n4 6GGG (001) / GdIG (100 nm) YAG (111) / DyIG (30 nm)\n(a) (b)\nFigure S9. Magnetization of (a) GdIG and (b) DyIG as a function of external magnetic field at\ndifferent temperatures.\n25S4. TYPICAL SSE TEMPERATURE DEPENDENCE MEASUREMENTS\nFor a comprehensive characterization of the spin Seebeck (SSE) temperature dependence\ndifferent measurements can be performed, here shown exemplarily for the GGG(001)/GdIG(100 nm)/Pt(5 nm)\nsample measured at JGU Mainz.\nFig. S10 (a) shows the temperature dependence of the recorded SSE voltage VLSSE for\ndifferent heating currents. As expected, VLSSE increases with increasing heating current.\nUsing the temperature calibration method described in the main text one can extract the\ndependence of VLSSE on the temperature difference ∆ Tapplied across the sample, revealing\na linear dependence (Fig. S10 (b)). This corresponds to the behavior expected for the SSE\nand is observed for both regimes above and below the magnetization compensation point\nwhereVLSSE changes its sign. Generally, the position of the first and second sign change of\nVLSSE is independent on the applied heating power, emphasizing their physical significance\nand their dependence on intrinsic material properties.\nUsing the evaluated temperature difference ∆ Tacross the sample (plotted in Fig. S10 (c),\nagain for the different applied heating currents) one can deduce the temperature dependence\nof the spin Seebeck coefficient σLSSE, see Fig. S10 (d). Within the error the different curves\nobtained for different heating currents superimpose, demonstrating the robustness of the\nmeasurement method.\n26Figure S10. Typical data obtained from the GGG/GdIG (100 nm) sample during measurements\nusing the setup at JGU Mainz. (a) Spin Seebeck voltage VLSSE as a function of temperature\nfor different applied heating currents. (b) VLSSE as a function of temperature difference between\nsample top and bottom at different temperatures, revealing a linear dependence. (c) Temperature\ndifference between sample top and bottom as a function of temperature for different applied heating\ncurrents. (d) Spin Seebeck coefficient σLSSE =VLSSE/∆Tas a function of temperature.\n27"    },    {        "title": "1905.12771v1.Spin_wave_propagation_in_ferrimagnetic__Gd__x_Co__1_x__.pdf",        "content": "                    \n1 Spin wave propagation in ferrimagnetic Gd xCo1-x \nShinsaku Funada1, Tomoe Nishimura1, Yoichi Shiota1*, Shuhei Kasukawa1, Mio Ishibashi1, \nTakahiro Moriyama1, and Teruo Ono1,2* \n1Institute  for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto 611 -0011, Japan  \n2Center  for Spintronics Research Network, Graduate School of Engineering Science, Osaka \nUniversity,  Toyonaka, Osaka 560 -8531, Japan  \n*E-mail: shiota -y@scl.kyoto -u.ac.jp , ono@scl.kyoto -u.ac.jp  \n \nAbstract  \nRecent advances in antiferromagnetic spin dynamics using rare -earth (RE) and transition -\nmetal (TM) ferrimagnets have attracted much interest for  spintronic  device s with a high \nspeed  and density . In this study, the spin wave properties in the magnetostatic backward \nvolume mode  and surface mode  in RE -TM ferrimagnetic GdxCo1-x films with various  \ncomposition  x are investigated using  spin wave spectroscopy. The obtained group velocity \nand attenuation  length are well explained by the ferromagnet -based  spin wave theory when  \nthe composition of GdxCo1-x is far from the compensation point . \n                      \n2 In magnonic s where spin w aves are used for information transport and processing ,1–4) \nyttrium iron garnet  (Y3Fe5O12) and permalloy  (Ni 80Fe20) have been  widely used , because \nthey exhibit  a low Gilbert  damping  parameter  and soft magnetic properties . However, i n \nsuch material s, the magnetization precesses in  right -handed chirality because they only have  \none type of magnetic lattice . In contrast, the polarization of spin wave s and its manipulation \nhave been theoretically predicted using antiferromagnetic spin waves .5,6) \nThe recent progress in the development of antiferromagnetic spin tronic s has enabled \nspintronic devices with high  speed and density .7–9) In particular, rare -earth (RE) and \ntransition -metal (TM) ferrimagnets  are one of most fascinating  materials  to investigate \nantiferromagnetic spin dynamics . As the magnetic moments of TM elements (Fe, Co, Ni, \nand their alloys) and RE elements (Gd, Tb, etc.) can be antiferromagnetically coupled , the \ncompensation points of the net magnetization TM or the net  angular momentum  TA can be \nreached by varying the temperature and/or the composition  of each element .10–14) These \nproperties make these materials interest ing for investigating phenomena such as ultrafast all -\noptical switching ,15–17) current -induced switching ,18,19) spin-orbit torque ,20–23) and fast \ndomain wall motion .24–26) These findings motivated us to study the spin wave properties in \nferrimagnetic GdxCo1-x alloy  with various composition x. Note that magnetic properties in \nYIG show ferromagnetic -like behavior unlike the RE -TM ferrimagnets even though YIG is \nalso a ferrimagnet. This is because that the magnetism in YIG is governed by an \nuncompensated magnetic moment of Fe3+ ions in octahedral and tetrahedral coordinate sites.  \nIn addit ion, we can expect large group velocity in RE -TM ferrimagnets due to large \nsaturation magnetization than that in YIG. Therefore,  ferrimagnetic Gd xCo1-x alloy has a \npotential to  investigat e the antiferromagnetic spin waves.  \nThe samples were  prepared on thermally oxidized Si substrates using direct current \nmagnetron sputtering. The film stack consisted of GdxCo1-x (20 nm)/Pt  (2 nm)/Ta  (5 nm), \nwhere in the Pt/Ta bilayer was used as  the capping layer . Because Gd is  a very reactive \nelement and ea sily oxidized, we selected Pt/Ta bilayer for capping layer rather than an oxide \ncapping layer, such as Al 2O3 or SiO 2. The enhancement of the Gilbert damping originated \nfrom the spin pumping effect  is not negligible, even though it  should be small due to th e \nlarge thickness of Gd xCo1-x.27,28) Therefore, the estimated ma gnetic properties of Gd xCo1-x \nin the following experiments include the effect of metal capping layer. However, it should \nbe emphasized that the metal capping layer does not affect  discussions on the composition \ndependence of  magnetic properties in  GdxCo1-x. The GdxCo1-x alloy was deposited using the \nco-sputtering method from Gd and Co targets with different sputtering powers. The                     \n3 compositions of the Gd and C o atoms were calculated from the deposition rates, which were \ncalibrated in advance with X -ray reflec tivity. We selected GdxCo1-x compositions with x = \n0.22, 0.30, 0.40, and 0.59 to investigate the spin wave propagation in both Co -rich and Gd -\nrich GdxCo1-x alloys. All the samples exhibited an  in-plane magnetic easy axis.  \nFirst, the static magnetic properties were measured using superconducting quantum \ninterference device magnetometry.  Figure 1 (a) shows t he saturation  magnetization  Ms as a \nfunction of the temperature for the samples with x = 0.22, 0.30, 0.40, and 0.59. In the \ntemperature range of 20–300 K, the magnetic moment of Co (Gd) was domina nt for the \nsamples with x = 0.22, 0.30 ( x = 0.59). We found  the magnetization compensation \ntemperature TM to be  approximately  170 K for the sample w ith x = 0.40 , which deviates from \nthe previous reported compensation point of around x = 0.2.12) This might be due to the \ndeviation between expected and actual comp osition of Gd xCo1-x and/or the thickness \ndependence of the compensation point in RE -TM ferrimagnets.14) Then, the magnetization \ncurves under perpendicular magnetic field were  investigated for the samples with x = 0.22, \n0.30, and 0.59, as shown in Figs. 1(b)-(d). All the samples e xhibit an in -plane magnetic easy \naxis. The effective saturation magnetization Ms,eff = Ms – H⊥, where H⊥ is the perpendicular \nmagnetic anisotropy field,  was estimated from the perpendicular saturation magnetic field.  \nTable I  lists the  obtained values of  Ms (300 K ) and Ms,eff. The reduction in Ms,eff compared \nwith Ms (300 K) results from the  perpendicular magnetic anisotropy , which originates from \nthe bulk perpendicular magnetic anisotropy.29,30) \nNext, the spin wave spectr a were  measured using a vector network analyzer (VNA) at \nroom temperatu re. Figure 2 shows the top view of the spin wave device. The films were \nstructured into spin wave waveguides of sizes  50 × 100 μm2 (x = 0.22, 0.30 , and 0.40) and \n100 × 100 μm2 (x = 0.59)  using  electron -beam lithography and Ar ion milling. After \ndepositing an 80 nm -thick S iO2 layer for electrical isolatio n, shortened coplanar waveguides \n(CPWs) were patterned on top of the spin wave waveguide s using  electron -beam lithography \nand lift -off process of sputter -deposited Ti  (5 nm)/Au  (100 nm). Two CPWs were designed \non a signal line (2 μm) and two ground lines (1 μm) with a gap of 1 μm, which effectively \nexcited or detected the spin waves with a wavenumbe r of 1.2 μm−1, as shown in Fig. 2(b) .31) \nThe injected microwave power of the VNA was −5 dBm , which was low  enough to maintain  \nthe linear response region of magnetization dynamics . Depending on the direction of the  \napplied magnetic field,  the spin waves in the  magnetostatic backward volum e wave \n(MSBVW) and magnetostatic surface wave (MSSW) configurations were excited ,32) as \nshown in Fig. 2. From the self-scattering parameters S11 and S22 and the mutual -scattering                     \n4 parameters S12 and S21, we extract ed the local spin wave resonance under the CPWs and the \npropagation characteristics of the spin waves between the two CPWs, respectively.  Note that \nwe could not  observe  the resonance peaks for the sample with x = 0.40 because of the  low \nnet saturation magnetization . Hereafter , we discuss the spin wave propertie s of the sample s \nwith x = 0.22, 0.30, and 0.59.  \nWe measured the local spin wave resonance under the CPWs, not the propagating spin \nwave resonance,  by means of S11 measurements  in the MSBVW configuration, where the \nmagnetic field was applied along the spin wave propagation direction. To suppress the \nfrequency -dependent background, including circuit resonances and losses, we measured the \nspectra by sweeping the magnetic fiel d. Figure 3(a) shows the Re[ S11] spectra in the \nMSBVW configuration under various frequenc ies for the sample with x = 0.22 . Figures  3(b) \nand 3(c) show the resonance frequency and linewidth. Given  the small dispe rsion in the \nMSBVW configuration in the limit of kt << 1, where k and t denote  the wavenumber of the \nspin waves and the thickness of the GdxCo1-x layers, respectively,  the resonance frequency \n𝑓MSBVW  and full width at half maximum of the resonance peak Δ𝐻 can be expressed as \nfollows .33) \n 𝑓MSBVW=𝜇0𝛾\n2𝜋√𝐻(𝐻+𝑀s,eff1−𝑒−𝑘𝑡\n𝑘𝑡)≃𝜇0𝛾\n2𝜋√𝐻(𝐻+𝑀s,eff) (1) \n Δ𝐻=Δ𝐻0+4𝜋𝛼\n𝜇0𝛾𝑓MSBVW  (2) \nwhere γ = (gμB)/ℏ is the gyromagnetic ratio, g is the Land é g-factor, μB is the Bohr magneton,  \nℏ is the Dirac ’s constant, μ0 is the permeability of free space, H is the applied magnetic field, \nΔH0 is the inhomogeneous linewidth broadening, and α is the Gilbert damping parameter. \nFrom the fittings obtained  using Eqs. (1) and (2), g, Ms,eff, and α were estimated for the \nsamples with various composition s, as summarized in Table I I. Ms,eff obtained from M-H \ncurves and MSBVW measurements are more or less consistent . The g in Co -rich sample ( x \n= 0.22) shows higher value than that in Gd -rich sample ( x = 0.59), and g and α takes \nmaximum for the sample with x = 0.30, which is near the angular  momentum compensation \npoint  in the range of composition that we examined . This compositional dependence of g \nand α can be  explained  with the simple mean field model ,11,34) where 𝑔𝜇𝐵ℏ⁄=𝑀net𝐴net⁄  \nand 𝛼=𝐴0𝐴net⁄  . Theoretically t hese two values diverge at the vicinity of the angular \nmomentum compensation point , and  the tendenc ies we observed in this study  are consistent. \nM. Binder et al.12) previously reported t he composition dependence  of g and α in \nferrimagnetic GdCo, which has a large difference  from our results . Because  they only \nexamine d x = 0.2 - 0.25 composition range of GdCo,  which is the vicinity of the angular                      \n5 momentum  compensation point,  simple comparisons cannot be made.  However we find that \nour results are  comparable to  the reported  values in other RE -TM ferrimagnetic systems, e.g. \nGdFeCo .11,13) \nThe propagating spin wa ves in the MSSW configuration , where the magnetic field was \napplied perpendicular to the spin wave propagation direction,  were investigated. Figures \n4(a)–(c) show the Re[S21] and Im[ S21] spectra for samples with x = 0.22, 0.30, and 0.59, \nrespectively, under a magnetic field of 10 mT. The distance d between the two CPWs for \neach sample is indicated  in Fig. 4. The oscillating signatures indicate the spin wave \npropagation between the two CPWs .33) We estimated the spin wave group velocity vg using \nthe relationship 𝑣g=Δ𝑓∙𝑑, where Δf/2 corresponds to the frequency difference between \nthe positive and negative peaks in the Im[S21] spectra, as shown in Fig. 4(b). The insets in \nFigs. 4(a) and (c) show the intensity of the transmittance signal | S21| as a function of d. The \nfittings of the exponential decay function yield the spin wave attenuation length Latt. Because \nof the  low signal intensity and high damping constant , we could not  evaluate Latt from d \ndependence of | S21| for the sample with x = 0.30.  \nFrom the theore tical dispersion relation ship in the MSSW configuration , 𝑓MSSW=\n𝜇0𝛾\n2𝜋√𝐻(𝐻+𝑀s,eff)+𝑀s𝑀s,eff\n4(1−𝑒−2𝑘𝑡), vg and Latt can be calculated as follows , \n 𝑣g=(𝜇0𝛾)2𝑀s,eff∙𝑀s𝑑\n8𝜋𝑓 (3) \n 𝐿att=𝑣g\n𝜇0𝛾(2𝐻+𝑀s,eff)𝛼 (4) \nwhere, 𝑣g=2𝜋𝜕𝑓MSSW𝜕𝑘⁄  and (𝜇0𝛾(2𝐻+𝑀s,eff)𝛼)−1  represent life -time of the \nmagnetization precession. Table I I lists the experimentally and theoretically obtained vg and \nLatt values.  The t heoretically  calculated vg and Latt qualitatively reproduce the experimental \nresults . The o verestimation  of vg in the experiments  may have result ed from the finite widths \nof the CPWs .35) According to Table I I, the spin wave  observed  in this study can be explained \nby the typical magnetostatic spin wave theory based on  ferromagnets . \nIn conclusion, we investigated the spin wave properties in RE-TM ferrimagnetic \nGdxCo1-x films with various composition x. In the MSBVW configuration, g, Ms,eff, and α \nwere estimated. In  the MSSW configuration, the obtained vg and Latt were  qualitatively  \nconsistent with  the ferromagnet -based  spin wave theory when the composition of GdxCo1-x \nwas far from  the compensation point . \n \nAcknowledgments                      \n6 This work was partially supported by JSPS KAKENHI (Grant Numbers 26103001, \n15H05702, 16H05977 , 18K19021 ).                      \n7 References  \n1) V. V Kruglyak, S.O. Demokritov, and D. Grundler, J. Phys. D. Appl. Phys. 43, 264001 \n(2010).  \n2) A.A. Serga, A.V. Chumak, and B. 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N’Diaye, G. Chen, K. Liu, \nand S.S.P. Parkin, Nat. Commun. 9, 4984 (2018).  \n27) S. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl. Phys. 40, 580 (2001).  \n28) Y. Liu, Z. Yuan, R.J.H. Wesselink, A.A. Starikov, and P.J. Kelly, Phys. Rev. Lett. 113, \n207202 (2014).  \n29) R.J. Gambino, J. Ziegler, and J.J. Cuomo, Appl. Phys . Lett. 24, 99 (1974).  \n30) R.C. Taylor and A. Grangulee, J. Appl. Phys. 47, 4666 (1976).  \n31) V. Vlaminck and M. Bailleul, Phys. Rev. B 81, 014425 (2010).  \n32) R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961).  \n33) M. Bailleul, D. Olligs, and C. Fermon, Appl. Phys. Lett. 83, 972 (2003).  \n34) R.K. Wangsness, Phys. Rev. 91, 1085 (1953).  \n35) K. Yamanoi, S. Yakata, T. Kimura, and T. Manago, Jpn. J.  Appl. Phys. 52, 083001 \n(2013).  \n \n                      \n9 Figure Captions  \nFig. 1. (a) Temperature dependen ce of magnetization  under in -plane magnetic field of 100 \nmT for GdxCo1-x with various values of  x. Right figures show the magnetic moments of \nsublattice, Co (blue) and Gd (yellow), with various composition at room temperature. The \ndominated element of the net magnetization transitions between Co and Gd depending on \nthe composition  x. (b)-(d) Perpedicular magnetization curves for the sample with ( b) x = 0.22, \n(c) x = 0.30, and ( d) x = 0.59. Red lines represent the linear fittings of the magnetization in \nbelow and above saturation magnetic field, and Ms,eff for each sample was extracted from the  \nintersection of two linear fittings.  \n \nFig. 2. (a) Top view of a spin wave device. The distance between the two CPW s is defined \nas d. Depending on the direction of the magnetic field, spin waves with magnetostatic \nbackward volume wave (MSBVW) and magnetostatic surface wave (MSSW) configuration s \nare investigated.  (b) Fourier transform of the current distribution in the designed CPWs.  \n \nFig. 3. (a) Frequency dependence of Re[ S11] spectra under external magnetic field in the \nMSBVW configuration for Gd 0.22Co0.78. (b) Resonance frequency as a function of the \nmagnetic field , and (c) full width at half maximum of the resonance peak as a function of \nthe frequency for GdxCo1-x with vari ous x values . The solid lines in (b) and (c) indicate the  \nfittings obtained  using Eqs. (1) and (2), respectively.  \n \nFig. 4. (a)–(c) Propagating spin wave spectra under an external magnetic field of 10 mT for \n(a) Gd 0.22Co0.78 with d = 10 μm, (b)  Gd0.30Co0.70 with d = 8 μm, and (c)  Gd0.59Co0.41 with d \n= 10 μ m. The i nsets in (a) and (c) show the intensity of  the transmittance signal | S21| as a \nfunction of d. \n                      \n10   Table I.   Experimentally determined Ms (300 K) and Ms,eff of Gd xCo1-x measured \nfrom SQUID measurements.  \n \nx Ms (300 K)  \n[MA/m]  Ms,eff \n[MA/m]  \n0.22 1.06 0.66 \n0.30 0.67 0.49 \n0.59 0.55 0.52 \n \nTable II.  Experimentally determined magnetic properties of Gd xCo1-x measured \nfrom spin wave measurements. The values in parentheses under vg and Latt were \ntheoretically calculated using Eqs. (3) and (4), respectively.  \n \nx Ms,eff \n[MA/m]  g α vg \n[km/s]  Latt \n[μm] \n0.22 0.68 2.26 0.016 10.4 (7.5)  2.13 (2.60)  \n0.30 0.32 2.68 0.032 7.5 (3.2)  (0.98)  \n0.59 0.45 1.90 0.016 3.2 (2.8)  1.65 (1.85)  \n                     \n11  \nFig. 1 \n  \n0 50 100 150 200 250 3000.00.20.40.60.81.01.21.4x = 0.22\nx = 0.30\nx = 0.40\nx = 0.59Ms (MA/m)\nTemperature (K)Mco,MGd\nx= 0.22 (Co -rich)Mnet\nx= 0.30 (Co -rich)\nx= 0.40 ( Gd-rich)\nx= 0.59 ( Gd-rich)\n0 500 1000 15000.00.30.60.91.2Magnetization (MA/m)\nMagnetic field (mT)Gd0.22Co0.78m0Ms,eff\n0 500 1000 15000.00.30.60.91.2\nGd0.30Co0.70Magnetization (MA/m)\nMagnetic field (mT)m0Ms,eff数式 y = a + b*x\nプロット Magnetization\n重み 重み付けなし\n切片 0.06994 ± 0.00798\n傾き 9.87513E-4 ± 2.40613E-5\n残差平方和 1.73685E-4\nピアソンのr 0.99911\nR二乗(COD) 0.99822\n補正Ｒ二乗 0.99763\n0 500 1000 15000.00.30.60.91.2\nGd0.59Co0.41Magnetization (MA/m)\nMagnetic field (mT)m0Ms,eff\n(b) (c) (d)(a)                    \n12   \nFig. 2 \n  \n1 μm2 μm\n5 μm\nHext\nMSSW configuration\ndμm\n+kHext\nMSBVW configuration\n0 2 4 6Intensity\nk (mm-1)1.2 mm-1(a) (b)                    \n13  \nFig. 3 \n  \n0 50 100 150 200 250Re[S11]\nMagnetic field (mT)Gd0.22Co0.78\n 7 GHz\n 8 GHz\n 9 GHz\n 10 GHz\n 11 GHz\n 12 GHz\n0 50 100 150 200 2500246810121416Frequency (GHz)\nMagnetic field (mT) Gd0.22Co0.78\n Gd0.30Co0.70\n Gd0.59Co0.41\n4 6 8 10 12 14 1651015202530\n Gd0.22Co0.78\n Gd0.30Co0.70\n Gd0.59Co0.41Linewidth (mT)\nFrequency (GHz)(a) (b) (c)                    \n14  \nFig. 4 \n2 3 4 5 6-2.0x10-40.02.0x10-44.0x10-4Re[S21], Im[ S21]\nFrequency (GHz) Re[S21]\n Im[S21)]Gd0.22Co0.78, d = 10 μm\n8 12 16 2010-4|S21|\nd (mm)\n2 3 4 5 6-2.0x10-40.02.0x10-44.0x10-4Gd0.30Co0.70, d = 8 μmRe[S21], Im[ S21]\nFrequency (GHz) Re[S21]\n Im[S21)]\nDf/2\n2 3 4 5 6-2.0x10-40.02.0x10-44.0x10-4Gd0.59Co0.41, d = 10 μmRe[S21], Im[ S21]\nFrequency (GHz) Re[S21]\n Im[S21)]\n6 7 8 91010-410-3|S21|\nd (mm)(a) (b) (c)"    },    {        "title": "0706.2726v2.Probing_the_Role_of_the_Barrier_Layer_in_Magnetic_Tunnel_Junction_Transport.pdf",        "content": "arXiv:0706.2726v2  [cond-mat.str-el]  5 Dec 2007Probing the role of the barrier layer in magnetic tunnel junc tion\ntransport\nB. B. Nelson-Cheeseman,1,∗R. V. Chopdekar,2,1L. M. B.\nAlldredge,2,1J. S. Bettinger,1E. Arenholz,3and Y. Suzuki1\n1Department of Materials Science and Engineering,\nUniversity of California, Berkeley, California 94720, USA\n2School of Applied and Engineering Physics,\nCornell University, Ithaca, New York 14853, USA\n3Advanced Light Source, Lawrence Berkeley National\nLaboratory, Berkeley, California 94720, USA\nAbstract\nMagnetic tunnel junctions with a ferrimagnetic barrier lay er have been studied to understand\nthe role of the barrier layer in the tunneling process - a fact or that has been largely overlooked until\nrecently. Epitaxial oxide junctions of highly spin polariz ed La0.7Sr0.3MnO3and Fe 3O4electrodes\nwithmagneticNiMn 2O4(NMO)insulatingbarrierlayersprovideamagnetictunnelj unctionsystem\nin which we can probe the effect of the barrier by comparing junc tion behavior above and below\nthe Curie temperature of the barrier layer. When the barrier is paramagnetic, the spin polarized\ntransport is dominated by interface scattering and surface spin waves; however, when the barrier\nis ferrimagnetic, spin flip scattering due to spin waves with in the NMO barrier dominates the\ntransport.\nPACS numbers: 75.47.-m,85.75.-d,72.25.Mk,75.70.Cn\n1The magnetic tunnel junction (MTJ) is one of the most simple spin pola rized devices\nwhose transport behavior has yet to be completely understood. I ts basic device charac-\nteristics have been explained in terms of a simple quantum mechanical model of two spin\npolarized reservoirs separated by a potential barrier over two de cades ago.1In this picture,\ndepending on the relative orientation of the magnetization in the two ferromagnetic elec-\ntrodes, a high or low resistance results. However, this simplified pict ure does not accurately\ndescribe the behavior of real MTJs. Improvements in the electrod e-barrier interfaces have\nnow led to the observation of magnetic tunnel junction behavior at room temperature2and\nthe realization that the transport through these structures is e xtremely sensitive to the in-\nterface scattering and spin polarized interface density of states of the electrodes.3It was not\nuntil recently, however, that the importance of understanding t he role of the barrier layer\nin the tunneling process was recognized in experimental and theore tical studies of MTJs\nwith MgO barriers.4In these junctions, a number of factors dictated by the barrier la yer\nwere recognized to be important, including the chemical bonding bet ween the atoms at the\nelectrode-barrier interface, interface resonance states, and the symmetries of the propagat-\ning states in the electrodes and evanescent states in the barrier la yer. In order to develop\na complete picture of spin polarized transport in MTJs, the role of th e barrier layer in the\ntunneling process as well as electrode-barrier interface effects m ust be generalized.\nThere have been some efforts to elucidate the role of the barrier lay er by introducing\nmagnetism into the barrier through dilute doping of a nonmagnetic ba rrier with magnetic\nionsortheuseofparamagneticbarriers. Jansen et al.foundthat δ-dopingofAl 2O3barriers\nwith Fe ions increased the junction magnetoresistance (JMR) as a f unction of Fe content in\nthe barrier.5The enhanced JMR was explained in terms of spin filtering of the tunne ling\nelectrons, whereby a preferential transmission (increased cond uctance) exists for one type of\nspin due to the polarized barrier states.6Other studies of MTJs with paramagnetic barriers\nsuggest that the presence of paramagnetism in the barrier layer d oes not preclude large\nJMR values or distinct switching characteristics.7,8,9These results suggest that magnetism\nin the barrier can be spin preserving rather than adding to spin-flip s cattering via spin\nwaves. If weak magnetism in the barrier is spin preserving, then inco rporating long-range\nferromagnetic order in the barrier layer of a MTJ should provide a sy stem where the role of\nthe barrier layer can be readily probed as long as the barrier does no t magnetically couple\nthe two ferromagnetic electrodes.\n2In this paper, we demonstrate magnetic tunnel junction behavior in junctions with ferro-\nmagnetic electrodes and a barrier layer with long range ferrimagnet ic order. Comparison of\njunction behavior when the barrier layer is ferrimagnetic to when it is paramagnetic enables\nus to probe the role of the barrier layer in one of the simplest spin pola rized devices. In\nparticular, we focus on Fe 3O4/NiMn 2O4(NMO)/LSMO (spinel/spinel/perovskite) junctions\nto preserve the magnetic decoupling of the electrodes in the prese nce of a magnetic bar-\nrier layer. Magnetization measurements of the layers as well as at t he interfaces indicate\nthat we have two ferromagnetic electrodes that switch independe ntly despite the presence of\na ferrimagnetic-paramagnetic barrier layer. Spin polarized transp ort measurements reveal\njunction magnetoresistance values as high as -30% (normalized to 8 kOe). Two different\nconduction mechanisms are observed which directly highlight the pas sive or active role of\nthe barrier layer in the spin transport. Above the T Cof the NMO barrier, when the barrier\nlayer is paramagnetic, the electrode-barrier interfaces dominate the spin transport, resulting\nin anasymmetric bias dependence ofthe JMR andinelastic tunneling sp ectra (IETS). Below\nthe TCof the NMO barrier, the ferrimagnetism in the barrier dominates the spin transport,\nresulting in a transition to a symmetric bias dependence of the JMR an d IETS.\nThe trilayers of Fe 3O4/NMO/LSMO were grown by pulsed laser deposition on SrTiO 3\n(STO) (110) substrates following Ref. 9. LSMO and Fe 3O4were chosen for their highly\nspin polarized nature as both have been theorized and shown to be h alf- metallic.8Since\nisostructural barrier layers have been shown to greatly increase the JMR values for epitaxial\nFe3O4MTJs,8a ferrimagnetic spinel, NiMn 2O4, was selected. NMO films were grown at 550\n◦C in 10 mTorr of a 99% N 2/1% O 2gaseous mixture. For these deposition conditions, the\nNMO film was insulating with an onset of ferrimagnetism around 60 K, a la rge coercive field\nof 1.6 T at 30 K, a magnetization of 0.8 µBper formula unit, and an inverse spinel cation\nsite distribution.10Barrier thicknesses were 20, 30, and 45 ˚A. The films grow epitaxially on\nthe STO substrate with excellent crystallinity as confirmed by x-ray diffraction and Ruther-\nford backscattering channeling measurements. The MTJs were fa bricated by conventional\nlithography and Ar ion milling.\nThe bulk magnetism of the samples was investigated by a supercondu cting quantum\ninterference device (SQUID) magnetometer, while the interface m agnetization was inves-\ntigated by X-ray magnetic circular dichroism (XMCD) at the Advance d Light Source on\nBL4.0.2. The transport measurements were conducted in magnetic fields up to 8kOe and\n3temperatures from 15 to 300 K. The bias and temperature depend ence of the JMR were\ncalculated in accordance with Julliere’s model by the following equation :△R\nR(P)×100, where\n△R=RAP-RP,1and the reference (parallel magnetization) resistance was taken at 8 kOe.\nMagnetic hysteresis loops of thetrilayers show distinct switching of the two ferromagnetic\nlayers. We observe large coercive field differences between the LSM O and Fe 3O4, thus creat-\ning well-defined parallel and antiparallel magnetization states at all t emperatures. However,\nthe magnetization of the NMO barrier cannot be probed with bulk met hods due to its small\nmagnetic signal compared to that of the ferromagnetic electrode s.\nIn order to probe the magnetic behavior at the interfaces and with in the barrier layer,\nelement-specific XMCD was used in the total electron yield mode. Sinc e the total electron\nyieldmethodissurfacesensitive, westudiedheterostructuresof LSMO(40nm)/NMO(5nm)\nand LSMO (40 nm)/NMO (5 nm)/Fe 3O4(5 nm) to probe both interfaces. Figure 1 shows\nthe Mn XMCD spectra for the LSMO/NMO bilayer sample for various te mperatures. A\npronouncedvalley(peak)intheXMCDisevident at640eV(642.5eV)a tlowertemperatures\nresembling XMCD spectra of NMO single films.10This feature disappears above 60 K,\nresulting in Mn XMCD spectra at 80 and 100 K resembling that of LSMO t hin films.11In\norder to probe the origins of the Mn magnetism, hysteresis loops we re taken at 640 and\n642.5 eV. Hysteresis loops taken at 642.5 eV show magnetically soft M n with coercive fields\n<0.5 kOe over the entire temperature range. Those taken at 640 eV show a transition from\nthe soft loops above 60 K (similar to those at 642.5 eV) to magnetically harder loops that\ndo not reach saturation by 8 kOe, resembling SQUID hysteresis loop s of single NMO films.\nThis transition indicates the onset of ferrimagnetism in the magnetic ally hard NMO barrier\nat 60 K.\nFrom the XMCD hysteresis loops in Fig. 1, it is clear that the Mn magnet ism in the\nNMO and LSMO is probed at 640 eV whereas only the LSMO magnetism is p robed at 642.5\neV. The observation of soft hysteresis loops at all temperatures for Mn at 642.5 eV is direct\nevidence that the LSMO is not exchange coupled to the magnetically h arder NMO whose\nonset of ferrimagnetism is verified at 640 eV. A comparison of the hy steresis loops taken at\n640and642.5eVat55Kisfurtherevidence thatonlyameresuperpo sitionoftheMnsignals\nfrom the LSMO and NMO takes place, and not a correlated magnetic c oupling effect. The\nlack of magnetic coupling at the LSMO-NMO interface likely arises from misfit dislocations\ndue to the lattice mismatch between the perovskite and spinel stru ctures, providing for easy\n4FIG.1: (Coloronline) SurfacesensitiveXMCDinvestigatin g theLSMO-NMOinterfacemagnetism.\nNormalized Mn XMCD of the NMO-LSMO interface, and Mn hystere sis loops taken at 640.0 eV\nand 642.5 eV as a function of temperature.\n180◦domain wall creation at the LSMO-NMO interface, anddecoupling the magnetic layers.\nThislackofcoupling atthisinterfaceenablestheindependent switch ing ofthetwo electrodes\nbelow themagneticordering temperature oftheNMO. This independ ent switching isfurther\nverified in symmetric minor loops of LSMO.\nXMCD hysteresis loops were also taken at various temperatures on the trilayer sample\nto probe the magnetic behavior of the NMO-Fe 3O4interface. A distinct increase in coercive\nfield of the Fe 3O4layer is seen below the NMO T C. This increase, as well as coincident Fe,\nMn, and Ni loops, reveals the presence of strong ferromagnetic c oupling between the two\nspinel layers, while still maintaining the onset of NMO ferrimagnetism.12\nNow that we have established the magnetic decoupling of the ferrom agnetic electrodes\ndespite the magnetic barrier, we focus on the spin transport in the junction above and\nbelow the T Cof the barrier layer. As shown in Fig. 2(a), transitions in the magnet ization\nhysteresis loops coincide well with large and abrupt transitions in the JMR. The antiparallel\nLSMO-Fe 3O4magnetization configuration is the low resistance state, resulting in a negative\nmagnetoresistance. This is due to the opposite spin polarizations of LSMO and Fe 3O4\nelectrodes, which are majority and minority spin polarized, respect ively.8\nThe temperature dependence of the JMR is studied to investigate t he effect of the barrier\nlayer magnetism on the magnetotransport of the junctions [Fig. 2( b)]. The maximum JMR\n5FIG. 2: (Color online) (a) Switching characteristics of Fe 3O4/NMO/LSMO junctions: magnetic\nhysteresis loop and resistance vs applied magnetic field at 7 5 K. (b) Junction resistance and max-\nimum JMR as a function of temperature, with bias dependence o f JMR for each regime.\nincreases in magnitude with decreasing temperature until reaching a maximum around 45K,\nandthendecreaseswithsubsequent decreaseintemperature. A dramaticincreaseinjunction\nresistance is coincident with the decrease in JMR. A change in JMR bias dependence also\noccurs as a function of temperature. At high temperatures, the JMR drops off gradually\nat higher voltages and has a skewed, asymmetric behavior in which th e JMR maximum is\nobserved at a finite voltage. A transition occurs around 60 K, in whic h the bias dependence\ntransitions to a symmetric bias dependence, characterized by a JM R maximum at zero\nbias, a highly symmetric shape, and a rapid decrease in JMR for V >50-100 mV. At low\ntemperatures, the bias dependence continues to be symmetric ab out zero bias, although the\nmaximum JMR decreases with decreasing temperature. At these low er temperatures, the\nchange in resistance △R due to the application of a magnetic field continues to increase\nwith decreasing temperature despite the fact that the JMR decre ases. Similar behavior is\nseen for all barrier layer thicknesses.\nIn order to investigate the inelastic tunneling processes in our junc tions, we plot the\nsecond derivative of the IV curves, known as the IETS. The IETS illu strates the inelastic\ntunneling processes due to phonons and spin waves, or magnons. I n order to isolate the\neffects of magnetism on the inelastic transport, IETS from the two distinct magnetization\n6configurations, (IETS) Pand (IETS) AP, are subtracted from one another. The resulting\nIETS (Fig. 3) shows inelastic tunneling events related solely to magno ns as all other factors\nareheld constant between bothmagnetizationconfigurations.13Thed2I/dV2=0value occurs\nat a finite bias for T >60 K, but moves towards zero bias below 60 K. The peak locations\n(30-50 mV) of the IETS of our junctions for all temperatures are well within the range of\n12-100 mV commonly associated with magnons.14,15\nThe observation of asymmetric bias dependence above the orderin g temperature of NMO\nand symmetric bias dependence below the ordering temperature of NMO in both the JMR\nand IETS suggests that different conduction mechanisms are domin ant depending on the\nmagneticstateofthebarrier. Therehavebeennumerousstudies ofMTJswhereasymmetries\nin the JMR bias dependence have been attributed to two different int erface density of states\nat the two electrode-barrier interfaces. Magnon-assisted tunn eling current from electrode\nsurface magnons has also been predicted to be asymmetric with res pect to bias. Moodera\net al.proposed that part of the large decrease in JMR with bias can be att ributed to the\nexcitation of magnons with increasing bias, which randomize the tunn eling electron spins.16\nThus an asymmetry in JMR could also be caused by the unequal numbe r of magnons\ncreated at each electrode surface as a result of the inevitable diffe rence in the two electrode-\nbarrier interfaces.13,15In our junctions, the Fe 3O4-NMO interface is isostructural since both\ncrystallize inthespinel structure; theLSMO-NMO interface isspine l/cubic perovskite where\nantiphase boundaries inevitably result in a more disordered interfac e. These two different\ninterfaces give rise to two different barrier heights, two different in terface density of states,\nandtwodifferentmagnondensityofstates. Thustheasymmetryin theJMRbiasdependence\ncan be attributed to a number of mechanisms, all of which are assoc iated with the interface.\nHowever, while the observation of an asymmetric bias dependence d ue to the electrode-\nbarrier interfaces is not unusual, the observation of a symmetric a nd asymmetric bias de-\npendence of the JMR at different temperatures in the same junctio n is a surprising result\nand sheds light on the role of the barrier in the tunneling process. Su ch changes in sym-\nmetry are also observed in the IETS data. The asymmetric bias depe ndence of the IETS\nabove the magnetic ordering temperature of the NMO is indicative of a difference in the\nforward and reverse magnon-assisted conduction and is consiste nt with transport dominated\nby electrode-barrier interface magnons. As the temperature is lo wered below the magnetic\n7FIG. 3: (Color online) Subtracted IETS as a function of tempe rature, showing shift towards\nsymmetry about zero bias below 65 K.\nordering temperature of the NMO (60 K), the bias dependence of t he JMR and IETS be-\ncomes symmetric, thus suggesting that the dominant conduction m echanism is no longer\nsensitive to interface effects but to the bulk of the barrier. Gajek et al.postulated that the\nsymmetric drop of JMR with bias for their spin filter junctions with fer romagnetic barrier\nlayers was due to magnonexcitations inside the barrier layer.17Recalling that the subtracted\nIETShighlights theinelastic transportduetoonlymagnonsinthejun ction, thefactthatthe\nIETS becomes symmetric with respect to bias at around 60 K sugges ts that magnons within\nthe barrier are beginning to influence the spin dependent transpor t. Such magnons from the\nbulk of the barrier would manifest themselves in the same way regard less of the direction of\nthe current and thus give rise to a symmetric bias dependence both in the IETS and JMR.\nFurthermore, a closer look at the bias dependence of the JMR indica tes a dramatic decrease\nin the JMR as a function of increasing bias. Below the magnon excitatio n energy of about\n30-50 mV ascertained from the IETS data, spin polarized transpor t is preserved, resulting\nin significant JMR; above this voltage, magnons are excited and spin p olarized transport is\nsuppressed.\nAt even lower temperatures, theJMR biasdependence remainssym metric but themagni-\ntude of the JMR begins to decrease around 45 K. This decrease is at tributed to the dramatic\nincrease in the junction resistance that is due to the suppressed m etal-insulator transition\nof the Fe 3O4electrode, causing the electrode resistance to rival the barrier r esistance.9As a\nfunction of decreasing temperature, the increase in electrode re sistance along with a compa-\nrable△R results in an overall decrease in JMR. Similar junctions with nonmagn etic barriers\n8also show peaks in the JMR, precluding the possibility that the decrea se in the JMR is re-\nlated to the magnetic transition of the barrier.9The symmetric bias dependence of the\nJMR and the increase in △R in this temperature range suggests that the magnetotranspor t\ncontinues to be dominated by bulk magnon excitations in the NMO barr ier layer.\nIn summary, spin polarized transport in our Fe 3O4/NMO/LSMO junctions reveals that\nthe barrier plays an important role in the tunneling process. In thes e junctions, we have\nshown that spin transport is dominated by the interfaces or by the barrier itself depending\non whether NMO is paramagnetic or ferrimagnetic. This behavior is ob served via a distinct\nchange in the bias dependence of the MTJs which coincides with the on set of ferrimagnetism\nin the NiMn 2O4barrier layer as confirmed by XMCD. When the barrier layer is parama g-\nnetic at high temperatures, the electrode-barrier interfaces de termine the asymmetric JMR\nbias dependence due to their different electrode-barrier surface magnon properties and dif-\nferent barrier heights. As the temperature is lowered, the JMR bia s dependence becomes\nhighly symmetric, suggesting a conduction mechanism involving magno ns within the fer-\nrimagnetic NMO barrier layer. Such hybrid behavior in a single junction is promising for\nfuture spintronic applications.\nThis work was supported in full by the Office of Basic Energy Sciences , Division of\nMaterials Sciences and Engineering, of the U.S. Department of Ener gy under Contract No.\nDE-AC02-05CH11231. Thanks to J.S. Moodera for fruitful discus sion. Processing was\nperformed in the University of California - Berkeley Microlab.\n∗Electronic address: bbnelsonchee@berkeley.edu\n1M. Julliere, Phys. Lett. 54A, 225 (1975).\n2J.S. Moodera, L.R. Kinder, T.M. Wong, and R. Meservey, Phys. Rev. Lett. 74, 3273 (1995).\n3G.T. Woods, R.J. Soulen, Jr., I.I. Mazin, B. Nadgorny, M.S. O sofsky, J. Sanders, H. Srikanth,\nW.F. Egelhoff, and R. Datla, Phys. Rev. B. 70, 054416 (2004).\n4W.H. Butler, X.-G. Zhang, T.C. Schulthess, and J.M. MacLare n, Phys. Rev. B 63, 054416\n(2001).\n5R. Jansen, and J.S. Moodera, Appl. Phys. Lett. 75, 400 (1999).\n6R. Jansen, and J.C. Lodder, Phys. Rev. B 61, 5860 (2000).\n97M-H. Jo, N.D. Mathur, N.K. Todd, and M.G. Blamire, Phys. Rev. B61, R14905 (2000).\n8G. Hu, and Y. Suzuki, Phys. Rev. Lett. 89, 276601 (2002), and references therein.\n9L.M.B. Alldredge, R.V. Chopdekar, B. Nelson-Cheeseman, an d Y. Suzuki, Appl. Phys. Lett.\n89, 182504 (2006).\n10B.B. Nelson-Cheeseman, R.V. Chopdekar, M.F. Toney, J.S. Be ttinger, E. Arenholz, and Y.\nSuzuki. (Unpublished).\n11R. Stadler, Y.U. Idzerda, Z. Chen, S.B. Ogale, and T. Venkate san, Appl. Phys. Lett. 75, 3384\n(1999).\n12B.B. Nelson-Cheeseman, R.V. Chopdekar, J.S. Bettinger, E. Arenholz, and Y.Suzuki.\narXiv:0709.3518 [cond-mat.str-el]\n13J.S. Moodera, J. Nowak, and R.J.M. van de Veerdonk, Phys. Rev . Lett.80, 2941 (1998).\n14D.C. Tsui, R.E. Dietz, and L.R. Walker, Phys. Rev. Lett. 27, 1729 (1971).\n15Y. Ando, J. Murai, H. Kubota, and T. Miyazaki, J. Appl. Phys. 87, 5209 (2000).\n16J.S. Moodera, and G. Mathon, J. Magn. Magn. Mater. 200, 248 (1999).\n17M. Gajek, M. Bibes, A. Barthelemy, K. Bouzehouane, S. Fusil, M. Varela, J. Fontcuberta, and\nA. Fert, Phys. Rev. B 72, 020406(R) (2005).\n10"    },    {        "title": "1201.3465v1.Weak_ferrimagnetism_and_multiple_magnetization_reversal_in_α_Cr3_PO4_2.pdf",        "content": "Weak ferrimagnetism and multiple magnetization reversal in α-Cr 3(PO 4)2 \nA.N. Vasiliev1, O.S. Volkova1, E. Hammer2, R. Glaum2, J.-M. Broto3, M. Millot3,4, G. Nénert5, \nY. T. Liu6, J.-Y. Lin6, R. Klingeler7, M. Abdel-Hafiez8, Y. Krupskaya8, A.U.B. Wolter8, B. \nBüchner8 \n \n1Low Temperature Physics Department, Moscow State University, Moscow 119991, Russia \n2Institute for Inorganic Chemistry, Bonn University, D-53121 Bonn, Germany \n3Université de Toulouse; UPS, INSA, 143 Avenue de Rangueil, F-31400 Toulouse, France and \nLaboratoire National des Champs Magnétiques Intenses (LNCMI) -- CNRS UPR 3228, 143 \nAvenue de Rangueil, F-31400 Toulouse, France \n4Department of Earth and Planetary Science, Un iversity of California-Berkeley, Berkeley, \nCalifornia 94720, USA \n5Laue-Langevin Institute, Grenoble 38042, France \n6Institute of Physics, National Chiao-Tung University, Hsinchu 30076, Taiwan \n7Kirchhoff Institute for Physics, Heidelberg University, Heidelberg D-69120, Germany \n8Leibniz Institute for Solid State and Mate rials Research, Dresden D-01069, Germany \n \n The chromium(II) orthophosphate α-Cr 3(PO 4)2 is a weak ferrimagnet with the Curie \ntemperature T C = 29 K confirmed by a λ-type peak in specific heat. Dominant antiferromagnetic \ninteractions in this system are characterized by the Weiss temperature Θ = - 96 K, indicating an \nintermediate frustration ratio | Θ|/TC ~ 3. In its magnetically ordered states α-Cr 3(PO 4)2 exhibits a \nremarkable sequence of temperature-induced magnetization reversals sensitive to the protocol of \nmeasurements, i.e. either field-cooled or zer o-field-cooled regimes. The reduction of the \neffective magnetic moment 4.5 μB/Cr2+, as compared to the spin-only moment 4.9 μB/Cr2+, \ncannot be ascribed to the occurence of the low-spin state in any crystallographic site of the Jahn-Teller active 3d\n4 Cr2+ ions. X-ray absorption spectra at the K-edge indicate divalent chromium \nand unravel the high-spin state of these ions at the L 2,3-edges. Weak ferrimagnetism and multiple \nmagnetization reversal phenomena seen in this compound could be ascribed to incomplete \ncancellation and distortion of partial s pontaneous magnetization functions of Cr2+ in its six \ncrystallographically inequivalent positions. \n \nIntroduction \n The chromium (II) orthophosphate, α-Cr 3(PO 4)2, belongs to the vast family of anhydrous \nphosphates of divalent metals M 3(PO 4)2 with M = Mg, Ca, Cr - Zn. The numerous crystal \nstructures met in this multitude differ in the interconnection patterns of the metal-oxide polyhedra being related to those in the naturally occurring minerals farringtonite [1], graftonite \n[2], and sarcopside [3]. The basic motif in their structures is that of chains of edge-sharing \noctahedra similar to the olivine structure type but with every fourth octahedron missing. This \nleaves trimers of octahedra containing two inequivalent cation sites, the central M1 site with a rather regular symmetry and two distorted M2 sites of the terminal octahedra.  \n Except, probably, manganese(II) orthophosphate, Mn\n3(PO 4)2 [4], the magnetic properties \nof the transition metals orthophosphates are well documented.  Iron(II) orthophosphate, \nFe3(PO 4)2 (sarcopside ), orders antiferromagnetically (or ferrimagnetically) at T N = 44 K, with the \nFe2 sites in each chain having opposite spin directions along the [100] axis, leaving the central \nFe1 ion frustrated with no net magnetic  moment [5]. Cobalt(II) orthophosphate, Co 3(PO 4)2, \nshows the onset of antiferromagnetism at T N = 30 K, its magnetic structure being commensurate \nwith the chemical unit cell with the magnetic cell doubled along the a-axis [6]. Nickel(II) \northophosphate, Ni 3(PO 4)2, exhibits a three-dimensio nal magnetic ordering at T N = 17 K. \nAssumingly, there are ferromagnetic interactions within the Ni 3O14 trimers which are coupled \nantiferromagnetically between them, giving rise to a purely antiferromagnetic structure [7]. \nFinally, copper(II) orthophosphate, Cu 3(PO 4)2, was found to be antiferromagnetic with the Néel \ntemperature T N = 22.5 K. The magnetic propagation vector (0 0 1/2) is referred to the triacute \nreduced chemical unit cell and the magnetic structure is collinear with equal moments of about 0.68 μ\nB on each of the two Cu1 and Cu2, cr ystallographically inequivalent Cu2+ ions [8]. Besides \ncited data on magnetism in monometallic orthophosphates M 3(PO 4)2, an extensive information is \navailable on the magnetic properties of the mixed metal orthophosphates, e.g. CuNi 2(PO 4)2 [9] or \nSrFe 2(PO 4)2 [10]. All of them exhibit long-range anti ferromagnetic ordering at low temperatures, \nbut the complexity of crystal structures hampers, usually, the parameterization of the magnetic \nsubsystem. \n One of the least studied members of this family, chromium(II) orthophosphate, Cr 3(PO 4)2, \ncan be found in two distinct crystallographic modifications [11, 12]. The crystal structure of the high-temperature P2\n1/n monoclinic phase β-Cr 3(PO 4)2, stable in the temperature range 1250 - \n1350 C, is close to that of farringtonite [1]. It contains interconnected zigzag chains of corner-\nsharing - edge-sharing elongated octahedra in the Cr2 – Cr1 – Cr1 – Cr2 sequence [12]. This \nphase experiences long-range magnetic ordering at T C = 36 K which is preceded by short-range \ncorrelation maximum at T ~ 60 K. The paramagnetic Weiss temperature in β-Cr 3(PO 4)2 is \nnegative, Θ = –165 K, indicating strong predominance of antiferromagnetic interactions. At \nvariance with “standard” antiferromagnetic behavior, the magnetization in the magnetically ordered state of β-Cr\n3(PO 4)2, i.e. at T < T C, rises with lowering temperature. This may indicate either incomplete cancellation of primarily antip arallel sublattice magnetizations or their canting \ndue to the effects of magnetocrystalline anisot ropy or Dzyaloshinskii – Moriya interaction. \n The orthorhombic phase α-Cr 3(PO 4)2 (P212121, Z = 8, a = 8.4849(10) Å, b = 10,3317(10) \nÅ, c = 14.206(2) Å) is stable in the temperature range between 1100 and 1250 °C, at lower \ntemperature decomposition into CrP, Cr 2O3, and Cr 2P2O7 is observed [11]. The unit cell \nparameters were determined at room temper ature from X-ray powder diffraction data of a \nquenched sample. The local environment of the Cr2+ ions is rather diverse as compared to the \nstructure of β-Cr 3(PO 4)2. Following the numbering scheme in [11] Cr1 to Cr5 show fourfold, \ndistorted (roof-shaped) square-planar coordination with 1.96 ≤ d(Cr-O) ≤ 2.15 Å. Cr6 shows five \noxygen ligands at 1.97 ≤ d(Cr-O) ≤ 2.29 Å.  \n The arrangement of the structural units in α-Cr 3(PO 4)2 can be rationalized in terms of \nclose-packed tubes (parallel to the crystallographic b-axis) with Cr2+ on their inner surface and \nPO 4 tetrahedra on the outer surface as well as in the tube centers Fig. 1a [11]. The spatial \narrangement of Cr2+ ions is that of a double helix remini scent of DNA molecule, as shown in \nFig. 1b. The atomic arrangement within one tube is related to the one in adjacent tubes by pseudo \n31-screw axes. The local oxygen coordination of Cr2+ ions exhibits five distorted square-planar \ngroups Cr1O 4 to Cr5O 4 and a slightly distorted square-pyramidal unit Cr6O 5 (Fig. 1b). One chain \nformed by vertex-sharing of alternating PO 4 and CrO 4 units (containing Cr1, Cr2 and Cr3) winds \nalong the pseudo 3 1 axis. By edge-sharing between Cr2O 4 and Cr5O 4 this one is linked to a \nsecond type of chain. The latter consists of “dimers” Cr4O 4-Cr6O 5, which are linked via Cr5O 4 \ngroups and PO 4 tetrahedra by vertex-sharing. This second chain winds around the 2 1 screw axes \nparallel to the b-axis at the centre of the “tubes” descri bed above (Fig. 1b). An unusual structural \nfeature is the vertex-sharing between the almost orthogonal square-planes Cr4O 4 and Cr5O 4. \nThus, a rather short distance d(Cr4-Cr5) = 3.08 Å) is formed. An unusually low average \nmagnetic moment μeff = 4.28 μB per Cr2+ ions and negative Weiss temperature Θ = –55 K were \nreported for α-Cr 3(PO 4)2 [11]. In this paper we present the first extensive experimental study to \nelucidate the unexpected behavior of this compound.  Experimental \n The present study of α-Cr\n3(PO 4)2 includes the sample preparation, the K and L 2,3 edges X-\nray absorption spectroscopy (XAS), measurements  of specific heat in the range 2 - 100 K, \nmagnetic susceptibility measurements in the range 2 - 300 K, and pulsed magnetic field \nmeasurements up to 50 T. \n \n Sample preparation The α-modification of chromium(II) orthophosphate Cr 3(PO 4)2 has been synthesized according \nto Ref. [11] from mixtures of CrPO 4 and Cr metal in the ratio 2:1 in evacuated silica ampoules at \n1200°C (4 days) and quenched to room temperature. By chemical vapor transport (transport \nagent I 2, 1200 → 1100 °C, quartz ampoule) deep blue-violet single crystals of α-Cr 3(PO 4)2 with \nedge-lengths up to several tenths of a millimeter have been obtained.  \n To avoid irreproducibility of results in di fferent measurements due to the effects of \nmagnetocrystalline anisotropy the necessary amount s of small single crystals were crushed into \npowder in the agate mortar and pressed into pellets. The material used for all measurements was \nselected under a microscope from crystals deposited by chemical vapour transport. Its X-ray \npowder diffraction pattern showed  no traces of impurities.  \n \n X-ray absorption spectroscopy \n The K edge and L 2,3 edges of chromium in α-Cr 3(PO 4)2 were recorded at at EXAFS and \nHSGM beam lines, respectively, of the National Synchrotron Radiation Research Center in \nTaiwan. The metal K edge corresponds to excita tion of 1s electrons to valence bond states \nlocalized on the metal. The energy and the shape of the X-ray absorption near edge structure characterize the local symmetry and the oxidation state of the metal ions. To the first \napproximation, the correlation between the ener gy of the K-edge and the valence state of \nchromium is linear [13]. The XAS spectra in α-Cr\n3(PO 4)2 and several reference compounds in \ndifferent oxidation states [14] are shown in Fi g. 2. The XAS K-edge in various compounds was \ndefined as an inflection point in corresponding spectra. The removal of one electron from the \nvalence shell of chromium results in ~ 3 eV shift of the K- edge to higher energy. The \ndependence of the K-edge energy on the oxidation state of chromium is shown in Fig. 3. As \nexpected from its chemical formula, the observed value of the K-edge in α-Cr 3(PO 4)2 is a clear \nsignature of divalent chromium. \n The L 2,3 edges in XAS spectra correspond to excita tions of 2p electrons to the partially \nunfilled 3d shell. XAS spectra taken at the transition metal L 2,3 edges are highly sensitive to both \nthe valence and the spin states [15]. An increase  of the valence state of the metal ion by one \ncauses a shift of XAS L 2,3 edges by ~ 1 eV toward higher energy. This shift is due to a final state \neffect in the X-ray absorption process. The energy difference between a 3dn (3d4 for Cr2+) and a \n3dn-1 (3d3 for Cr3+) configurations is \neV U U dp dpE dp dpEEdd pdn n n n≈ − ≈ → − → =Δ+ −) 32 32( )32 32(1 5 6 5 1 6 \nwhere U dd (resp. U pd ) is the Coulomb repulsion energy between two 3d electrons (between a 3d \nelectron and the 2p core hole) [16]. Both energies are sensitive to the arrangement of the \nelectrons on the d shell, i.e. to the spin state of metal.   In Fig. 4, a shift of the L 3 edge in divalent α-Cr 3(PO 4)2 s y s t e m  t o  l o w e r  e n e r g y  b y  \napproximately 1 eV as compared to Cr 2O3 containing trivalent chromium is evidenced. \nMoreover, the detailed analysis of the Cr L 2,3 edges in α-Cr 3(PO 4)2 allows one to suggest that it is \nvery similar to the behavior of CrF 2 where the high-spin S = 2 ground state for the 3d4 electron \nconfiguration was firmly established [17]. The fine structure of the L 2,3 spectral features deserves \na complementary analysis. In particular, we do not propose any interpretation for the appearance \nof extra peaks of moderate amplitudes CT 1-3 between L 3 and L 2 edges which were attributed to \ncharge transfer from the ligand valence orbitals to the Cr 3d orbitals in CrF 2.  \n    Specific heat \n The temperature dependence of specific heat C\np in α-Cr 3(PO 4)2 is shown in Fig. 5. The \nobvious λ-peak at T C = 29 K indicates a second-order phase transition from the paramagnetic to \nthe magnetically ordered phase. Although fluctuations yield a seemingly enlargement of the \nspecific heat jump at T C so that the experimentally observed anomaly clearly overestimates the \nmean field result, the observed value ΔCp = 10.5 J/mol K is far smaller than expected in the \nmean field theory: \nmolKJ\nS SS nRSCp 5.57)1()1( 5\n2 2=+ ++= Δ  \nwhere n = 3 is the number of magnetically active ions in the α-Cr 3(PO 4)2 chemical formula, R = \n8.314 J/mol K is the universal gas constant and S = 2 is the spin-only moment in presumably \nhigh-spin state of Cr2+ ions. This fact indicates that a large amount of magnetic entropy is \nreleased above T C due to short-range magnetic correlations. Under a magnetic field of 9 T, the λ-\npeak somewhat broadens and slightly seems to shift to higher temperatures. The upward shift of \nthis anomaly could be associated with the presence of ferromagnetic exchange interactions in the \nsystem stabilized by the external magnetic field. The influence of the magnetic field on the \nspecific heat in α-Cr 3(PO 4)2 is illustrated by the inset to Fig. 4, with a C p/T vs. T2 plot of  the \nexperimental data. Such a presentation usually allows one to separate ~ T3 terms, assigned to \nphonons and three-dimensional antiferromagnetic magnons, from any other terms of lower \ndimensionality. Here however in the case of α-Cr 3(PO 4)2, this procedure is hampered by a large \nSchottky-type anomaly, hardly sensitive to the magnetic field. The indifference of this feature to \nsignificantly strong magnetic field signals its non-magnetic origin. \n  Magnetic susceptibility  The temperature dependence of the magnetic susceptibility χ of α-Cr 3(PO 4)2 taken at B = \n0.1 T is shown in Fig. 6. The smooth increase of χ seen  under decreasing temperature is \nfollowed by the abrupt jump of the signal at T C = 29 K in agreement with earlier observations \n[11], typical of a compound whose magnetization in the magnetically ordered state contains a \nferromagnetic component. Below T C however, the magnetic susceptibility evidences a \nremarkable sequence of temperature-induced magnetization reversals sensitive to the protocol of \nmeasurements, i.e. either field-cooled (FC) or zero-field-cooled (ZFC) regimes, as shown in the \ninset of Fig. 6. Note that in the FC regime the magnetic susceptibility even shows a \n“diamagnetic” response at lowest temperatures, which rapidly disappears as the external field is \nincreased above 0.1 T.  \n The temperature dependence of the reciprocal magnetic susceptibility χ-1(T), shown in Fig. \n7, indicates the predominance of antiferromagnetic interactions in α-Cr 3(PO 4)2. In the 200 – 300 \nK range the experimental data can be fitted by the Curie-Weiss law with inclusion of the \ntemperature independent term χ0, i.e. \n) (3)1(2 2\n0 0Θ−++ =Θ−+ =TkSSgNnTС\nBB A μχ χχ , \nwhere C and Θ are the Curie and Weiss constants, N A, μB, and k B are the Avogadro, Bohr and \nBoltzmann constants, g is the g-factor. In this framework, the paramagnetic χ0 = 1.6 ×10-4 \nemu/mol is that of summation of diamagnetic and van Vleck contributions, the Weiss temperature Θ = –96 K is large and negative, and the effective magnetic moment μ\neff = \n[ng2S(S+1)]μB = 4.50 ±0.05 μB is significantly smaller than the spin-only value 4.9 μB per Cr2+ \nion. The reduced value of the effective magnetic moment μeff is hardly attributable to the \npresence of an unquenched orbital magnetic mo ment which might reduce the effective g-factor. \nWhile there are neither X-band (~ 9 GHz) nor Q-band (~35 GHz) electron spin resonance (ESR) \nstudies of “ESR-silent” non-Kramers 3d4 Cr2+ ions, the measurements at very high frequencies \n(~ 90 – 440 GHz) provided an value of g = 1.98 for the g-factor of Cr2+ in frozen aqueous \nsolutions [18].  \n Below 200 K, the reciprocal susceptibility χ-1(T) in α-Cr 3(PO 4)2 deviates from linearity and \nunravels ferromagnetic interactions in the system through the temperature dependence of the \neffective Curie constant C = ( χ – χ0)×(T – Θ) shown in Fig. 7. Evidently, the short-range \nmagnetic correlations develop in this compound far above the magnetic ordering temperature in \ncorrespondence with a significant reduction of the jump ΔCp in specific heat at T C. \n \n High-field magnetization  The field dependences of magnetization ta ken at several temperatures both below and \nabove the magnetic ordering temperature T C = 29 K are shown in Fig. 8. Note, that the remanent \nmagnetization at T < T C strongly depends on the protocol of measurements, i.e. why the M(B) \ncurve taken at 27 K shows negative magnetization at low fields. The magnetization loop in α - \nCr3(PO 4)2 taken at 2 K is shown in the inset to Fig. 8. At 2 K, the remanence is 0.024 μB/f.u. and \nthe coercivity is 0.2 T signaling presence of weak ferromagnetism. \n The field dependence of magnetization M(B) in α – Cr 3(PO 4)2 has been measured in pulsed \nmagnetic field up to B = 50 T at T = 2 K at LNCMI-Toulouse using concentric inductive pick-up \ncoils on a powdered sample ( Fig. 9). The M(B) curve shows two almost linear segments at B < \nB1 ~ 5 T and B > B 2 ~ 30 T, along with rather pronounced  variations of M vs. B rate at B 1 < B < \nB2. While the overall behavior of the magnetization is reminiscent of that at spin-flop and spin-\nflip transitions in antiferromagnets, the value of M ~ 6 μB at B = 50 T indicates that the system is \nfar from the saturation. The extrapolation of the linear segment of the M(B) curve to the \nsaturation magnetization value  \nB B sat gS M μ μ 9.11= =  \nfor g = 1.98 provides an estimate of the saturation field B sat ~ 130 T. This is in rough \ncorrespondence with the estimation of the antiferromagnetic exchange interaction parameters \nwhich can be deduced from the negative value of the Weiss temperature Θ = –96 K. From mean \nfield theory B sf ~ (2B aBsat)1/2 we can hence obtain an estimation of the effective field B a of \nmagnetocrystalline anisotropy B a ~ 0.1 T. We note, however, that the high-field magnetization \ndata do not resemble a typical spin-flop phase behavior but indicate a more complex \nferromagnetic-like phase since it does extrapolate to the finite magnetization M(B=0) ~ 3 μB. \n \n Discussion \n Up to now, it is generally accepted that the presence of the Jahn-Teller ion Cr2+ in an oxide \nenvironment is restricted to the case of the trirutile chromium(II) tantalate, CrTa 2O6, which is an \nantiferromagnet with complex magnetic structure, the Néel temperature T N = 10.3 K [19], and an \neffective magnetic moment μeff = 4.44 μB [20] (or 4.72 μB [19]). On the one hand, the \ndiscrepancy regarding the magnetic moment might already be taken as an indication on some \nuncertainty in the purity of the previously studied powder samples. On the other hand, a large \nseries of very well defined, well crystallized and pure phosphates of divalent chromium are \neasily accessible by chemical vapour transport (e. g.: α-/β-Cr 3(PO 4)2 [11, 12], Cr 2P2O7 [26]) or \nsolid state reactions (e. g.: SrCrP 2O7 [27], BaCrSi 4O10 [28]).  These compounds are well suited to \nstudy the electronic structure, chemical bonding an d cooperative magnetic behavior related to the \npresence of Cr2+ ions. Ligand-field spectra of the afor ementioned compounds have already been reported [12, 28]. Since chemical bonding behavior of the polyatomic phosphate ion, (PO 4)3-, is \nquite different from that of the oxide ion, one should expect for phosphates and silicates of Cr2+ \nsignificant deviations from the physical and chemical properties reported for CrTa 2O6 [19, 20]. \nThe magnetic properties of α-Cr 3(PO 4)2, i.e. weak ferrimagnetism and multiple magnetization \nreversals, are evidences for this expectation and of considerable interest themselves. \n The phenomena of ferrimagnetism are associated with a partial cancellation of \nantiferromagnetically aligned magn etic sublattices with different values of magnetic moments \nand/or different temperature dependences of magnetization. It is frequently observed in \ncompounds containing different magnetic ions and can be seen also in materials containing only \none type of magnetic ions, which are in different valence states or crystallographic positions [21, \n22]. In the latter case, the origin of weak ferrima gnetism lies in the difference of molecular fields \nacting on inequivalent magnetic sites [23].  \n In some ferrimagnets (in Néel’s classifi cation N-type ferrimagn ets [24]) the total \nmagnetization of a substance vanishes at a ce rtain compensation temperature. Both above and \nbelow this temperature the magnetization of different sublattices prevails. In this case \nmagnetization reversal can be observed. In a weak magnetic field (less than the field of \ncoercitivity) the magnetization changes sign at the compensation temperature. Even the \nmetastable “diamagnetic” state can be fixed by the magnetocrystalline anisotropy in a certain temperature range [25].   A similar, but even more complicated, situation is seen in α-Cr\n3(PO 4)2 containing six \ncrystallographically independent positions for the magnetically active Cr2+ ions. While the \noverall arrangement of magnetic moments in this compound is essentially antiferromagnetic, \nslight variations in temperature dependences of  the six inequivalent spontaneous magnetization \nfunctions produce remarkable effects of weak  ferrimagnetism and multiple magnetization \nreversals. The estimation of the effective field of magnetocrystalline anisotropy B a ~ 0.1 T \nallows the observation of “diamagnetic” metastable states, as seen at lowest temperatures in the FC curve measured at B = 0.1 T. Unfortunately, the well developed procedure for the description \nof magnetization reversal phenomena in two-subl attices ferrimagnets [23, 25] can not be applied \nunequivocally to α-Cr\n3(PO 4)2 with six magnetic sublattices. Similarly, the significant reduction \nof the effective magnetic moment, i.e. 4.5 μB per Cr2+ ion, in the paramagnetic state cannot be \nassociated with the complex magnetization beha vior in the magnetically ordered state. A \npossible explanation might be the assumption of direct overlap of the d(z2) orbitals of adjacent \nsquare-planar units Cr4O 4 and Cr5O 5. As already pointed out these two square-planar units are \nalmost perpendicular to each other and linked by an  oxygen atom. Via the rather short distance \nd(Cr-Cr) = 3.08 Å chemical bonding might become  possible, as it is well documented for Cr2SiO 4 [29]. Thus, one should expect for Cr4 and Cr5 a reduced paramagnetic moment which \ncorresponds to the three remaining unpaired elec trons on these ions. Actually, a rough estimate \nof the average paramagnetic moment for a system containing four Cr2+ ions with S = 2 and two \nwith only S = 3/2 would lead to µ = 4.5 μB per Cr2+. At present, it is unclear whether the charge \ntransfer effect seen in the XAS spectra of α-Cr 3(PO 4)2 might also be related to the reduced \nparamagnetic moment. Further investigation by ne utron diffraction experiments may help to gain \na better understanding of these peculiar magnetic properties. \n \n Summary \n The chromium (II) orthophosphate α-Cr 3(PO 4)2 is a rare case of a weak ferrimagnet based \non a single transition metal in one oxidation state Cr2+. The magnetic ordering at the Curie \ntemperature T C = 29 K is confirmed by a λ-type peak in specific heat. Dominant \nantiferromagnetic interactions in this system are characterized by the Weiss temperature Θ = –96 \nK, indicating a frustration ratio Θ/TC ~ 3 reasonable for a three-di mensional magnetic entity. In \nthe magnetically ordered state α-Cr 3(PO 4)2 exhibits a remarkable sequence of temperature-\ninduced magnetization reversals sensitive to the protocol of measurements, i.e. either field-\ncooled or zero-field-cooled regimes. The signif icant reduction of the effective magnetic moment \n4.5 μB/Cr2+, as compared to the spin-only moment 4.9 μB/Cr2+, cannot be ascribed to the \nformation of the low-spin state in any crystallographic site of the Jahn-Teller active 3d4 Cr2+ \nions. . X-ray absorption spectra at the K-edge indicate divalent chromium and unravel the high-spin state of these ions at the L\n2,3-edges. Weak ferrimagnetism and multiple magnetization \nreversal phenomena seen in this compound could be ascribed to incomplete cancellation and \ndistortion of partial spontaneous magnetization functions of Cr2+ in its six crystallographically \ninequivalent positions. \n \n Acknowledgements \n We acknowledge the support of the present work by Deutsche Forschungsgemeinschaft Grants DFG 486 RUS 113/982/0-1 and WO 1532/3-1, Russian Foundation for Basic Research \nGrants RFBR 09-02-91336, 10-02-00021, 11-02-00083. Part of this work was supported by \nEuroMagNET II at LNCMI-T facility and National Science Council of Taiwan (\nNSC98-2112-\n009-005-MY3) . \n \n References \n1. A. Nord and T. Eriksson, Am. Mineral., 70, 624 (1985). \n2. E. Kostiner and J.R. Rea, Inorg. Chem., 13, 2876 (1974). 3. P.B. Moore, Am. Mineral., 57, 24 (1972). \n4. W. Massa, O.V. Yakubovich, O.V. O.V. Dimitrova, Solid State Sciences, 7, 950 (2005). \n5. J.K. Warner, A.K. Cheetham, A.G. Nord, R.B. von Dreele, and M. Yethira, J. Mater. \nChem., 2, 191 (1992). \n6. J.B. Forsyth, C. Wilkinson, S. Paster, and B.M. Wanklyn, J. Phys. C: Solid State Phys., 21, \n2005 (1988). \n7. J. Escobal, J.L. Pizarro, J.L. Mesa, J.M. Roj o, B. Bazah, M.I. Arriortua, T. Rojo, J. Solid \nState Chem., 178, 2626 (2005). \n8. J.B. Forsyth, C. Wilkinson, S. Paster, and H. Effenberger, J. Phys.: Condens. Matter, 2, \n1609 (1990). \n9. J. Escobal, J.L. Pizarro, J.L. Mesa, A. Larrana ga, J. Rodriguez-Fernandez, M.I. Arriortua, \nT. Rojo, J. 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Krzystek, and L.-C. Brunel, Inorg. Chem., 37, 5769 (1998). \n19. M. Saes, N.P. Raju, and J.E. Greedan, J. Solid State Chem., 140, 7 (1998). \n20. V. Guillen-Viallet, J.F. Marucco, and M. Ghysel, J. Alloys and Compounds, 317-318 , 127 \n(2001). \n21. N. Shirakawa and M. Ishikawa, Jpn. J. Appl. Phys., 30, 755 (1991). \n22. A.V. Mahajan, D.C. Johnston, D.R. Torgesen, F. Borsa, Phys. Rev. B, 46, 10966 (1992). \n23. J.W. Culvahouse, J. Magn. Magn. Mater., 21, 133 (1980). \n24. L. Néel, Ann. Phys. (Leipzig) 3, 137 (1948). 25. H. Kageyama, D.I. Khomskii, R.Z. Levitin, A.N. Vasil’ev, Phys. Rev. B, 67, 224422 \n(2003).  \n26. L. Palatinus, M. Dusek, R. Glaum, B. El Bali, Acta. Crystallogr.  B, 62, 556 (2006). \n27. K. Maaß, R. Glaum, Acta. Crystallogr.  C, 56, 404 (2000). \n28. R. Miletich, D. R. Allan, R. J. Angel, Amer. Miner., 82, 697 (1997).  \n29. R. Miletich, M. Novak, F. Seifert, R. J. A ngel, G. Brandstätter, Phys. Chem. Minerals, 26, \n446 (1999).  \n \n  \nFig. 1. Structural features of α-Cr 3(PO 4)2. a) Close-packed arrangement of tubes from phosphate \ntetrahedra and Cr2+ ions. b) Helical arrangement of the six independent Cr2+ ions within a tube \n(right) and the same tube section with coordination polyhedra [CrO n] (left). The phosphate \ngroups PO 4 are represented by the light-yellow tetrahedra.  5980 5990 6000 6010 6020 6030 6040 60506001.7CrCl2(Cr2+)Cr K-edge\nCuCrO4CrO3(Cr6+)CrF3(Cr3+)Cr2O3(Cr3+)CrCl3(Cr3+)CrF2(Cr2+)CdCr2S4Cr3(PO4)2Cr(foil)(Cr0+)Normalized Absorption (arb. units)\nEnergy (eV)Cr2S3\n5997.95989.0\n5995.0\n5995.3\n6007.05995.9\n5999.65996.3\n6007.15995.1\n \nFig. 2. The Cr K-edge X – ray absorption spectra in various chromium compounds taken at T = \n300 K. 012345659885990599259945996599860006002600460066008  Standard samples\n Data of Ref.[14]\n Fitting line\n Cr3(PO4)2\n CdCr2S4\n Cr2S3\n CuCrO4\nCrF2(Cr2+)Energy (eV)\n  \n CrO3(Cr6+)\nCrCl3(Cr3+)\nCrCl2(Cr2+)Cr2O3(Cr3+)\nCr foil(Cr0+)\nOxidation state(μ-C7H8)[Cr(Nacnac)]2(Cr1+)X-ray absorption K-edge energy vs oxidation states of Cr\n \nFig. 3. The “linear” dependence of the K-edge energy on the oxidation state of chromium. The \nsize of symbols corresponds roughly to the error bars. \n \nFig. 4. The Cr L 2,3 edges X - ray absorption spectra in α-Cr 3(PO 4)2 and Cr 2O3 taken at T = 300 K. \nThe CT 1-3 spectral features are ascribed tentativel y to the charge transfer transitions. 02 0 4 0 6 0 8 0 1 0 00204060801001200 400 800 1200\n0,81,01,21,4C (J/mol K)\nT (K)TC = 29 K T2 (K2)\n C/T (J/molK2)\n \nFig. 5. The temperature dependence of the specific heat C p in α-Cr 3(PO 4)2. Inset :  Cp/T vs. T2 \ncurves taken at B = 0 and B = 9 T.  \n0 5 10 15 20 25 30 35-5051015\n0 100 200 300051015\nFCχ, 10-2 (emu/mol)\nT (K)ZFCχ, 10-2 (emu/mol)\nT (K)\n \nFig. 6. The temperature dependence of magnetic susceptibility in α-Cr 3(PO 4)2 taken at rising \ntemperature at B = 0.1 T after cooling in zero field. Inset: enlarged portions of χ(T) curves taken \nwith rising temperature after cooling at B = 0 (ZFC regime) and B = 0.1 T (FC regime). -100 0 100 200 30001020304050\n7,58,08,5\nμeff = 4.5 μB/Cr2+1/(χ-χ0) (mol/emu)\nT (K)TC = 29 K\nΘ = -96 K\n (χ-χ0)⋅(T-Θ) (emu K/mol)\n \nFig. 7. The inverse magnetic susceptibility in α-Cr 3(PO 4)2. The product ( χ - χ0)⋅(T - Θ) indicates \nincreasing relevance of ferromagnetic interactions when approaching the critical temperature T C \n= 29 K from above.  \n \n-3 -2 -1 0 1 2 3-0.30-0.150.000.150.30\n0.0 0.1 0.2 0.3-0.030-0.0150.0000.0150.0300.045\nM (μB/f.u.)\nB (T)432M (μΒ)\nB (T)1\n \nFig. 8. The field dependences of magnetization in α-Cr 3(PO 4)2 taken at selected temperatures \nboth below and above the critical temperature T C = 29 K (1 – 25 K, 2 – 27 K, 3 – 29 K, 4 – 50 \nK). The inset represents the magnetization loop taken at 2 K.  04 0 8 0 1 2 0024681012\nMsat ~ 11.9 μB\nBsat ~ 130 T\n B (T)M (μB/f.u.)\nBsf ~ 5 T\n \nFig. 9. The field dependence of magnetization in α-Cr 3(PO 4)2 taken at T = 2 K. The solid lines \nare the extrapolations of the linear segments of the M(B) curve at B < B 1 and B > B 2. The \nhorizontal line denotes value of saturation magnetization M sat for g – factor g = 1.98 μB [19]. "    },    {        "title": "2403.08112v1.Ferrimagnetic_Heusler_tunnel_junctions_with_fast_spin_transfer_torque_switching_enabled_by_low_magnetization.pdf",        "content": " 1 Ferrimagnetic Heusler tunnel junction s with fast spin-transfer torque switching  enabled \nby low magnetization  \n \nChirag Garg1+*, Panagiotis Ch. Filippou1*, Ikhtiar3*, Yari Ferrante1, See -Hun Yang1, Brian \nHughes1, Charles T. Rettner1, Timothy Phung1, Sergey Faleev1, Teya Topuria1, Mahesh G. \nSamant1+, Jaewoo Jeong3+,, and  Stuart S. P. Parkin2+ \n1IBM Research - Almaden, San Jose, California 95120, USA.  \n2Max Plank Institute for Microstructure Physics, Weinberg 2, 06120 Halle (Saale), Germany.  \n3Samsung Semiconductor, Inc. , San Jose, California  95134 , USA  \n* These authors contributed equally to this work.  \n+ Emails of corresponding authors : chirag.garg1@ibm.com , j.jeong1@samsung.com , \nstuart.parkin@mpi -halle.mpg.de , mgsamant@us.ibm.com  \n \nMagnetic random access memory that uses magnetic tunnel junction  memory cells is a \nhigh performance,  non-volatile memory technolog y that goes beyond traditional charge -\nbased memories.  Today its speed  is limited by the high magnet ization  of the memory \nstorage layer.  Here we show that fast  and highly reliable  switching is possible using a \nvery low magnetization  ferrimagnetic  Heusler alloy, Mn 3Ge. Moreover, the tunneling \nmagnetoresistance  is the highest yet  achieved  for a ferrimagnetic material  at ambient \ntemperature . Furthermore, the  devices  were  prepared on technologically relevant \namorphous substrates using a novel combination of a nitride  seed layer and a chemical \ntemplating layer. These results show a clear path to the lowering of switching currents \nusing  ferrimagnetic  Heusler materials and, therefore, to the scaling of high performance \nmagnetic random access memories  beyond those nodes possible with ferromagnetic  \ndevices.  \n \nSpintronic magnetoresistive  devices have powered the reading elements in hard disk drives for \nmore than two decades, thereby enabling a ~10,000  fold increase in storage capacity.  The \noriginal spin valve devices that were introduced in 19971 were based on spin -dependent \ninterfacial resistive scattering2,3. These were superseded by devices that are based on spin -\ndependent tunneling4-7. These latter devices were also proposed for solid -state non -volatile \nmagnetic memory cells as early as 1995 . The first demonstration was made in 1999 when very \nfast reading and writing (~ 10 ns) was demonstrated  in magnetic tunnel junctions (MTJs) that \nused amorphous Al 2O3 tunnel barriers, which exhibited up to ~ 40% tunneling  2 magnetoresistance (TMR)8-10.  A major breakthrough in 2004 was the demonstration of much \nhigher TMR in MTJs that used crystalline MgO(100) tunnel barriers in conjunction with bcc \nCoFe (or amorphous CoFeB) electrodes11,12. Subsequently, MTJs with thinner CoFeB -based \nelectrodes that exhibited perpendicular magnetiz ation  were shown to  reduce the electrical \ncurrent that is used to write the memory state by the phenomenon of spin -transfer torque \n(STT)13-16.  These  materials have since been incorporated into MTJs , forming the basis of \nrecently introduced embedded MRAM products17 at the 28  nm node.  The speed, however, is \nlimited by the use of the relatively high magnetization CoFeB material. This is due to the spin-\ntransfer angular momentum conservation18-20. Replacing the CoFeB storage layer with a \nmagnetic material that has a significantly lower magnetization would enable a wider use , in \nparticular for  cache memories , that require  much higher  speeds.   At the same time the smaller \nthe current needed, the smaller can be the access transistor, thereby enabling higher densities.  \nSome longstanding challenges  have prevented the adoption of  low magnetization  \nmaterials . While many ferromagnetic materials exhibit low magnetization, most of them  have  \nlow Curie temperatures (<400  K)21,22. A class of ferrimagnets composed of Rare-Earth \nelements  (Gd, Tb)  and transition metals (Co,  Fe), while easy to grow , are unsuitable  not only  \nowing to the low Curie temperature  of the Rare -Earth component,  but also because of their low \nthermal structural stability23,24. In contrast, certain ferrimagnetic materials  such as Mn 3Ga25 \nand Mn 3Ge26 have both low magnetization and much higher Curie temperatures, thanks to their \ncompensated yet strongly exchange coupled spin structure . The challenges  with these \ncompounds  are the growth of ultra -thin layers with bulk -like properties on technologically \nrelevant substrates a s well as  achieving high TMR at a  low enough resistance -area (RA) \nproduct which is suitable  to facilitate  STT-switching27. In such ferrimagnets, the spin \npolarization of the tunneling current may be compromised due to the compensated nature of  \nthe two antiferromagnetically coupled Mn sub -lattices , especially at the interface28. \nIn our current work, we address these issues to demonstrate  an STT -switchable MTJ  \nformed from Mn 3Ge, a tetragonal ferrimagnetic Heusler alloy  with low magnetization29-33, with \nhigh TMR  and a low RA = 11.4 Ωµm2. We find these  MTJs  have much lower switching \ncurrents at high write speeds than for MTJs  formed from conventional ferromagnetic electrodes  \nfor otherwise the same thermal stability.  Moreover, these Mn 3Ge-based  MTJs are formed on \namorphous SiO x and are thermally stable at temperatures above 400  °C making them \ncompatible with typical CMOS back -end of line processing.  A particular advantage of Heusler \nferrimagnets is that the two magnetic sub -lattices are formed from transition metals and thus \nthe net moment is weakly dependent on temperature for the operational temperature window  3 of the MTJ, in contrast to, for example, rare -earth – 3d transition metal ferrimagnetic alloys \nwhose net magnetization varies considerably with temperature34. This makes tetragonal \nferrimagnetic Heusler alloys  ideal candidates for high -speed magnetic memor y applications .   \nA recently developed chemical templating layer (CTL) technique35,36 was used to grow \nthe Mn 3Ge-based MTJ stacks but here we refine the technique to allow for growth on Si/SiO x \nsubstrates , which is required for CMOS compatibility (see below). The film stacks were  \npatterned into circularly shaped  devices , 30-40 nm in diameter, using  electron -beam \nlithography and Ar ion -beam milling. A typical ~36 nm diameter device  is shown in the bright -\nfield transmission electron microscopy  (BFTEM)  cross -sectional  image presented in Fig. 1a. \nThe Mn 3Ge forms the lower electrode , i.e. the free layer (FL). The top electrode, the reference \nlayer (RL), is  formed from  CoFeB that is ferromagnetically exchange -coupled, via a thin Ta \nspacer layer, to a synthetic antiferromagnetic ( SAF) structure composed of two distinct [Co/Pt] \nmultilayers separated by a thin Ir layer37,38.  The upper of these [Co/Pt] multilayers is designed \nto have higher  moment than the lower  CoFeB/Ta/[Co/Pt]  multilayers of the film  structure. The \ntwo memory states of the MTJ correspond to the moment of the Mn 3Ge-FL being parallel  (P) \nor antiparallel (AP) to the moment of the lower layer of the RL, with corresponding resistance \nvalues, 𝑅𝑃 and  𝑅𝐴𝑃, respectively . \nFigures 1b -c show the switching  between the P and AP states of the MTJ, driven by \nfield (b) and current (c). The MTJ is switched reversibly by using an out -of-plane (OOP) \nmagnetic field (Fig. 1b) resulting in  two well defined, non -volatile states in the absence of an \nexternal  magnetic field. By initializing the  net moment of the  SAF along +𝑧, we infer  from the \nmagnetic hysteresis loop  that the higher resistance belongs to the AP configuration. Therefore, \nthe resulting TMR (𝑅AP−𝑅P\n𝑅P)  is positive , exhibiting  a value of +45%.    Interestingly, the sign of \nthe TMR was found to be negative  for thick er Mn 3Ge films  in an earlier study28. In Fig. 1c, the \nspin transfer -torque driven  switching of this device is demonstrated by applying voltage pulses \nof increasing amplitude. Reversible current induced switching back and forth between the P \nand AP configurations is clearly observed. From the resistance versus voltage ( R-V) hysteresis \nloop,  it can be confirmed that the Mn 3Ge-FL is switch ing.  \nCTLs are binary compounds with the CsCl structure which have been shown to promote \nchemical ordering in ultra -thin Heusler alloy films deposited on them35 even at single unit cell \nthicknesses and at ambient temperature . They also enable  the desire d perpendicular magnetic \nanisotropy (PMA)  in the Heusler layer35.  This previous work used single crystalline MgO  \n(001) substrates. In order to promote the growth of the CTL on amorphous SiO 2 layers we  4 carried out a wide -ranging materials explorat ory search . We found that binary nitrides with a \nNaCl structure readily grow with a (001) texture on such amorphous surfaces. The nitrides \nTiN, VN, CrN, TaN , MnN and ScN  are all effective for this purpose  (see Supplementary Table \n1) and allow for a wide range of lattice constants for epitaxial matching with the CTL. Two of \nthese nitrides, MnN (see supplementary Figure 1) and ScN, were chosen for further study. It \nwas determined that m etallic MnN is not as thermally stable as semiconducting ScN  at ~400  \nºC, the annealing temperature required for back -end-of-line (BEOL) integration.  \nCross-sectional high -angle annular dark -field scanning transmission electron \nmicroscopy (HAADF -STEM) images that are presented in Figs. 2a and 2b show the epitaxial \ngrowth o f a Mn 3Ge layer for an MTJ stack grown on Si/SiO 2. The CTL was grown over  a ScN \nseed layer, here only ~1 Å thick. The detailed structure of  the MTJ is as follows  Si/SiO 2/ 50Ta/  \n5CoFeB / 1ScN / 400Cr/  50IrAl/  150CoAl/  19Mn 3Ge/ 14MgO /  14.5CoFeB/  50Ta/  100Ru  \n(thickness  values  in angstroms) . A 400Å Cr layer is included to allow for Current In -Plane \nTunneling  (CIPT) measurements.  The CTL is formed from a bilayer of IrAl and CoAl , both \nof which exhibit  a CsCl structure with  a (001) texture , thereby  templating the growth of a \nchemically ordered  Mn 3Ge (001) layer on top  (see Fig. 2b ). In particular note that the  Mn 3Ge \nexhibits  alternating ferromagnetic ally aligned  layers of Mn-Mn and Mn -Ge whose \nmagne tizations  are coupled antiferromagnetically39. This structure  gives rise to a significant \nvolume perpendicular magnetic anisotropy (PMA) along the tetragonal (001) axis30.  A CTL \nfrom a single layer of CoAl also is effective but we found that higher TMR was possible by \nusing the bilayer CTL. Similarly, ScN can be replaced by other nitrides ( Supplementary  Table \n1). To further  demonstrate the role of ScN we present x-ray diffractograms (XRD s) of the \nabove stack and the related stack without any nitride layer (Si/SiO 2/ 50Ta/ 5CoFeB  /400Cr \n/150CoAl /19Mn 3Ge /14MgO /20Ta ) in Fig. 2c.  In the former case well defined Cr (002) and \nCoAl (001) peak s are observed, whereas in the latter case no such peaks are found . This \ncomparison shows that the CoAl CTL has the desired (001) texture only when grown on nitride \nor nitride/Cr sublayers .  \nMTJs grown in this way show Mn 3Ge free layers with the desired magnetic properties,  \nas seen in the resistance (measured by CIPT) versus out -of-plane field ( R-H) hysteresis loop  in \nFig. 2d.  CIPT-TMR measured in this MTJ stack is  69% but when patterned into devices give s \nrise to higher TMR values  (see Supplementary Figure 2 ). Note that the  TMR is positive , i.e. \n𝑅𝐴𝑃>𝑅𝑃, which is opposite to previous ly reported result s for MTJs with  much thicker ( 300 Å) \nMn 3Ge layers grown without a CTL  (TMR~ -35%)28. We also find negative TMR values for \nthick ( 50Å) Mn 3Ge layers grown using CTL layers with TMR values as high as ~ -109%  (see  5 Supplementa ry Figure 3 ) but these layers were not current switchable due to their very high \nmagnetic anisotropy . Nevertheless, this is the highest TMR value yet reported for any \nferrimagnetic material.  \nThe sign difference  in TMR for strained and rela xed Mn 3Ge layer s reflects  the larger \nin-plane lattice constant  and reduced tetragonality of the Mn 3Ge layer for the MTJ grown with \nthe CTL . Indeed, t he TEM images (Figs. 2a and 2b) show  that the  thin Mn 3Ge layer assumes \nthe in-plane lattice parameter (~4.03Å) of the CoAl CTL.  A positive TMR was found from \nDensity Functional Theory (DFT)  calculations ( see Supplementary  Figure 4) for MTJs that \ninclude a Mn 3Ge layer with the same in -plane lattice constant as that found here for the CoAl \nCTL. For Mn 3Ge layers with the bulk lattice constant previous DFT calculations show a \nnegative TMR28.  Note that the TMR sign is determined by  which of the two Mn -Mn or Mn -\nGe ferromagnetic  layers have the larger magnetic moment  which in turn depend s on the in -\nplane  tensile  strain  according to  DFT  calculations . Furthermore , the spin polarization of the \ntunneling current depend s sensitively  on this strain so that , surprisingly,  the spin polarization \nfrom each of the  Mn layers can have the same sign and , therefore , higher TMR . Indeed, our \nDFT calculations show the possibility of very high TMR values of up to +400 % (as discussed \nin Supplementary Figure 5).  The largest TMR that we find for current switchable  Mn 3Ge layers \nis +87 %  (Supplementary Figure 2). \nThe magnetic properties of  Mn 3Ge thin films were studied by growing stacks identical \nto MTJ stacks but without the RL. The magnetic OOP hysteresis loops for a series of  Mn 3Ge \nthin film s with varying thickness  𝑡Mn3Ge  are shown in Fig. 3a.  All the  films exhibit PMA with \nhigh remanence . We find that the magnetization ( 𝑀s) and coercivity ( 𝐻c) are very sensitive to \nsmall changes in 𝑡Mn3Ge  (Fig. 3b) .  𝐻c more than doubles for every 2 Å increase  in thickness \nreach ing a very high value of  ~42 kOe for 𝑡Mn3Ge  = 21 Å. Being able to attain such high values \nof 𝐻𝑐 can be beneficial for making devices impervious to external fields.  Thicker films exceed \n70 kOe , the maximum field in our measurement apparatus.  For  𝑡Mn3Ge  = 11 Å, 𝑀s ~165 \nemu/cc, the bulk value33, but decreases as 𝑡Mn3Ge  is further increased . This observation  likely \narise s from a subtle shift in the delicate balance of magnetic  moments  in the MnGe and Mn \nlayers . The 𝐻k values  were  extracted from in -plane magnetic hysteresis loops (see Fig. 3c) . For \n𝑡Mn3Ge >17 Å, 𝐻k was too large (>70 kOe) to be  measu red (see Supplementary Note 3 ).  \nA useful parameter is the  thermal stability factor  that is given by  ∆ =𝐸𝐵\n𝑘𝑇 where 𝐸B is \nthe energy barrier to switch the  magnetic  FL volume ( 𝑉m) between the P and AP states  and 𝑘 \nis the Boltzmann’s constant .  For MRAM applications ∆ should exceed ~ 50 to meet  data  6 retention requirements . For single magnetic domain  reversal  (macrospin approximation) , ∆ = \n𝑀s𝑉m𝐻k\n2𝑘𝑇 but typically  smaller values are found , except in small devices  (<30 nm)40,41. \nExperimentally, ∆  can be extracted from  STT-driven MTJ switching experiments when the \nswitching voltage  𝑉𝑠𝑤 is less than 𝑉C0, the threshold  voltage for switching in the absence  of \nthermal fluctuations42,43. 𝑉𝑠𝑤 is extracted from R -V scans (see Fig. 1c for the case of 0.5 ms \nlong pulses).  𝑉𝑠𝑤 is plotted versus the voltage  pulse  length ( 𝑡𝑝) ranging from  0.5 to 100 ms in \nFig. 3e  for a 30 nm diameter  device with 𝑡Mn3Ge  = 17 Å. Within the macrospin approximation42-\n44, the slope of 𝑉SW vs ln(𝑡𝑝\n𝜏0) gives 1\n∆. Values of  ∆ thus obtained for both P→AP and AP→P \nswitching processes is then averaged  and a  value of ~60  is estimated for this device . The mean  \n∆ averaged over a set of 30 to 60 devices  with a diameter of 35 nm  is plotted  versus 𝑡Mn3Ge  in \nFig. 3f.  ∆ increases, as would be expected, as 𝑡Mn3Ge  is increased.  Moreover , the values  are in \ngood agreement  with estimates  based on  the values of  𝑀s and 𝐻k  deduced  from Fig. 3b,d. \nOur results show that  Mn 3Ge-FLs may enab le scaling the diameter  of such MTJ s to below  ~10 \nnm.  \nHaving established  that Mn 3Ge-FLs have high thermal stability  from millisecond STT -\nswitching characteristics , we now discuss  their current -driven  switching  properties  at much \nshorter time scales ( ≤10 ns ), where thermal fluctuations play  little role . Instead, the  reversal \ntakes place above  the threshold current ( 𝐼C0 = 𝑉C0\n𝑅p) at which the STT surpasses the intrinsic \ndamping torque45,46. In this  precessional  regime the switching becomes faster with increasing \ncurrent . At a given  temperature, the reversal starts from a thermally distributed initial state .  \nThe switching current  𝐼𝑐 for a given write -error rate ( 𝑊𝐸𝑅 ) is given by the following \nequation19,44,46: \n \n                                               𝐼𝐶 =4𝑒\n𝜇𝐵𝑔𝑃(𝛼𝛾𝐸𝐵+ 𝑀𝑠𝑉m\n4𝑡𝑝𝑙𝑜𝑔 (𝜋2∆\n4𝑊𝐸𝑅))       (1) \nor alternatively expressed as  \n                                               𝐼𝐶=𝐼C0+𝐼overdrive                                       (2) \n \nHere, 𝜇𝐵, 𝑔, 𝑃, 𝛼, 𝛾 are the Bohr magneton, g -factor, spin -polarization, Gilbert damping \nconstant  and gyromagnetic ratio , respectively.  In the long -pulse limit  𝐼𝐶 reaches a saturation \nlower bound value that is given by  𝐼C0=4𝑒𝛼𝛾 𝐸𝐵\n𝜇𝐵𝑔𝑃 and which  is governed  by 𝐸𝐵 and 𝛼.  The  7 efficiency of the switching process is often characterized by the term ∆\n𝐼C0, which should be as \nhigh as possible.   ∆\n𝐼𝐶0 can be increase d, for example,  by suppression of spin -pumping by using \nan MgO dielectric cap layer  on top of the FL47,48 or making use of two tunnel barriers49 to \nincrease the spin torque. In the  short -pulse limit,  the switching current is increased above the \nsaturation lower bound value by the term  𝐼overdrive =4𝑒\n𝜇𝐵𝑔𝑃𝑀𝑠𝑉m\n4𝑡𝑝𝑙𝑜𝑔 (𝜋2∆\n4𝑊𝐸𝑅). This term  scales \nwith 𝑡𝑝−1 and the magnetic moment of the free layer.  \n It is clear from Eq. (1)  that a high 𝑀𝑠 is detrimental for current -driven switching at short \npulses although , for typical ferromagnets , it contributes towards  a high 𝐸B.  In contrast,  \nferrimagnets that have a low 𝑀𝑠 and high 𝐻k, such as Mn 3Ge used here,  can attain a high 𝐸𝐵 \nwhilst reducing  𝐼overdrive . We now consider the overdrive term by studying the ratio  𝐽𝐶\n𝐽C0 where \n𝐽𝐶 is the switching current density  and 𝐽𝐶0 is the threshold saturation lower bound current \ndensity . For a given 𝑡𝑝 Eqn. (1) can be rewritten  as:  \n   𝐽𝐶\n𝐽C0=[1+ 𝜏𝐷\n𝑡𝑝(1\n2𝑙𝑜𝑔(𝜋2∆\n4𝑊𝐸𝑅))]                       (3) \n \nHere 𝜏𝐷 is the characteristic timescale of switching , that equals 1\n𝛼𝛾𝐻𝑘 .  Calculated values \nof 𝐽𝐶\n𝐽𝐶0 are plotted in Fig. 4a  as a function of 𝐻𝑘 using Eq. (3). Note that as 𝐻𝑘 is varied , 𝑀𝑠 is \ncorrespondingly adjusted to  maintain the value of  ∆ to be 60. The value of 𝛼 was set at 0.01. \nFor longer pulse lengths  (~10 ns)  𝐽𝐶\n𝐽C0  is small and relatively insensitive to 𝐻𝑘. As clearly \nshown in Fig. 4a, f or low 𝐻k and short 𝑡𝑝, 𝐽𝐶 is calculated to increase to more  than an order of \nmagnitude higher than 𝐽𝐶0 but, o n the other hand, ferrimagnetic layers  with a low 𝑀𝑠 and high \n𝐻𝑘 will dramatically reduce  𝐽𝐶\n𝐽C0 even at sub -nanosecond speeds.   \nAs shown in Eqn. 3 there is a finite probability that a device will not switch for a given \nintensity and length of the current pulse  that is given by the parameter WER. The WER was \ndetermined by first setting the MTJ to either the AP or P state using an appropriate current \npulse and then applying a switching current pulse. The state of the MTJ was then read with a \nmuch smaller current sensing pulse. This procedure was then repeated up to 1 07 times for \ncurrent pulses with varying magnitudes and durations. Re sults for the same Mn 3Ge-FL device  \nas in Fig. 3 e are shown in Fig. 4 b and c.   Fig. 4b shows how  𝐽𝐶\n𝐽C0 varies as the 𝑡p is increased at  8 a fixed WER  = 0.5 for switching from P to AP and AP to P.   E xperimental data  for both \nswitching processes are fitted with Eqn. 3 to obtain 𝐽C0 and 𝜏𝐷. These values are shown in the \nFigure. Based on the  fitted data  the STT switching efficiency ∆\n𝐼c0 is 1.37. This value is \ncomparable to those reported in the literature for conventional ferromagnetic materials16,50. For \n1 ns long current pulses  we ob serve  the increase in 𝐽𝐶 over 𝐽𝐶0 to be 38  % (at WER=0.5) .  For \nMTJs with CoFeB  FLs this increase is typically more  than 200 %51,52. This is simply due to  the \nlarge difference in 𝑀𝑠 for these  two materials.  \n The values of WER are plotted versus 𝐽C for several values of 𝑡p in Fig. 4c .  To obtain \nlower WER values 𝐽C must be increased.  Fits to the WER vs 𝐽C plots for each 𝑡p are shown as \nsolid lines  (see Methods) . For some 𝐽𝐶 no errors were detected (hollow points) for  the \nmaximum number of switching attempts  (107) so they are shown at  𝑊𝐸𝑅  = 10-7.  From these \ndata a plot of  𝐽𝐶\n𝐽𝐶0 vs 𝑡p for 𝑊𝐸𝑅  = 10-6 is shown in  Fig. 4d . The values of  𝐽𝐶\n𝐽𝐶0 agree well with \nthose calculated using the value of  𝜏𝐷 obtained  from the data in Fig. 4b.  In particular 𝐽𝐶\n𝐽𝐶0 for 1 \nns long current pulse  at WER = 10-6 is ~1.85 which is again  considerably smaller than that \nreported  for conventional ferromagnetic free layers51.   \n In summary, we find that the switching current needed to set the state of an MTJ \nmemory element at high speeds is significantly lowered by using a ferrimagnetic Heusler free \nlayer that has a very low saturation magnetization compared to the ferromagnetic CoFeB -based \nfree layers used in today’s MRAM products. Moreover, these MTJ devices  that were prepared \non CMOS compatible amorphous substrates  show reliable switching up  to very low write error \nrates as well as high TMR , high thermal stability , and high field immunity . The low switching \ncurrents that we have demonstrated will allow the use of minimum size drive transistors, \nthereby, enabling a s ubstantial  reduction in the footprint of the memory cell and overcoming a \nfundamental roadblock to the widespread application of  spintronic technologies  based on MTJ \ndevices .  \n \n \n \n \n \n \n  9 Figure captions:  \nFigure 1  \nStack details and switching characteristics of a Mn 3Ge-FL MTJ device. a, BFTEM (bright -\nfield transmission electron microscopy) cross -section image of a Mn 3Ge-FL MTJ device with \na diameter of ~ 36 nm. The MTJ film stacks had the following structure: Si(001)/ 250 SiO 2/ 50 \nTa/ 5 CoFeB / 1ScN / 400 Cr/ 100 ScN / 10 CoAl/ 17 Mn 3Ge (in -situ annealed at 355 oC)/ 13 \nMgO/ 13.5 CoFeB/  2.4 Ta/ 6 Co/ 10 Pt/ [2. 5 Co/ 5 Pt]x3/ 7 Co/ 5.7 Ir/ 6Å Co/ 5.5 Pt/ [5.5 Co/ \n5 Pt]x4 / 100 Ru. Thickness values are in angstroms. For scale, a white horizontal bar near the \nbottom corresponds to 10 nm. The bottom and top electrode are connected to signal, ground \nrespectively.  b, 𝑅 vs. (out -of-plane) 𝐻 c, and 𝑅 vs 𝑉 where the length of applied voltage pulse \nis 0.5 millisecond . 𝑅 is measured with a small bias of 50 mV.  Note that a static magnetic field \nequal to the offset field derived from the R -H loop is applied for the  R-V measurements to \ncompensate for the fringing dipole field from the top electrode . \n \nFigure 2 .  \nGrowth of crystalline chemical templating layer on silicon substrates. a, HAADF cross -\nsectional STEM  image of a representative Mn 3Ge-based  MTJ stack grown on silicon substrate. \nThe stack consists of: Si/SiO 2/ 50Ta/  5CoFeB/  1ScN/  400Cr/  50IrAl/  150CoAl/  19Mn 3Ge/ \n14MgO /  14.5CoFeB/  50Ta/  100Ru. b, high resolution HAADF imaging of the indicated region \nin a, showing the highly epitaxial growth attained even on silicon substrates. c, Out -of-plane \nθ-2θ XRD scans of the stack shown in (a) (in red) , showcasing the CsCl structure of CoAl CTL \nas seen from the (001) and (002) peaks as indicated  and the exemplary stack  without any nitride \nlayer  (in black ). d, CIPT R -H loop showing high TMR from Mn 3Ge grown on ScN/CoAl CTL \non silicon substrate. The white scale bars in (a-b) correspond to 2 nm.  \n \nFigure 3 . \nMagnetic measurements and estimation of E B for various 𝒕𝐌𝐧𝟑𝐆𝐞 . a, out-of-plane M-H \ncurves for Mn 3Ge-FL film without any RL. b, Extracted 𝑀s (black) and 𝐻c (red) from M-H \ncurves. Blue dashed line is reference for bulk Mn 3Ge 𝑀s. c, saturation magnetic moment, 𝑚s \nnormalized to a sample area of 0.2 cm2. d,  𝐻k extracted from in -plane M-H curves (see \nSupplementary Note 3 ). e, Linear fit to the variation of switching voltage  (𝑉SW) with ln (𝑡p\n𝜏0) as \nper the macrospin model for a device with diameter  ~30 nm and  𝑡Mn 3Ge= 17 Å. 𝜏0 = 1 ns. 𝑉SW \nis calculated  as an average from 20 repeated measurements . f, experimentally obtained 𝐸B  10 (black) from a set of 35  nm diameter devices and theoretically calculated estimate (red) based \non 𝑀s and 𝐻k from 3b,d. Error bars in the figure correspond to one standard deviation.  The 35 \nnm group includes any devices in the size range 32.5 – 37.5 nm.  \n \nFigure 4.  \nScaling of switching currents at high speeds. a , 𝐽𝐶\n𝐽𝐶0 calculated for WER = 0.5, ∆ = 60 and 𝛼 \n= 0.01 using Eq. (3). b, 𝐽𝐶\n𝐽𝐶0 vs 𝑡p for Mn 3Ge-FL MTJ with 𝑡Mn 3Ge= 17 Å (same device from \nFig. 3e). Measurement data for P -AP (blue) and AP -P (red)  and the average  (black) is shown \nusing fits to Eq. (3). c, Measured 𝐽𝐶 vs WER for different 𝑡p (solid points) with fits to the WER  \n(solid lines) for the same device. The open symbols are measurements where no error was \ndetected for 107 switching events and a WER of 10-7 was assumed as the upper bound. d, 𝐽𝐶\n𝐽𝐶0 \nfrom c (black) and based on the Eq. (3) estimate (red) are plotted for different 𝑡p. \n \nAcknowledgements  \nWe thank Holt Bui, Eugene Delenia , Andrew Kellock and Kevin P. Roche for their help.  \n  \nCompeting interests:  The authors declare no competing interests.  \n \n \nREFERENCES:  \n1 Parkin, S. S. P. in Ann. Rev. 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Applied Physics Letters  115, 182404 (2019).  \n \n \n \n \nMethods:  \nMagnetic measurements :  \nThe magnetic measurements of the Mn 3Ge film were conducted using a SQUID -VSM \nmagnetometer by Quantum Design.   \nDevice size s:  \nThe device size of the MTJs is estimated using  𝑅𝑃 and the value of resistance -area (RA) product \ndetermined from CIPT (Current In -Plane Tunneling) measurements . We find that these \nestimates agree well with the sizes obtained from TEM  cross -sectional  images . \nFilm growth:  \nThe samples used in these studies were prepared using an ultra -high vacuum chamber with a \nbase pressure of ~ 10-9 Torr. While the Ta and Mn 3Ge layers were deposited using ion beam \ndeposition (with Kr gas) at a pressure 10-4 Torr, all the other layers were deposited by DC \nmagnetron sputtering at an Ar sputter gas pressure of 3 mTorr (the MgO tunnel barrier was \ndeposited by RF sputtering at the same pressure). All the films were deposited at room \ntemperature.  \nWrite -error rate (WER) measurements :  \nWER measurements were  performed by  connecting a Keithley 2602A multimeter and a pulse \ngenerator through a biased tee to the MTJ device. The 2602a unit was used for sending the  \nreset pulse  to set the initial state of the MTJ and reading the resistance  of the MTJ . Either the  \nPicosecond Pulse Generator 10070A or Tektronix AWG610 was used for  sending the writ e \npulse. The 10070A has a rise time of 55  ps and was used for the shorter writing pulses . Both  14 the reset and write  resistance  states were measured after the state of the MTJ was set . 𝐽𝐶 for \nWER = 0.5 is obtained after accumulating the WER for different  𝐽𝐶 and using that data to \nobtain the fit for  𝐽𝐶 at WER = 0.5  (see Fig. 4b) . For deeper error rate measurements (from Fig. \n4c), measurement at a particular 𝐽𝐶 is run until any of these conditions are satisfied; either 10 \nerrors are accumulated or 107 trials have been performed. The WER then obtained  is passed \nthrough the  inverse of the standard normal cumulative distributive function  and then linearly \nfitted against 𝐽𝐶 (solid lines) in Fig. 4c. The smallest 𝜀 we can measure is only limited by \nmeasurement time . \n   15 z Figure 1.  \n \n  \n 16 Figure 2.  \n \n  \n 17 Figure 3.  \n \n  \n 18 Figure 4.  \n \n  \n 19 Supplementary Information:  Ferrimagnetic Heusler tunnel junctions with \nfast spin -transfer torque switching enabled by low magnetization  \nChirag Garg1+*, Panagiotis Ch. Filippou1*, Ikhtiar3*, Yari Ferrante1, See -Hun Yang, Brian \nHughes1, Charles T. Rettner1, Timothy Phung1, Sergey Faleev1, Teya Topuria1, Mahesh G. \nSamant1+, Jaewoo Jeong3+,, and  Stuart S. P. Parkin2+ \n1IBM Research - Almaden, San Jose, California 95120, USA.  \n2Max Plank Institute for Microstructure Physics, Weinberg 2, 06120 Halle (Saale), Germany.  \n3Samsung Semiconductor, Inc., San Jose, California 95134, USA  \n* These authors contributed equally to this work.  \n+ Emails of corresponding authors: chirag.garg1@ibm.com , j.jeong1@samsung.com , \nstuart.parkin@mpi -halle.mpg.de , mgsamant@us.ibm.com  \n \nSupp lementary Table 1 : Exploration of different nitrides as the seed layer for the \ngrowth of Mn 3Ge free layer magnetic tunnel junction  \n \nNitride  TMR (%)  RA (Ω·μm2) \nScN 56.6 5.58 \nTiN 60.8 7.75 \nVN 58.8 7.13 \nCrN 51.9 4.88 \nMnN  62.3 19.4 \nTaN 56.9 9.11 \n \nTable shows  current in -plane tunneling derived tunneling magnetoresistance  (CIPT -TMR ) \nvalues for Mn 3Ge layer stacks  grown on Si substrates utilizing the nitride/CTL  (chemical \ntemplating layer)  concept.  These nitrides may not be stoichiometric.  All film stacks are \ncomprised of: 50  Ta/ 5 CoFeB/  Nitride /  400 Cr/ 50 IrAl/  150 CoAl/  13-19 Mn 3Ge/ 14-17 MgO/  \n13-14.5 CoFeB / 50 Ta/ 100 Ru, all thicknesses are in Å. Mn 3Ge is annealed after deposition at \n~390°C and there is a second  annealing step at ~300 °C after all layers are deposited  to set  the  20 perpendicular magnetic anisotropy ( PMA ) in the CoFeB layer . The nitride thicknesses range \nfrom 1-10 Å, ex cept for  MnN which is  300Å  thick . \n \nSupplementary Note 1: Selection and growth of suitable nitrides  \n \nWe show that nitrides extend the  CTL technique of ordered growth of Heusler films from single \ncrystal substrates to amorphous SiO x. Here we discuss in detail the case of  a metallic Mn xN \nlayer  (other nitrides are also similar) . Mn xN films with different chemical compositions , varied \nfrom MnN to Mn 4.8N, were prepared by reactive magnetron sputtering using varying mixtures \nof Ar – N2 sputter gas.  This allows the lattice constant of the Mn xN layer to closely match that \nof the CTL . \nWe illustrate the growth of ultra -thin, 10 Å thick Heusler layers formed from Mn 3Sb, using a \ncombination of Mn xN and CoAl CTL underlayers in Supplementary Figure 1 .  The detailed \nstructure is as follows: Si(001)/ 250 SiO 2/ 50 Ta/ 3  CoFeB/ 300 Mn xN/ 300 CoAl/  Mn 3Sb or \nMn 3Ge / 20 MgO/ 20 Ta , all thicknesses are in Å .  The variation of the lattice constant of the \nMn xN layer as a function of nitrogen content is shown in Supplementary Figure 1 a and b. θ-2θ \nx-ray diffraction (XRD) scans in  Supplementary Figure 1 a show that the (002) Mn xN peak \nshifts to lower 2θ angles with increasing nitrogen content.  Thus, the Mn xN out -of-plane lattice \nparameter can be varied considerably from ~3.76 to ~4.26 Å with increasing nitrogen content, \nas summarized in Supplementary Figure 1 b.  When the nitrogen content is ~ 2.5 the lattice \nconstant of the Mn xN layer matches closely with that of the CoAl CTL. However, we find that \nwell-ordered CoAl can be prepared for a wide range of nitrogen content within the Mn xN layer.  \nThe chemically ordering within the CoA l layer gives rise to the (001) peak shown  in \nSupplementary Figure 1 a.  As can be seen in the figure , a strong (001) peak, that varies little \nin 2θ is observed for a wide range of x between ~2 and ~4.  For the same range of nitrogen \ncontent,  the surface of the film stacks was found to be very smooth. As can be seen in \nSupplementary Figure 1c, the root -mean -square roughness ( Rrms) of the surface topography, \nfound from atomic force microscopy (AFM) studies, was less than 3Å for x between ~2 and \n~4. \nUsing this combination of Mn xN and CTL underlayers very thin Heusler films with excellent \nmagnetic properties were obtained. Exemplary Magneto -optical Kerr effect (MOKE) \nperpendicular magnetic hysteresis loops are shown in the inset to  Supplementary Figure  1d for  \n10 Å -thick Mn 3Sb and 8 Å -thick Mn 3Ge films. Both the Mn 3Sb and Mn 3Ge layers were \ndeposited at room temperature (RT) but the Mn 3Ge layer was in -situ annealed at ~ 340 oC in  21 ultra-high vacuum for 30 min before the capping layers were deposited. These both show \nexcellent PMA with square hysteresis loops.  This shows that the Heusler layers are highly \nthermally stable. The dependence of film coercivity ( 𝐻𝑐) on the thickness of the Mn 3Ge layer \n(𝑡𝑀𝑛 3𝐺𝑒) is shown in Supplementary Figure 1 d. \n \n \nSupplementary Figure 1.  \n \nGrowth of crystalline chemical templating layer on amorphous substrates. a, Out-of-plane θ -\n2θ XRD scans of CoAl films grown on Mn xN with varying compositions. The dotted lines \nindicate the positions of the (001) and (002) CsCl -CoAl peaks. b, Dependence of the Mn xN \nout-of-plane lattice constant on the Mn:N ratio extracted from (a). The dotted line indicates the \nvalue of the CoAl lattice constant after a 45° in -plane rotation. c, Surface roughness of the \nfilms shown in (a) as measured by atomic force microscopy. d, 𝐻𝑐 dependence of Mn 3Ge films \nwith thickness 𝑡𝑀𝑛3𝐺𝑒 grown on top of the optimized CoAl layer. Inset shows strong PMA in \nthe magnetic hysteresis loops of 8 Å Mn 3Ge and 10 Å Mn 3Sb films measured using p -MOKE. \nThe stack structure used for a -d is: Si(001)/ 250 SiO 2/ 50 Ta/ 3 CoFeB/ 200 Mn 3N/ 300 CoAl/ \n[Mn 3Ge or Mn 3Sb]/ 20 MgO/ 20 Ta , all thicknesses are in Å .  \n 22  \n \n \nSupp lementary  Figure  2:  \nResistance versus field (R -H), magnetic  hysteresis loop for a 35 nm MTJ device with Mn 3Ge \nfree layer, showing a TMR of 87% . The stack is as follows: 50Ta/  5CoFeB 20/ 300Mn xN/ \n400Cr/  50IrAl/  150CoAl/  13Mn 3Ge/ anneal at ~340° C/ 17MgO/  13CoFeB/  50Ta/  100Ru / \nanneal at ~300° C, all thicknesses are in Å .  \n \n  \n 23  \n \nSupplementary Figure 3: \nIn red, h igh TMR from Mn 3Ge of 50Å where Mn 3Ge assumes its bulk properties of MnMn \nmoment aligning with the total moment and TMR is negative . For comparison, in black, \nprevious ly measured TMR on 300 Å bulk Mn 3Ge [1]. \n  \n 24 Supp lementary  Note  2: Density functional theory  calculations of Mn 3Ge \n \nIn order to study the electronic structure and transport properties of Mn 3Ge deposited on CoAl \nsubstrate we performed density functional theory (DFT) calculations of the Mn 3Ge crystal \nstructure with fixed in-plane lattice constant a = 4. 03 Å (which equals to the  lattice constant of \nCoAl) using the VASP program [ 2] with projector augmented wave (PAW) potentials [ 3,4] \nand Perdew -Burke -Ernzerhof (PBE) GGA /DFT functional [ 5]. We found that for  fixed in -\nplane lattice constant  a = 4.03 Å the relaxed out -of-plane lattice constant equal c = 5.96 4 Å \n(that corresponds to the dimensionless  out-of-plane lattice constant c’ = c/(2a) = 0.74 ). Note \nthat a = 4.03 Å is just 1% smaller than the in -plane lattice constant ac = 4.06 Å of the cubic \nphase of Mn 3Ge [6] (that corresponds to the dimensionless out -of-plane lattice constant c’  = \nc/(2a) = 1/√2  ≈ 0.707 ). The convergence of the results was verified by varying the number of \ndivisions in reciprocal space from 10 ×10×10 to 18 ×18×18 and the energy cutoff from 400 to \n520 eV.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSupplementary Figure 4: The density of states of Mn 3Ge with in -plane lattice constant a = \n4.03 Å calculated by LDA and QSGW methods . \n \nThe electronic structure of Mn 3Ge with in-plane lattice constant a = 4.03 Å and out -of-plane \nlattice constants  c’ = 0.74  was calculated using the full -potential all -electron  linear muffin -tin \norbital  (LMTO ) approach [ 7] with Barth -Hedin LDA/DFT functional [ 8], and also using the \nquasiparticle self -consistent GW (QSGW) method that is known to describe band gaps and \n 25 other properties of materials with moderate e-e correlations significantly better than DFT [ 9–\n11]. The density of states (DOS) calculated by LDA and QSGW methods is presented \nSupplementary Figure 4 . One can see that in both approache s the minority DOS has a valley \nnear the Fermi energy resulting in large spin polarization (SP) of Mn 3Ge for a = 4.03 Å. In \nparticular, the spin polarization obtained by LDA is SP LDA = 0.92, and spin polarization \nobtained by more accurate QSGW method is SP QSGW  = 0.88.  The magnet ic moment was found \nto be 1.02 𝜇𝐵 in LDA and 1.03 𝜇𝐵 in QSGW, that is close to the magnetic moment of Mn 3Ge \nin cubic phase m c = 1.00 𝜇𝐵 [6]. \n \nThe transport properties of Mn 3Ge/MgO/Fe  magnetic transport junction (MTJ) device  with \nfixed  in-plane lattice constant a = 4.03 Å  were  calculated using a tight -binding linear muffin -\ntin orbital method in the atomic sphere approximation (TB-LMTO -ASA) with the local density \napproximation of DFT for the exchange -correlation energy  [12,13]. Relaxed positions of atoms \nat the Mn 3Ge/MgO interfaces (for both, the Mn -Mn and Mn -Ge terminations  of the interface ) \nwere determined using the VASP molecular dynamic program  [2]. The O -top configuration \nwas found to be the most stable configuration (as compared with Mg -top and hollow) for both \nterminations at the Mn 3Ge/MgO interface (in agreement with Ref. [14]). For Fe/MgO interface \nthe atomic positions from Ref. [1 5] were used.  \nEven though the Mn 3Ge/MgO interface can be very smooth ( see, e.g., Ref. [1]) inevitably there \nwill be atomic scale fluctuations in the morphology of the Mn 3Ge layer that gives rise to regions \nwith Mn –Mn and Mn –Ge terminations, due to the fundamental underlying structure of the \nHeusler compound . The simplest way to model such fluctuations is to average the transmission \nfunctions over the different terminations separately for parallel (P)  and antiparallel  (AP) \nconfigurations  of magnetization of the Mn 3Ge and Fe electrodes in the Mn 3Ge/MgO/ Fe MTJ , \nassuming that the MgO thickness is the same across the device. The tunneling magneto \nresistance ( TMR ) is this simple model is calculate d as TMR = (𝑇𝑃−𝑇𝐴𝑃) /𝑇𝐴𝑃, where  \ntransmission in the parallel configuration is given by 𝑇𝑃=[𝑇↑↑(𝑀𝑛𝑀𝑛 )+𝑇↓↓(𝑀𝑛𝑀𝑛 )+\n𝑇↑↑(𝑀𝑛𝐺𝑒 )+𝑇↓↓(𝑀𝑛𝐺𝑒 )]/2 and transmission in the antiparallel configuration is given by \n𝑇𝐴𝑃=[𝑇↑↓(𝑀𝑛𝑀𝑛 )+𝑇↓↑(𝑀𝑛𝑀𝑛 )+𝑇↑↓(𝑀𝑛𝐺𝑒 )+𝑇↓↑(𝑀𝑛𝐺𝑒 )]/2. (Here two arrows denote \ndirection of the magnetization of Mn 3Ge and Fe, correspondingly , and MnMn or MnGe denote \nthe termination at the Mn 3Ge/MgO interface) . The calculated TMR  is shown in Supp lementary  \nFigure 5.  26  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSupplementary Figure 5: TMR calculated for Mn 3Ge/MgO/Fe MTJ with in -plane lattice \nconstant a = 4.03 Å and with an assumption of equal areas occupied by Mn –Ge and Mn –Mn \nterminations at the Mn 3Ge/MgO interface  shown as a function of the number of MgO layers, \nNMgO. \n \nOne can see that TMR is 100% for N MgO = 2 and  60% for NMgO = 4 and varies from  360% to \n430% for N MgO ranging from 6 to 12.  The high TMR values (~ 400%)  at N MgO >= 6 is a \nconsequence of the high spin polarization  of the  near-cubic crystal structure of Mn 3Ge at a = \n4.03 Å. Lower TMR values (~100%) at N MgO <= 4 is a consequence of the  presence of  the \ninterfac e resonance states localized near  the Fe/MgO interface at the Fermi energy  in Fe \nminority channel that lead s to enhanced AP transmission (and therefore lower TMR) at small \nvalues of N MgO. \n \n \n  \n 27 Supplementary Note  3: Extraction of anisotropy field  magnetic properties of Mn 3Ge \nthin film  \n \nThe effective anisotropy field  (𝐻𝑘) of the Mn 3Ge film used in our study was estimated by \nobtaining the area enclosed (light green) between the OOP (black) and IP (blue) measurements \nof M -H (magnetization vs field) . M-H was measured using a Quantum Design VSM -SQUID \nmagnetometer  capable of applying maximum field of 7 Tesla . The area enclosed gives the \nenergy density difference between the OOP and IP configuration which when normalized by \nthe saturation magnetization gives us the value of the effective anisotropy field.  The Mn 3Ge \nfilms used for  these measurements  have been described earlier in the main text a nd their stack \nis: Si(001)/ 250Å SiO 2/ 1ScN/  10CoAl/  ‘𝑡𝑀𝑛 3𝐺𝑒’ Mn 3Ge/ 20 MgO/ 20 Ta , all thicknesses are \nin Å. For  𝑡𝑀𝑛 3𝐺𝑒 = 11 Å, 15 Å, 17 Å, the M -H data and the enclosed curves  illustrating our \nprocedure are shown in Supp lementary  Figure 6a-c. We are not able to measure the anisotropy \nfield for higher 𝑡𝑀𝑛 3𝐺𝑒  as the field range of our magnet (7T) is not sufficient to completely \nsaturate the  magnetization  during the IP measurement.  \n \nSupplementary Figure 6. Normalized magnetization  (𝑚) vs H curves for OOP (black) and IP \nconfigurations (blue) . The enclosed area is shaded  in light -green.  (a-c) correspond to  𝑡𝑀𝑛 3𝐺𝑒 = \n15 Å b, 𝑡𝑀𝑛 3𝐺𝑒 = 11 Å and c, 𝑡𝑀𝑛 3𝐺𝑒 = 17 Å, respectively.  \n  \n \n  \n 28 References  \n \n[1] J. Jeong, Y. Ferrante, S. V. Faleev, M. G. Samant, C. Felser, and S. S. P. Parkin, Nature \nComm. 7, 10276 (2016)  \n[2] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).  \n[3] P. E. Blochl, Phys. Rev. B 50, 17953 (1994).  \n[4] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).  \n[5] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).  \n[6] S. V. Faleev, Y. Ferrante, J. Jeong, M. G. Samant, B. Jones, and S. S. P. Parkin, Phys. \nRev. Applied 7, 034022 (2017).  \n[7] M. Methfessel et al. , in Lecture Notes in Physics , edited by H. Dreysse (Springer -Verlag, \nBerlin, 2000), Vol. 535.  \n[8] U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972).  \n[9] S. V. Faleev, M. van Schilfgaarde, and T. Kotani, Phys. Rev. Lett. 93, 126406 (2004).  \n[10] M. van Schilfgaarde, T. Kotani, and S. V. Faleev, Phys. Rev. Lett. 96, 226402 (2006).  \n[11] T. Kotani, M. van Schilfgaarde, and S. V. Faleev, Phys. Rev. B 76, 165106 (2007).  \n[12] Turek, I., Drchal, V., Kudrnovsky, J., Sob, M. & Weinberger, P. Electronic structure of \ndisordered alloys, surfaces and interfaces  (Kluwer, 1997).  \n[13] Schilfgaarde, M. v. & Lambrecht, W. R. L. in Tight -binding approach to computational \nmaterials science  vol. 491 (eds Colombo, L., Gonis, A. & Turchi, P.) 137 (MRS, 1998).  \n[14] Y. Miura, and M.  Shirai, IEEE Trans. Magn. 50, 1400504 (2014).  \n[15] D. Wortmann, G. Bihlmayer and S. Blugel , J. Phys. Condens. Matter 16, S5819 –S5822 \n(2004).  \n \n  \n "    },    {        "title": "0904.3843v3.Factorized_ground_state_for_a_general_class_of_ferrimagnets.pdf",        "content": "arXiv:0904.3843v3  [cond-mat.stat-mech]  11 May 2009Factorized ground state for a general class of ferrimagnets\nM. Rezai and A. Langari\nDepartment of Physics, Sharif University of Technology, Te hran 11155-9161, Iran\nJ. Abouie\nDepartment of physics, Shahrood University of Technology, Shahrood 36199-95161, Iran and\nSchool of physics, Institute for Research in Fundamental Sc iences (IPM), Tehran 19395-5531, Iran\n(Dated: December 4, 2018)\nWe havefound the exact (factorized) ground state of a genera l class of ferrimagnets in thepresence\nofamagnetic fieldwhichincludesthefrustrated, anisotrop ic andlongrangeinteractions for arbitrary\ndimensional space. In particular cases, our model represen ts the bond-alternating, ferromagnet-\nantiferromagnet and also homogenous spin smodel. The factorized ground state is a product of\nsingle particle kets on a bipartite lattice composed of two d ifferent spins ( ρ,σ) which is characterized\nby two angles, a bi-angle state. The spin waves analysis around the exact ground state show two\nbranch of excitations which is the origin of two dynamics of t he model. The signature of these\ndynamics is addressed as a peak and a broaden bump in the speci fic heat.\nPACS numbers: 75.10.Jm, 03.67.-a, 64.70.Tg\nIntroduction.- Spin models are the building blocks of\nthe theory of quantum magnetism and strongly corre-\nlated electron systems. In addition, they have been con-\nsidered as an effective model to describe the behavior\nof a system in several disciplines. Recently, the im-\nplementation of quantum notions in quantum devices\nhas attracted much attentions both in research labs\nand demanding applications like nanotechnology, quan-\ntum computation[1] and particularly optical lattices[2].\nQuantum spin models are prototype realization of many\nrelevant properties of quantum implementation in such\ndevices. Therefore, different aspects of a quantum\nphase is of utmost importance for scientists and engi-\nneers. Quantum phases are characterized by the ground\nstate (GS) properties of the corresponding many body\nsystem.[3]\nExcept of a few particular cases such as 1D bond al-\nternating Heisenberg [4], anisotropic Heisenberg model\n(XYZ), XXZ in a longitudinal magnetic field and Ising\nmodel in transverse field which are exactly solvable[5],\nthe GS of a general spin model is not known. However,\nat some special values of the model parametersthe quan-\ntum correlations are vanishing and the GS is a product\nof single particle states. The factorized state (FS) man-\nifests zero entanglement which is necessary to be iden-\ntified for reliable manipulating of quantum computing.\nA FS (unentangled) which is associated with an entan-\nglement phase transition can be also a quantum critical\npoint in certain condition which is discussed in this arti-\ncle. This information is also attractive for the study of\nquantum phase transitions. Moreover, finding an exact\nground state (as a FS) even at particular values of the\nparameter space of a many body spin model leads to the\nidentificationofthatphaseinadditiontomoreknowledge\nabout the properties of the model close to the factorized\npoint via implementing an approximate method.In a seminal work, Kurmann, Thomas and M¨ uller [6]\nidentified the factorized state of a homogeneous spin- s\nXYZ chain at a magnetic field of arbitrary direction.\nFactorized GS has been also observed in the two di-\nmensional lattice through Quantum Monte Carlo sim-\nulation in terms of entanglement estimators.[7] Recently,\nGiampaolo, Adesso and Illuminati [8] have introduced a\ngeneral analytic approach to find the factorized ground\nstates in a homogenous translational invariant spin-s\nquantum spin model for arbitrary long range interaction\nand any dimensional space. Their study is based on the\nsingle-spin unitary operation and the factorized point is\ndetermined at the position where the associated entan-\nglement excitation energy becomes zero. More elaborate\nexplanations came up with some generalizations quite\nrecently.[9] The factorized GS of the dimerized XYZ spin\nchain in a transversemagnetic field has been investigated\nand reported that the factorized point in the parameter\nspace of the Hamiltonian corresponds to an accidental\nground state degeneracy.[10] However, in this article we\nwill present (i) the FS of an inhomogeneous (ferrimag-\nnetic) spin model whichis composedoftwospins( ρ,σ) in\nthepresenceofamagneticfieldonabipartitelattice with\narbitrary long range interaction and dimensional space,\n(ii) the Hamiltonian is not necessarily translational in-\nvariant and (iii) the exchange couplings can be compet-\ning antiferromagnetic and ferromagnetic arbitrarily be-\ntween different sublattices to build many practical mod-\nels such as frustrated, dimerized and tetramerized ma-\nterials. Moreover, our results recover the previous ones\nforσ=ρand a particular configuration of the couplings.\n[6, 7, 8, 10] In addition, we will address on the existence\nof two energy scales which lead to a surprising dynam-\nics of the model close to the factorizing point and its\nfingerprint as a double peak in the specific heat versus\ntemperature. As an enclosure, the results have been ap-2\nplied to the 1D ferrimagnetic XXZ ( ρ,σ) spin chain in\nthe presence of a transverse magnetic field which is real-\nized as a bimetallic substance. [11] We will also address\nthe cases where the factorizing field coincides the critical\npoint.\nFactorized state.- Let us consider a two sites model\nwhich is composed of two spins σ=1\n2andρ= 1 with\nthe following Hamiltonian\nH′=Jxσxρx+Jyσyρy+Jzσzρz+h′(σz+ρz),(1)\nwhereJµ,µ=x,y,zare the exchange couplings in dif-\nferent directions and h′is proportional to the magnetic\nfield. We are looking for a factorized state which is sat-\nisfied byH′|σ/angbracketright|ρ/angbracketright=ǫ|σ/angbracketright|ρ/angbracketright,in which |σ/angbracketrightand|ρ/angbracketrightare\nthe single particle states. It is appropriate to choose |σ/angbracketright\nand|ρ/angbracketrightto be the eigenstates of /vector σ·ˆn′and/vector ρ·ˆn′′with\neigenvalues +1\n2and +1; respectively, where ˆ n′(θ,ϕ) and\nˆn′′(β,α) are unit vectors in Bloch sphere. The solution\nofH′|σ/angbracketright|ρ/angbracketright=ǫ|σ/angbracketright|ρ/angbracketrightgives the factorized state at h′=h′\nf\nanditscorrespondingenergy( ǫ)[12]. Moreover,wefound\nthat the angles θandβare fixed by the couplings ( Jµ,h′;\nseeEq.(2))while αandϕaregivenbyoneofthesechoices\n(I)α= 0,ϕ= 0, (II)α= 0,ϕ=π, (III)α=π\n2,ϕ=−π\n2,\n(IV)α=π\n2,ϕ=π\n2. The spins are located in the xz-plane\nfor choices I and II while they have projections only in\nthe yz-plane for III and IV. Without loss of generality we\ncan assume the spins are located in the xz-plane. In fact,\nthe spins of yz-plane will fall to xz-plane by interchange\nofJx↔Jy. Moreover, the coordinates ( θ,ϕ= 0) and\n(−θ,ϕ=π) are representing the same direction, there-\nfore case (I) α= 0,ϕ= 0 is able to describe all possibil-\nities.\nThe two spin model ( σ=1\n2,ρ= 1) is now general-\nized to arbitrary ( σ,ρ) spins.[6] To find the factorized\nstate of a general two site ferrimagnet we consider a ro-\ntation onσandρspins such that− →σand− →ρpoint in\n(θ,ϕ= 0) and ( β,α= 0) directions, respectively, The\nrotation operator is D=Dσ(0,θ,0)Dρ(0,β,0) where\nDρ(0,β,0) =D(α= 0,β,γ= 0) =Dz(α)Dy(β)Dz(γ)\nis defined in terms of Euler angles and a similar expres-\nsion is considered for Dσ(0,θ,0). Then, We impose the\ncondition to have a factorized (fully polarized) eigenstate\nfor this Hamiltonian which fixes the following relations\nfor the model parameters\ncosθ=−h′2\nfJy+Jx(Jz2−Jy2)ρσ+h′\nfJz(Jyρ+Jxσ)\nh′2\nfJx+Jy(Jz2−Jx2)ρσ+h′\nfJz(Jxρ+Jyσ),\ncosβ=−h′2\nfJy+Jx(Jz2−Jy2)ρσ+h′\nfJz(Jyσ+Jxρ)\nh′2\nfJx+Jy(Jz2−Jx2)ρσ+h′\nfJz(Jxσ+Jyρ),\nh′\nf=/radicalbigg\n1\n2/parenleftbig\n2JxJyρσ+(ρ2+σ2)Jz2+CJz/parenrightbig\n,\nC≡/radicalBig\n4ρσ(ρJx+σJy)(σJx+ρJy)+(ρ2−σ2)2Jz2,\nǫ=JxJy\nJzσρ−h′2\nf\nJz. (2)Therefore, for arbitrary ( σ,ρ) and at the above value for\nh′=h′\nfwe have a fully polarized eigenstate which is a\nfactorized state. The ordering of this state is defined by\ntwo angels ( θ,β) which show the orientations of (− →σ,− →ρ),\nrespectively.\nNow, we intend to find the condition for having a fac-\ntorized state for a ferrimagnetic lattice in a magnetic\nfield. We will then show that the factorized state is the\nground state of lattice implementing a spin wave theory\ntostudythequantumfluctuations. Weconsiderageneral\nHamiltonian of ferrimagnets on a bipartite lattice where\nsublattice ( Aσ) contains σspins and the other sublat-\ntice (Bρ) includesρspins. The interaction can be long\nranged between different sublattices but no interaction\nin the same sublattice. The ferrimagnetic Hamiltonian\nfor such case can be written as\nH=/summationdisplay\ni,r/bracketleftBig\nζiˆζi+r(Jx\nrσx\niρx\ni+r+Jy\nrσy\niρy\ni+r)+Jz\nrσz\niρz\ni+r/bracketrightBig\n+h/summationdisplay\ni(σz\ni+ρz\ni), (3)\nwherei= (i1,i2,i3) andr= (r1,r2,r3) are repre-\nsenting the three dimensional index on the lattice and\nζi,ˆζi+r=±1 which realize both ferromagnetic (F) and\nantiferromagnetic (AF) exchange interactions. A remark\nis in order here, the Hamiltonian in Eq.(3) is a sum of\ntwo sites Hamiltonian defined in Eq.(1) where the two\nspins can be far from each other. However, the inter-\naction between each couple of ( σi,ρi+r) can depend on\ndistance (r) with different strength and also be F or AF\narbitrarily defined by ζi,ˆζi+r. A factorized eigenstate for\nthe Hamiltonian of Eq.(3) can be written as\n|FS/angbracketright=/circlemultiplydisplay\ni∈Aσ,j∈Bρ|σ′\ni/angbracketright|ρ′′\nj/angbracketright (4)\nwhere|σ′\ni/angbracketrightand|ρ′′\nj/angbracketrightare the eigenstates of /vector σi·ˆn′\niand/vector ρj·ˆn′′\nj\nwith largest eigenvalue where ˆ n′\niand ˆn′′\njare unit vectors\npointing in ( ζiθ,ϕ= 0) and ( ˆζjβ,α= 0), respectively.\nHowever, the factorized state ( |FS/angbracketright) is an eigenstate of\nthe Hamiltonian if the angle ζiθ(ˆζjβ) be consistent with\nall pair of interactions originating from σi(ρi) on sublat-\nticesAσ(Bρ). According to Eq.(2) the former condition\nis satisfied if the interaction between each pair ( σi,ρi+r)\nis the same for all directions while depending on distance\n(r), i.eJµ\nr=λ(r)Jµ,µ=x,y,z,λ(r)>0. Under these\nconstraints the factorized state (Eq.(4)) is an eigenstate\nofHwiththecharacteristicangles( θ,β)definedinEq.(2)\nand the factorizing field is\nhf=h′\nfNr/summationdisplay\nr=0λ(r), (5)\nwhereNris the number of spins on each sublattice. To\nshow that |FS/angbracketrightis the ground state of Hathfwe first3\nimplement a rotation on the Hamiltonian. The rotated\nHamiltonian ( ˜H) is the result of rotations on all lattice\npoints ofH,\n˜H=˜D†H˜D,˜D=/circlemultiplydisplay\ni∈Aσ,j∈BρDσ\ni(0,ζiθ,0)Dρ\nj(0,ˆζjβ,0).\n(6)\nIn the next step the rotated Hamiltonian is bosonized\nusing the Holstein-Primakoff (HP) transformation, σ+\ni=/radicalBig\n2σ−a†\niaiai,σz′\ni=σ−a†\niai,ρ+\nj=/radicalBig\n2ρ−b†\njbjbj,ρz′′\nj=\nρ−b†\njbj,whereai(a†\ni) andbj(b†\nj) are two types anni-\nhilation (creation) boson operators. Using HP trans-\nformations the anisotropic ferrimagnetic spin model is\nmapped to an interacting system of bosons. The Hamil-\ntonian in the momentum ( k) space and in the linear\nspin wave theory is diagonalized via the rotation, χk=\nakcosηk−eiδbksinηk;ψk=eiδbkcosηk+aksinηk, and\na shift atk= 0 where δis defined by/summationtext\nre−ik·rJy\nr=\n|/summationtext\nre−ik·rJy\nr|eiδ. The diagonalized Hamiltonian is\n˜H=Egs+/summationdisplay\nk/parenleftbigg\nω−(k)χ†\nkχk+ω+(k)ψ†\nkψk/parenrightbigg\n,(7)\nwhereEgsis the ground state energy [12] and ω±(k)≥0\nare normal modes parallel and perpendicular to the field\ndirection.\nω±(k) =D+±D−+√σρtan(2ηk)|/summationtext\nre−ik·rJy\nr|/radicalbig\n1+tan2(2ηk),\ntan(2ηk) =√ρσ|/summationtext\nre−ik·rJy\nr|\nD−, (8)\nin which\nD±≡h2\nf\n2τz(1\nρ±1\nσ)+hf(σ\n2ρcosθ±ρ\n2σcosβ)\n−τxτy\n2τz(σ±ρ)+hf−h\n2(cosβ±cosθ),\nτµ=Jµ/summationdisplay\nrλ(r). (9)\nThe bosons number /angbracketlefta†a/angbracketrightis proportional to ( hf−h)\nwhich states that at h=hfthe bosons number in the\nground state is exactly zero [12], i.e /angbracketlefta†\niai/angbracketright|h=hf= 0 =\n/angbracketleftb†\nibi/angbracketright|h=hf. It is also true for /angbracketlefta†\nibi/angbracketright|h=hf= 0. Thus,\nthe spin wave theory is exact in all orders at the factoriz-\ning field (h=hf) and|FS/angbracketrightis the corresponding ground\nstate ofH.\nTo visualize the configuration of a factorized ground\nstate of a general frustrated model we have plotted an\nexample in Fig.(1) with the assumption Jx\nr,Jy\nr>0 and\nJz\nr<0 whereζiandˆζi+rdefine the sign of interac-\ntions. The solid-lines represent antiferromagnetic inter-\naction and the dash-dotted ones are the ferromagnetic\ncounterparts. As shown in Fig.(1) the interactions can\nFIG. 1: The configuration of a factorized state on a one-\ndimensional lattice for arbitrary frustrated couplings. S olid\nlines (dash-dotted) represent antiferromagnetic (ferrom ag-\nnetic) couplings which are defined by ζiˆζi+ras depicted by\n±on each site. Each color belongs to equal distance interac-\ntion (same r).\nbe frustrated and long ranged without a translational in-\nvariance. However, the factorized state is defined by two\nangles (θ,β) while each σ(ρ) spin is directed in θ(β)\nor−θ(−β) directions. We call this a bi-angleordering.\nIn a special case the bi-angle ordered state can configure\na ferromagnet ( θ=β) or antiferromagnet ( θ=π−β)\nfactorized state.\nDiscussions.- In a spin model when the magnetic field\nis strong enough all spins will align in the direction of the\nmagnetic field which characterizesthe saturated phase as\nfar ash≥hs. In our notation, the saturated phase ap-\npears when all σ(ρ) spins get θ=π(β=π). Thus, the\nsaturating field ( hs) is a factorizing one when Jx=Jy.\nIn case ofJx/negationslash=Jythe saturation can only appear at in-\nfinite value of the magnetic field while a finite factorizing\npoint (hf) still exists. For Jx=Jy, the lower excitation\nband becomes gapless ( ω−(k= 0) = 0) which confirms\nthat the factorizing point ( hf=hs) is the critical point\nwhich separates the non-saturated phase ( h < h s) from\nthe saturated one ( h>hs). It is worth to mention that\natJx=Jytherotationalsymmetryaroundthe magnetic\nfield is restored where the quantum fluctuations around\nthe field axis are suppressed. In the absence of rotational\nsymmetry the saturation in the field direction does not\nhappen at a finite value of magnetic field while the model\napproaches saturation asymptotically at infinity.\nA general feature of our result is that it can simply\nrecover the previous study of homogenous systems by\nreplacingσ=ρ=s. In that case the restriction of\nbipartite lattice is promoted to arbitrary lattice and the\ninteraction between any pair of spins can exist. However,\nour Hamiltonian is not restricted to the translational in-\nvariant symmetry or bond-alternating ones which is wit-\nnessed by the example given in Fig.(1). This can also\nbe generalized to any dimension. We claim that the gen-\neral Hamiltonian which can possess a nontrivial factor-\nized ground state should be of the form Eq.(3) with the\nrestrictionJµ\nr=λ(r)Jµ,µ=x,y,z.\nIn the absence of magnetic field, the spin-( ρ,σ) ferri-\nmagnet turns into the spin- σantiferromagnetin the limit\nρ→σ, whereasit lookslikethe spin- ρferromagnetin the4\nother extreme limitρ\nσ→ ∞. In this sense, the difference\nρ−σcan be regarded as the ferromagnetic contribution.\nThe analysis of our result shows that the value of hfis a\ndecreasing function ofρ\nσconverging to the ferromagnet\nfeature as this ratio becomes large. More investigations\non different applications of our approach in several mod-\nels are in progress.[12]\nLet us now be more concrete by concentrating on the\none-dimensional nearest neighbor XXZ ferrimagnet in\nthe presence of transverse magnetic field. Suppose that\nJx=Jz=JandJy=J∆ where ∆ represents the easy\naxis anisotropy. At zero magnetic field the quantum fluc-\ntuations are large and the ground state of the model is\nstronglyentangled. Uponaddingthetransversemagnetic\nfield the U(1) symmetry of the XXZ model is lost and\nthe entanglement of the GS is decreased. In the mapped\nbosonic system the magnetic field is served as a chemical\npotential, thus the number of bosons ( /angbracketlefta†a/angbracketrightand/angbracketleftb†b/angbracketright) is\ndependent on the magnetic field. An enchantment of the\nmagneticfieldcausesdeducingofthebosons’numberand\nthe quantum correlations decrease. At factorizing field\nh=hf, the number of bosons is zero and the quantum\nfluctuations become completely uncorrelated. Moreover,\nour calculations show that the factorizing field in fer-\nrimagnetic model depends on the anisotropy parameter\n(∆) similar to a homogeneous antiferromagnetic Heisen-\nberg model. For ∆ = 0 (ferrimagnetic XY model) the\nin-plane magnetic field completely breaks the rotational\nsymmetry. Increasing ∆ suppresses the effect of the field\nand try to evoke the rotational symmetry to the sys-\ntem. Thus, by increasing ∆ the factorizing Ne´ el field\napproaches to the saturation field. At ∆ = 1, the rota-\ntional symmetry is completely repayed and hfis exactly\nlied onhs.\nA benefit of identification a factorized state is that we\ncan work out an approximate method around the factor-\nized point to get some information on the properties of\nthat phase. This help us to calculate the magnetic prop-\nerties of the ferrimagnetic XXZ model in the presence of\na transverse magnetic field. We have implemented the\nlinear spin wave theory around h=hfforσ=1\n2and\nρ= 1. Our results for magnetization ( Mx,My) and stag-\ngered magnetization ( SMx,SMy) in bothxandydirec-\ntions are plotted in Fig.(2) where the magnetic field is in\nxdirection. The approach can be extended to derive the\nthermodynamic of this model which will be done in near\nfuture.\nIt is also worth to mention that our results are ap-\nplicable to the homogenous XXZ Heisenberg spin-1/2\nchains in the presence of a transverse magnetic field\n(hx). This model has been studied intensively in the\nliterature.[13, 14, 15, 16] On the onset of transversemag-\nneticfieldaperpendicularanti-ferromagneticorderissta-\nbilized by promoting a spin-flop phase. At the classi-\ncal (factorizing) field ( hf< hc) in the spin-flop phase\nthe ground state is factorized to single spin states and\nFIG. 2: The magnetization andstaggered magnetization ofan\nanisotropic ferrimagnetic ( σ= 1/2,ρ= 1) spin chain versus\ntransverse field and for ∆ = 0 .25.\nthe staggered magnetization along the ydirection has a\nlarge value. Very close to the critical field ( hc) the anti-\nferromagnetic order becomes unstable and the staggered\nmagnetizationfalls sharplyto vanishat the criticalpoint.\nForh>hcthe spins are almost aligned in field ( x) direc-\ntion and the factorized state as a fully polarized phase\nwill be appeared for h→ ∞. The excitation energies\naround the factorizing point have the following form:\nω±(k) = (1+∆)h\nhf+∆(±cos(k\n2)−1).(10)\nThus, we have two branches of magnon energies as two\nscales which impose two dynamics in the system. The\nmost interesting feature is that around the factorizing\nfield both scales show up. These dynamics correspond\nto the coexistence of two different features of the model.\nIn other words, by increasing the transverse magnetic\nfield, spins try to align in the xdirection which is the\nferromagnetic feature of the system. In the intermediate\nvalues ofh, we have both anti-ferromagnetic and ferro-\nmagnetic behaviors since the anti-ferromagnetic order in\nydirection has already been stabilized. The finger print\nof these features appear in the thermodynamic functions\nsuch as specific heat and internal energy. As it is seen\nfrom Ref.[16] the second feature can be seen as a shoul-\nder at the right side of specific heat curve. By further\nincreasing of h, the ferromagnetic behavior is seen as a\nbroaden peak in the curve. This point is almost near the\nclassical field where the ground state of the system has\nbeen factorized.\nThis work was supported in part by the Center of\nExcellence in Complex Systems and Condensed Matter\n(www.cscm.ir).\n[1] M. A. Nielsen, I. L. Chuang, Quantum Computation\nand Quantum Information (Cambridge University Press,\nCambridge, 2000).5\n[2] G. K. Brennen, C. M. Caves, P. S. Jessen, I. H. Deutsch,\nPhys. Rev. Lett, 82, 1060 (1999).\n[3] S. Sachdev, Quantum Phase Transitions (Cambridge\nUniversity Press, Cambridge, 2000).\n[4] H. Suzuki and K. Takano, J. Phys. Soc. Jpn. 77, 113701\n(2008).\n[5] M. Takahashi, Thermodynamics of one-dimensional\nsolvable models , (Cambridge University Press: Cam-\nbridge, 1999 ).\n[6] J. Kurmann, H. Thomas, and G. Muller, Physica 112 A,\n235 (1982).\n[7] T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V.\nTognetti, Phys. Rev.Lett. 93167203 (2004); T. Roscilde,\nP. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Phys.\nRev. Lett. 94147208 (2005).\n[8] S. M. Giampaolo, G. Adesso, and F. Illuminati, Phys.Rev. Lett. 100, 197201 (2008).\n[9] S. M. Giampaolo, G. Adesso, and F.\nIlluminati,arXiv:0904.1213.\n[10] G. L. Giorgi, Phys. Rev. B 79, 060405(R) (2009).\n[11] P. J. Koningsbruggen, O. Khan, K. Nakatani, Y. Pei, J.\nP. Renard, M. Drillon, and P. Legoll, Inorg. Chem. 29,\n3325 (1990).\n[12] More details of the calculations will appear elsewhere .\n[13] M. Kenzelmann, et al., Phys. Rev. B 65, 144432 (2002).\n[14] A. Langari, Phys. Rev. B 69, 100402(R) (2004).\n[15] M. Siahatgar and A. Langari, Phys. Rev. B, 77054435\n(2008).\n[16] J. Abouie, A. Langari and M. Siahatgar,\narXiv:0809.1987/cond-mat (2008)."    },    {        "title": "2309.14140v1.Magnetic_States_and_Electronic_Properties_of_Manganese_Based_Intermetallic_Compounds_Mn__2_YAl_and_Mn__3_Z__Y___V__Cr__Fe__Co__Ni__Z___Al__Ge__Sn__Si__Pt_.pdf",        "content": " Magnetic States and Electronic Properties of \nManganese-Based Intermetallic Compounds Mn 2YAl and \nMn3Z \n(Y = V, Cr, Fe, Co, Ni; Z = Al, Ge, Sn, Si, Pt) \nVyacheslav V. Marchenkov * and Valentin Yu. Irkhin \nM.N. Mikheev Institute of Metal Physics, UB RAS, 620108 Ekaterinburg, Russia;  \n*Correspondence: march@imp.uran.ru; valentin.irkhin@imp.uran.ru \nAbstract: We present a brief review of experimental and theoretical papers on studies of electron transport \nand magnetic properties in manganese-based compounds Mn 2YZ and Mn 3Z (Y = V, Cr, Fe, Co, Ni, etc.; Z = Al, Ge, \nSn, Si, Pt, etc.). It has been shown that in the electronic subsystem of Mn 2YZ compounds, the states of a \nhalf-metallic ferromagnet and a spin gapless semiconductor can arise with the realization of various magnetic states, \nsuch as a ferromagnet, a compensated ferrimagnet, and a frustrated antiferromagnet. Binary compounds of Mn 3Z \nhave the properties of a half-metallic ferromagnet and a topological semimetal with a large anomalous Hall effect, \nspin Hall effect, spin Nernst effect, and thermal Hall effect. Their magnetic states are also very diverse: from a \nferrimagnet and an antiferromagnet to a compensated ferrimagnet and a frustrated antiferromagnet, as well as an \nantiferromagnet with a kagome-type lattice. It has been demonstrated that the electronic and magnetic properties of \nsuch materials are very sensitive to external influences (temperature, magnetic field, external pressure), as well as \nthe processing method (cast, rapidly quenched, nanostructured, etc.). Knowledge of the regularities in the behavior \nof the electronic and magnetic characteristics of Mn 2YAl and Mn 3Z compounds can be used for applications in \nmicro- and nanoelectronics and spintronics. \nKeywords: manganese based intermetallic compounds; antiferromagnetism; compensated ferrimagnetism; \nfrustrated magnets; half-metallic ferromagnets; spin gapless semiconductors; topological semimetals; anomalous \nHall effect; kagome lattice  \n1. Introduction \nThe exploration and advancement of novel materials with distinct magnetic and \nelectronic properties, along with the experimental and theoretical investigation of their electronic \nenergy spectrum, structural configurations, and magnetic states, hold significant importance from \nboth a fundamental and a practical standpoint. Heusler compounds, discovered over 100 years \nago by German chemist F. Heusler [1], continue to be a subject of great interest in current \nresearch. These compounds exhibit a wide range of unique multifunctional properties [2], such \nas half-metallic ferromagnetism [3–5], spin gapless semiconductor state [6–8], topological \ninsulator and semimetal behavior [8–11], shape memory effect [12,13], magnetocaloric effect \n[14,15], and many others (see, for example, [2,16,17] and references therein). \nAmong these materials, intermetallic compounds based on manganese, such as Mn 2YZ \nand Mn 3Z (Y = V, Cr, Fe, Co, Ni, etc.; Z = Al, Ga, Ge, Sn, etc.), are particularly noteworthy. In \naddition to the functional properties mentioned above, these compounds exhibit a range of \nunique magnetic characteristics, including the ability to manifest states of antiferromagnetism, \ncompensated ferrimagnetism, and frustrated magnets. They possess unconventional magnetic  2 and electronic properties that are highly susceptible to external influences, making them \npromising candidates for practical applications in fields such as spintronics, microelectronics, \nand nanoelectronics. \nManganese-based Heusler compounds, specifically Mn 2YZ, possess several remarkable \nproperties. They exhibit characteristics of half-metallic ferromagnets (HMF), spin gapless \nsemiconductors (SGS), and potentially topological semimetals (TSM). These compounds also \ndemonstrate a significant magnetocaloric effect and shape memory. Moreover, they offer the \nopportunity to realize unique magnetic states, including ferromagnetic (FM), antiferromagnetic \n(AFM), compensated ferrimagnetic (CFIM) ones, etc. (see, e.g., refs. [4,8,18]). \nIn previous studies, the intermetallic compounds Mn 3Z (Z = Ge, Sn, Ga, Ir, Rh) were \nfound to exhibit a strong anisotropic anomalous Hall effect (AHE) and spin Hall effect in their \nantiferromagnetic state [19]. Researchers reported [20] a zero magnetic moment in thin films of \nthe Mn 3Al alloy, which was attributed to a compensated ferrimagnetic state. This CFIM state \ndiffers from AFM due to the distinct crystallographic positions of manganese. Another study [21] \nobserved a zero magnetic moment in the cast Mn 3Al alloy and suggested that it may also be \nindicative of compensated ferrimagnetism. \nThe combination of noncollinear magnetic structure and Berry curvature can result in a \nnonzero topological anomalous Hall effect, as demonstrated in antiferromagnets Mn 3Sn and \nMn3Ge [9,22]. These compounds, along with their noncollinear magnetic structures, also exhibit \ntopological states in real space in the form of magnetic anti-skyrmions. The ability to manipulate \nthe Berry curvature highlights the significance of understanding both the electronic and magnetic \nstructures of Mn 3Z compounds. \nFurthermore, among these intermetallic compounds, there are topological systems that \npossess unique surface states and display anomalous transport phenomena due to their \nunconventional bulk electronic topology. For instance, Mn 3Ge and Mn 3Sn compounds with a \ndistorted D019 structure in the ab plane form a lattice of Mn atoms resembling a highly frustrated \nkagome lattice [22,23]. Further theoretical and experimental investigations of such structures and \ntheir impact on electronic and magnetic properties remain pertinent and in demand. \nFigures 1–3 schematically show the models of an antiferromagnet and a compensated \nferrimagnet (Figure 1), a half-metallic ferromagnet and a spin gapless semiconductor (Figure 2), \nand topological materials (Figure 3). \nThe magnetic structure of D03 compounds in the antiferromagnetic and compensated \nferrimagnetic states is presented in Figure 1. As can be seen from Figure 1, the difference \nbetween antiferromagnetic and ferrimagnetic states is that the crystallographic positions of \nmanganese with oppositely directed magnetic moments are completely different.  3 \n \n    (a)             ( b) \nFigure 1. Schematic view of the magnetic structure of D03 compounds. ( a) G-type \nantiferromagnetic structure of V 3Al, ions in Mn( X) positions being nonmagnetic. ( b) \nMagnetic structure of compensated ferrimagnet Mn 3Al [20]. Up-directed magnetic \nmoments are shown by red arrows, and down-directed ones by blue. \nFigure 2 shows the states of a half-metallic ferromagnet and a spin gapless \nsemiconductor. The electronic structure of a half-metallic ferromagnet has the following features: \nthere is a gap at the Fermi level for spin-down electronic states, which is absent for spin-up ones \n(Figure 2a). In the case of a spin gapless semiconductor (Figure 2b), the situation is similar, but \nthere is a significant difference. As in a half-metallic ferromagnet, there is a finite gap for the \nspin-down spin projection, but the gap is zero for spin-up ones (Figure 2b). This is similar to the \ncase of classical gapless semiconductors [24]. \n(a) (b) \nFigure 2. Simple models of half-metallic ferromagnets ( a) and spin gapless \nsemiconductors ( b) [8]. The valence band for the electron states with spin up (blue \narrows) is marked by green, and for ones with spin down (red arrows) by blue.  \nIn topological materials (Figure 3), an inversion of the conduction and valence bands can \noccur because of a strong spin–orbit interaction. This leads to the appearance of a nontrivial \ntopology of the electronic band structure, which is observed in topological insulators, Dirac and  4 Weyl semimetals. Topological insulators have a characteristic energy gap in bulk and “metallic” \nstates on the surface (Figure 3a). The Dirac and Weyl semimetals also have a gap in the bulk \nresulting from strong spin–orbit coupling, except for the band intersection at Dirac (Figure 3b) \nand Weyl (Figure 3c) points, respectively. \nThe methods used to prepare intermetallic compounds, specifically those dealing with \nMn3Z, can have a significant impact on their structural properties (see, e.g., refs. [25–29]). This, \nin turn, influences their electronic and magnetic states. Understanding the role of the structural \nstate in the formation and behavior of intermetallic compounds based on manganese is an \nimportant and fascinating problem. \n \nFigure 3. Schematic representation of the band structure of ( a) massive Dirac, ( b) \n“massless” Dirac, and ( c) Weyl fermions. The latter arises during the decay of the Dirac \npoint. Two-color and one-color curves and lines represent doubly degenerate and \nnondegenerate zones, respectively [8]. \nBoth experimental and theoretical studies in this area hold great relevance and scientific \nsignificance (see papers [30–32] and references therein). They allow for a comprehensive \ndescription of the evolution of the electronic structure, magnetic state, and properties of Mn 2YZ \nand Mn 3Z compounds. These studies aim to understand the characteristics and differences in the \nmanifestation of various states, such as antiferromagnetism, compensated ferrimagnetism, \nfrustrated magnetism, half-metallic ferromagnetism, spin gapless semiconductors, topological \nsemimetals, etc. This work provides a concise overview of the research on electron transport and \nthe magnetic state in compounds based on manganese, specifically Mn 2YZ and Mn 3Z (Y = Sc, Ti, \nV, Cr, Fe, Co, etc.; Z = Al, Ge, Sn, Si, Pt, etc.). \n2. Electronic and Magnetic Properties of Mn 2YZ (Y = V, Cr, Fe, Co, Ni, etc.; Z = Al, \nGe, Sn, Si, Pt, etc.) Compounds \nThe HMF state was theoretically predicted in 1983 in ref. [33] using band calculations of \nthe NiMnSb half-Heusler compound. Later, the HMF theory was significantly supplemented and  5 extended [3,34]. After 40 years, band calculations are still extensively used to predict the HMF \nstate in various compounds (see, for example, refs. [4,5] and references therein). \nElectronic structure calculations performed on manganese-based Heusler alloys have \nshown that many of them can exhibit HMF properties. These are Mn 2VZ (Z = Al, Ga, In, Si, Ge, \nSn) compounds [35], Mn 2FeZ (Z = Al, Ga, Si, Ge, Sb) [36], Mn 2CrZ (Z = Al, Ga, Si, Ge, Sb) \n[37], Mn 2TiZ (Z = Al, As, Bi, Ga, Ge, Sb, Si, Sn) [38], Mn 2ZrZ (Z = Ga, Ge, Si) [39,40]. It is \nworth emphasizing that the HMF state predicted theoretically and/or as a result of band \ncalculations is by no means always realized in real compounds. \nUsing density functional theory, the electronic, magnetic, and structural properties of \nferrimagnetic Mn 2VAl and Mn 2VSi Heusler alloys were studied in ref. [41]. It was shown that \ntwo states can exist in studied compounds: one state with low magnetization and a HMF \nbehavior with almost 100% spin polarization and the second state with high magnetization and a \nmetallic character. \nThe temperature dependences of the electrical resistivity ρ(T) of the Mn 2CrAl alloy were \nstudied in [21]. It can be observed (see Figure 1 in ref. [21]) that the residual resistivity ρ0 \nreaches a large value of ≈250 μΩ cm, and the resistivity ρ decreases with temperature increasing \nup to room temperature. A similar behavior ρ(T) was observed for Mn 2FeAl in [42]. The large \nvalues of ρ0 and the “semiconductor” dependence ρ(T) were explained by the structural disorder. \nThere is a so-called Mooij rule [43], according to which in metallic systems with the static \ndisorder, i.e., with resistivity ρ > (150–200) μΩ cm, a negative temperature coefficient of \nresistance is usually observed. The estimates of the concentration n of current carriers from \nmeasurements of the Hall effect give large values: n ≈ 1.6∙1022 cm−3 for Mn 2CrAl [21], and n ≈ 2 \n× 1022 cm−3 for Mn 2FeAl [42]. \nStudies of the magnetic properties of Mn 2CrAl [21] and Mn 2FeAl [42,44] alloys showed \nthat the behavior of the magnetization M(H) indicates a total moment close to zero (see, e.g., \nFigure 4). Such a state can be characterized as (1) a compensated ferrimagnet, which retains the \nnature of a half-metallic ferromagnet state [20,45], or as (2) an antiferromagnet. \nThe distinction between them can be explained as follows. In the case of \nantiferromagnetism, the opposite magnetic moments of manganese are situated in positions that \nare equivalent from a crystallographic perspective. Conversely, in the case of compensated \nferrimagnetism, these opposite moments of manganese occupy crystallographically distinct \npositions. In CFIM, the compensation of magnetic moments is not solely reliant on crystal \nsymmetry but is also influenced by a unique band structure. The utilization of CFIM and/or AFM \nstates in such materials holds promise for applications in spintronics as they can exhibit high spin \npolarization of current carriers.  6 It is shown in ref. [44] that a frustrated antiferromagnetic state is observed in Mn 2FeAl \nalloy with a Neel temperature TN = 48 K and a Curie–Weiss temperature θ CW ≈ −230 K. In this \ncase, large antiferromagnetic spin fluctuations, caused by the geometric frustrations, lead to the \nappearance of an unusually large electronic heat capacity. According to ref. [44], the \ncorresponding value of the T-linear heat capacity coefficient is γ = 210 mJ/molK2. \n \nFigure 4. Field dependence of magnetization for Mn 2FeAl at different temperatures [42]. \nThe Mn 2CoAl compound exhibits even higher residual resistivity values [46,47] (Figure \n5), belonging to a novel class of quantum materials known as spin gapless semiconductors. The \ntheoretical prediction of SGS materials was made in 2008 [6]. SGS materials possess an unusual \nband structure, wherein there exists an energy gap for the spin-down electron subsystem near the \nFermi level, while the spin-up electronic states have the top of the valence band touching the \nbottom of the conduction band. Under these circumstances, the kinetic properties of SGS \nmaterials are expected to resemble those of “classical” gapless semiconductors [24], including \nhigh residual resistivity with weak temperature dependence, relatively low current carrier \nconcentration, and low thermoelectric power. Due to the prevalence of spin-up current carriers, a \nhigh degree of spin polarization, a significant magnetization, and an anomalous Hall effect are \nanticipated. Experimental evidence of this spin gapless semiconductor state with unusual \nmagnetic and magnetotransport properties has been observed in Mn 2CoAl [46,47] and Ti 2MnAl \nHeusler alloys [48]. \nFigure 5 shows the temperature dependencies of the electrical resistivity, the Seebeck \ncoefficient, and the concentration of current carriers of the Mn 2CoAl compound [47]. It can be \nseen that the residual resistivity is very large and amounts to ~445 μΩ cm, and the resistivity \nitself slightly decreases with temperature, reaching a value of ~410 μΩ cm at room temperature.  7 In this case, the thermoelectric power is small and weakly changes with temperature. Finally, the \nconcentration of charge carriers is relatively low, monotonically increasing with temperature: n ≈ \n1.65 × 1020 cm−3 at 2 K and ~3 × 1020 cm−3 at room temperature. Finally, measurements of the \nHall effect in Mn 2CoAl [47] show that the value of the anomalous Hall conductivity σxy is \nrelatively small and at 2 K is σxy = 21.8 S cm−1, which can be explained by the symmetry features \nof the Berry curvature (Figure 5). These facts confirm the realization of the state of a spin \ngapless semiconductor in the Mn 2CoAl compound. \n \n \n(a) ( b) \nFigure 5. Mn2CoAl compound [47]. ( a) Temperature dependence of the electroresistivity ρ, \nSeebeck coefficient S, and charge carriers concentration n. (b) Field dependence of Hall \nconductivity at T = 2 K. \nIt is important to highlight that the characteristics of a spin gapless semiconductor (SGS) \nare not limited to electron transport but also extend to other properties, including optical \ncharacteristics. In a study referenced as [49], the optical properties of the Mn 2CoAl compound, \nwhich exhibits SGS behavior, were investigated. The study also involved the calculation of its \nelectronic structure. The findings revealed distinct optical properties of Mn 2CoAl, including \npositive values of the real part of the dielectric constant ( ε1) and the absence of the Drude \ncontribution to the optical conductivity in the infrared (IR) region of the spectrum within the \nexamined range. These observations suggest a slight deviation or “deterioration” of the metallic \nproperties of Mn 2CoAl.  8 Besides, the authors of the study observed [49] intense interband absorption in the IR \nrange and concluded that the band spectrum had a complex structure with a high density of \nd-states near the Fermi level EF. The observed features of the optical properties allowed the \nauthors [49] to explain the band spectrum characteristic of spin gapless semiconductors. \nThe electronic structure and magnetic properties of Heusler alloys, specifically Mn 2YZ, \ncan be significantly altered by modifying their composition through the substitution of atoms and \napplying external influences, like hydrostatic pressure and mechanical processing. In ref. [50], \nthe authors investigated the structure, mechanical, electronic, and magnetic properties of \nMn2−xFe1+xSi Heusler alloys (where x = 0, 0.25, 0.5, 0.75, 1) using density functional theory. By \nreplacing Mn atoms with Fe atoms, it was observed that all the studied alloys maintain \nmechanical and dynamic stability, adopting the Hg 2CuTi structure for x = 0, 0.25 and 0.5 and the \nCu2MnAl structure for x = 0.75 and 1. Furthermore, as the iron content increases, the alloys \nexhibit improved ductility. All the studied alloys display half-metallic ferromagnetic behavior, \nwith the Fermi levels gradually shifting toward the middle of the bandgap in the spin-down \ndirection with increasing iron content. \nIn the study [51], researchers successfully synthesized the Mn 2FeSi Heusler alloy using a \nhigh-energy planetary ball mill. The resulting compound was found to be an inverse Heusler \nalloy with an X A structure. Through the use of Mössbauer spectroscopy and magnetization \nmeasurements, they discovered that the synthesized material exhibits a heterogeneous magnetic \nstructure at room temperature. This structure is composed primarily of the paramagnetic phase, \nwith small contributions from ferro- and ferrimagnetic phases. The Neel temperature TN = 67 K \nwas determined from the temperature dependences of the magnetization. \nUsing ab initio calculations and the Monte Carlo simulation method in [52], the \nstructural, magnetic, and electronic properties of Mn 2YSn (Y = Sc, Ti, and V) Heusler alloys were \nstudied under applied hydrostatic pressure. The coexistence of two magnetic states with a low \nand a high magnetic moment was found for a small and large volume of the crystal lattice, \ncorrespondingly. These states coexist together due to almost equal energy at an applied pressure \nof 3.4, −2.9, and −3.25 GPa for Mn 2ScSn, Mn 2TiSn, and Mn 2VSn, respectively. A positive \npressure corresponds to a uniform compression of the lattice, and a negative one corresponds to a \nuniform expansion of the lattice. It was demonstrated that for the studied compounds, the \nlow-magnetic state (LMS) was characterized by an almost half-metallic behavior, while the \nhigh-magnetic state (HMS) acquired a metallic character. It was shown that the parameters of \nmagnetic exchange and Curie temperatures were significantly higher for the HMS than for the \nLMS. The authors proposed a mechanism for switching between the half-metallic state with the \nLMS and the metallic state with the HMS using applied pressure.  9 In addition to the electronic and magnetic states discussed above, Heusler alloys based on \nmanganese can exhibit other thermodynamic phenomena, such as the shape memory effect and \nthe magnetocaloric effect. Moreover, unlike the well-known Heusler-like alloys based on \nNi-Mn-Z (Z = Ga, In, Sn, etc.) with strong deviations from stoichiometry (of the type \nNi2±xMn1±xZ), SME and MCE can also be observed in full Heusler alloys Mn 2YZ, where Y = Sc, \nTi, V, Co, Ni, etc.; Z = Ga, Ge, Sn, etc. (see, e.g., [53–55] and references therein). \nSimilar magnetic and electronic states can be realized in binary intermetallic compounds \nbased on manganese Mn 3Z (Z = Al, Ge, Ga, etc.) as well. \n3. Features of the Electronic Transport and Magnetic State of Mn 3Z (Z = Al, Ge, Si, \nSn, Pt, etc.) Compounds \nCompensated ferrimagnets possess a magnetic moment that totals zero, but their density \nof electronic states for opposite spin directions allows for a significant spin polarization of \ncurrent carriers. A study [20] indicates that the compensated ferrimagnetic state is present in \nMn3Al, and this half-metallic state can persist at room temperature. In the same study, the \nmagnetic properties of thin films of Mn 3Al were examined. The results reveal that the films \ndemonstrate the state of a compensated ferrimagnet, with a total magnetization value of M = 0.11 \n± 0.04 μ B/f.u. \nResults [20] are in good agreement with refs. [28,29] where the electronic transport and \nmagnetic characteristics of cast and rapid melt-quenched (RMQ) Mn 3Al alloys were measured \nand compared. \nIn Figure 6, the field dependence of the magnetization ( M = f(H)) is shown for both the \ncast and RMQ alloy Mn 3Al at a temperature of 4.2 K, as described in reference [29]. It can be \nobserved from Figure 6a that the cast alloy exhibits a small magnitude of magnetization, which \nincreases linearly with the applied field. At a field strength of 70 kOe, the magnetization reaches \napproximately 1.1 emu/g. In contrast, the Mn 3Al RMQ alloy (Figure 6b) displays a significantly \ndifferent shape of the M(H) dependence. In this case, even in weak magnetic fields, there is an \nincrease in magnetization, followed by a tendency toward saturation. \nIn ref. [29], the difference in M(H) behavior for Mn 3Al is explained as follows. In the \ncase of a cast compound having the β-Mn structure, a frustrated antiferromagnetic state can arise, \nwhich manifests itself in the M(H) dependences (Figure 6a) and in the temperature dependences \nof the susceptibility (Figure 7). The results obtained in [29] are in good agreement with the \nconclusions of ref. [42] obtained on Mn 2FeAl with the same β-Mn structure.  10 \n \n  \n(a) ( b) \nFigure 6. Field dependence of magnetization for Mn 3Al at Т = 4.2 K: ( a) cast and ( b) RMQ samples \n[29]. \n \nFigure 7. The magnetic susceptibility of the Mn 3Al alloy measured at a magnetic field \nstrength of 100 Oe during the cooling process. The results are represented by two sets of data \npoints: (1) open circles that indicate measurements conducted without the application of a \nmagnetic field and (2) solid red circles that correspond to the presence of a magnetic field. \nThe inset shows the temperature dependence of the inverse magnetic susceptibility χ−1 [29]. \nThe Curie–Weiss law with the temperature θ CW ≈ –710 K is shown by dashed line. \nAccording to ref. [29], a decrease in grain size and an increase in “disorder” are observed \nin Mn3Al after RMQ treatment. As shown in [56], the magnetic state of the Mn 3Al alloy is very \nsensitive to the filling of interstices and degree of ordering. According to first-principles \ncalculations [18] in Mn 3Al with the D03 structure, a compensated ferrimagnetic state with a zero  11 moment can arise, which is observed in the RMQ compound Mn 3Al (Figure 6b) according to \n[29]. \nFigure 7 displays the temperature dependence of the susceptibility for the Mn 3Al alloy \n[29]. The inset graph illustrates the temperature dependence of the inverse susceptibility at a \nmagnetic field strength of 100 Oe. At high temperatures, the data approximately follow the \nCurie–Weiss law with a temperature θ CW of around −700 K. This behavior suggests the possible \npresence of an antiferromagnetic state. The Neel temperature TN can be estimated by observing \nthe point where the magnetic susceptibility curve breaks. In our case, TN is determined to be 35 \nK (as shown in Figure 7). The large ratio |θ CW|/TN indicates a frustrated antiferromagnetism in \nthe cast Mn 3Al compound [29]. \nThe cast and RMQ alloys investigated in [28,29] have the β-Mn structure. This consists \nof two distinct sublattices, one of which is made up of triangles arranged perpendicular to the \n[111] axes and forming a three-dimensional kagome-like lattice. Recent experimental studies \nhave shown that in systems with strong frustration caused by competing exchange interactions, \nnot only a quantum state of spin liquid can arise, but also antiferromagnetism with a significantly \nreduced, but still finite, Neel temperature can occur. These systems are characterized by a \nfrustration parameter, the ratio |θ CW|/TN; in the intermediate temperature range TN < T < |θCW|, the \nsystem can exhibit unusual spin-liquid properties. High values of the frustration parameter have \nbeen observed in the PdCrO 2 compound where TN = 37 K and θ CW ≈ −500 K [57]. This behavior \nis not explained by the standard Heisenberg model and is thought to be due to correlation effects \nin the subsystem of itinerant electrons [58]. A similar behavior of the magnetic susceptibility was \nrecently discovered in the Mn 2FeAl compound with the β-Mn structure in [42,44] ( TN = 42 K, \nθCW ≈ −230 K according to [44]). \nAccording to calculations [56], the Mn 3Al compound in the β-Mn structure exhibits a \nferrimagnetic state, where the magnetic moment of the sublattices is not significantly \ncompensated. However, ab initio calculations [20,59] in the D03 structure suggest that a \ncompensated ferrimagnetic state with a zero moment and a half-metallic structure arises. The \nresults from [60] are not fully supported by experimental data [28,29]. These data suggest that \nboth the cast and RMQ Mn 3Al alloys exhibit a frustrated antiferromagnetic state and an almost \ncompensated ferrimagnetic state, respectively. In the case of a cast Mn 3Al alloy with the β-Mn \nstructure, the absence of a long-range magnetic order can manifest itself as the Hall effect in the \nform of a zero anomalous contribution. On the contrary, in the case of the RMQ alloy with the \nβ-Mn structure in the state of a compensated ferrimagnet, an anomalous Hall effect should be \nobserved.  12 Figure 8 illustrates the temperature dependence of the electrical resistivity ρ(T) for cast \nand rapidly melt-quenched Mn 3Al alloys [29]. In the cast alloy (Figure 8a), it can be observed \nthat the residual resistivity ρ0 is relatively high, reaching a value of approximately 307 μΩ cm. \nThe resistivity follows a semiconductor-like trend, decreasing with increasing temperature. \nHowever, after quench hardening (Figure 8b), there is a significant reduction in the residual \nelectrical resistivity ρ0 by more than an order of magnitude, down to 12 μΩ cm. Additionally, a \nminimum point appears on the temperature dependence ρ(T) at around 60 K. \nTypically, rapid melt quenching results in the formation of fine-grained structures, which \ncan lead to increased resistivity due to the scattering of current carriers at grain boundaries. \nHowever, this phenomenon was not observed in the Mn 3Al alloy studied [28,29]. Instead, the \npresence of manganese sulfide (MnS) precipitates at the grain boundaries occurred, explaining \nthe high electrical resistivity values in the cast alloy. The rapid melt-quenching process caused \nthe dissolution of manganese sulfide within the volume of the grains, resulting in grain \nboundaries free of MnS. As a result, the resistivity of the rapidly melt-quenched alloy decreased \ncompared with the cast alloy. Similar behavior was also observed in the Hall effect \nmeasurements [28,29].  \n \n  \n(a) ( b) \nFigure 8. Temperature dependence of the resistivity for cast ( a) and RMQ ( b) Mn3Al [29]. \nFigure 9 shows the dependence of the Hall resistivity  ρxy = f(H) at Т = 4.2 K for the cast \nand RMQ Mn 3Al alloy. In the first case, a linear increase in ρxy is observed, and in the second \ncase, the behavior of ρxy(H) is typical for alloys with an anomalous Hall effect [28,29]. These \ndata also indicate the absence of spontaneous magnetization in the case of cast Mn 3Al and the \nrealization of a compensated ferrimagnet state in the RMQ Mn 3Al alloy [28,29]. \nTraditionally, it was believed that the anomalous Hall effect (AHE) was exclusive to \nferromagnetic materials [61]. However, it has been discovered that the AHE can manifest in  13 various magnetic materials as a result of broken time-reversal symmetry. In reference [62], ab \ninitio calculations were conducted, taking into consideration the symmetry properties of the \nnontrivial AHE in the Mn 3Al compensated ferrimagnet. The nonzero elements of the anomalous \nHall conductivity were determined based on the magnetic space group of Mn 3Al. The \ncalculations revealed that the value of the anomalous Hall conductivity was σxy = −320 (Ω cm)−1. \nThe study also delved into the nature of the Berry curvature, which is responsible for the internal \norigin of the AHE, using group theory. Furthermore, the overall behavior of the Berry curvature \nacross the entire Brillouin zone was illustrated. \n \n  \n(a) ( b) \nFigure 9. Field dependence of Hall resistivity for Mn 3Al at T = 4.2 K [29]: ( a) cast and ( b) \nRMQ samples. \nThe observation of a nonzero topological anomalous Hall effect in antiferromagnets \nMn3Sn and Mn 3Ge [9] can be attributed to their noncollinear magnetic structure and Berry \ncurvature. A study conducted in [63] focused on the magnetization and anomalous Hall effect of \na single-crystal hexagonal chiral antiferromagnet Mn 3Ge. Remarkably, it was discovered that this \nmaterial exhibited a significant anomalous Hall conductivity of approximately 60 (Ω cm)−1 at \nroom temperature and ~380 (Ω cm)−1 at 5 K in zero field, approaching half the anticipated value \nfor a quantum Hall effect on an atomic layer with a Chern number of one. Furthermore, the sign \nof the anomalous Hall effect was found to change with a magnetic field vector direction change \nof less than 0.1 T or with a rotation of a small magnetic field of less than 0.1 T. This intriguing \nbehavior could have implications in the development of switching and storage devices based on \nantiferromagnets. \nIt is worth noting that recent studies have reported new antiferromagnetic materials with \nrapid response times, low power consumption, and high immunity to external magnetic fields. \nThese materials show promise in the field of spintronics [64–69].  14 In materials exhibiting noncollinear antiferromagnetism, the magnetic moments of atoms \nare not strictly aligned along a single axis. One such material is Mn 3Ge, which falls into the \ncategory of noncollinear antiferromagnetic materials. In Mn 3Ge, the manganese atoms form a \nhexagonal sublattice, and their magnetic moments arrange themselves in a kagome lattice \nstructure (Figure 10). This unique arrangement gives rise to a noncollinear antiferromagnetic \nstate and several other unusual effects, including a large anomalous Hall effect, a new spin Hall \neffect, and a significant spin Nernst effect [70–72].  \n \nFigure 10. D019 structure for Mn 3Ge. Red arrows show the Mn magnetic moments [23]. \nA study conducted in [73] focused on the magnetic structure of antiferromagnetic Mn 3Ge \ncrystals with a kagome lattice, using polarized neutron diffraction. The study has revealed that in \nMn3Ge, the magnetic order is characterized by a coplanar state represented by k = 0, which \nbelongs to the two-dimensional irreducible representation of the double group Γ 9, which is the \nonly irreducible representation consistent with the observed diffraction pattern of Mn 3Ge [73]. \nThis coplanar state exhibits a perfect 120° antichiral structure with an angle of 120° and a \nmagnetic moment of 2.2 µ B/Mn. Additionally, a weak collinear ferromagnetism is observed. A \nphenomenological spin Hamiltonian is proposed to describe the manganese-based magnetism, \nincorporating exchange interactions, Dzyaloshinskii–Moriya interactions, and single-ion crystal \nfield terms. These interactions contribute to spin wave damping and an extended range of \nmagnetic interactions indicate an itinerant magnetism compatible with anomalies of transport \nproperties. \nIn [74], a study was conducted on the electrical, magnetic, galvanomagnetic, and thermal \nproperties of a single crystal of the noncollinear antiferromagnet Mn 3Ge. The researchers \ndiscovered that at very low temperatures, the Wiedemann–Franz law, which establishes a \nconnection between the electronic and thermal Hall effects, holds true. However, deviations from \nthis law were observed at temperatures above 100 K. The investigation revealed that the carriers \nin Mn3Ge have a short mean free path, similar to the distance between Mn antisite defects. As a \nresult, the elastic scattering of current carriers becomes the dominant factor. The researchers  15 proposed that the deviation from the Wiedemann–Franz law is attributed to a disparity between \nthe thermal and electrical contributions of the Berry curvature along the Fermi surface. \nTheoretical calculations supported this interpretation, demonstrating that the Berry spectra in the \ntwo systems are not identical. Importantly, the study also confirmed that the Bridgman relation, \nwhich links the anomalous Nernst and Ettingshausen coefficients, is valid across the entire \ntemperature range investigated. \nExternal pressure and doping can alter the Berry curvature, as well as the corresponding \nphysical properties. In a study [75], the anomalous Hall effect in a single crystal of \nantiferromagnetic Mn 3Ge was investigated under hydrostatic pressure up to 2.85 GPa. It was \nobserved that the Hall signal becomes zero at a pressure of 1.53 GPa, and changes sign at higher \npressures. Although the sign of the Hall conductivity changes with increasing pressure, its \nsaturation value at room temperature remains relatively high, around 23 (Ω cm)−1 at 2.85 GPa, \nwhich is comparable to the saturation value at atmospheric pressure 40 (Ω cm)−1. The authors \nsuggest that this change in the Hall conductivity may be attributed to gradual modifications in \nthe in-plane components of the Mn moments within a noncollinear triangular magnetic lattice. \nThese findings provide insights into the possibility of manipulating and controlling the \nanomalous Hall effect in chiral antiferromagnetic Mn 3Ge through pressure-induced changes. \nAuthors of [76] investigated the magnetic and electronic structure of the \n(Mn0.78Fe0.22)3Ge hexagonal single crystal, which is known as a Weyl semimetal, using electron \ntransport, magnetic properties, and neutron diffraction experiments. The researchers conducted \ntemperature measurements of the magnetization and observed two magnetic transitions at TN1 = \n242 K and TN2 = 120 K. In this case, the anomalous Hall effect is observed in the intermediate \nrange at temperatures between TN1 and TN2, disappearing at T < TN2. Neutron diffraction \nexperiments were carried out to determine the magnetic structures of the (Mn 0.78Fe0.22)3Ge single \ncrystal. Neutron diffraction studies made it possible to conclude that the sample has a collinear \nantiferromagnetic structure below the temperature TN2. In this case, the magnetic structure is \nnoncollinear antiferromagnetic in the intermediate temperature range. The authors conclude that \nthe observation of an anomalous Hall effect and a noncollinear magnetic structure in \n(Mn0.78Fe0.22)3Ge in this temperature range indicates the existence of Weyl points. At \ntemperatures below TN2, there is no anomalous Hall effect, and the magnetic structure changes to \na collinear antiferromagnetic one. This indicates a strong coupling between the magnetic and \nelectronic structures of the (Mn 0.78Fe0.22)3Ge compound. \nThe influence of pressure, reaching up to 2.2 GPa, on the electrical resistivity ρ and \nthermoelectric power S of the Mn 3Si single-crystal compound was investigated in a study \nreferenced as [77]. The Neel temperature TN was determined by analyzing the temperature  16 dependencies of ρ and S, revealing an increase in TN with rising pressure. Notably, the resistivity \nand thermoelectric power displayed significant changes at around P ≈ 1 GPa at a temperature of \n2 K, indicating the occurrence of a phase transition. \nAnother research, referenced as [78], presents findings on the structural and magnetic \nproperties of noncollinear antiferromagnetic Mn 3Sn films that adopt the D019 hexagonal \nstructure. These films were observed to exhibit weak ferromagnetism, characterized by an \nuncompensated in-plane magnetization of 34 kA/m and a coercive force μ0Hc of 4.0 mT at room \ntemperature. Additionally, the study investigated the phenomenon of exchange bias in Mn 3Sn/Py \nbilayers, revealing the potential to achieve exchange bias fields of up to μ0HEB = 12.6 mT at 5 K. \nThese results highlight the attractiveness of Mn 3Sn films for applications in antiferromagnetic \nspintronics. \nA study in [79] reports on the unusual behavior of the anomalous Hall effect in the \ncompound Mn 3Sn. The authors discovered a linear dependence on the magnetic field, which they \nattributed to the Berry curvature of the wave function. Interestingly, the magnitude of the Hall \nsignal in this case exceeds what would be expected in the semiclassical model. The authors \npropose that the magnetic field induces a nonplanar spin canting, resulting in a nontrivial \nchirality of the spins on the kagome lattice. This leads to changes in the band structure, \nspecifically gapping out previously unknown Weyl nodal lines, which explains the observed \nbehavior of the anomalous Hall effect. The findings suggest a connection between the Berry \nphase in real space, arising from spin chirality, and the curvature of the Berry momentum space \nin the kagome lattice. \nThe paper [80] investigates the electronic transport and magnetic properties of Mn 3Sn \nfilms. The findings reveal that these films exhibit a weak uncompensated magnetic moment of \napproximately 0.12 μ B/f.u. and an electrical resistance of about 3.8 μΩ m at room temperature. \nThese results closely match the data obtained for bulk Mn 3Sn, indicating high purity and \nperfection in the synthesized films, with a resistance ratio RRR of 3.92. The Mn atoms in the \nMn3Sn compound are arranged in a kagome-type lattice. The study demonstrates that at room \ntemperature when the external magnetic field is perpendicular to the kagome planes, a weak \nanomalous Hall effect is observed along with a Hall resistance that varies linearly with the field. \nThe researchers identify three distinct magnetic phases in the investigated films of the chiral \nantiferromagnet Mn 3Sn: an inverse triangular spin state occurring above 250 K, a helical phase \nstabilized between 100 and 250 K, and a spin glass phase formed below 100 K. Based on their \nfindings, the authors suggest that these films may support topologically protected skyrmions, \nwhere the fictitious effective magnetic field is estimated to be around 4.4 T. This indicates the \npotential for unique magnetic phenomena in the Mn 3Sn films.  17 Currently, there is active research on the physics of Berry curvature and related nontrivial \nmagnetic transport in two-dimensional spin-lattice kagome structures of noncollinear \nantiferromagnets. The focus has mainly been on hexagonal chiral antiferromagnets like Mn 3Sn, \nwhile studies on face-centered cubic (fcc) noncollinear antiferromagnetism are limited. One such \nexample of fcc noncollinear antiferromagnets is Mn 3Pt. \nIn reference [81], the researchers examined single crystals of Mn 3Pt. They discovered that \napplying uniaxial stress to Mn 3Pt crystals decreased their coercive force. This allowed them to \nobserve the anomalous Hall effect in bulk materials of the Mn 3Z family with a cubic structure. \nInterestingly, the anomalous Hall effect remained even after the stress was removed, suggesting \nthat stress-induced ferromagnetic moments had minimal impact on the effect. The study found \nthat longitudinal stress reduced the coercive force more than transverse stress, indicating that the \ndominant response to applied stress was a rotation of spins within the plane. However, further \ninvestigation is needed to verify this assumption, particularly through direct observation of \npiezomagnetism. \nIn reference [82], the authors systematically studied the effect of deformation on the \nmagnetic properties and the anomalous Hall effect of fcc noncollinear antiferromagnetic Mn 3Pt \nfilms. They found that the ferromagnetic characteristics of Mn 3Pt films showed a similar \nresponse to strain, in contrast to hexagonal chiral antiferromagnets. The ferromagnetic signal in \nthese films was attributed to the magnetic moment slope of Mn atoms in kagome structures, \nwhile the anomalous Hall resistivity was linked to the nonzero Berry curvature. By adjusting the \nthickness of the Mn 3Pt film, they achieved an anomalous Hall conductivity exceeding 100 (Ω \ncm)−1. Additionally, the relationship between the Berry phase, magnetic properties, and AHE in \nMn3Pt was confirmed through a study on crystal growth orientation. The researchers concluded \nthat this compound holds promise as a candidate for room-temperature antiferromagnetic \nspintronics. \n4. Physical Mechanisms of the Magnetic States Formation in Mn-Based \nCompounds: Half-Metallic Ferromagnets and Spin Gapless Semiconductors \nThe existence of distinct spin-up and spin-down states in half-metallic ferromagnets \npresents a complex challenge to the general theory of itinerant magnetism [34]. The process of \nachieving a half-metallic state in Heusler alloys like XMnZ and X2MnZ, which have C1b and L21 \nstructures, can be explained as follows [33,34,83,84]. When we disregard the hybridization of \natomic states X and Z, the d band of manganese displays a significant energy gap between \nbonding and antibonding states. In a ferromagnetic state, the strong intra-atomic exchange \n(Hund’s exchange) among manganese ions results in a notable separation of spin subbands for  18 up and down spins. One of these spin subbands closely interacts with the ligand’s p band, \nleading to a partial or complete blurring of the corresponding energy gap due to p-d \nhybridization. Meanwhile, in the other spin subband, the energy gap remains intact and has the \npotential to align with the Fermi level, thus giving rise to the HMF state.  \nThe C1b structure exhibits a real energy gap, whereas the L21 structure has a pronounced \npseudogap. This difference is attributed to significant alterations in p-d hybridization, \nparticularly between the p and t2g states, in the absence of an inversion center, which is a \ncharacteristic feature of the C1b structure. As a result, the C1b structure is more favorable for the \nformation of the HMF state. The stability of the ferromagnetic state is a result of variations in p-d \nhybridization between states with opposite spin orientations, as elaborated in [84].  \nIn a study [85], researchers conducted calculations for 54 ternary Heusler compounds \nwith the composition X2YZ. In this context, X represents a 3 d transition metal (specifically Mn, \nFe, Co), while Y and Z represent elements Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Al and Si, \nrespectively. The findings from this study revealed that seven of these compounds–namely, \nMn2NbAl, Mn 2ZrSi, Mn 2RhSi, Co 2ZrAl, Co 2NbAl, Co 2YSi, and Co 2ZrSi–displayed 100% spin \npolarization, which classifies them as half-metallic ferromagnets (HMFs). Furthermore, five \nother alloys, specifically Mn 2TcAl, Mn 2RuAl, Mn 2NbSi, Mn 2RuSi, and Fe 2NbSi, exhibited high \nspin polarization levels (above 90%) with a gap present for one of the spin directions near the \nFermi level. These alloys were categorized as “almost HMF” in the study [85]. Importantly, their \nFermi levels were found to shift under pressure, resulting in the alignment of the Fermi level \nwith the gap and the emergence of the HMF state. \nAn intriguing development in the field of half-metallic magnetism is represented by \nelectron-deficient full Heusler alloys. Reducing the number of valence electrons to 24 per \nformula unit leads to either a nonmagnetic semiconductor or a half-metallic antiferromagnet. \nRemarkably, the reduction in the number of valence electrons can continue, entering a range of \nhalf-metals but with a bandgap for the majority spin direction. This is best exemplified by the \ncase of Mn 2VAl, which is a half-metallic ferrimagnet, as calculated using the generalized \ngradient exchange-correlation potential [86]. \nIn addition to the mentioned study, other works (referenced as [35,86,87]) explored the \nexchange interactions in Ni 2MnZ (Z = Ga, In, Sn, Sb) and Mn 2VZ (Z = Al, Ge) alloys. Ni 2MnZ \nwas found to be non-half-metallic, while Mn 2VZ was identified as half-metallic. These studies \nemphasized the significance of intersublattice exchange interactions. In the case of Mn 2VZ (Z = \nAl, Ge), it was observed that the ferrimagnetic coupling between V and Mn moments stabilized \nthe ferromagnetic alignment of Mn moments. The V-based Heusler alloys Mn 2VZ (Z = Al, Ga, \nIn, Si, Ge, Sn) were predicted to exhibit half-metallic ferrimagnetism in these studies [35,86,88].  19 Half-metallicity in the Mn 2VAl ferrimagnet was detected using resonant inelastic soft \nX-ray scattering (SX-RIXS) in the presence of a magnetic field [89]. When V L-edge excitation \nwas applied, the findings revealed that the partial density of states for the V 3 d states around the \nFermi energy was minimal, and these states exhibited a relatively localized character. \nConversely, when Mn L-edge excitation was used, the spectra were dominated by fluorescence \nand displayed clear magnetic circular dichroism, which showed significant dependence on the \nexcitation photon energy. These experimental results were compared with theoretical predictions \nbased on density-functional-theory band structure calculations, confirming the itinerant, \nspin-dependent nature of the Mn 3 d states and the decay of the Mn 2 p core states. This \nconsistency in the experimental data aligns with the half-metallic behavior of the Mn 3 d states. \nThe electronic structure and magnetic state of SGS materials are quite similar to HMF \ncompounds in many ways. “Transition” from the HMF to the SGS state can occur during doping. \nUsing the first-principles calculations within density functional theory, the authors of ref. [90] \ndescribed such a transition by studying the magnetism and electronic structure of Co 2−xMn1+xAl \n(0 ≤ x ≤ 1) alloys, i.e., during the transition from HMF Co 2MnAl to SGS Mn 2CoAl. \nAs reported in [90], the electronic spectrum of the Co 2MnAl compound ( x = 0) exhibits \nan intersection of its spin-up and spin-down bands with the Fermi level EF, indicating metallic \nproperties for both spin directions. However, Co 2MnAl displays a low density of states for \nspin-down electrons, resulting in a pseudogap rather than a true gap. The calculated spectra show \na spin polarization of approximately 75%. As the manganese content increases, the spin-down \nvalence band of Co 2MnAl shifts downward, causing the density of states to approach zero. \nConsequently, the degree of spin polarization increases. In the case of the Co 1.5Mn1.5Al \ncompound, as Mn content further increases, the top of the spin-down band aligns with the Fermi \nlevel, leading to the emergence of a genuine gap and achieving a maximum spin polarization of \n100%. Despite that, Co 1.5Mn1.5Al faces a vulnerability as its Fermi level is situated at the edge of \nthe spin-down bandgap, making it susceptible to external influences, especially structural \ndefects. \nAs Mn content continues to rise in Co 2−xMn1+xAl, the spin-down valence band moves \nfurther away from the Fermi level, resulting in an enlarged band gap. At x = 1.875, the width of \nthe spin-down gap reaches approximately 0.4 eV, with the Fermi level nearly positioned at the \nmidpoint of the gap. This leads the authors of reference [90] to classify the Co 1.125Mn1.875Al \ncompound as an “ideal” half-metallic ferromagnet. With increasing Mn content to x = 2, the \ncompound transforms into an inverse Heusler alloy Mn 2CoAl. In this compound, a genuine \nenergy gap is preserved for spin-down states. The electronic structure of Mn 2CoAl exhibits a \nunique characteristic: for spin-up electronic states, the conduction band and valence band edges  20 closely approach the Fermi level, resulting in a minimal energy gap. This distinctive feature \ncategorizes the inverse Mn 2CoAl compound as a spin gapless semiconductor. \nTable 1 presents information on Mn 2YZ and Mn 3Z (Y = Cr, Fe, Co; Z = Al, Ge, Si, Sn, Pt), \nfor which the electronic state, magnetic state, and anomalous Hall conductivity are given.  \nTable 1. Manganese-based compounds of Mn 2YZ and Mn 3Z (Y = V, Cr, Fe, Co, Ni; Z = \nAl, Ge, Si, Sn, Pt, Ir). Electronic and magnetic states and anomalous Hall conductivity. \nCompound Electronic State Magnetic State Anomalous Hall \nConductivity σAHE (Ohm \ncm)−1 \nMn2VAl HMF 1 [41] FIM 2 [41]  \nMn2VSi HMF [41] FIM [41]  \nMn2CrAl HMF [91,92] CFIM 3 [21]  \nMn2FeAl HMF [36] CFIM [42] \nFrustrated AFM 4 [44]  \nMn2NiAl  AFM [93]  \nMn2CoAl SGS 5 [47] FM 6 [47] 21.8 [47] \nMn3Al HM-AFM 7 [94] CFIM (cast); frustrated \nAFM (RMQ) 8 [28,29] \nAFM [94] 320 [62] \nMn3Ge HM-FIM 9 [94]  \nTSM 10 [9] FIM [94]  \nNoncollinear AFM [9,63]  60 at RT; ~380 at 5 K [63] \nMn3Si HM-FIM [94] FIM [94]  \nMn3Sn TSM [79] Noncollinear AFM [78,81]  \nkagome AFM [79] ~20 at RT; ~100 at 100 K \n[95]  \nMn3Pt TSM [82]  kagome AFM [82] 100 [82] \nMn3Ir  AFM [96] 40 at RT [96] \n1 HMF is a half metallic ferromagnet. 2 FIM is a ferrimagnet. 3 CFIM is a compensated \nferrimagnet. 4 AFM is an antiferromagnet. 5 SGS is a spin gapless semiconductor. 6 FM is a \nferromagnet. 7 HM-AFM is a half-metallic antiferromagnet. 8 RMQ is rapid melt quenched. 9 \nHM-FIM is a half-metallic ferrimagnet. 10 TSM is a topological semimetal. \n5. Conclusions \nIntermetallic compounds based on manganese, specifically Mn 2YZ and Mn 3Z (where (Y = \nSc, Ti, V, Cr, Fe, Co, etc.; Z = Al, Ge, Sn, Si, Pt, etc.), exhibit unique characteristics of the \nelectronic structure. These compounds have the potential to exhibit various electronic and \nmagnetic states, such as antiferromagnet, compensated ferrimagnet, topological semimetal, and \nfrustrated antiferromagnet. Furthermore, their magnetic and electronic properties are highly \nsensitive to external influences. \nIn the electronic subsystem of Mn 2YAl compounds, different magnetic states can lead to \nthe emergence of half-metallic ferromagnet and spin gapless semiconductor states. For instance,  21 Mn2CoAl can exhibit a ferromagnetic spin gapless semiconductor state, while Mn 2FeAl can \ndisplay a frustrated antiferromagnet state. \nCompounds of Mn 3Z manifest as a half-metallic ferromagnet or a topological semimetal, \ndisplaying notable phenomena such as a large anomalous Hall effect, spin Hall effect, spin \nNernst effect, and thermal Hall effect. The magnetic subsystem of these compounds can exhibit \nstates such as ferrimagnetism, antiferromagnetism, compensated ferrimagnetism, frustrated \nantiferromagnetism, and antiferromagnetism with a kagome-type lattice. Overall, Mn 3Z \ncompounds offer a rich spectrum of electronic and magnetic properties. \nWhen comparing the properties of Mn compounds Mn 2YAl and Mn 3Z, it becomes evident \nthat there are both similarities and differences between them, as shown in Table 1. In the \nelectronic structure of Mn 2YAl alloys, the predominant state is that of a half-metallic \nferromagnet, whereas in the Mn 2CoAl compound, it exhibits characteristics of a spin gapless \nsemiconductor. It is worth noting that Mn 2CoAl is among the first Heusler alloys where the SGS \nstate is prominently observed. In terms of magnetism, Mn 2YAl alloys display a range of \nbehaviors, including ferromagnetism, compensated ferrimagnetism, and frustrated AFM state. \nOn the other hand, the Mn 3Z compounds exhibit more diverse electronic and magnetic \nstates. Many of them demonstrate the characteristics of a topological semimetal with significant \nanomalous Hall conductivity, even at room temperature. Similar to Mn 2YAl alloys, they also \nexhibit a variety of magnetic states. Moreover, due to their richer electronic structure, they can \nrealize states such as noncollinear AFM states and AFM states typical for a kagome-type lattice. \nIt should be emphasized that in actual materials, the presence and behavior of electronic \nand magnetic states, as well as transitions between them, can vary depending on various factors. \nThese factors include the composition of the sample, external parameters such as temperature, \nmagnetic field, and external pressure, as well as the dimensionality of the material (bulk, film, \nnanostructure), and the processing method used (cast, rapidly quenched, nanostructured samples, \netc.). The electronic and magnetic states discussed, as well as the transitions between them, can \nbe implemented and put into practice. \nBesides the practical applications, all the states discussed, especially frustrated \nantiferromagnetism and half-metallic ferromagnetism, are of general scientific interest. Some \nsystems considered can demonstrate anomalous electronic properties (particularly, large T-linear \nheat capacity) and are highly intriguing from a physical standpoint, posing an exciting challenge \nfor future theoretical advancements. \nIt should be noted that a significant number of studies on manganese-based Heusler \ncompounds exploit band calculations, which may not always be reliable. This is especially true \nwhen treating the size of the energy gap and the degree of spin polarization in the case of  22 possible HMF and SGS compounds, thereby overestimating the stability of these states. \nHowever, there are ongoing improvements in electronic structure calculation methods, and new \nmodern techniques are emerging. It is essential to constantly compare these calculations with \nexperimental data. For example, in the case of the HMF, this could involve comparing the results \nof band calculations and direct experiments to determine the gap and spin polarization in situ by \nultraviolet-photoemission spectroscopy, taking advantage of a multichannel spin filter [97]. \nNMR experiments could also be conducted to search for the vanishing of a linear contribution to \nthe temperature dependence of the nuclear spin-lattice relaxation rate 1/ T1 (violation of \nKorringa’s law), which is theoretically predicted for the HMF [34]. \nFinally, a promising direction for exploring new electronic effects and magnetic state \nfeatures is a comprehensive study of quaternary Heusler alloys containing manganese atoms. \nAlthough the present review did not cover such compounds, there are many studies in this area, \nwhich can be found in related reviews [31,98,99] and references therein. \nThis opens up further opportunities for fine control of the magnetic and electronic \ncharacteristics of such compounds for their possible practical use in spintronics and micro- and \nnanoelectronics. \nAuthor Contributions: Conceptualization, V.V.M. and V.Y.I.; methodology, V.V.M. and \nV.Y.I.; formal analysis, V.V.M. and V.Y.I.; investigation, V.V.M.; resources, V.V.M. and V.Y.I.; \ndata curation, V.V.M. and V.Y.I.; writing—original draft preparation, V.V.M. and V.Y.I.; \nwriting—review and editing, V.V.M. and V.Y.I.; visualization, V.V.M. and V.Y.I.; supervision, \nV.V.M. and V.Y.I.; funding acquisition, V.V.M. All authors have read and agreed to the published \nversion of the manuscript. \nFunding:  The research was carried out within the state assignment of Ministry of Science \nand Higher Education of the Russian Federation (themes «Spin», № 122021000036-3 and \n«Quantum», № 122021000038-7). Section 3 “Features of the electronic transport and magnetic \nstate of Mn 3Z (Z = Al, Ge, Si, Sn, Pt etc.) compounds” was prepared with the financial support of \nthe Russian Science Foundation within the framework of research project No. 22-22-00935. \nAcknowledgments:  The authors consider it their pleasant duty to thank their colleagues \nand co-authors Lukoyanov, A.V., Skryabin, Y.N., Marchenkova, E.B. for valuable discussions, \nand Perevozchikova, Y.A., Perevalova, A.N., Fominykh, B.M., Semiannikova, A.A., Emelyanova, \nS.M. for help with the design of the review. \nConflicts of Interest: The authors declare no conflict of interest. \n  23 References \n1. Heusler, F. 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Miyazaki1\n1)WPI Advanced Institute for Materials Research, Tohoku Univ ersity,\nSendai 980-8577, Japan\n2)Department of Applied Physics, Tohoku University, Sendai 9 80-8579,\nJapan\n(Dated: 23 July 2018)\nLaser-inducedmagnetizationprecessional dynamicswasinvestiga tedinepitaxialfilms\nof Mn 3Ge, which is a tetragonal Heusler-like nearly compensated ferrimag net. The\nferromagnetic resonance (FMR) mode was observed, the preces sion frequency for\nwhich exceeded 0.5 THz and originated from the large magnetic anisot ropy field of\napproximately 200 kOe for this ferrimagnet. The effective damping c onstant was\napproximately 0.03. The corresponding effective Landau-Lifshitz c onstant of approx-\nimately 60 Mrad/s and is comparable to those of the similar Mn-Ga mate rials. The\nphysical mechanisms for the Gilbert damping and for the laser-induc ed excitation of\nthe FMR mode were also discussed in terms of the spin-orbit-induced damping and\nthe laser-induced ultrafast modulation of the magnetic anisotropy , respectively.\na)Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp\n1Among the various types of magnetization dynamics, coherent mag netization precession,\ni.e.,ferromagneticresonance(FMR),isthemostfundamentaltype, andplaysamajorrolein\nrf spintronics applications based on spin pumping1–5and the spin-transfer-torque (STT).6,7\nSpin pumping is a phenomenon through which magnetization precessio n generates dc and rf\nspin currents in conductors that are in contact with magnetic films. The spin current can be\nconverted into anelectric voltage throughthe inverse spin-Hall eff ect.8The magnitude of the\nspin current generatedvia spinpumping is proportionaltothe FMRf requency fFMR;4,5thus,\nthe output electric voltage is enhanced with increased fFMR. In the case of STT oscillators\nand diodes, the fFMRvalue for the free layer of a given magnetoresistive devices primarily\ndetermines the frequency range for those devices.9,10An STT oscillator and diode detector\nat a frequency of approximately 40 GHz have already been demonst rated;11–13therefore, one\nof the issues for consideration as regards practical applications is the possibility of increasing\nfFMRto hundreds of GHz or to the THz wave range (0.1-3 THz).11,14\nOnesimple methodthroughwhich fFMRcanbeincreased utilizes magneticmaterials with\nlarge perpendicular magnetic anisotropy fields Heff\nkand small Gilbert damping constants\nα.13,15,16This is because fFMRis proportional to Heff\nkand, also, because the FMR quality\nfactor and critical current of an STT-oscillator are inversely and d irectly proportional to α,\nrespectively. The Heff\nkvalue is determined by the relation Heff\nk= 2Ku/Ms−4πMsfor thin\nfilms, where KuandMsare the perpendicular magnetic anisotropy constant and saturat ion\nmagnetization, respectively. Thus, materials with a small Ms, largeKu, and low αare\nvery favorable; these characteristics are similar to those of mate rials used in the free layers\nof magnetic tunnel junctions integrated in gigabit STT memory applic ation.17We have\npreviously reported that the Mn-Ga metallic compound satisfies the above requirements,\nand that magnetization precession at fFMRof up to 0.28 THz was observed in this case.18\nA couple of research groups have studied magnetization precessio n dynamics in the THz\nwave range for the FePt films with a large Heff\nk, and reported an αvalue that is a factor of\nabout 10 larger than that of Mn-Ga.19–21Thus, it is important to examine whether there are\nmaterialsexhibiting properties similartothoseofMn-Gaexist, inorde r tobetter understand\nthe physics behind this behavior.\nIn this letter, we report on observed magnetization precession at fFMRof more than 0.5\nTHz for an epitaxial film of a Mn 3Ge metallic compound. Also, we discuss the relatively\nsmall observed Gilbert damping. Such THz-wave-range dynamics ca n be investigated by\n2means of a THz wave22or pulse laser. Here, we use the all-optical technique proposed\npreviously;23therefore, the mechanism of laser-induced magnetization preces sion is also dis-\ncussed, because this is not very clearly understood.\nMn3Ge has a tetragonal D0 22structure, and the lattice constants are a= 3.816 and\nc= 7.261˚A in bulk materials [Fig. 1(a)].24,25The Mn atoms occupy at two non-equivalent\nsites in the unit-cell. The magnetic moment of Mn I(∼3.0µB) is anti-parallel to that of\nMnII(∼1.9µB), because of anti-ferromagnetic exchange coupling, and the net magnetic\nmoment is ∼0.8µB/f.u. In other words, this material is a nearly compensated ferrima gnet\nwith a Curie temperature Tcover 800 K.26The tetragonal structure induces a uniaxial\nmagnetic anisotropy, where the c-axis is the easy axis.24The D0 22structure is identical to\nthat of tetragonally-distorted D0 3, which is a class similar to the L2 1Heusler structure;\nthus, D0 22Mn3Ge is also known as a tetragonal Heusler-like compound, as is Mn 3Ga.27\nThe growth of epitaxial films of D0 22Mn3Ge has been reported quite recently, with these\nfilms exhibiting a large Kuand small Ms, similar to Mn-Ga.28–30Note that Mn 3Ge films\nwith a single D0 22phase can be grown for near stoichiometric compositions.29,30Further, an\nextremely large tunnel magnetoresistance is expected in the magn etic tunnel junction with\nMn3Ge electrodes, owing to the fully spin-polarized energy band with ∆ 1symmetry and the\nBloch wave vector parallel to the c-axis at the Fermi level.29,31These properties constitute\nthe qualitative differences between the Mn 3Ge and Mn 3Ga compounds from the material\nperspective.\nAll-optical measurement for the time-resolved magneto-optical K err effect was employed\nusing a standard optical pump-probe setup with a Ti: sapphire laser and a regenerative\namplifier. The wavelength and duration of the laser pulse were appro ximately 800 nm and\n150 fs, respectively, while the pulse repetition rate was 1 kHz. The p ulse laser beam was\ndivided into an intense pump beam and a weaker probe beam; both bea ms weres-polarized.\nThe pump beam was almost perpendicularly incident to the film surface , whereas the angle\nof incidence of the probe beam was ∼6◦with respect to the film normal [Fig. 1(b)].\nBoth laser beams were focused on the film surface and the beam spo ts were overlapped\nspatially. The probe and pump beams had spot sizes with 0.6 and 1.3 mm, respectively.\nThe Kerr rotation angle of the probe beam reflected at the film surf ace was analyzed using\na Wollaston prism and balanced photodiodes. The pump beam intensity was modulated\nby a mechanical chopper at a frequency of 360 Hz. Then, the volta ge output from the\n3FIG. 1. (a) Illustration of D0 22crystal structure unit cell for Mn 3Ge. (b) Diagram showing\ncoordinate system used for optical measurement and ferroma gnetic resonance mode of magnetiza-\ntion precession. The net magnetization (= MII−MI) precesses about the equilibrium angle of\nmagnetization θ, whereMI(MII) is the magnetization vector for the Mn I(MnII) sub-lattice. (b)\nOut-of-plane normalized hysteresis loop of the Kerr rotati on angle φkmeasured for the sample.\nphotodiodes was detected using a lock-in amplifier, as a function of d elay time of the pump-\nprobe laser pulses. The pump pulse fluence was ∼0.6 mJ/cm2. Note that the weakest\npossible fluence was used in order to reduce the temperature incre ase while maintaining the\nsignal-to-noise ratio. A magnetic field Hof 1.95 T with variable direction θHwas applied\nusing an electromagnet [Fig. 1 (b)].\nThec-axis-oriented Mn 3Ge epitaxial films were grown on a single-crystalline (001) MgO\nsubstrate with a Cr seed layer, and were capped with thin MgO/Al lay ers at room tempera-\nture using a sputtering method with a base pressure below 1 ×10−7Pa. The characteristics\nof a 130-nm-thick film with slightly off-stoichiometric composition (74 a t% Mn) deposited\nat 500◦C are reported here, because this sample showed the smallest coer civity (less than\n1 T) and the largest saturation magnetization (117 emu/cm3) of a number of films grown\nwith various thicknesses, compositions, and temperatures. Thes e properties are important\nto obtaining the data of time-resolved Kerr rotation angle φkwith a higher signal-to-noise\nratio, because, as noted above, Mn 3Ge films have a large perpendicular magnetic anisotropy\n4field and a small Kerr rotation angle.30Figure 1(c) displays an out-of-plane hysteresis loop\nofφkobtained for a sample without pump-beam irradiation. The loop is norm alized by the\nsaturation value φk,sat 1.95 T. The light skin depth is considered to be about 30 nm for the\nemployed laser wavelength, so that the φkvalue measured using the setup described above\nwas almost proportional to the out-of-plane component of the ma gnetization Mzwithin the\nlight skin depth depth. The loop shape is consistent with that measur ed using a vibrating\nsample magnetometer, indicating that the film is magnetically homogen eous along the film\nthickness and that value of φk/φk,sapproximates to the Mz/Msvalue.\nFigure 2(a) shows the pump-pulse-induced change in the normalized Kerr rotation angle\n∆φk/φk,s(∆φk=φk−φk,s) as a function of the pump-probe delay time ∆ twith an applied\nmagnetic field Hperpendicular to the film plane. ∆ φk/φk,sdecreases quickly immediately\nafter the pump-laser pulse irradiation, but it rapidly recovers within ∼2.0 ps. This change\nis attributed to the ultrafast reduction and ps restoration of Mswithin the light skin depth\nregion, and is involved in the process of thermal equilibration among t he internal degrees of\nfreedom, i.e., the electron, spin, and lattice systems.32. After the electron system absorbs\nlight energy, the spin temperature increases in the sub-ps timesca le because of the heat\nflow from the electron system, which corresponds to a reduction in Ms. Subsequently, the\nelectron and spin systems are cooled by the dissipation of heat into t he lattices, which have\na high heat capacity. Then, all of the systems reach thermal equilib rium. This process is\nreflected in the ps restoration of Ms. Even after thermal equilibrium among these systems is\nreached, the heat energy remains within the light skin depth region a nd the temperature is\nslightly higher than the initial value. However, this region gradually co ols via the diffusion\nof this heat deeper into the film and substrate over a longer timesca le. Thus, the remaining\nheat causing the increased temperature corresponds to the sma ll reduction of ∆ φk/φk,safter\n∼2.0 ps.\nWith increasing θHfrom out-of-plane to in-plane, a damped oscillation becomes visible\nin the ∆φk/φk,sdata in the 2-12 ps range [Fig. 2(b)]. Additionally, a fast Fourier tran sform\nof this data clearly indicates a single spectrum at a frequency of 0.5- 0.6 THz [Fig. 2(c)].\nThese damped oscillations are attributed to the temporal oscillation ofMz, which reflects\nthe damped magnetization precession,23because the zcomponent of the magnetization\nprecession vector increases with increasing θH. Further, the single spectrum apparent in\nFig. 2(c) indicates that there are no excited standing spin-waves ( such as those observed in\n5thick Ni films), even though the film is thicker than the optical skin de pth.23\nFerrimagnets generally have two magnetization precession modes, i.e., the FMR and\nexchange modes, because of the presence of sub-lattices.33In the FMR mode, sub-lattice\nmagnetization vectors precess while maintaining an anti-parallel dire ction, as illustrated in\nFig. 1(b), such that their frequency is independent of the exchan ge coupling energy between\nthe sub-lattice magnetizations. On the other hand, the sub-lattic e magnetization vectors\nare canted in the exchange mode; therefore, the precession fre quency is proportional to the\nexchange coupling energy between them and is much higher than tha t of the FMR mode.\nAs observed in the case of amorphous ferrimagnets, the FMR mode is preferentially excited\nwhen the pump laser intensity is so weak that the increase in tempera ture is lower than the\nferrimagnet compensation temperature.34No compensation temperature is observed in the\nbulk Mn 3Ge.25,26Also, the temperature increase in this experiment is significantly sma ller\nthanTcbecause the reduction of Msis up to 4 %, as can be seen in Fig. 2(a). Therefore,\nthe observed magnetization precession is attributed to the FMR mo de. Further, as the\nmode excitation is limited to the light skin depth, the amplitude, freque ncy, and etc., for\nthe excited mode are dependent on the film thickness with respect t o the light skin depth.\nThis is because the locally excited magnetization precession propaga tes more deeply into\nthe film as a spin wave in cases where fFMRis in the GHz range.23Note that it is reasonably\nassumed that such a non-local effect is negligible in this study, becau se the timescale of the\ndamped precession discussed here ( ∼1-10 ps) is significantly shorter than that relevant to a\nspin wave with wavelength comparable to the light skin depth ( ∼100 ps).\nThe FMR mode in the THz-wave range is quantitatively examined below. When the ex-\nchangecouplingbetween thesub-latticemagnetizationsissufficient ly strongandthetemper-\nature is well below both Tcand the compensation temperature, the magnetization dynamics\nfor a ferrimagnet can be described using the effective Landau-Lifs hitz-Gilbert equation35\ndm\ndt=−γeffm×/bracketleftbig\nH+Heff\nk(m·z)z/bracketrightbig\n+αeffm×dm\ndt, (1)\nwheremis the unit vector of the net magnetization parallel (anti-parallel) to the magnetiza-\ntion vector MII(MI) for the Mn II(MnI) sub-lattice [Fig. 1(b)]. Here, the spatial change of\nmis negligible, as mentioned above. Heff\nkis the effective value of the perpendicular magnetic\nanisotropyfieldincluding thedemagnetizationfield, even thoughthe demagnetizationfieldis\nnegligibly small for thisferrimagnet (4 πMs= 1.5 kOe). Further, γeffandαeffaretheeffective\n6FIG. 2. Change in Kerr rotation angle ∆ φknormalized by the saturation value φk,sas a function\nof pump-probe delay time ∆ t: (a) for a short time-frame at θH= 0◦and (b) for a relatively long\ntime-frame and different values of θH. The solid curves in (a) and (b) are a visual guide and values\nfitted to the data, respectively. The data in (b) are plotted w ith offsets for clarity. (c) Power\nspectral density as a function of frequency fand magnetic field angle θH.\n7values of the gyromagnetic ratio and the damping constant, respe ctively, which are defined\nasγeff= (MII−MI)/(MII/γII−MI/γI) andαeff= (αIIMII/γII−αIMI/γI)/(MII/γII−MI/γI),\nrespectively, using the gyromagnetic ratio γI(II)and damping constant αI(II)for the sub-\nlattice magnetization of Mn I(II). In the case of Heff\nk≫H,fFMRand the relaxation time of\nthe FMR mode τFMRare derived from Eq. (1) as\nfFMR=γeff/2π/parenleftbig\nHeff\nk+Hz/parenrightbig\n, (2)\n1/τFMR= 2παefffFMR. (3)\nHere,Hzis the normal component of H. Figure 3(a) shows the Hzdependence of the\nprecession frequency fp. This is obtained using the experimental data on the oscillatory\npart of the change in ∆ φk/φk,svia least-square fitting to the damped sinusoidal func-\ntion, ∆φk,p/φk,sexp(−t/τp)sin(2πfp+φp), with an offset approximating the slow change\nof ∆φk/φk,s[solid curves, Fig. 2(b)]. Here, ∆ φk,p/φk,s,τp, andφpare the normalized am-\nplitude, relaxation time, and phase for the oscillatory part of ∆ φk/φk,s, respectively. The\nleast-square fitting of Eq. (2) to the fpvs.Hzdata yields γeff/2π= 2.83 GHz/kOe and\nHeff\nk= 183 kOe [solid line, Fig. 3(a)]. The γeffvalue is close to 2.80 GHz/kOe for the free\nelectron. The value of Heff\nkis equal to the value determined via static measurement (198\nkOe)30within the accepted range of experimental error. Thus, the analy sis confirms that\nthe THz-wave range FMR mode primarily results from the large magne tic anisotropy field in\nthe Mn 3Ge material. The αeffvalues, which are estimated using the relation αeff= 1/2πfpτp\nfollowing Eq. (3), are also plotted in Fig. 3(a). The experimental αeffvalues are indepen-\ndent ofHzwithin the accepted range of experimental error, being in accorda nce with Eq.\n(3); the mean value is 0.03. This value of αefffor D0 22Mn3Ge is slightly larger than the\npreviously reported values for for D0 22Mn2.12Ga (∼0.015) and L1 0Mn1.54Ga (∼0.008).18\nIn the case of metallic magnets, the Gilbert damping at ambient tempe rature is primarily\ncaused by phonon and atomic-disorder scattering for electrons a t the Fermi level in the\nBloch states that are perturbed by the spin-orbit interaction. Th is mechanism, the so-\ncalled Kambersky mechanism,36,37predicts α∝M−1\ns, so that it is more preferable to use\nthe Landau-Lifshitz constants λ(≡αγMs) for discussion of the experimental values of α\nfor different materials. Interestingly, λeff(≡αeffγeffMs) for Mn 3Ge was estimated to be 61\nMrad/s, which is almost identical to the values for D0 22Mn2.12Ga (∼81 Mrad/s) and L1 0\nMn1.54Ga(∼66Mrad/s). The λfortheKamberkymechanism isapproximatelyproportional\n8FIG. 3. (a) Normal component of magnetic field Hdependence on precession frequency fpand\neffective dampingconstant αeffforMn 3Gefilm. (b)Oscillation amplitudeoftheKerrrotation angle\n∆φk,p/φk,scorresponding to the magnetization precession as a functio n of the in-plane component\nofH. The solid line and curve are fit to the data. The dashed line de notes the mean value of αeff.\ntoλ2\nSOD(EF), whereλSOisthespin-orbitinteractionconstant and D(EF)isthetotaldensity\nof states at the Fermi level.37The theoretical values of D(EF) for the above materials are\nroughly identical, because of the similar crystal structures and co nstituent elements, even\nthough the band structures around at the Fermi level differ slight ly, as mentioned at the\nbeginning.18,29Furthermore, the spin-orbit interactions for Ga or Ge, depending on the\natomic number, may not differ significantly. Thus, the difference in αefffor these materials\ncan be understood qualitatively in terms of the Kambersky mechanis m. Further discussion\nbased on additional experiments is required in order to obtain more p recise values for αeff\nand to examine whether other relaxation mechanisms, such as extr insic mechanisms (related\nto the magnetic inhomogeneities), must also be considered.\nFinally, the excitation mechanism of magnetization precession in this s tudy is discussed\nbelow, in the context of a previously proposed scenario for laser-in duced magnetization\n9precession in Ni films.23The initial equilibrium direction of magnetization θis determined\nby thebalance between HandHeff\nk[Fig. 1(b)]. Duringtheperiodinwhich thethree internal\nsystems are not in thermal equilibrium for ∆ t <∼2.0 ps after the pump-laser irradiation\n[Fig. 2(a)], not only the value of Ms, but also the value of the uniaxial magnetic anisotropy,\ni.e.,Heff\nk, is altered. Thus, the equilibrium direction deviates slightly from θand is restored,\nwhich causes magnetization precession. This mechanism may be exam ined by considering\nthe angular dependence of the magnetization precession amplitude . Because the precession\namplitudemaybeproportionaltoanimpulsive torquegeneratedfro mthemodulationof Heff\nk\nin Eq. (1), the torque has the angular dependence |m0×(m0·z)z|, wherem0is the initial\ndirection of the magnetization. Consequently, the z-component of the precession amplitude,\ni.e., ∆φk,p/φk,s, is expressed as ∆ φk,p/φk,s=ζcosθsin2θ∼ζ/parenleftbig\nHx/Heff\nk/parenrightbig2, whereζis the\nproportionalityconstant and Hxisthe in-plane component of H. The experimental values of\n∆φk,p/φk,sare plotted as a function of Hxin Fig. 3(b). The measured data match the above\nrelation, which supports the above-described scenario. Although ζcould be determined via\nthe magnitude and the period of modulation of Heff\nk, it is necessary to consider the ultrafast\ndynamics of the electron, spin, and lattice in the non-equilibrium stat e in order to obtain a\nmore quantitative evaluation;38,39this is beyond the scope of this report.\nIn summary, magnetization precessional dynamics was studied in a D 022Mn3Ge epitaxial\nfilm using an all-optical pump-probe technique. The FMR mode at fFMRup to 0.56 THz\nwas observed, which was caused by the extremely large Heff\nk. A relatively small damping\nconstant of approximately 0.03 was also obtained, and the corresp onding Landau-Lifshitz\nconstant for Mn 3Ge were shown to be almost identical to that for Mn-Ga, being in quali-\ntatively accordance with the prediction of the Kambersky spin-orb it mechanism. 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J.ˇSaf´ arik University, Park Angelinum 9, 041 54 Koˇ sice, Slov akia\n(Dated: July 16, 2018)\nAbstract\nMixed spin-1/2 and spin-1 Ising ferrimagnets on a triangula r lattice with sublattices A, B and\nC are studied for two spin value distributions ( SA,SB,SC) = (1/2,1/2,1) and (1 /2,1,1) by Monte\nCarlo simulations. The non-bipartite character of the latt ice induces geometrical frustration in\nboth systems, which leads to the critical behavior rather di fferent from their ferromagnetic coun-\nterparts. We confirm second-order phase transitions belong ing to the standard Ising universality\nclass occurring at higher temperatures, however, in both mo dels these change at tricritical points\n(TCP) to first-order transitions at lower temperatures. In t he model (1 /2,1/2,1), TCP occurs on\nthe boundary between paramagnetic and ferrimagnetic ( ±1/2,±1/2,∓1) phases. The boundary\nbetween two ferrimagnetic phases ( ±1/2,±1/2,∓1) and (±1/2,∓1/2,0) at lower temperatures is\nalways first order and it is joined by a line of second-order ph ase transitions between the paramag-\nnetic and the ferrimagnetic ( ±1/2,∓1/2,0) phases at a critical endpoint. The tricritical behavior\nis also confirmed in the model (1 /2,1,1) on the boundary between the paramagnetic and ferrimag-\nnetic (0,±1,∓1) phases.\nPACS numbers: 05.50.+q, 64.60.De, 75.10.Hk, 75.30.Kz, 75.50.Gg\nKeywords: Mixed-spin system, Frustrated Ising ferrimagnet, Tr iangular lattice, Monte Carlo simulation,\nTricritical point, Critical endpoint\n1I. INTRODUCTION\nMixed-spin Ising systems have been mostly investigated as possible m odels of some\ntypesofferrimagneticandmolecular-basedmagneticmaterials. Th eusedapproachesinclude\nanexacttreatmentinspecialcases1–5,mean-fieldapproximation6,7,effective-fieldtheorywith\ncorrelations8–14, Monte Carlo simulations15–22and some other methods23–28. The main focus\nwere their phase diagrams as well as technologically interesting comp ensation behavior with\npossibility to achieve zero total magnetization by tuning of tempera ture below the critical\npoint. Most of the studies considered the simplest models consisting of two sublattices one\nof which is occupied with spins S= 1/2 and the other with S= 1. Such a mixed-spin model\ncan be described by the Hamiltonian\nH=−J/summationdisplay\n/angbracketlefti,j/angbracketrightσiSj−D/summationdisplay\njS2\nj, (1)\nwhereσi=±1/2 andSj=±1,0 are spins on different sublattices, ∝angbracketlefti,j∝angbracketrightdenotes the sum\nover nearest neighbors, J <0 is a antiferromagnetic exchange interaction parameter and D\nis a single-ion anisotropy parameter. Negative values of the parame terDfavor nonmagnetic\nstates with Sj= 0 and positive values magnetic states with Sj=±1.\nDue to persisting ambiguities majority of the investigations focused on the simplest lat-\ntices, i.e., the square in two and cubic in three dimensions. We note tha t a long standing\ncontroversy regarding the critical and compensation behaviors e ven for the most studied\ncase of the model on a square lattice was solved only recently by Mon te Carlo simulation\nthat has convincingly shown22that there are neither tricritical nor compensation points, as\nhad been suggested by some previous approximative approaches6,8,9,28. On the other hand,\nin the same study the presence of both the tricritical point and a line of compensation\npoints was confirmed in the three-dimensional model on a simple cubic lattice. This finding\nmight suggest that the increased dimensionality is responsible for th e appearance of the tri-\ncritical and compensation behaviors. Nevertheless, our recent s tudy on a triangular lattice\nferromagnet29demonstrated that the tricritical point can also appear in a two-dim ensional\nlattice as long as the coordination number is sufficiently high. The effec t of the coordination\nnumber in the presence of bond disorder on tricritical behavior of a two dimensional system\nwas also recently studied in a random Blume-Capel model on a triangu lar lattice30.\nWe point out that the previous studies were performed on bipartite lattices, in which\n2A BC(a)\nAC(b)\nB\nFIG. 1: (Color online) Mixed-spin S= (SA,SB,SC) models on a triangular lattice consisting of\nsublattices A, B and C, with (a) S= (1/2,1/2,1) mixing - model I and (b) S= (1/2,1,1) mixing\n- model II. Small and large circles denote spin-1/2 and spin- 1 sites, respectively.\ncase the sign of the exchange interaction is irrelevant to the therm odynamic and critical\nproperties of the model in the absence of an external field. On the other hand, the present\nmixed-spin model is considered on a non-bipartite triangular lattice, in which case the sign\nof the exchange interaction matters. Namely, in contrast to the f erromagnetic case, the\nferrimagnetic interaction will induce geometrical frustration, whic h can be expected to have\nsome impact on the critical behavior. As shown in Fig. 1, the lattice co nsists of three\nsublattices A, B and C, occupied with spins S= (SA,SB,SC). This allows to further study\nthe model in two different mixing modes. We can consider a mixed-spin S= (1/2,1/2,1)\nmodel I, as schematically depicted in Fig. 1(a), in which one sublattice is occupied with\nspinS= 1 sites and the remaining two sublattices with spin S= 1/2 sites. Thus, each\nspin-1 site is surrounded by z= 6 nearest neighbors with spin S= 1/2. The other way\nof the spin-mixing is realized in a S= (1/2,1,1) model II, shown in Fig. 1(b), which is\nobtained when the spin-1/2 and spin-1 sites in the model I are swapp ed. The two models\nwere shown to display qualitatively different critical behaviors even f or the ferromagnetic\nexchange interactions29.\nThe goal of the present study is to examine effects of the geometr ical frustration on the\ncritical behavior of the above defined ferrimagnetic mixed-spin sys tems, to determine their\nphase diagrams and to confront them with their ferromagnetic cou nterparts as well as the\npure spin-1/2 and spin-1 antiferromagnetic systems.\n3100102104106−1−0.8−0.6−0.4−0.200.20.40.60.81\nMCSe/|J|                                  mO\n  \nmodel II: ms3, kBT/|J|=1.0model I: ms1, kBT/|J|=0.7\nmodel I: e/|J|, kBT/|J|=0.7\nmodel II: e/|J|, kBT/|J|=1.0\nFIG. 2: (Color online) Time evolutions of the internal energ y per site e/|J|and the staggered mag-\nnetizations ms1,ms3(see definitions below), starting from thedisordered phase at the temperatures\nkBT/|J|= 0.7 in model I and 1.0 in model II, for D/|J|= 0 and L= 120.\nII. MONTE CARLO SIMULATION\nIn order to study the behavior of various thermodynamic quantitie s in the parameter\nspace and to determine the phase diagrams we use Monte Carlo (MC) simulations with the\nMetropolis update rule and employ the periodic boundary conditions. We consider lattices\nwith the size L×L, withLranging from 24 up to 120. We perform N= 2×105up\nto 106MCS (Monte Carlo sweeps), the first 20% of which are used to bring t he system\nto equilibrium and then discarded, and the remaining data are used to estimate thermal\naverages and statistical errors. In order to demonstrate that the used MCS is sufficient to\nensure equilibrium conditions, in Fig. 2 we present MC time evolutions of some relevant\nquantities, such as the order parameters ms1(model I) and ms3(model II) and the internal\nenergy per site e/|J|, starting from the disordered phase. The simulations are perform ed\nin both models for the largest of the considered lattice sizes L= 120, which is the most\ndifficult to equilibrate, at the temperatures in the vicinity of the resp ective critical points\nforD/|J|= 0. The plots show that in these cases the equilibrium is reached in less than\n104MCS.\nThe phase boundaries are roughly determined from the maxima of so me thermodynamic\nfunctions, such as the specific heat, for a selected fixed value of L. We chose L= 48, as\n4a compromise value above which the specific heat maxima positions do n ot change con-\nsiderably and the phase diagrams can be determined in a relatively wide parameter space\nin a reasonable computational time. In the region where the critical line as a function of\nthe single-ion anisotropy parameter Dis more or less horizontal it is convenient to obtain\ntemperature dependencies of the calculated quantities at a fixed v alue ofD. In such a\ncase simulations start from the paramagnetic phase using random in itial configurations with\nthe temperature gradually decreased and a new simulation starting from the final config-\nuration obtained at the previous temperature. On the other hand , if the phase boundary\nshape changes to vertical we obtain variations of the quantities as functions of the single-ion\nanisotropy parameter Dat a fixed temperature. Then simulations start from appropriately\nchosen states (i.e., not necessarily random), expected in the cons idered region of the param-\neter space. Following the above described approach we ensure tha t the system is maintained\nclose to the equilibrium in the entire range of the changing parameter and thus considerably\nshortens thermalization periods. In order to estimate statistical errors, we perform three\nindependent simulations at all considered parameter values.\nAt some selected points of the phase boundaries we perform a more thorough finite-\nsize scaling (FSS) analysis in order to determine more precisely the loc ation of the critical\npoints and the corresponding critical exponents. In such a case w e perform more extensive\nsimulations using up to N= 107MCS and apply the reweighing techniques31. For more\nreliableestimationofstatistical errors, inthiscase weusedtheΓ-m ethod32. Having obtained\nthe maxima of the relevant quantities, we apply the linear fitting proc edure for logarithms of\ndata with errors, following the method in York et al.33. In order to assess the quality of the\nfitting, asameasureofgoodnessoffitweevaluatedanadjustedc oefficient ofdetermination34\nof the linear fit R2. The critical points and the exponents are then extracted from t he FSS\nanalysis, using the linear sizes L= 24,48,72,96 and 120.\nWe calculate the following quantities: the internal energy per spin e=∝angbracketleftH∝angbracketright/L2, the\nrespective sublattice magnetizations per site mX, (X = A, B or C), as order parameters on\nthe respective sublattices, which for the model I are given by\nmA(B)= 3∝angbracketleft|MA(B)|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈A(B)σi/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (2)\nmC= 3∝angbracketleft|MC|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈CSi/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (3)\n5and for the model II by\nmA= 3∝angbracketleft|MA|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈Aσi/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (4)\nmB(C)= 3∝angbracketleft|MB(C)|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈B(C)Si/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (5)\nwhere∝angbracketleft···∝angbracketrightdenotes thermal average. Based on the ground-state consider ations (see below),\nfortheidentified orderedphaseswe additionallydefinethefollowingo rder parametersforthe\nentire system, which take values between 0 in the fully disordered an d 1 in the fully ordered\nphase. For the model I we introduce two order parameters (stag gered magnetizations per\nsite)ms1andms2given by\nms1=∝angbracketleft|Ms1|∝angbracketright/L2=/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle2/summationdisplay\ni∈Aσi+2/summationdisplay\nj∈Bσj−/summationdisplay\nk∈CSk/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (6)\nand\nms2=∝angbracketleft|Ms2|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈Aσi−/summationdisplay\nj∈Bσj/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2. (7)\nFor the model II we define the order parameter ms3as\nms3=∝angbracketleft|Ms3|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈BSj−/summationdisplay\nk∈CSk/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/2L2. (8)\nUnlike in ferrimagnetic systems on bipartite lattices, the three subla ttices in the present\nsystem facilitate spin arrangements in such a way that the total ne t magnetization is always\nzero and, therefore, of no practical use.\nFurther, we calculate the susceptibilities pertaining to the respect ive order parameters\nO=MX(X = A, B, C) and also O=Msi(i= 1,2 or 3)\nχO=∝angbracketleftO2∝angbracketright−∝angbracketleftO∝angbracketright2\nNOkBT, (9)\nthe specific heat per site c\nc=∝angbracketleftH2∝angbracketright−∝angbracketleftH∝angbracketright2\nNOkBT2, (10)\nwhereNOis the number of sites on the (sub)lattice on which Ois defined. Further, we\ndefine the logarithmic derivatives of ∝angbracketleftO∝angbracketrightand∝angbracketleftO2∝angbracketrightwith respect to β= 1/kBT,\nD1O=∂\n∂βln∝angbracketleftO∝angbracketright=∝angbracketleftOH∝angbracketright\n∝angbracketleftO∝angbracketright−∝angbracketleftH∝angbracketright, (11)\n6D2O=∂\n∂βln∝angbracketleftO2∝angbracketright=∝angbracketleftO2H∝angbracketright\n∝angbracketleftO2∝angbracketright−∝angbracketleftH∝angbracketright, (12)\nand finally the fourth-order Binder cumulant UOcorresponding to the quantity O\nUO= 1−∝angbracketleftO4∝angbracketright\n3∝angbracketleftO2∝angbracketright2. (13)\nThe above standard thermodynamic quantities (9-10), as well as t he less traditional ones\ndefined by Eqs. (11-13), serve to obtain estimates of the respec tive critical exponents by FSS\nanalysis35,36. In particular, we use the following scaling relations, applied to the ma ximum\nvalues of the following functions:\nχO,max(L)∝LγO/νO, (14)\ncmax(L)∝Lα/νO, (15)\nD1O,max(L)∝L1/νO, (16)\nD2O,max(L)∝L1/νO, (17)\nwhereαis the critical exponent of the specific heat and νO,γOare the critical exponents\nof the correlation length and susceptibility, respectively, pertainin g to the quantity O. In\ncase of a first-order phase transition, the above quantities (14- 17) are expected to scale as\n∝Ld, whered= 2 is the system dimension. The order parameter cumulant, defined by\nEq. (13), can also serve for a simple yet relatively precise location of the phase transition\npoint as a point at which the cumulant curves obtained for different s ystem sizes intersect\nat a universal value, e.g., UO(Tc) = 0.611 for a two-dimensional Ising model37.\nIII. RESULTS\nA. Ground state\nLet us first identify all the possible ground states (GS) for entire r ange of the single-ion\nanisotropy parameter D. Considering the lattice system consisting of three interpenetrat ing\nsublatticesA,BandC,asschematicallydepictedinFig.1, theHamilton iansoftherespective\nmodels I and II can be defined as\nHI=−J/parenleftBig/summationdisplay\ni∈A,j∈Bσiσj+/summationdisplay\ni∈A,k∈CσiSk+/summationdisplay\nj∈B,k∈CσjSk/parenrightBig\n−D/summationdisplay\nk∈CS2\nk, (18)\n7TABLE I: Ground state configurations and the respective ener gies for different ranges of the single-\nion anisotropy parameter.\nModel I II\nD/|J| State Energy e/|J|State Energy e/|J|\n(−∞,−3/2) FRI\n2: (±1/2,∓1/2,0) −1/4 P: (0 ,0,0) 0\n(−3/2,∞) FRI\n1: (±1/2,±1/2,∓1)−3/4−D/3|J|FRII: (0,±1,∓1)−1−2D/3|J|\nHII=−J/parenleftBig/summationdisplay\ni∈A,j∈BσiSj+/summationdisplay\ni∈A,k∈CσiSk+/summationdisplay\nj∈B,k∈CSjSk/parenrightBig\n−D/parenleftBig/summationdisplay\nj∈BS2\nj+/summationdisplay\nk∈CS2\nk/parenrightBig\n.(19)\nFocusing on a triangular elementary unit cell consisting of the spins SA,SB,SC, from\nthe Hamiltonians (18) and (19) one can obtain expressions for the r educed energies per\nspine/|J|of different spin arrangements as functions of D/|J|. Then the ground states\nare determined as configurations corresponding to the lowest ene rgies for different values of\nD/|J|, as tabulated in Table I. There are two long-range order (LRO) fer rimagnetic (FR)\nstates FRI\n1and FRI\n2in the model I and one LRO ferrimagnetic state FRIIand one disordered\nparamagnetic (P) phase in the model II. We note that while in FRI\n2zero means nonmagnetic\nstates (Sj= 0) of spins on sublattice C, in FRIIzero means magnetic states ( σi=±1/2) of\nspins on sublattice A but equally in states +1 /2 and−1/2, thus giving zero net sublattice\nmagnetization. The critical value of the single-ion anisotropy param eter separating the\nrespective phases is the same for both models Dc/|J|=−3/2.\nB. Monte Carlo\n1. Model I: S= (1/2,1/2,1)\nPhase boundaries between the disordered paramagnetic and the r espective ordered ferri-\nmagnetic phases are determined from the specific heat maxima for a fixedL= 4848and the\nnature of the ordered phases is established from the introduced o rder parameters ms1and\nms2. TheseareshowninFigs.3(a)and3(b)forselectedvaluesofthep arameter D/|J|below,\nclose to and above the critical value Dc/|J|. All the specific heat curves show pronounced\nsharp peaks, signifying phase transitions to the low-temperature ferrimagnetic states. How-\never, the (pseudo)transition temperatures appear to be a nonm onotonic functions of D/|J|.\n80.20.30.40.50.60.70.80.900.511.522.5\n  \nkBT/|J|c(a)\nD/|J|=0 D/|J|=−2D/|J|=−1.48\n0.20.30.40.50.60.70.80.900.10.20.30.40.50.60.70.80.91\nkBT/|J|ms1, ms2\n  \n(b)\nms1, D/|J|=0\nms2, D/|J|=0\nms1, D/|J|=−2ms2, D/|J|=−2ms1, D/|J|=−1.48\nms2, D/|J|=−1.48\nFIG. 3: (Color online) Temperature dependencies of (a) the s pecific heat and (b) the order param-\neters, for selected values of D/|J|and a fixed lattice size L= 48. In (a) additionally the results\nforL= 72 are shown (squares).\nMore specifically, the transition temperature for D/|J|=−1.48, i.e., close to the critical\nvalueDc/|J|, is lower than for the other two values below and above Dc/|J|. Moreover, the\ncorresponding specific heat maximum has a spike-like shape and its ma gnitude is one order\nhigher then the other maxima, which is typical for a first-order pha se transition. In order\nto support our claim, that the peaks’ positions do not significantly c hange above L= 48,\nin Fig. 3(a) we also included the results obtained for L= 72. This will also become evi-\ndent later on in the respective phase diagrams in which for some selec ted points the phase\ntransition temperatures estimated for L= 48 will be compared with those determined for L\nextrapolated to infinity. The order parameters depicted in Fig. 3(b ) demonstrate that the\ntransition for D/|J|=−2 is to the state ( ±1/2,∓1/2,0), characterized by a finite values\nofms2, while for D/|J|=−1.48 and 0 the system tends to the state ( ±1/2,±1/2,∓1),\ncharacterized by a finite values of ms1. The discontinuous behavior of the order parameter\nforD/|J|=−1.48 corroborates the first-order nature of the transition.\nBy FSS analysis at two representative values of D/|J|= 0 and −2, selected on either\nside of the critical value Dc/|J|=−3/2, we confirmed that the disorder-to-order phase\ntransitions to both ( ±1/2,±1/2,∓1) and (±1/2,∓1/2,0) ferrimagnetic phases are indeed\nsecond order and belong to the standard Ising universality class. T he slopes of the fitted\n93 3.5 4 4.5 522.533.544.555.566.5\nln(L)  \n γM\ns1/νM\ns1 = 1.756 ± 0.006\n 1/νM\ns1 = 0.989 ± 0.011\n 1/νM\ns1 = 0.988 ± 0.013ln(χM\ns1,max), ln(D1M\ns1,max), ln(D2M\ns1,max)ln(χM\ns1,max)\nln(D1M\ns1,max)\nln(D2M\ns1,max)(a)\n3 3.5 4 4.5 51234567\nln(L)ln(χM\ns2,max), ln(D1M\ns2,max), ln(D2M\ns2,max)\n  \n γM\ns2/νM\ns2=1.753 ± 0.003\n 1/νM\ns2 = 0.982 ± 0.008 1/νM\ns2 = 0.985 ± 0.008ln(χM\ns2,max)\nln(D1M\ns2,max)\nln(D2M\ns2,max)(b)\n0.745 0.75 0.7550.520.540.560.580.60.620.64\nkBT/|J|UM\ns1\n  \nL=24\nL=48\nL=72\nL=96\nL=120(c)\n0.362 0.364 0.366 0.368 0.370.3720.450.50.550.60.65UM\ns2\nkBT/|J|  \nL=24\nL=48\nL=72\nL=96\nL=120(d)\n−1−0.5 00.511.522.50.20.250.30.350.40.450.50.550.60.65\nL1/ν\nM\ns1(T−Tc)/TcUM\ns1\n  \nL=24\nL=48\nL=72\nL=96\nL=120(e)\n−1−0.5 00.511.522.50.20.250.30.350.40.450.50.550.60.65\nL1/ν\nM\ns2(T−Tc)/TcUM\ns2\n  \nL=24\nL=48\nL=72\nL=96\nL=120(f)\nFIG. 4: (Color online) FSS analysis of the critical exponent s ratios 1 /νOandγO/νO(a,b), the\nfourth-order cumulant UOtemperature dependencies for different L(c,d) and the UOdata collapse\nanalysis (e,f), for O=Ms1atD/|J|= 0 (left column) and for O=Ms2atD/|J|=−2 (right\ncolumn). The coefficients of determination R2for the respective fits (from top to bottom) are\n0.9999, 0.9998, 0.9997 in (a) and 0.9999, 1.0000, 1.0000 in ( b).\n10−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.100.20.40.60.81\nD/|J|ms1\n  \nD/|J| down\nD/|J| upkBT/|J|=0.150.20.250.30.35\nFIG. 5: (Color online) Order parameter ms1as a function of the increasing ( ⊲) and decreasing ( ⊳)\nsingle-ion anisotropy parameter D/|J|at various temperatures and L= 48. The double-headed\narrows mark the hysteresis widths.\ncurves in Figs. 4(a) and 4(b) represent ratios of the critical expo nents 1/νOandγO/νO,\nfollowing from the scaling relations (14)-(17), where O=Msi(i= 1,2). We also checked\nthat for both D/|J|= 0 and −2 the specific heat maxima follow the logarithmic scaling\ncmax=c0+c1ln(L), as expected for the Ising universality class in two dimensions (not\nshown). Having performed the FSS analysis, these two critical poin ts can be determined\nwith a higher accuracy from the Binder cumulant crossing method38, as an intersection of\nthe Binder parameter UOcurves for different lattice sizes LandO=Msi(i= 1,2) (see\nFigs.4(c)and4(d)). Thecriticaltemperaturesweredetermined askBTc/|J|= 0.7485±0.001\natD/|J|= 0 and kBTc/|J|= 0.3663±0.001 atD/|J|=−2. Also the critical values of\nthe Binder cumulants UO(Tc) = 0.61137confirm the Ising universality class. Furthermore, in\nFigs. 4(e),4(f)we show thatfor thecritical temperatures deter mined by theBinder cumulant\ncrossing method the data for different lattice sizes indeed collapse o n a single curve. The\napparent first-order character of the phase transition at D/|J|=−1.48 will be discussed\nbelow.\nLet us now examine the transition between the two low-temperatur e ferrimagnetic phases\nFRI\n1and FRI\n2. Since the expected phase boundary is almost vertical to the x-ax is, instead\nof the temperature dependencies of various thermodynamic func tions it is more convenient\nto look into their single-ion parameter dependencies at a fixed tempe rature. By plotting\n11−0.25 −0.2 −0.15 −0.1 −0.0500.511.522.533.544.55x 105\ne/|J|  \nL=24\nL=48\nL=96\nL=120(a)\n3 3.5 4 4.5 5123456789\nln(L)ln(cmax)                ln( χM\ns1,max)\n  \nD/|J|=−1.47\nD/|J|=−1.47D/|J|=−1.48, αt/νt=1.60D/|J|=−1.4825, α/ν=2D/|J|=−1.48, γt/νt=1.85(b)\nD/|J|=−1.4825, γ/ν=2\n3 3.5 4 4.5 51234567\nln(L)  \n γM\ns1/νM\ns1 = 1.744 ± 0.008\n 1/νM\ns1 = 0.996 ± 0.009\n 1/νM\ns1 = 0.999 ± 0.007ln(χM\ns1,max), ln(D1M\ns1,max), ln(D2M\ns1,max)ln(χM\ns1,max)\nln(D1M\ns1,max)\nln(D2M\ns1,max)(c)\nFIG. 6: (Color online) (a) Energy distributions for D/|J|=−1.47 and different L. The respective\ntemperatures are tuned by the reweighing technique to achie ve approximately equal peak heights.\n(b) FSS analysis of the susceptibility χMs1and the specific heat c, forD/|J|=−1.47,−1.48 and\n−1.4825. For D/|J|=−1.48and−1.4825thelog-log plotsarerespectivelyfittedtotheexactva lues\nof the tricritical exponents ratios γt/νt= 1.85,αt/νt= 1.60 and the exponent 2, corresponding\nto the system volume. (c) FSS analysis of χMs1,D1Ms1,D2Ms1forD/|J|=−1.46, with the\ncoefficients of determination R2for the respective fits (from top to bottom) 0.9999, 1.0000, 0 .9999.\nthe order parameters as increasing and decreasing functions of D/|J|one can observe their\ndiscontinuous character and the appearance of hysteresis loops , the widths of which increase\nwith decreasing temperature. This behavior is demonstrated in Fig. 5 for the order parame-\nterms1andL= 48. Such behaviorsignalsfirst-orderphasetransitions. Theyse em topersist\n124 4.5 5 5.5 6−0.500.51\n105 MCSsublattice magnetizations\n  \nm1\nm2\nm3(a)\n3.5 4 4.5 5 5.5−1−0.8−0.6−0.4−0.200.20.40.60.81\n105 MCSsublattice magnetizations\n  \nm1\nm2\nm3(b)\nFIG. 7: (Color online) Time evolution of the sublattice magn etizations at the points (a)\n(D/|J|,kBT/|J|) = (−1.487,0.33) and (b) ( −1.47,0.35), forL= 48.\neven to higher temperatures at which the hysteretic behavior in no t apparent any longer.\nThe highest temperatures at which we still could observe some signs of first-order phase\ntransitions, such as bimodal energy distribution, was kBT/|J|= 0.35 and the corresponding\nvalue ofD/|J|=−1.47 (see Fig. 6(a)).\nNevertheless, the energy barrier separating the two peaks upon initial increase seems\nto decrease for larger L, which indicate that the phase transition may not be truly first\norder. Indeed, when we checked whether the specific heat and su sceptibility scale with the\nsystem volume, as it should be in case of a first-order transition, we found that within\nthe used lattice sizes the linear ansatz could not be established for D/|J|=−1.47, as\nshown in Fig. 6(b). We note that such a behavior that can lead to misin terpretation of a\nsecond-order transition as first order was also observed in the Blu me-Capel model as well\nas in the frustrated J1−J2Ising antiferromagnet on a square lattice39,40. Clearly first-order\nscaling is observed only at D/|J|=−1.4825. Nevertheless, fairly good linear fits were also\nachieved in the case of D/|J|=−1.48 with the exponents that are between 2 and the exact\ntricritical values (1.85 for γ/νand 1.6 for α/ν)41,42, suggesting that the system is close to\nthe tricritical point. On the other hand, for D/|J|=−1.46, the FSS presented in Fig. 6(c)\nclearly indicates a second-order phase transition. Thus, we rough ly estimate the tricritical\npoint at ( Dt/|J|,kBTt/|J|) = (−1.47±0.01,0.35±0.01).\nIt is interesting to notice that the phase transition at this point is no longer between\n13−3 −2 −1 0 1 200.10.20.30.40.50.60.70.80.9\nD/|J|kBTc/|J|P\nCEFR1I\nFR2ITCP\nfirst order\nFIG. 8: (Color online) Phase diagram of the model I in ( kBT/|J|−D/|J|) parameter space. The\nempty circles represent thephasetransition temperatures kBTc/|J|between theparamagnetic state\nP and the ferrimagnetic states FRI\n1(±1/2,±1/2,∓1) and FRI\n2(±1/2,∓1/2,0), estimated from the\nspecific heat peaks for L= 48, the empty triangles mark the hysteresis widths at first- order\ntransitions between the phases FRI\n1and FRI\n2with the expected phase transition boundary marked\nby the dash-dot line. The filled symbols at finite temperature s show more precise values obtained\nfrom the FSS analysis and the Binder cumulant crossing, wher e the diamond is the tricritical point\n(TCP), the hexagon is the critical endpoint (CE) and the squa re at (Dc/|J|,kBTc/|J|) = (−3/2,0)\nrepresents the exact value of the GS transition point.\nthe two ferrimagnetic phases FRI\n1and FRI\n2but between the phase FRI\n1and the param-\nagnetic phase. This is in line with the above observation of the first-o rder-like features\nof the phase transition at D/|J|=−1.4825 between the paramagnetic and FRI\n1phases.\nThe fact that there are first-order phase transitions between F RI\n1and FRI\n2as well as FRI\n1\nand paramagnetic phases can be verified by looking at the relevant o rder parameters. For\ndemonstration, in Figs. 7 we show segments of sublattice magnetiza tion time series ob-\ntained at ( D/|J|,kBT/|J|) = (−1.487,0.33) (Fig. 7(a)) and ( −1.47,0.35) (Fig. 7(b)). In\nboth cases we can see discontinuous switching between two phases , however, in either\npoint those phases are different. Namely, at ( −1.487,0.33) the sublattice magnetizations\n(mA,mB,mC) switch between the values characteristic for the states ( ±1/2,∓1/2,0)\nand (±1/2,±1/2,∓1), while at ( −1.47,0.35) they switch between the values characteris-\n14tic for the states (0 ,0,0) and (±1/2,±1/2,∓1). This finding implies the existence of a\ncritical endpoint (CE) at which the second-order transition bound ary between the para-\nmagnetic and FRI\n2phases joins the above discussed first-order transition line at abo ut\n(Dce/|J|,kBTce/|J|) = (−1.485±0.005,0.335±0.005).\nThe resulting phase diagram is presented in Fig. 8. The empty circles r epresent the\npseudo-critical points determined from the specific heat maxima at L= 48 and the empty\ntriangles mark the metastable branches of the first-order phase transitions obtained from\nthe order parameter hysteresis loops. The filled circles and square s at finite temperatures\nrepresent respectively second- and first-order transition point s obtained from the FSS anal-\nysis and the filled diamond marks approximate location of the tricritica l point. The filled\nsquare at ( Dc/|J|,kBTc/|J|) = (−3/2,0)shows the exact locations ofthe ground-statephase\ntransition. The expected first-order phase transition boundary , marked by the dash-dot line,\nis obtained by a simple linear interpolation between the estimated tricr itical and the exact\nGS transition points and only serves as a guide to the eye. We note th at in this highly sup-\npressed mixed-phase region the first-order phase boundaries ca n be located quite precisely,\nfor example, by multicanonical MC simulations43,44. However, in our case, we are only in-\nterested in approximate location of the phase boundaries. Then, c onsidering the exact value\nof the GS transition point, the estimate location of the tricritical po int and assuming no\nanomalous behavior, such as reentrance, the low-temperature p art of the phase boundary\nmust be practically vertical to the D/|J|axis.\n2. Model II: S= (1/2,1,1)\nAlso for the model II the phase boundary as a function of the single -ion anisotropy\nparameter D/|J|is estimated from the specific heat maxima for L= 48, except for D/|J|\nclose to the critical value of Dc/|J|=−3/2, where the boundary becomes almost vertical.\nThe only identified LRO phase is the ferrimagnetic phase FRII: (±1/2,±1,∓1), present\nforD > D c, and the phase transition is second order complying with the standa rd Ising\nuniversality class. This is illustrated in Fig. 9(a) for a selected value of D/|J|= 0, in which\nwe show that the obtained ratios of the critical exponents 1 /νMs3andγMs3/νMs3are in a\ngood agreement with the 2D Ising universality class values 1 and 7 /4, respectively. We also\nchecked the consistency of the specific heat critical exponent va lueα= 0 by verifying the\n153 3.5 4 4.5 522.533.544.555.5\nln(L)ln(χM\ns3,max), ln(D1M\ns3,max), ln(D2M\ns3,max)\n  \nln(χM\ns3,max)\nln(D1M\ns3,max)\nln(D2M\ns3,max)(a)\nγM\ns3/νM\ns3 = 1.757 ± 0.006\n1/νM\ns3 = 1.000 ± 0.013\n1/νM\ns3 = 1.001 ± 0.015\n1.055 1.061.065 1.071.075 1.080.580.60.620.640.660.68\nkBT/|J|UM\ns3\n  \n0 0.02 0.041.061.081.11.121.141.161.18\nL−1  \nkBTmax,χ\nM\ns3/|J|\nkBTmax,D1\nM\ns3/|J|\nkBTmax,D2\nM\ns3/|J|\nL=24\nL=48\nL=72\nL=96\nL=120(b)\n−1−0.500.511.520.20.250.30.350.40.450.50.550.60.65\nL1/ν\nM\ns3(T−Tc)/TcUM\ns3\n  \nL=24\nL=48\nL=72\nL−96\nL=120(c)\nFIG. 9: (Color online) (a) FSS analysis of the critical expon ent ratios 1 /νMs3andγMs3/νMs3, (b)\nthe fourth-order cumulant UMs3temperature dependencies for different L, and (c) the UMs3data\ncollapse analysis at D/|J|= 0. In (a) the coefficients of determination R2for the respective fits\n(from top to bottom) are 0.9998, 0.9999 and 0.9999. Inset in ( b) shows an alternative way of the\ncritical temperature estimation from the FSS analysis.\nlogarithmic scaling (not shown). The critical temperature for D/|J|= 0 estimated by the\nBinder cumulant method (Fig. 9(b)) and FSS analysis (inset in Fig. 9(b )) takes the value\nkBTc/|J|= 1.0635±0.0015 and the cumulant curves for different Lintersect at the universal\nvalue ofUMs3(Tc) = 0.611.\nOn approach to the critical value Dc/|J|=−3/2 the phase boundary rapidly drops and\n16−1.6 −1.55 −1.5 −1.45 −1.4 −1.3500.10.20.30.40.50.60.70.80.91\nD/|J|ms3kBT/|J|=0.3\nkBT/|J|=0.05kBT/|J|=0.2\nFIG. 10: (Color online) Order parameter ms3as a function of the increasing ( ⊲) and decreasing\n(⊳) single-ion anisotropy parameter D/|J|at various temperatures and L= 48.\nbecomes almost vertical. Therefore, in order to locate the critical temperatures in this re-\ngion, it is more convenient to measure the physical quantities at a fix ed temperature as\nfunctions of the parameter D/|J|. At sufficiently low temperatures the measured quantities\nshow some properties typical for first-order phase transitions. Namely, as the anisotropy\nparameter D/|J|is decreased and increased at the fixed temperature the sublattic e mag-\nnetizations, the order parameter ms3and the internal energy show discontinuities at some\nvalues of D/|J|, as demonstrated in Fig. 10 for ms3. Nevertheless, it is interesting to notice\nthat, in contrast to the strongly hysteretic behavior of the mode l I at the transition between\nthe two ferrimagnetic phases, no apparent hysteresis can be obs erved in the present model.\nDiscontinuous character of the transition reflected in bimodality of the relevant observables,\nsuch as the internal energy, disappears at higher temperatures but is still evident at temper-\natures slightly above kBT/|J|= 0.2. In Fig. 11(a) it is demonstrated for D/|J|=−1.48 and\nthe temperatures kBT/|J| ≈0.215 tuned by the reweighing technique for each Lto achieve\ndistributions with the two modes of about the same heights. The inse t gives us some idea\nabout the characteristic tunneling times between the coexisting ph ases forL= 48. However,\nsimilar tothesituationinthemodelIpresented inFig.6, for D/|J|=−1.48thespecific heat\nand staggered susceptibility maxima do not scale with volume. Such a s caling is observed\nonly at slightly lower temperatures for D/|J|=−1.4825 (at least for L≥72), as shown in\nFig. 11(b). At higher temperatures the second-order phase tra nsition with standard Ising\n17−0.02−0.015 −0.01−0.005 00.005 0.0100.511.522.53x 105\ne/|J|  \nL=48\nL=72\nL=96\nL=1208 9−505x 10−3\n105 MCSe/|J|(a)\n3 3.5 4 4.5 5−20246810\nln(L)ln(cmax)             ln( χs3,max)\n  \nD/|J|=−1.47\nD/|J|=−1.48D/|J|=−1.47D/|J|=−1.48\nD/|J|=−1.4825, α/ν=2(b)\nD/|J|=−1.4825, γ/ν=2\n3 3.5 4 4.5 50123456\nln(L)ln(χM\ns3,max), ln(D1M\ns3,max), ln(D2M\ns3,max)\n  \nln(χM\ns3,max), γI/νI=1.75\nln(D1M\ns3,max), 1/νI=1.00ln(D2M\ns3,max), 1/νI=1.00(c)\nFIG. 11: (Color online) (a) Energy distributions for D/|J|=−1.48 and different L. The respective\ntemperatures are tuned by the reweighing technique to achie ve approximately equal peak heights.\nTime evolution of the internal energy shown in the inset demo nstrates tunneling between the\ncoexisting phases for L= 48. (b) FSS analysis of the susceptibility χMs3and the specific heat\nc, forD/|J|=−1.47,−1.48 and−1.4825. For D/|J|=−1.4825 the log-log plots are fitted to\nthe exponent 2, corresponding to the system volume. (c) FSS a nalysis of χMs3,D1Ms3,D2Ms3for\nD/|J|=−1.46withthefittedIsingvaluesofthecritical exponentsrati osγI/νI= 1.75,1/νI= 1.00.\ncritical exponents is recovered for D/|J|=−1.46, although again larger system sizes are\nrequired to reach the linear asymptotic regime (Fig. 11(c)). Thus, the tricritical point in\nthe model II is roughly located at ( Dt/|J|,kBTt/|J|) = (−1.47±0.01,0.27±0.04).\nThe above results can be summarized into the phase diagram shown in Fig. 12. As in\n18−1.5−1−0.500.511.5200.20.40.60.811.21.4\nD/|J|kBTc/|J|P\nTCPFRII\nfirst order\nFIG. 12: (Color online) Phase diagram of the model II in ( kBT/|J|−D/|J|) parameter space. The\nempty circles represent the phase transition temperatures kBTc/|J|between the paramagnetic P\nand the ferrimagnetic phase FRII(0,±1,∓1) estimated from the specific heat peaks for L= 48,\nthe filled circle shows a more precise value obtained from the FSS analysis and Binder cumulant\ncrossing, the filled diamond is the tricritical point and the filled squares represent a first-order\ntransition point determined from FSS analysis at D/|J|=−1.48 and the exact value of the GS\ntransition point at Dc/|J|=−3/2. The empty triangles mark the first-order transition relat ed\ndiscontinuities in the order-parameter ms3in theD/|J|increasing ( ⊲) and decreasing ( ⊳) processes.\nthe phase diagram of the model I, the empty circles represent the pseudo-critical points\ndetermined from the specific heat maxima at L= 48 and the empty triangles mark the first-\norder phase transitions located from the jumps in the order param eter loops obtained by\nincreasing and decreasing of the single-ion parameter D/|J|. The filled circle at D/|J|= 0,\nthe square at D/|J|=−1.48 and the filled diamond represent respectively second-order,\nfirst-order and tricritical points, obtained from the FSS analysis. As in Fig. 8, the filled\nsquare at ( Dc/|J|,kBTc/|J|) = (−3/2,0) represents the exact value of the ground-state\nphase transition point.\nIV. CONCLUSIONS\nWe have studied the mixed spin-1/2 and spin-1 Ising ferrimagnets on a triangular lattice\nwithsublattices A, BandC,intwo mixing modes: ( SA,SB,SC) = (1/2,1/2,1)(model I)and\n19(SA,SB,SC) = (1/2,1,1)(model II). The pure spin-1/2 and spin-1 Ising antiferromagne ts on\na triangular lattice show respectively no LRO45and partial LRO for some range of a single-\nion anisotropy parameter at low temperature with quasi-LRO of the Berezinskii-Kosterlitz-\nThouless type at higher temperatures46. In comparison with these models in the present\nferrimagnetic models the frustration is partially accommodated by d ifferent ferrimagnetic\nspin arrangements and thus their critical behavior is rather differe nt from the pure systems.\nOn the other hand, the net magnetization of both ferrimagnetic mo dels is always zero and\nthus they canshow no compensation points. Therefore, fromthis point ofview, the behavior\nof the present mixed-spin models is typical for antiferromagnets r ather than ferrimagnets.\nAs for the critical properties, the model I shows two ferrimagnet ic phases FRI\n1:\n(±1/2,±1/2,∓1) and FRI\n2: (±1/2,∓1/2,0), which may be compared with the ferromag-\nnetic case displaying two ferromagnetic phases ( ±1/2,±1/2,±1) and (±1/2,±1/2,0)29. We\nnote that on bipartite lattices the thermodynamic behavior of the s ystems with ferrimag-\nnetic and ferromagnetic interactions is the same and thus the phas e diagrams would be\nidentical. However, there are substantial differences between th e two phase diagrams for the\ntriangular lattice ferromagnetic and ferrimagnetic models. First of all, due to frustration\nthe transition temperatures for the ferrimagnetic case are signifi cantly reduced and collapse\nwith the ferromagnetic boundary only in the large negative D/|J|limit, when the partial\nfrustration inducing magnetic states on the C-sublattice are comp letely suppressed and the\ncritical temperature for either case tends to the exact spin-1/2 Ising value on a honeycomb\nlatticekBTc/|J|= 0.379747. The frustration is further increased close to the boundary be-\ntween the two ferrimagnetic phases FRI\n1and FRI\n2, which is reflected in the depression in the\norder-disorder phase boundary that is absent in the ferromagne tic model. Nevertheless, the\nfrustration did not seem to affect the standard Ising values of the critical exponents. We\nnote that besides the above presented points of D/|J|=−2 and 0, we also performed the\nFSS analysis in this region of an increased frustration at the P-FRI\n1branch of the phase\ndiagram for D/|J|=−1.6 (not shown) but did not find any deviation larger than statis-\ntical errors from the standard values. However, the most consp icuous difference from the\nferromagnetic case is the presence of the strongly discontinuous phase transition between\nthe ferrimagnetic phases ( ±1/2,±1/2,∓1) and (±1/2,∓1/2,0), which is completely absent\nbetween the ferromagnetic phases ( ±1/2,±1/2,±1) and (±1/2,±1/2,0). This can be ex-\nplained by the fact that in the ferrimagnetic case dramatic changes occur when the two\n20spin-1/2 sublattices A and B, forming a connected honeycomb back bone, switch their mag-\nnetizations between 1 /2 and−1/2 and the spin-1 sublattice C between magnetic ±1 and\nnonmagnetic 0 states. On the other hand, in the ferromagnetic ca se nothing happens in\nsublattices A and B and in sublattice C the change between magnetic ±1 and nonmagnetic\n0 states occurs only gradually in the isolated (mutually directly nonint eracting) spins.\nThe model II has been shown to display only one ordered ferrimagne tic state FRII:\n(0,±1,∓1) with no LRO on sublattice A and an antiferromagnetic LRO on the re maining\nsublattices B and C forming a honeycomb lattice. Similar phase occurs in the pure spin-\n1 triangular antiferromagnet for the single-ion anisotropy parame ter−3/2< D/|J|<0,\nhowever, it is destabilized for D/|J|>046. On the other hand, in the present model II\nthe increasing D/|J|stabilizes the ferrimagnetic phase FRIIand the critical temperature for\nD/|J| → ∞tends to kBTc/|J|= 1.518847, i.e., the exact value of the spin-1/2 Ising model\non a honeycomb lattice when the spin states ±1 are considered in the Hamiltonian instead\nof±1/2. In comparison with the ferromagnetic model II, again the critica l temperatures\nare lowered due to frustration but qualitatively the phase diagrams look similar. As the\nparameter D/|J|is decreased both models show change of the phase transition natu re from\nsecond to first order at a tricritical point. Nevertheless, interes tingly, the strong hysteretic\nbehavior observed intheferromagneticmodel accompanying thefi rst-orderphase transitions\nis not evidenced in the ferrimagnetic one.\nAcknowledgments\nThis work was supported by the Scientific Grant Agency of Ministry o f Education of\nSlovak Republic (Grant Nos. 1/0234/12 and 1/0331/15). The auth ors acknowledge the\nfinancialsupportbytheERDFEU(EuropeanUnionEuropeanRegion alDevelopmentFund)\ngrant provided under the contract No. ITMS26220120047 (activ ity 3.2.).\n∗Electronic address: milan.zukovic@upjs.sk\n1L.L. Goncalves, Phys. Scripta 32, 248 (1985).\n2A. Lipowski, T. Horiguchi, J. Phys. A: Math. Gen. 28, L261 (1995).\n3M. 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Prog. Phys. 30, 615 (1967).\n48Hence, in fact these are only pseudo-critical points\n23"    },    {        "title": "1404.7058v2.High_Curie_temperature_ferrimagnetism_and_ferroelectricity_in_Bi2FeMoO6.pdf",        "content": "arXiv:1404.7058v2  [cond-mat.mtrl-sci]  17 Oct 2014High-Curie-temperature ferrimagnetism and ferroelectri city in Bi 2FeMoO 6\nPeng Chen and Bang-Gui Liu∗\nBeijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n(Dated: August 29, 2018)\nBiFeO 3is the most famous multiferroic material, but its G-type ant iferromagnetism is highly\ndesirable to be replaced by strong macroscopic magnetism be yond room temperature. Here we\nobtain double perovskite Bi 2FeMoO 6with R3 (#146) space group by substituting Mo for 50% Fe\nin BiFeO 3. Our first-principles calculated results show that it is a se miconductor with gap reaching\nto 0.725 eV, its net magnetic moment is 2 µBper formula unit, and its ferroelectric polarization\nis 85µC/cm2. This ferroelctricity is comparable with that of BiFeO 3, but here the magnetism\nis a strong ferrimagnetism with Curie temperature of 650 K. O ur first-principles phonon spectra\nestablishes that this R3 phase is stable. Electric polariza tion and magnetic easy axis are shown to be\nin pseudo-cubic [111] axis. Our further analysis shows that the multiferroic mechanism is similar to\nthat in BiFeO 3. Therefore, this Bi 2FeMoO 6can be used to achieve strong macroscopic magnetism\nand ferroelectricity well above room temperature, being us eful for designing new multifunctional\nmaterials and devices.\nPACS numbers: 75.85.+t, 75.30.-m, 77.80.-e, 75.10.-b\nMultiferroic materials have great potentials to achieve\nnew multifunctional devices for spintronics and data\nstorage[1–5]. Recent years witnesses that perovskite\nBiFeO 3becomesthemostfamousmultiferroicmaterial[4,\n6–19]. It can show strong ferroelectricity with Curie\ntemperature 1140K and antiferromagnetism with mag-\nnetic Neel temperature 640K[4, 8, 14]. In order to real-\nize high-performance applications, one hopes to achieve\nstrong macroscopic magnetism (ferromagnetism or fer-\nrimagnetism) above room temperature instead of an-\ntiferromagnetism in the BiFeO 3, keeping its strong\nferroelectricity[4, 12–14]. For this purpose, it is a big\nstep forward to predict and then synthesize multiferroic\nBi2FeCrO 6whose double perovskite structure can be ob-\ntained from perovskite BiFeO 3by substituting Cr for\n50% Fe in the B site and forming alternate Fe and Cr or-\ndering along the [111] axis[20–26]. However, it is difficult\nto keep the Fe and Cr ordering at a very high level be-\ncause of the small difference between Fe and Cr[25, 26].\nTherefore, it is highly desirable to realize much higher\nB-site ordering and achieve much better materials with\nstrong macroscopic magnetism and ferroelectricity.\nHere, we present a promising multiferroic Bi 2FeMoO 6\nby substituting Mo for 50% Fe in perovskite BiFeO 3and\nkeepingalternateorderofFe and Mo alongthe [111]axis.\nIt is a double perovskite phase with R3 (#146) space\ngroup. Considering that excellent Fe-Mo ordering has\nbeen acieved in double perovskite Sr 2FeMoO 6, a well-\nknown half-metallic ferrimagnet[27–29], we believe that\nthissubstitutioncanleadtogoodferrimagnetismbecause\nthe moments at Fe and Mo sites cannot completely com-\npensate any more. Our systematic first-principles calcu-\nlations and structural analyses reveal that double per-\novskite Bi 2FeMoO 6has a semicinductor gap of 0.725 eV\nand can show both strong ferroelectricity and strong fer-\nrimagnetismwith highCurietemperature. Moredetailedresults will be presented in the following.\nResults\nStructure optimization. Following Glazers famous\nwork on perovskite structures[30, 31], a systematic\ngroup analysis[32–34] reveals possible crystal structures\nfor a specific double perovskite material with formula\nA2BB′O6. These group analyses can be used to sim-\nplify our structural optimizations. Here, we consider\nonly those tilted BO6 and B′O6 octahedra allown by\ngroup theory[34]. For optimizing Bi 2FeMoO 6, we start\nwithacubicperovskitestructurewithspacegroupFm ¯3m\n(#225). This corresponds to a0a0a0in Glazor’s nota-\ntion. The optimization leads to rhombohedral lattice\nconstant ac=5.558˚Aandangle αc=60◦, primitivecellvol-\numeVc=121.41˚A3, and total energy Ec=−67.3366eV. In\nthis structure, the O octahedra do not rotate around the\n[111] axis. Allowing the O octahedra to able to rotate\naround the [111] axis, we can obtain an optimized struc-\nture with R ¯3 (#148) space group. It has rhombohe-\ndral lattice constant a1=5.644˚A and angle α1=61.63◦,\nprimitive cell volume V1=131.76˚A3, and total energy\nE1=−69.5888eV.Clearly, this symmetry breaking makes\nthe total energy 2.2522 eV lower than the cubic phase.\nThis structure still keeps inverse symmetry and there-\nforehas no ferroelectricity, and in contrastwith the cubic\nphase, the O octahedraofFe and Modo rotateby +15.7◦\nand-17.4◦aroundthe [111]axis. This isa−a−a−tilt (an-\ntiphaseR+\n4rotation) in Glazor’s notation. For searching\nfor the ground-state phase of Bi 2FeMoO 6, it is useful to\ncompareit with BiFeO 3[7, 14, 16, 17] and Bi 2FeCrO 6[20–\n24] whose ground state phases are R3c (#161) and R3\n(#146) phases, respectively. Because %50 Fe is substi-\ntuted by Mo, there is no R3c structure for Bi 2FeMoO 6\nand we should turn to the R3 structure for possible can-\ndidate of the ground-state phase of Bi 2FeMoO 6. For2\nthis purpose, we allow the Fe ion to have a Γ−\n4dis-\nplacement along the [111] direction[17, 35, 36]. This in-\nverse symmetry breaking makes the total energy move\nfurther downward when we carefully optimize the struc-\nture. Consequently, we obtain a rhombohedral structure\nwith space group R3 (#146). This optimized structure\nhas rhombohedral lattice constant a2=5.743˚A and angle\nα2=59.76◦, primitive cell volume V2=133.24˚A3, and to-\ntal energy E2=−69.8186eV. This energy is 0.2298 and\n2.4820 eV lower than the R ¯3 (#148) and Fm ¯3m (#225),\nrespectively. Considering its similarity with BiFeO 3and\nBi2FeCrO 6, we believe that this R3 structure is the\nground-state phase for Bi 2FeMoO 6. Our analysis results\nshow that the tilting of oxygen octahedra of Fe and Mo\ncan be described with rotation angles of +17.8◦and -\n17.0◦. It is clear that the three-fold rotational symmetry\naround the [111] axis is kept, but the calculated bond an-\ngles of Fe-O-Mo (150.0◦), O-Fe-O (163.3◦), and O-Mo-O\n(168.6◦) deviate substantially from 180◦. These reveal\nthat the oxygen octahedra are substantially deformed.\nWe illustrate the crystal structure and local octahedral\ntilting and deformation in Fig. 1 and summarize our\ncalculated parameters in Table I. Although we have let\nonly the Fe ion move along the [111] direction with re-\nspect to its oxygen environment in the initial structure,\nthe optimization makes all the four cations coherently\nmove along the [111] direction with respect to their oxy-\ngen environments, and we shall show that these coherent\ndisplacements are favorable to form ferroelectricity.\nTABLE I: Main parameters of the fully optimized double per-\novskite Bi 2FeMoO 6with the #146 space group: rhombohe-\ndral lattice constant ( a) and angle ( α), relative coordinates of\nthe different ions, and rotational angles of the O octahedron\nof Fe (∆ θFe) and Mo (∆ θMo).\na 5.743˚A α 59.76◦\nBi 0.2244 0.2244 0.2244\n0.7289 0.7289 0.7289\nFe 0.0000 0.0000 0.0000\nMo 0.5089 0.5089 0.5089\nO 0.8432 0.6774 0.2913\n0.2913 0.8432 0.6774\n0.6774 0.2913 0.8432\n0.1765 0.3511 0.7917\n0.7917 0.1765 0.3511\n0.3511 0.7917 0.1765\n∆θFe 17.8◦∆θMo -17.0◦\nMagnetic moment and electronic energy gap.\nPresented in Fig. 2 is our GGA calculated spin-\ndependent density of states (DOS) and energy bands of\nthe optimized double perovskite Bi 2FeMoO 6(R3 phase).\nOur analysis indicates that in the spin-up channel, the\nthree filled Fe t2g bands merges into the oxygen p bands\nFIG. 1: (Color online.) Crystal structure (a,b) and lo-\ncal changes of O octahedra of Fe and Mo (c-e) of double\nperovskite Bi 2FeMoO 6with the #146 space group. (a) A\nschematic crystal structure, showing the alternate occupa tion\nof Fe and Mo at the B-sites of the simple perovskite structure\nalong the [111] direction. (b) The primitive cell of the doub le\nperovskite structure with the O octahedron of Mo shown. (c)\nThe relative rotation projected on the (111) plane of an O\noctahedron of Fe with respect to three of the nearest O octa-\nhedra of Mo. (d) The two neighboring O octahedra of Fe and\nMo projected on the plane of the Fe-O-Mo triangle, reflecting\nthe Fe-O-Mo bond angle. (e) the same two O octahedra of Fe\nand Mo projected on the perpendicular plane including the\nFe and Mo ions, showing local deformation.\nand the two filled Fe eg bands are separated from them,\nand the empty bands, from Mo 4d states, are above 0.65\neV. It can be seen that there is a narrow semiconduc-\ntor gap, 0.059 eV, which is formed between the three\nfilled Mo 4d bands and the three empty Fe 3d ones in\nthe spin-down channel. This narrow GGA gap remains\nopen as long as the optimization is done with high force\nconvergence standard of 1 meV/ ˚A or smaller, or else it\ncan become a pseudo-gap indicating a metallic electron\nphase[37]. Totally, wehavefiveFe d electronsin thespin-\nup channel and three Mo d electrons in the spin-down\nchannel, and thus the magnetic moment per formula unit\nis equivalent to 2 µB, which is consistent with the calcu-\nlated data. The small gap value is due to the well-known\nunder-estimation of GGA on semiconductor gaps. In or-\nder to make an accuratecalculation of the semiconductor\ngap, we use a modified Becke-Johnson (mBJ) exchange\nfunctional to replace the GGA one. Our mBJ calculated\ngap, 0.725 eV, should be a much better value for the true\ngap because mBJ has been proved to give accurate semi-\nconductor gaps for most of semiconductors. We also use\notherexchange-correlationschemesto calculatethe semi-\nconductorgapandconcludethatthereiscertainlyafinite\nsemiconductorgapforthedoubleperovskiteBi 2FeMoO 6.\nIonic displacements towards ferroelectricity. In\nFig. 3 we shows the relative distances between the3\nFIG. 2: Electronic structure of the double perovskite\nBi2FeMoO 6calculated with GGA. Upper: the spin-resolved\ndensity of states (DOS, state/eV per primitive cell), in whi ch\nthe upper part are spin-up DOS and the lower part spin-\ndown DOS. Lower: the spin-resolved energy band structures,\nin which the left part shows spin-up bands and the lower part\nspin-down ones.\ncations and their nearest O environment along the [111]\ndirection for the three structures: #225, #148, and\n#146. The vertical line, ∆ 1-∆4, indicates the position\nof three O atoms forming a regular triangle perpendic-\nular to the [111] axis. The O-O bond lengthes and ro-\ntation angles of the O octahedron are presented at the\ntop of vertical lines. The lengthes of the primitive cells\nalong [111] direction are given at the right-most sides of\nthe [111] lines. Comparing the #148 structure with the\n#225, Fe and Mo are still at the centers of their O octa-\nhedra, respectively, and the local asymmetry at Bi1 and\nBi2 ions in the #148structure becomes much largerthan\nin the latter. However, the displacements of Bi1 and Bi2\nwith respectto theirO environmentsareoppositetoeach\nother along the [111] axis, which means that there is still\nno ferroelectricity, but an Bi-induced antiferroelectricity,\nin the #148 structure. In contrast, the #146 structure\nis very different in that all its cations have in-phase dis-\nplacements in the same [111] direction with respect to\ntheir O environment. In order to get a clearer insight,\nwe further investigate the displacements of cations with\nrespect to their O environments along the [111] directionin terms of two cation-related quantities, Plus and Mi-\nnus. The Plus (Minus) value of a cation is defined as the\nsummation (subtraction) of the two distances from the\ncation to its left O triangle and right triangle along the\n[111] axis. The Plus value indicates the distance between\nthe two O triangles on the two sides of the cation, and\nthe Minus value describes the displacement of the cation\nwith respect to its two O triangles along the [111] direc-\ntion. We present the Plus and Minus values of the four\ncations in the #146 phase of the Bi 2FeMoO 6in Table\nII. The Minus values of all the four cations are negative\nand thus in-phase, and therefore indicate ferroelectric-\nity in the Bi 2FeMoO 6, being the same as those of the\nBi2FeCrO 6which has a strong ferroelectricity along the\n[111] direction[20–25, 38].\nElectron density distributions implying ferro-\nelectricity. We present in Fig. 4 the electron den-\nsity distributions in a typical plane for the #148 and\n#146 structures of the Bi 2FeMoO 6. The plane, namely\nA plane, is defined to include the Fe-Bi-Mo-Bi line along\nthe [111] axis and the two nearest O atoms on the two\nsides of the Mo atom. The charge and spin density dis-\ntributions can be compared between the two structures.\nAn asymmetry can be seen near the two Bi atoms for\nthe charge density distribution between -8 and 0 eV for\nthe two structures, and it is, however, in phase for the\n#146 structure but out of phase for the other. Further-\nmore, we can easily see an in-phase asymmetry in the Bi\n6s charge density distribution in the #146 structure, but\ncannot in the other. As for the spin density distribution,\na difference can also be found between the two Bi atoms\nfor the #146 structure. These electron density proper-\nties are consistent with the in-phase displacements of the\ncations in the [111] direction.\nDFT calculations of ferroelectric polarization.\nFinally, we quantitatively investigate the ferroelectric-\nity of the Bi 2FeMoO 6through modern DFT-based\ncalculations[39]. Since the R ¯3 (#148) as the reference\nstructure has no semiconductor gap under GGA, we\nmake quantitative calculations of the electric polariza-\ntion by using GGA+U[40, 41] and HSE06[42, 43] meth-\nods which have been implemented in VASP package. For\nthe GGA+U scheme, we use three sets of ( U,J) param-\neters for Fe, keeping the same ( U,J)=(2.5eV,0.5eV) for\nMo. With these schemes and parameters, the semicon-\nductor gaps range from 0.23 to 0.91 eV, but it should\nbe kept in mind that accurate semiconductor gap needs\nto be calculated with mBJ scheme. We summarize the\ncalculational parameters and electric polarization results\nin Table II. These results means that the Bi 2FeMoO 6\n(#146) has a large spontaneous ferroelectric polarization\n85µC/cm2, which is almost independent of calculational\nschemesand parameters. Thereforethe Bi 2FeMoO 6is an\nexcellent ferroelectric material.\nFerroelectric switching and phonon spectra. It\nshould be pointed out that the R3 (#146) phase is de-4\nFIG. 3: (Color online.) The relative distances (tilted numb ers) between the positive ions (Bi1, Fe, Bi2, Mo) and the cent ers of\nthe nearest O triangles (∆ 1, ∆2, ∆3, ∆4) and the relative rotational angles of the O triangles aroun d the [111] axis, compared\namong the three crystal structures with the space groups: #2 25, #148, and #146. The values in ˚A above the O triangles are\ntheir edge lengthes (O-O distances), respectively. The thr ee values at the rightmost side are the lengthes of the primit ive cells\nalong the [111] direction. Notice: there is a rotational ang le of 60◦between ∆ 1and ∆ 2, and between ∆ 3and ∆ 4.\nFIG. 4: (Color online.) The electron density distributions in the A plane of the double perovskite Bi 2FeMoO 6with the #148\n[left: (a), (c), (e)] and #146 [right: (b), (d), (f)] space gr oups in comparison. (a) and (b) show the charge density distr ibutions\nof the valence electrons between -8 and 0 eV, ranging from 0 to 3|e|/˚A3with a contour increment of 0.15 |e|/˚A3; (c) and (d)\nthe charge density distribution of Bi 6s and O 2s electrons be tween -21 and -9 eV, ranging from 0 to 2 |e|/˚A3with a contour\nincrement of 0.1 |e|/˚A3; and (e) and (f) the spin density distributions of the valenc e electrons between -8 and 0 eV, ranging\nfrom 0 to 2 µB/˚A3with a contour increment of 0.1 µB/˚A3.\ngenerate in total energy. For example, one of them, with\nthe polarization P, can be changed into another one with\nthepolarization −P. Usingalinearinterpolationmethod\nand performing full optimization on the basis of the two\nwith±P, we can obtain a reversalpath connecting them,\nwith the R ¯3 (#148) phase naturally appearing as an in-termediate structure. The calculated energy curve along\nthe path[44, 45] is shown in Fig. 5(a). Most importantly,\nwehaveperformedfirst-principlescalculationsonphonon\nspectra for these two phases. The calculated phonon\nspectra are presented in Fig. 5 (b) and (c). These re-\nsults show that both of them are dynamically stable, in5\nFIG. 5: (Color online.) Polarization reversal path (a) and\nstructural stability (b,c). (a) A reversal path of the ferro elec-\ntric phase of #146 through the intermediate phase of #148.\n(b) Phonon spectra of the intermediate phase (#148). (c)\nPhonon spectra of the ferroelectric phase (#146).\naddition to the static stability of the R3 phase in terms\nof formation heat calculations[37].\nRobust multiferroicity. All our above analyses of\nthe ion displacements and electronic density distribu-\ntions indicate that the double perovskite Bi 2FeMoO 6in\nthe #146 structure should have a strong ferroelectricity\nin the pseudocubic [111] direction. More importantly,\nour DFT-based Berry phase calculation establishes that\nit indeed has strong ferroelectricity Palong the direc-\ntion. Its electric polarization, reaching to 85 µC/cm2,\nis comparable with that of the BiFeO 3, which indicates\nthat it should have a high ferroelectric Curie tempera-\nture. In addition, our analysis indicate that there is an-\nother degenerate ground-state phase with ferroelectricity\n−P, which implies that the polarization can be reversed\nthrough the #148 structure as an intermediate structure\nin between. On the other hand, we also calculate spin\nexchange energies and Curie temperatures for the mul-\ntiferroic phase and the intermediate one. The simulated\ncurves for magnetization and susceptibility are presented\nin Fig. 6 (a) and (b), indicating a high magnetic Curie\ntemperature of 650 K[37]. Our magnetocrystalline cal-\nculations including the spin-orbit coupling are presented\nin Fig. 6(c). The curve for the #146 phase can be fitted\nwithE=E0+EMAsin2(π\n180ω), withEMA= 1.52±0.005\nmeV. This indicates that the easy axis is the pseudocu-\nbic [111] and the magneto-crystalline anisotropic energy\nis 1.52 meV. It is interesting that the #148 phase also\nhas ferrimagnetic order with a net magnetic moment of 2\nFIG. 6: (Color online.) The magnetism of the ferroelectric\nphase and the intermediate phase of the double perovskite\nBi2FeMoO 6. Monte Carlo simulated results for magnetization\n(a) and susceptibility (b) of the two phases. (c) The magne-\ntocrystalline energies of the two phases, defined as a functi ons\nof the polar angle (in degree) from the [111] direction.\nµBbelow 360 K and magnetocryslline energy 1.50 meV\nwith the easy axis long the [111]. For convenience, we\nalso summarize the key multiferroic parameters in Ta-\nble II. Therefore, it has been established that the double\nperovskite Bi 2FeMoO 6is a robust multiferroic material\nwith high Curie temperatures.\nDiscussion\nWe have shown robust ferrimagnetism and ferroelectric-\nityin thedoubleperovskiteBi 2FeMoO 6inthe R3(#146)\nstructure. The large ferroelectric polarization in Table\nII means that this Bi 2FeMoO 6is comparable with the\nBiFeO 3in ferroelectricity, and in addition the strong\nferrimagnetism is shown to be able to persist at high\ntemperature far beyond room temperature. Because the\nBi2FeMoO 6is stable in terms of both static and dynami-\ncalcalculations, it could be synthesized soon. When real-\nized experimentally, it should be a promising multiferroic\nmaterial, especially for spintronics applications. Because\nof its robust multiferroic properties, the Bi 2FeMoO 6can\nbe used to form some interesting composite structures\nsuch as multiferroic superlattices[46].\nAs for the mechanism of multiferroicity, it is similar\nto that of the BiFeO 3, except that here the Mo mo-\nment cannot be completely compensated by the Fe mo-6\nTABLE II: Calculated Plus and Minus values ( ˚A) of\ncations, DFT-calculated ferroelectric polarizations, an d sum-\nmarized key multiferroic parameters of the double perovski te\nBi2FeMoO 6.\nIonic displacement analysis\nBi1 Fe Bi2 Mo\nPlus 4.7337 2.3836 4.6978 2.3142\nMinus -1.1771 -0.6164 -1.3736 -0.3646\nBerry phase calculations of polarization\nscheme Fe( U,J) / Mo(U,J) (eV) P(µC/cm2)\nGGA+U (2.0,0.8) / (2.5,0.5) 85.4\n(4.0,0.8) / (2.5,0.5) 85.5\n(8.0,0.8) / (2.5,0.5) 84.7\nHSE06 - 85.4\nKey multiferroic parameters\nFerrimagnetism M 2µB Tc650 K\nFerroelectricity P85µC/cm2Tc high\nment, producing the finite net magnetic moment 2 µB\nper formula unit. It is interesting that the ferrimag-\nnetic Curie temperature is predicted to be also similar to\nthe Neel temperature of the BiFeO 3. The ferri-distortive\nrotations of the oxygen octahedra of Fe and Mo is fa-\nvorable to stabilize the Bi-dominated antiferroelectricity\nin the #148 structure of the Bi 2FeMoO 6. These ferri-\ndistortive rotations of the O octahedra of Fe and Mo\nare similar to antiferrodistortive rotations of O octahe-\ndra of Fe in the BiFeO 3[35]. It is also clear that these\nferri-distortive rotations coexist with the ferroelectricity\nin the #146 phase. It should be pointed out that in ad-\ndition to the the multiferroic #146 phase, there should\nexist some metastable phases for Bi 2FeMoO 6, with the\n#148 structure being one of them. This is also similar to\nBiFeO 3for which many metastable structures have been\nfound[14, 17–19, 36].\nIn summary, we have investigated double perovskite\nBi2FeMoO 6through systematic DFT calculations and\nstructural analyses. Our careful GGA optimization re-\nveals that the Bi 2FeMoO 6in the #146 structure is a\nferrimagnetic semiconductor. The net moment reaches\nto 2µBper formula unit because the magnetic moments\nof Fe and Mo cannot be canceled, and the semiconduc-\ntor gap corrected with mBJ potential reaches to 0.725\neV. The O octahedra of Fe and Mo ions have rota-\ntion angles of +17.8◦and -17.0◦around the [111] axis,\nrespectively. Our analyses of the electronic structure\nand ionic displacements indicate strong ferroelectricity\nin the Bi 2FeMoO 6. Our DFT-based electric polariza-\ntion calculations directly shows that the Bi 2FeMoO 6has\nstrong ferroelectricity in the [111] direction. Our first-\nprinciples phonon spectra show that this multiferroic\nphase is stable. Therefore, it has been established thatthe Bi 2FeMoO 6is a promising multiferroic material with\nstrong ferroelectricity and high ferrimagnetic Curie tem-\nperature far beyond room temperature. With the spin-\norbit effect into account, our calculation show that the\nmagnetic easy axis is also in the [111] axis. The mag-\nnetocrystalline anisotropy should be manipulated as has\nbeen done in Sr 2FeMoO 6film materials[29]. This mul-\ntiferroic material can be useful for designing new multi-\nfunctional materials and devices in the future.\nMethods\nWe use projector augmented-wave (PAW)[47] plus\npseudo-potential methods within the density functional\ntheory[48, 49], as implemented in the Vienna ab ini-\ntio simulation package (VASP) [44, 45], to optimize\nthe crystal structures and then study the electronic\nstructures, ferroelectricity, and magnetic properties.\nThe spin-polarized generalized-gradient approximation\n(GGA)[50], to the electronic exchange-correlation func-\ntional, is used to do structure optimization. We use a\nplane wave basis set with a maximum kinetic energy of\n500eVanda12 ×12×12k-pointmeshgeneratedwith the\nMonkhorst-Pack scheme[51]. During optimization, all of\nthe structures were fully relaxed until the largest force\nbetween the atoms become less than 1 meV/ ˚A.\nPhonon spectra are calculated with first-principles\nperturbation method with norm-conserving pseudo-\npotentials as implemented in package Quantum-\nESPRESSO [52]. The magnetic anisotropy energy\n(MAE) due to the spin-orbit coupling is determined by\nthe force theorem, and calculated in terms of the to-\ntal energy with respect to the [111] direction. We use\na modified Becke-Johnson exchange potential to accu-\nrately calculate semiconductor gaps[53]. We use modern\nBerry phase method[39] and take both GGA+U [40, 41]\nand HSE06 [42, 43] schemes to calculate electric polar-\nization. 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Lett. 102,226401 (2009).Acknowledgments\nThis work is supported by Nature Science Foundation of\nChina (Grant No. 11174359), by Chinese Department of\nScience andTechnology(GrantNo. 2012CB932302),and\nby the Strategic Priority Research Program of the Chi-\nnese Academy of Sciences (Grant No. XDB07000000).\nPC is grateful to Sai Gong for some technical helps.\nAuthor contributions\nBGL conceived and supervised the project, and PC car-\nried out all the calculations. PC and BGL both con-\ntributed to the analysis and discussions of the results,\nand BGL and PC wrote the manuscript.\nAdditional information\nCompeting financial interests: The authors declare no\ncompeting financial interests."    }]